A Numerical Method for Two Phase Flows

Transcription

1 A Numerical Method for Two Phase Flows with Insoluble and Soluble Surfactants by Shilpa Madangopal Khatri A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics New York University September 9 Anna-Karin Tornberg Advisor

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3 in honor of Tom iii

4 Acknowledgements First and most of all, I would like to thank my advisor, Professor Anna-Karin Tornberg for all her help and guidance. Our work together and academic discussions have been the key of my education during graduate school. I would also like to thank her for inviting me numerous times to visit KTH in Sweden. I have gained a lot academically from my visits and really enjoyed my time in Stockholm. I would also like to acknowledge my collaborator, Dag Lindbo. I appreciate all that I have learned from our adventures in building a Navier-Stokes solver together and our numerous other discussions. Professor Olof Widlund, Professor Charles Peskin, and Marco Kupiainen have all been very helpful during discussions about different topics within my graduate studies. I would like to thank them for providing insight into these various topics. Also, being part of the Micro- and complex fluids group within the Linné Flow Centre at KTH added a great deal to my education. It allowed me to get input on my problems and learn about problems different yet related to my work. The discussions within group meetings and with members outside of meetings have been extremely helpful. Professor Stephen Childress has been here for me since day one of my arrival to Courant. I really appreciate all the time, advice, and encouragement he has provided me with during my time here. I would also like to thank Professor Michael Shelley, Professor Marsha Berger, and Professor Leslie Greengard for always taking the time to ask me how I was doing and giving me advice. iv

5 I appreciate all that KTH and NADA have done for me. They accepted me into their department and welcomed me every time I came to visit. All my colleagues and friends at NADA have made Stockholm a second home for me the last few years. At Courant, the eleventh floor has been my base for the past six years. I have made many friends and had many great colleagues. My education would have been a much lesser experience without all of them. I would especially like to thank Saverio Spagnolie, Sam Stechmann, Tom Bringley, and Christel Hohenegger. Without their endless support, encouragement, and dear friendships I would not have gotten as far as I have. Finally, I would like to thank my parents, Madan and Raju Khatri, and my brother, Ajay Khatri. Without their direction and love, I would never have even started this journey and had the strength to succeed. I would like to express my gratitude for everything they have given me over the years. This work was funded in part by a National Defense Science and Engineering Graduate Fellowship and a Henry MacCracken Fellowship. Additional funding was provided by the Applied Analysis and Computational Mathematics DOE Grant, DE-FG-88ER553, and the Alfred P. Sloan Research Fellowship and the Microsoft Research New Faculty Fellowship awarded to Professor Anna-Karin Tornberg. v

6 Abstract In this thesis a model and numerical method to simulate insoluble and soluble surfactants in two phase flows is presented. In many practical multiphase flow problems, surfactants, or surface reacting agents, are present. Surfactants are absorbed from the bulk fluid as a monomolecular layer to the interfaces between fluids, which modifies the surface tension at these interfaces. The effect of surfactants is important in many real world applications, i.e. treatment of gas emboli, microfluidic applications, and electrical components. The surfactant concentration on an interface separating the fluids can be modeled with a time dependent differential equation defined on the moving and deforming interface. When the surfactants are soluble and therefore present in the bulk fluid, this equation is coupled with a partial differential equation modeling the soluble surfactant concentration in the bulk fluid on one side of the interface. The equations for the location of the interface and the surfactant concentration on the interface and in the bulk are coupled with the Navier-Stokes equations. These equations include the singular surface tension forces from the interface on the fluid, which depend on the interfacial surfactant concentration. We develop a second order numerical method based on an explicit yet Eulerian discretization of the interface and the interfacial surfactant concentration. For soluble surfactants, the bulk concentration is discretized on a uniform grid. The boundary of the bulk surfactant concentration domain is defined by the interface location, and an embedded boundary method is used to enforce the boundary vi

7 condition here. A finite difference method is used to solve the Navier-Stokes equations on a regular grid with the forces from the interface spread to this grid using a regularized delta function. Drop deformation in shear flows in two dimensions is studied. The deformation of clean drops and drops in the presence of insoluble and soluble surfactants are presented. We study the effects of surfactant concentration on the deformation of the drop. In the soluble case, we look at the effect of varying parameters dealing with the adsorption and desorption of surfactant to the interface. vii

8 Contents Dedication Acknowledgements Abstract List of Figures iii iv vi xii Introduction I Insoluble Surfactants 9 Mathematical Model. Equations Nondimensionalization Limitations of the Equation of State for the Surface Tension Coefficient 8 3 Numerical Method 3. Interface Representation: Segment Projection Method Advection of the Interface Surfactant on the Interface viii

9 3. Timestepping Discretization of the Interface Advection Equations Discretization of the Surfactant Concentration Equations Reconstruction Order of Convergence Discretization of the Navier-Stokes Equations Timestepping Spatial Discretization Coupling the Navier-Stokes Equations with the Interface and Surfactants Timestepping Surface Tension Forces Order of Convergence with the Navier-Stokes Solver Results Clean Interfaces Varying Domain Lengths Two Different Initial Conditions Varying Reynolds Numbers Varying Capillary Numbers Comparison to Previous Work Contaminated Interfaces Varying Initial Surfactant Concentrations Varying Reynolds Numbers ix

10 4..3 Varying Capillary Numbers Capillary Stress Comparison with Previous Work II Soluble Surfactants 9 5 Mathematical Model 3 5. Equations Nondimensionalization Numerical Method Timestepping Discretization of the Interfacial Surfactant Concentration Equations Discretization of the Bulk Surfactant Concentration Equations An Embedded Boundary Method for the Soluble Surfactants Enforcing the Boundary Condition Interpolation of C to New Points in the Domain Evaluating C at the Interface Order of Convergence Coupling the Navier-Stokes equations with the Interface and Soluble Surfactants Timestepping Order of Convergence with the Navier-Stokes Equations Results 8 x

11 7. Varying Peclet Numbers Varying Hatta Numbers Varying Langmuir Numbers Varying Initial Interfacial Surfactant Concentrations Conclusions 98 Bibliography xi

12 List of Figures. The interface is shown here with the normal and tangent vectors. Also, the direction of s is given Surfactant molecules are absorbed to the interface, Γ, as a monomolecular layer, aligned such that their hydrophobic ends are in one fluid and their hydrophilic ends are in the other [34] Plot of the maximum surfactant concentration, ρ, at which the surface tension coefficient σ is positive for varying values of the elasticity number, E. Here, the equations are nondimensionalized by ρ and σ c.the plot on the left is for the linear equation of state and the plot on the right is for the nonlinear equation of state Plot of σ versus ρ for E =.. The dashed line is the linear equation of state and the solid line is the nonlinear equation of state. Here, the equations are nondimensionalized by ρ and σ c xii

13 .5 Plot of the maximum surfactant concentration, ρ, at which the surface tension coefficient σ is positive for varying coverage, x, and fixed elasticity number, E =.. Here, the equations are nondimensionalized by ρ e and σ e. The plot on the left is for the linear equation of state and the plot on the right is for the nonlinear equation of state..6 Plot of σ versus ρ for x =.4 and E =.. The dashed line is the linear equation of state and the solid line is the nonlinear equation of state. Here, the equations are nondimensionalized by ρ e and σ e.. 3. An example of how an interface is divided into segments. The top figure shows the entire interface, the middle figure shows the x- segments and the bottom figure shows the y-segments. In the middle and bottom figures, the uniform grid on which these segments are defined is also shown An example of one x-segment and how ρ is a function of (x, t) that is defined at (x, f(x, t)). The top figure shows (x, f(x, t)), and the middle figure shows (x, ρ(x, t)). The bottom figure shows ρ(x, t) at (x, f(x, t)), where the shape of the curve gives the location of the interface and the color shows the surfactant concentration at that point on the interface xiii

14 3.3 This figure explains the domain decompsition scheme used on the segments to solve for the diffusion of the surfactants. The top figure shows all the segments forming Γ on which we want to solve for ρ. The middle figure shows segments and and the bottom figure shows segments 3 and 4. The black points are where we solve the system for ρ on each segment, and the blue points are where we apply the Dirichlet boundary conditions in the iterative scheme The interface at varying times with the oscillating ellipse given velocity field defined by equations 3.9 and The surfactant concentration along the interface at varying times with the corresponding Γ shown in Figure 3.4. Initially, the surfactant concentration was set to be a constant along the interface, ρ o =.5. The surfactants are plotted versus the arclength of the interface. The stars in Figure 3.4 correspond to when the arclength here is The error in the location of the interface for the oscillating ellipse case, equations 3.9 and 3.9. The green line in all the convergence plots in this section is a reference line with slope. The plot on the right shows the relative error in the area within the drop versus time when x = y =.5. In this plot, A(t), is the area in the drop at time t, measured numerically xiv

15 3.7 The error in ρ when ρ o =.5. The plots on the right show the relative error in the total amount of surfactant versus time. Here, ρ(t) is the total amount of surfactant along Γ at time t, computed numerically. The top plots correspond to P e Γ =, and the bottom plots correspond to P e Γ = The error for ρ when ρ o = (sin(θ)+)/3, θ = arctan(y.5, x.5). The plots on the right show the relative error in the total surfactants versus time. The top plots correspond to P e Γ = and the bottom plots correspond to P e Γ = The interface at varying times when advected with the rigid body rotation velocity field, given by equations 3.9 and The surfactant concentration at varying times with the corresponding Γ shown in Figure 3.9. Initially, the surfactants were set to be ρ o = (sin(θ) + )/3, where θ = arctan(y., x.5). The surfactants are plotted versus the arclength of the interface. The stars in Figure 3.9 correspond to when the arclength here is The error in the location of the interface for the rigid body rotation case, equations 3.9 and The plot on the right shows the relative error in the area inside the drop versus time when x = y = The relative error in the total surfactants versus time when ρ o =.5. The left plot corresponds to P e Γ = and the right plot corresponds to P e Γ = xv

16 3.3 The error in ρ when ρ o = (sin(θ) + )/3, θ = arctan(y., x.). The plots on the right show the relative error in the total surfactants versus time. The top plots correspond to P e Γ = and the bottom plots correspond to P e Γ = The interface at varying times with the rotating velocity field defined in equations 3.94 and We only show from time, t =, to time, t =.6, because the deformation of the drop is just repeated until time, t = The surfactant concentration at varying times with the corresponding Γ shown in Figure 3.4. Initially the surfactant concentration was set to a constant along the interface, ρ o =.5. The surfactants are plotted versus the arclength of the interface. The stars in Figure 3.4 correspond to when the arclength here is The error in the interface location for the rotating velocity field, equations 3.94 and The plot on the right shows the relative error in the area within the drop versus time when x = y = The error in ρ when ρ o =.5. The plots on the right show the relative error in the total surfactants versus time. The top plots correspond to P e Γ = and the bottom plots correspond to P e Γ = The error in ρ when ρ o = (sin(θ)+)/3, θ = arctan(y.3, x.3). The plots on the right show the relative error in the total surfactants versus time. The top plots correspond to P e Γ = and the bottom plots correspond to P e Γ = xvi

17 3.9 The staggered grid on which the Navier-Stokes equations are solved The staggered grid with all the terms used in the spatial discretization of the nonlinear terms The top two plots show the drop contour at the initial and final times when surfactants are included. The bottom two plots show the surfactant concentration on the interface at the same times. The stars on the interface correspond to zero arclength in the surfactant concentration plots The top plots show the velocity field inside and outside the drop at the final time, t =.5. For the plot showing the velocity field inside the drop, the maximum velocity is.34. The bottom plots show the pressure at the initial and final times Convergence plots for the interface, velocity field, and pressure, when only refining in space. In all the convergence plots in this section, the dotted black lines are the error measured using the maximum norm, the black solid lines are the error measured using the L norm, the green line is a reference line with slope, and the red line is a reference line with slope. Also, the relative error in the area inside the drop versus time is plotted. A(t) is the area inside the drop at time t measured numerically. The black line is the finest grid and the red is the coarsest grid xvii

18 3.4 Convergence plots for the interface, velocity field, and pressure with refinement in both space and time. Also, the relative error in the area inside the drop versus time is shown. The black line is the finest grid and the red is the coarsest grid Convergence plots for the interface, surfactants, velocity field, and pressure, when only refining in space. In this case, surfactants are included. The relative error in the area inside the drop versus time and in the total amount of surfactant versus time is shown. When including the relative error in the total surfactants on Γ, ρ(t) is the total amount of surfactant along Γ at time, t, computed numerically Convergence plots for the interface, surfactants, velocity field, and pressure with refinement in both space and time. Surfactants are included here. The relative error in both the area inside the drop versus time and the total surfactants versus time is shown Steady state contours of the drop with varying lengths. The red drop is when Lx = and the black drop is when Lx = 4, 6, 8, and Deformation plots for clean drops with varying domain lengths. The red line is when Lx =, the green line is when Lx = 4, the blue line is when Lx = 6, and the black line is when Lx = 8 and xviii

19 4.3 The velocity fields at the periodic boundary. The plot on the left shows the horizontal component of the velocity, and the plot on the right shows the vertical component of the velocity. The vertical component of the velocity is not solved for at the periodic boundary due to the staggered grid of the Navier-Stokes solver. We plot the average of the vertical component at points to the right and at points to the left of the periodic boundary, when we show the velocity field at this boundary. Again, the red dots are when Lx =, the green dots are when Lx = 4, the blue when Lx = 6, and the black when Lx = 8 and Deformation plots for clean drops with the two different initial conditions. The red line shows the deformation when initialized with a linear velocity field and the black line shows when initialized with a quiescent velocity field Drop contours with the two different initial conditions. The plot on the left is at time, t =, and the plot on the right shows that both cases reach the same steady state and gives the steady state contour. Again, the red is the contour when initialized with a linear velocity field and the black shows when initialized with a quiescent velocity field Deformation plots and steady state contours for clean drops with Reynolds numbers,.,,, and 5, which are black, blue, green, and red, respectively xix

20 4.7 The contours for the drops that do not reach steady state at time t = 5. The case with Re = 5 is magenta and the one with Re = is cyan The velocity fields at the periodic boundary. The plot on the left shows the horizontal component of the velocity, and the plot on the right shows the vertical component of the velocity. Again, the colors correspond to the different Reynolds numbers,. is black, is blue, is green, 5 is red, 5 is magenta, and is cyan The velocity field inside and outside the drop at time t = 5. The top plots are the case with Reynolds number and the drop is at steady state. For the plot with the velocity shown inside the drop, the maximum velocity is.36. The bottom case is with Reynolds number, and the maximum velocity in the interior of the drop is Deformation plot and steady state contours for the clean drops with capillary numbers.5,., and.4, which are black, blue, and green, respectively The contours for the drops that do not reach steady state at time t = 5. The capillary numbers are.7, which is magenta, and, which is cyan xx

21 4. The velocity fields at the periodic boundary. The plot on the left shows the horizontal component of the velocity, and the plot on the right shows the vertical component of the velocity. Again, the colors correspond to the different capillary numbers,.5 is black,. is blue,.4 is green,.7 is magenta, and is cyan On the left, the Ca =. and on the right, the Ca =.4. In each plot, a range of Reynolds numbers, Re =,, 5, and, which are black, blue, green, and red, respectively, is shown Here, the Reynolds number is fixed to, while the Ca =. and.4, and is black and red respectively. The steady state drop is shown for Ca =. but the drop with Ca =.4 does not reach steady state and is shown at time, t = Deformation plot and steady state contours for drops with varying amounts of surfactants. The initial surfactant concentration is set to ρ o =,.,.3,.4,.5,.7,.9, and.975 and is black, blue, green, red, magenta, cyan, yellow, and orange respectively The velocity fields at the periodic boundary. The plot on the left shows the horizontal component of the velocity, and the plot on the right shows the vertical component of the velocity. Again, the colors show the velocities for varying initial surfactant concentrations, as in Figure xxi

22 4.7 We show the velocity field inside and outside the steady state drop. The top plots are the case with ρ o =. For the plot with the velocity shown inside the drop, the maximum velocity is.36. The bottom case is with ρ o =.4, and here, the maximum velocity in the interior of the drop is The plot on the top shows how the surfactants have progressed in time, ρ ρ o versus arclength, for different initial surfactant concentration values on the drop. The plot in the middle is the surface tension difference, σ σ o, plotted versus arclength. And the plot on the bottom is the Marangoni stress versus arclength. The colors are the same as above, ρ o = is black, ρ o =. is blue, ρ o =.3 is green, ρ o =.4 is red, ρ o =.5 is magenta, ρ o =.7 is cyan, ρ o =.9 is yellow, and ρ o =.975 is orange. The stars on the contours in Figure 4.5 correspond to arclength here Deformation plot and steady state contours for the drops with Reynolds numbers,.,,, and 5, which are black, blue, green, and red respectively. Here, surfactants are included with initial surfactant concentration, ρ o = The contours for the drops that do not reach steady state at time t = 5. The Reynolds numbers are 5 for the magenta drop and for the cyan drop xxii

23 4. The velocity fields at the periodic boundary. Again, the colors correspond to the varying Reynolds numbers,. is black, is blue, is green, 5 is red, 5 is magenta, and is cyan Deformation plots and steady state contours for the drops with capillary numbers.5, which is black,., which is blue, and.4, which is green The contours for the drops that do not reach steady state at time t = 5. The capillary numbers are.7 for the magenta drop and for the cyan drop The velocity fields at the periodic boundary. Again, the colors correspond to the different capillary numbers,.5 is black,. is blue,.4 is green,.7 is magenta, and is cyan The capillary stress versus arclength is plotted for varying initial surfactant concentrations. These stresses go along with the results presented in Section The capillary stress for coverage, ρ o =.4, at varying times The capillary stress versus arclength for varying Reynolds numbers. These stresses go along with the results presented in Section The capillary stress versus arclength for varying capillary numbers. These stresses go along with the results presented in Section This figure shows the problem configuration. In this part of the thesis, the surfactants are soluble in the bulk fluid. The red represents the surfactants in the bulk and on the interface xxiii

24 6. The two stencils used in the update of C. The stencil on the left, the five point stencil, is used for the diffusion step, Step C(B), and the stencil on the right, the nine point stencil, is used for the advection step, Step C(A) Two examples when points in Ω n+ have points in their stencil which lie inside the drop, the red points. When this happens an embedded boundary method is used to evaluate a value of C at the point inside the drop. The red points are what is denoted as x in algorithm This figure explains how we extrapolate C at the red points using values of C in Ω n+, Γ, and ρ This figure shows when and why we have to interpolate values of C n. The red points are points where C n does not exist but where C n+ is evaluated. The red points are denoted x in the algorithm for the embedded boundary method This figure shows the same method as shown in Figure 6.3, except now we are interpolating C n to the red points, which are in Ω n This figure also shows the same embedded boundary method described in this section, but now the situation is slightly different, since C is being evaluated at the red points on Γ xxiv

25 6.7 The top two plots show the drop contour at the initial and final times when advected using (u, v) = (, ). The bottom two plots show the surfactant concentration on the interface versus arclength at the same times. The stars on the interface correspond to zero arclength for the surfactant concentration plots The top plots show the bulk surfactant concentration at the initial and final times. The bottom two plots show the bulk surfactant concentration at Γ at the same times Error plots for the interface and surfactants. The relative error in the area inside the drop versus time for x = y = 4 is also shown The top two plots show the drop contour at the initial and final times when advected using, (u, v) = ( + y, ). The bottom two plots show the surfactant concentration on the interface at the same times. The stars on the interface correspond to zero arclength for the surfactant concentration plots The top plots show the bulk surfactant concentration at the initial and final times. The bottom two plots show the bulk surfactant concentration at Γ at the same times Convergence plots for the interface. Also, the relative error in the area inside the drop versus time is shown for x = y = Error plots for the surfactants. The top plots are for P e =, the middle plots for P e = and the bottom plots for P e = xxv

26 6.4 The top two plots show the drop contour at the initial and final times of the convergence tests. The bottom two plots show the surfactant concentration on the interface at the same times. The stars on the interface correspond to zero arclength for the surfactant concentration plots The top plots show the velocity field inside and outside the drop at the final time, t =. For the plot showing the velocity field inside the drop the maximum velocity is.5. The bottom plots shows the pressure at the initial and final times The top plots show the bulk surfactant concentration at the initial and final times. The bottom two plots show the bulk surfactant concentration at the interface as a function of the arclength of the drop Convergence plots for the interface, surfactants, velocity field, and pressure. Here, the refinement is only in space. The relative error in the area inside the drop versus time is shown as well. The black line is the error for the finest grid and the red line is the error for the coarsest grid Convergence plots for the interface, surfactants, velocity field, and pressure. Here, both space and time are refined. The relative error for the area inside the drop versus time is shown xxvi

27 7. Deformation plots, Γ, and the total amount of surfactant along Γ for varying values of P e. In these plots, P e = is black, P e = is blue, P e = is green and P e =. is red The interfacial surfactant concentration and the bulk surfactant concentration at Γ versus arclength. Also, the surface tension and the Marangoni stresses along the interface are shown. The colors correspond to the varying P e numbers as described in Figure Deformation plots, Γ, and the total amount of surfactants along Γ for varying values of α. In these plots, α =.,.5,.5,,.5,,.5, and 3, which are black, blue, green, red, magenta, cyan, yellow, and orange, respectively The interfacial surfactant concentration and the bulk surfactant concentration at Γ versus arclength. Also, the surface tension and the Marangoni stresses along the interface are shown. The colors correspond to the varying Hatta numbers as described in Figure Deformation plots, Γ, and the total amount of surfactants along Γ for varying values of La. In these plots, La =, 4,, 4,, and.8, 3 which are black, blue, green, red, magenta, and cyan, respectively The interfacial surfactant concentration and the bulk surfactant concentration at Γ versus arclength. Also, the surface tension and the Marangoni stresses along the interface are shown. The colors correspond to the varying Langmuir numbers as described in Figure xxvii

28 7.7 Deformation plots, Γ, and the total amount of surfactants along Γ for varying values of ρ o. In these plots, ρ o =,.,.3,.4,.5,.7,.75,.8 and.85 which are black, blue, green, red, magenta, cyan, yellow, orange, and purple, respectively The change in the interfacial surfactant concentration and the bulk surfactant concentration at Γ from the initial values versus arclength. Also, the corresponding surface tension difference and the Marangoni stresses along the interface are shown. The colors correspond to the varying ρ o values as described in Figure The drop contour and deformation plots for insoluble (top) and soluble surfactants (bottom). In these plots, ρ o =,.,.3,.4,.5, and.7, which are black, blue, green, red, magenta, and cyan, respectively xxviii

29 Chapter Introduction In this thesis, we are presenting a numerical method to model insoluble and soluble surfactants on deforming interfaces in two phase flows. This problem consists of solving for the velocity of the fluid, the interface location, and the surfactant concentration along the interface. In the case of soluble surfactants, we solve for the surfactant concentration along the interface and in the bulk fluid. The effect of surfactants, surface reacting agents, is important in many real world applications, in which the surface tension plays a significant role. Examples include dealing with gas emboli, microfluidic applications, and electrical components. Decompression illness is often caused by gas bubbles entering the blood stream during surgery or if the pressure surrounding the body has changed drastically. Surfactants are being used to try and treat decompression illness by changing the interface properties of these bubbles [53, 6]. Surfactants can be used to control droplet size when forming emulsions, foams, suspensions, and pharmaceuticals []. Some different methods using surfactants for inkjet printing have been developed.

30 One use of surfactants in this field is to help hydrophobic surfaces absorb the ink [9]. Surfactants are used in so many versatile multiphase flow applications due to their affinity for the interface. Surfactants are usually organic molecules consisting of a hydrophobic head and a hydrophilic tail. This composition causes them to be absorbed as a monomolecular layer to the interface. This layer of molecules modifies the surface tension of the interface. Surfactants can either be soluble or insoluble in the fluid surrounding the interface. Insoluble surfactants exist only on the interface and are not present in the fluid surrounding the interface. Soluble surfactants are present both on the interface and in the surrounding fluid. The concentration on the interface is different from that in the bulk fluid due to the affinity the molecules have to the interface. In the case of soluble surfactants, adsorption and desorption of the molecules to and from the interface plays a role in the dynamics of the flow. Soaps and detergents are examples of surfactants which lower the surface tension of the interface [34]. Here, we will be presenting a model and numerical method to treat insoluble and soluble surfactants. The concentration of the surfactant on an interface separating the fluid phases can be modeled with a time dependent differential equation defined on the moving and deforming interface. We present a numerical method which has the benefit of having an explicit yet Eulerian discretization of the interface. The concentration of a soluble surfactant can be modeled with a similar differential equation along the deforming interface coupled with another time dependent differential equation defined on the deforming domain determined by the interface. We present a nu-

31 merical method to solve for the soluble surfactants, which continues to allow us to work on regular grids. Along with the interface and surfactants, the fluid flow itself must be modeled. We solve the Navier-Stokes equations for the flow velocity and pressure. The difficulty in coupling these equations with the interface is due to the singular force the interface places on the fluid. Much previous work has been done on the multiphase flow problem, where the interface between the phases is explicitly tracked. So called interface tracking and interface capturing methods allow us to have a representation of the interface and the exact location. These properties are essential for any method used to solve the problem presented here, since it requires solving an equation for the surfactants along the interface. The numerical methods for interface tracking most commonly found in the literature include front tracking and boundary integral methods, level set methods, and finite volume methods. More recently, work has been done on the study of material properties, specifically surfactants, along these interfaces. Studying multiphase flows with surfactants is an active area of research as can be seen by the number of published works in recent years. The numerical methods being developed to solve for insoluble and soluble surfactants use these common interface tracking methods to model the interface. Both in front tracking and boundary integral methods, Lagrangian points are used represent the interface and the surfactant concentration is solved for at these points. Work by Stebe, Pawar, and Eggleton [3, 3] uses Lagrangian points to 3

32 mark the interface. They combine this with a finite difference method to solve for the surfactants along the interface. These methods are coupled with a boundary integral method to solve the Stokes equations for the fluid flow. Lai, Tseng, and Huang [3] use a similar method for the interface and surfactants but couple these with a Navier-Stokes Immersed Boundary solver. Also, the front tracking method to represent the interface has been combined with a finite volume method to solve for the surfactants, as in [5, 4, 5, 8]. Yon and Pozrikidis [5] and Bazhlekov et al. [5] solve the Stokes equations using a boundary integral method for the fluid flow, while Lee and Pozrikidis [4] and Hameed et al. [8] solve the Navier- Stokes equations using the Immersed Boundary Method and a Lagrangian-Eulerian moving mesh, respectively. The use of Lagrangian points to represent the interface can make solving for the surfactants difficult because there is no regularity in the points of the interface. Level set methods define the interface as the zero level set of a function which is discretized on an Eulerian grid. Adalsteinsson and Sethian [] and Xu and Zhao [5] present numerical methods on how to solve for material quantities and surfactants, respectively, when the interface is represented using a level set function. They both extend the material quantity off the interface and then solve for the extension of the quantity. Xu et al. [49] use this method for surfactants and couple it with the Immersed Interface Method to solve the Stokes equations for the fluid flow. Level set methods provide a regular grid for the interface but require that the surfactant be extended to a higher dimension to be solved for along the interface. Finite volume methods represent the interface by tracking the volume of the 4

33 two fluid phases in cells. When the finite volume method is used to represent the interface, it also used to solve for the surfactants along the interface as in [, 35, ]. In implementations of this method, the interface is reconstructed to solve for the surfactant along it [] or the surfactants are extended off the interface as if the interface has some finite thickness [35]. The results shown in these works mentioned above usually consist of studying drop deformation in extensional or shear flows. The results that are most similar to what we present in this thesis are in Lai, Tseng, and Huang [3], where they implement a completely different numerical method, described above. This paper demonstrates their method by showing drop deformations in shear flow, where the flow is solved for using the Navier-Stokes equations. However, they only treat insoluble surfactants. The works listed above all model only insoluble surfactants. Work has just recently begun on developing numerical methods for soluble surfactants. Stebe et al. have presented theory with some experiments and simulations for soluble surfactants in [4,, 4]. They do not solve for the bulk surfactants, but rather assume that the adsorption and desorption of surfactants from the bulk is diffusion dominated and the bulk surfactant concentration is a constant. The front tracking method has been used to represent the interface coupled with a finite difference solver for the surfactants in methods developed by Muradoglu and Tryggvason [9] and Zhang, Eckmaann, and Ayyaswamy [53]. The front tracking method has also been combined with a finite volume method for the surfactants in [5]. All three of these methods solve the Navier-Stokes equations for the fluid flow. 5

34 In our numerical method, an interface is divided into overlapping segments, where each segment can be described as a single valued function of a specified coordinate plane. This allows the motion of the interface to be simulated by evolving each segment using a partial differential equation discretized on a uniform grid [44]. The concentration of surfactant on the interface is modeled using an advection diffusion equation with a source term, requiring the solution of a time dependent partial differential equation on the moving interface. The framework with overlapping segments is advantageous for solving this problem, allowing us to solve the differential equation for the surfactant concentration on uniform grids using standard discretizations. When the surfactants are soluble in the bulk fluid, we couple the above system with a separate advection diffusion equation for the bulk surfactant concentration. We model the exchange of surfactants between the bulk fluid and interface with an additional source term in the equation for the interfacial surfactant concentration and with a mixed Neumann Dirichlet boundary condition at the interface for the bulk surfactant concentration. We discretize the bulk surfactant concentration equation on a uniform two dimensional grid. The interface arbitrarily cuts through this grid, requiring a special treatment of the boundary condition at the interface. We use an embedded boundary method [], which allows us to continue using standard discretizations to solve for the bulk surfactant concentration while implementing the boundary condition to the desired order of accuracy. The equations for the location of the interface and the interfacial and bulk surfactant concentration are coupled with the Navier-Stokes equations. We use 6

35 a finite different method, based on Chorin s projection scheme [9], to solve the Navier-Stokes equations for the fluid flow. These equations include the surface tension forces from the interface, consisting of the capillary stress and the Marangoni stress. The interface can cut arbitrarily through the grid as occurs in interface tracking methods. We apply the singular force on the fluid by regularizing the force as is done in the Immersed Boundary method [33]. The strength of the surface tension forces depends on the surfactant concentration along the interface. We show that the convergence of our numerical method is second order with a given velocity field. We also present convergence tests when the interface and surfactants are coupled with the Navier-Stokes equations. In this case, a lower order of convergence is obtained due to the Navier-Stokes solver being a lower order scheme and due to the coupling itself. We present results of drop deformation in shear flows in two dimensions. The results shown here are for clean drops and contaminated drops, with both insoluble and soluble surfactants. For such flows there is a critical capillary number, defined as the ratio of viscous stresses in the flow to the surface tension of the interfaces, below which the drops attain a steady state. We investigate drop deformations at capillary numbers above and below this critical capillary number. We study the deformations that occur with varying individual parameter values while keeping the rest of the parameters fixed. We discuss the distribution of surfactant on the interface and in the bulk as the flow evolves and with this the capillary and Marangoni stresses that the interface is placing on the fluid. In Part I, we present the mathematical model, the numerical method, conver- 7

36 gence tests, and results for the problem with just insoluble surfactants. In Part II, we add in the solubility of the surfactants into the model presented in Part I. We once again present the mathematical model, the numerical method, convergence tests, and results. 8

37 Part I Insoluble Surfactants 9

38 Chapter Mathematical Model. Equations To model insoluble surfactants on evolving interfaces in a two dimensional domain, Ω, we must have an equation to solve for the fluid velocity, u, a method to track the interface, Γ, and an equation to solve for the surfactant concentration along the interface, ρ. We solve the incompressible Navier-Stokes equations for the fluid velocity, u, and the pressure, p, in the two dimensional domain, Ω, ρ Ω ( u t ) + (u )u + p µ u = in Ω (.) u = in Ω. (.) We describe how the force from the interface on the fluid is coupled with these equations below. In these equations, ρ Ω and µ are the density and the viscosity of

39 s ˆn ˆτ Figure.: The interface is shown here with the normal and tangent vectors. Also, the direction of s is given. Γ the fluid, respectively. We will be assuming the interface, Γ, separates two fluids with the same density and viscosity. The force applied on the fluid by the interface can be represented by a jump condition along the interface, Γ, [T ˆn] Γ = T ˆn outside Γ T ˆn inside Γ = F = (σˆτ) s = σκˆn + s σ, (.3) where T is the stress tensor, κ is the curvature along Γ which is positive for a circle, ˆn is the unit normal vector of Γ pointing into Γ, σ is the surface tension coefficient, and s is the surface gradient operator [3, 34]. The direction of ˆτ, ˆn, and s with respect to Γ are shown in Figure.. The surface gradient operator is defined to be the gradient along Γ as moving in the positive s direction. Generally, the operator is defined by s = (I ˆn ˆn) where the gradient operator,, is applied to a function on the entire domain, Ω. This function is any smooth extension of the function defined on Γ which s is to be applied to. When Ω is two dimensional, as in this thesis, s = (I ˆn ˆn) = ( ) x ˆτ(ˆτ ) = ˆτ + y = ˆτ d, so then no extension of the function to which s x s y ds

40 s is being applied is needed. We treat the jump condition by including it as a singular source term on the right hand side of the momentum equation, ρ Ω ( u t ) + (u )u + p µ u = δ (Γ, (σκˆn + s σ), x) in Ω, (.4) where δ(γ, F(X(s)), x) is a delta function as in [45], supported on Γ in Ω with strength F(X(s)). Here, x is a point in Ω, and X(s) is a point along Γ. The interface, Γ, is represented using the Segment Projection Method (SPM) [44]. This method allows us to have an explicit representation of the interface while still having a function representation to work with. We advect this interface with the fluid velocity, u. This method is described in detail in Chapter 3. Surfactants are surface reacting agents which are absorbed to the interface as a monomolecular layer as is shown in Figure.. These molecules tend to have a hydrophobic head and a hydrophilic tail, which causes them to be attracted to interfaces between fluids. The surface tension coefficient from above, σ, depends on the surfactant concentration, ρ, through a constitutive law [3, 7]. There are numerous different laws that can be used to relate σ and ρ. Here, we present two equations of state, one linear, σ = σ c RT ρ (.5) and one nonlinear, ( σ = σ c + RT ρ log ρ ). (.6) ρ

41 FLUID Γ FLUID Figure.: Surfactant molecules are absorbed to the interface, Γ, as a monomolecular layer, aligned such that their hydrophobic ends are in one fluid and their hydrophilic ends are in the other [34]. In these equations, σ c is the surface tension of the clean interface, R is the universal gas constant, T is the temperature, and ρ is the maximum surfactant concentration possible on Γ [3]. The nonlinear equation is the Langmuir equation of state and assumes that there are no adhesive or repulsive interactions between the surfactant molecules. In [3], they present theory and numerical simulations explaining and showing the effects of this assumption. The linear equation of state is only a good approximation when ρ << ρ. Further limitations of these models will be discussed in Section.3. For the surfactant, ρ, we solve an advection diffusion equation along the interface, Γ, Dρ Dt + ρ( s u) = D Γ sρ, (.7) where Dρ Dt is the material derivative which depends on how Γ is represented, D Γ is the diffusion coefficient along the interface, and s is the same surface gradient operator as used above in the force that Γ applies on the fluid (in equation.3). This equation is derived by following Stone [38]. The total surfactant along a 3

42 material element, S(t), of the interface Γ should be conserved, D ρ dx =. (.8) Dt S(t) Conducting a change of variables to Lagrangian coordinates, a, we are then able to take the derivative inside the integral, D ρ J da = Dt S o S o Dρ Dt J + ρd J Dt da =, (.9) where J is the Jacobian, J ij = X i a j, and S o is the interface element initially. Using D J Dt = ( s u) J where u is the velocity of the interface, we have S o Dρ Dt J + ρ( s u) J da =. (.) A change of variables back to Eulerian coordinates gives, S(t) Dρ Dt + ρ( s u) dx =, (.) for any arbitrary material element S(t). Therefore, Dρ Dt + ρ( s u) =. (.) In previous works, for example in Stone [38], and others [5, 49, 4], there is often a curvature term in the different ways this equation is expressed. This is due to how the different terms in equation. can be split and rewritten. Here, we 4

43 show one example of how this equation can be reformulated. The second term of equation. can be rewritten as, ρ( s u) = u s ρ + u s ρ + ρ( s u) (.3) = u s ρ + s (uρ) (.4) = u s ρ + s (u s ρ) + s (u n ρ), (.5) since u = u s + u n, where u s = ˆτ(u ˆτ) and u n = ˆn(u ˆn). Then, the last term is expressed as, s (u n ρ) = u n s ρ + ρ( s u n ) = + ρ( s ˆn)(u ˆn) = κρ(u ˆn), (.6) since ˆn s = κˆτ. Therefore, Dρ Dt + ρ( s u) = Dρ Dt u sρ + s (u s ρ) κρ(u ˆn) =. (.7) Even though it is possible to write this equation in different forms, we use.. Also, the surfactant diffuses along the interface. This is modeled using Fick s law of diffusion resulting in D Γ sρ being added to the right hand side of equation.7 [5].. Nondimensionalization We nondimensionalize the above equations following [49]. We let the characteristic length be L and the characteristic velocity be U, and therefore the characteristic 5

44 time is L U. We define L and U when we present the results for specific problems. For the above equations, we must also define a characteristic surface tension and surfactant concentration. There are two clear choices presented in the literature [49,, 35, 4, 3, 5]. The first choice is to let the characteristic surfactant concentration be ρ, the maximum surfactant concentration possible on Γ, and the characteristic surface tension coefficient be σ c, the surface tension of a clean interface. The second choice is to nondimensionalize with ρ e, the equilibrium concentration of surfactant on the interface initially, ρ e = Γ o Γ o ρ, and σ e, the surface tension coefficient which corresponds to ρ e. Here, Γ o is the initial interface. Nondimensionalizing using these characteristic values results in the following form of the Navier-Stokes equations, u t + (u )u + p Re u = ReCa δ (Γ, (σκˆn + sσ), x) in Ω (.8) u = in Ω. (.9) Here, Re and Ca are nondimensional parameters. The Reynolds number, Re = ρ Ω UL, is the ratio of advective to viscous forces. The capillary number, Ca, is the µ ratio of viscous to surface tension forces, defined as Ca = µu σ c and as Ca = µu σ e for the second. for the first choice For the first choice, where we are using ρ and σ c to nondimensionalize the equations, the linear equation of state for σ takes the form, σ = Eρ, (.) 6

45 and the nonlinear equation of state takes the form, σ = + E log( ρ), (.) where E is a nondimensional parameter. The elasticity number, E = RT ρ σ c, is how much the surface tension is affected by surfactants being absorbed to the interface. In the second choice, where the equations are nondimensionalized using ρ e and σ e, the linear equation of state for σ takes the form, σ = Exρ Ex, (.) and the nonlinear equation of state takes the form, σ = + E log( xρ) + E log( x), (.3) where E and x are both nondimensional parameters. The elasticity number is defined as in the first choice above and x = ρ e ρ (.4) is the coverage of the interface by surfactants. Finally, the nondimensional equation for the surfactant concentration, ρ, whether nondimensionalized using ρ or ρ e, is, Dρ Dt + ρ( s u) = P e sρ, (.5) Γ 7

46 where P e Γ = UL D Γ diffusive forces along Γ. is the interfacial Peclet number, the ratio of advective forces to We choose to nondimensionalize ρ by ρ and σ by σ c. When studying the effects of varying amounts of surfactant, it is ideal to not have the characteristic surface tension vary. The fixed characteristic surface tension makes it easier to understand the phenomena for a fixed capillary number, since the capillary number depends on the characteristic surface tension. Also, for soluble surfactants, ρ is a varying Γ quantity and therefore not ideal to use for the nondimensionalization of ρ..3 Limitations of the Equation of State for the Surface Tension Coefficient The equations of state for the surface tension coefficient, σ, are a model of the physical dependence the surface tension has on the surfactant concentration. The linear equation of state, when nondimensionalized by ρ, is a good approximation only when ρ <<. When nondimensionalized by ρ e, it is a good approximation only when ρx <<. The models presented above (equations. to.3) are problematic in that they can predict negative values of the surface tension coefficient. This issue is discussed in detail in [5]. For the situation where ρ is nondimensionalized using ρ and σ is nondimensionalized by σ c, negative values of σ do not occur for small values of E. In Figure.3, we show the maximum value ρ can have and σ still be positive for varying 8

47 elasticity numbers for this nondimensionalization. Since we are nondimensionalizing by ρ, the nondimensional maximum surfactant concentration possible on Γ is. Most of the results in this thesis use the elasticity number, E =.. Therefore in Figure.4 we show σ as a function of ρ for E =.. When σ is nondimensionalized by σ e and ρ by ρ e, it is xρ, where x is the coverage given by equation.4, that takes the role of ρ previously. In Figure.5, we show the maximum value that ρ can have and σ still be positive for varying coverage when E =.. In these figures, the dotted line shows the nondimensional ρ,. In the plot on the right, the dotted line is just slightly above the solid line, x so the maximum value of ρ that still gives a positive σ is just below the maximum surfactant concentration possible on the interface. In Figure.6, we show σ as a function of ρ for x =.4 as an example of one value of x and E. 9

48 ρ E ρ E Figure.3: Plot of the maximum surfactant concentration, ρ, at which the surface tension coefficient σ is positive for varying values of the elasticity number, E. Here, the equations are nondimensionalized by ρ and σ c.the plot on the left is for the linear equation of state and the plot on the right is for the nonlinear equation of state σ ρ Figure.4: Plot of σ versus ρ for E =.. The dashed line is the linear equation of state and the solid line is the nonlinear equation of state. Here, the equations are nondimensionalized by ρ and σ c.

49 ρ x ρ x Figure.5: Plot of the maximum surfactant concentration, ρ, at which the surface tension coefficient σ is positive for varying coverage, x, and fixed elasticity number, E =.. Here, the equations are nondimensionalized by ρ e and σ e. The plot on the left is for the linear equation of state and the plot on the right is for the nonlinear equation of state σ ρ Figure.6: Plot of σ versus ρ for x =.4 and E =.. The dashed line is the linear equation of state and the solid line is the nonlinear equation of state. Here, the equations are nondimensionalized by ρ e and σ e.

50 Chapter 3 Numerical Method 3. Interface Representation: Segment Projection Method We use the Segment Projection Method (SPM) to represent the interface [44]. This method gives an explicit representation of the interface defined on a uniform grid, which allows for the use of standard finite difference schemes to solve the equations numerically. The idea of the SPM is to have multiple segments which define the interface, where each segment is defined by a function. In two dimensions, the functions are of one variable. The segments are either x-segments with the curve segment described by (x, f(x, t)) or y-segments with the curve segment described by (g(y, t), y). The idea of splitting an interface into four segments is shown in Figure 3.. The functions, f(x, t) and g(y, t), are discretized on a uniform one dimensional grid as is

51 Figure 3.: An example of how an interface is divided into segments. The top figure shows the entire interface, the middle figure shows the x-segments and the bottom figure shows the y-segments. In the middle and bottom figures, the uniform grid on which these segments are defined is also shown. 3

52 shown here. We must keep track of the connectivity between segments and interpolate between the overlapping regions of segments so we have no discrepancies. Finally, as the segments are advected we may need to add and remove segments dynamically, so each segment can maintain a function representation. In this thesis, we have four fixed segments at all times. Therefore, we do not add or remove segments. Two of the segments are x-segments and two of the segments are y-segments as in Figure Advection of the Interface Given a velocity field, u = (u, v), defined on the segments, the equation to advect an x-segment given by (x, y) = (x, f(x, t)) is f t + uf x = v, (3.) and for a y-segment given by (x, y) = (g(y, t), y), it is g t + vg y = u. (3.) Following [44], the advection equation for an x-segment is derived by just looking at how that function changes with time, dy dt = f t + f dx x dt, (3.3) 4

53 or equivalently since dx dt = u and dy dt = v, v = f t + f x u, (3.4) as desired. The derivation is the same for a y-segment ((x, y) = (g(y, t), y)) but now we look at dx dt and get, dx dt = g t + g dy y dt, (3.5) or equivalently, u = g t + g y v. (3.6) 3.. Surfactant on the Interface From Chapter, the equation for the surfactants, ρ, on the interface, Γ is, Dρ Dt + ρ( s u) = P e sρ. (3.7) Γ When we use the SPM as the representation of the interface, we view ρ as a function of (x, t), defined at (x, f(x, t)) for an x-segment and as a function of (y, t), defined at (g(y, t), y) for a y-segment. This idea is shown for one x-segment in Figure 3.. As ρ is defined on the segment and solved for on the segments, we must interpolate ρ on the overlapping regions of the segments as we do for the interface location. 5

54 (x 5,f 5 ) Γ(t)=(x,f(x,t)) x 5 (x,ρ(x,t)) (x 5,ρ 5 ) x 5 ρ(x,t) on Γ(t)=(x,f(x,t)) (x 5,f 5,ρ 5 ) x 5 Figure 3.: An example of one x-segment and how ρ is a function of (x, t) that is defined at (x, f(x, t)). The top figure shows (x, f(x, t)), and the middle figure shows (x, ρ(x, t)). The bottom figure shows ρ(x, t) at (x, f(x, t)), where the shape of the curve gives the location of the interface and the color shows the surfactant concentration at that point on the interface. 6

55 The equation for ρ, when Γ is represented using the SPM, can be expressed as ( ) ρ t + uρ x + ρ (u x + v x f x ) = ρ x + f x P e Γ + f x + f x x, (3.8) for an x-segment and as ( ) ρ t + vρ y + ρ (v y + u y g y ) = ρ + g y P e Γ + g y + g y y y, (3.9) for a y-segment. There are two different ways to think about this derivation. Here, we will consider both. First, we will begin with equation 3.7 and rewrite the material derivative, s u, and the diffusion term using the SPM as the representation of the interface. First, we assume we are working with an x-segment where (x, y) = (x, f(x, t)). Then, the material derivative is, Dρ Dt = ρ + u ρ (3.) t = ρ ( ρ + (u, v) t x, ρ ) (3.) y = ρ t + u ρ x, (3.) since ρ is not a function of y. Similarly for a y-segment where (x, y) = (g(y, t), y), since ρ is not a function of x, we get Dρ Dt = ρ t + v ρ y. (3.3) 7

56 To derive the stretching term, s u, using this interface representation, we note that s u = = ( ˆτ d ) u (3.4) ds ( ) x u s x + x s y u s y + x s y v s x + ( ) y v s y, (3.5) where s is arclength along Γ. Since we can think of the velocity field, u = (u, v), as existing on the segment, in the case of an x-segment the velocity field is a function of (x, t) and in the case of the y-segment the velocity is a function of (y, t). So then the above simplifies to for an x-segment, since x y = s +g y ( ) x u s u = s x + x y v s s x ( ) x u = s x + x y x v s x s x = u + fx x + f v + fx x x (3.6) (3.7) (3.8) = (u x + v x f x ) + f x (3.9) = s +fx and y = f(x, t). and x = g(y, t), we similarly derive s u = (vy+uygy) +g y. For a y-segment, since Finally, to derive the diffusion term, we begin with showing sρ = d ρ ds, where 8

57 s is arclength as before. So, sρ = s s ρ (3.) = ˆτ d dρ ˆτ ds ds = x ( ) d x dρ + y ( d y s ds s ds s ds s ( ( x ) ( ) ) y d ρ = + s s ds + x s = d ρ ds + dρ ( x x ds s s + y ) y s s = d ρ ds + dρ ( ) ˆτ ds s ˆτ = d ρ ds + dρ ( ) (ˆτ ˆτ) ds s = d ρ ds + dρ ( ) ds s ( ˆτ ) ) dρ ds dρ x ds s + y s (3.) (3.) dρ y (3.3) ds s (3.4) (3.5) (3.6) (3.7) = d ρ ds. (3.8) Then the diffusion term can be written as, s ρ = d ρ P e Γ P e Γ ds = x P e Γ s for an x-segment and P e Γ s ρ = P e Γ d ρ ds = P e Γ y s for a y-segment. x ( x s ( y y s ρ x ) ( ) = ρ x P e Γ + f x + f x ) ( ρ = y P e Γ + g y ) ρ + g y y x, (3.9) y, (3.3) 9

58 The other method of deriving equations 3.8 and 3.9 begins with the conservation of the surfactant on the interface as in Chapter, equation.8, and rewriting that equation using the SPM representation. Then the diffusion is added in just as was done in the previous derivation [8]. We know that the surfactant on the interface must be conserved (assuming no sources or sinks to or from the interface) on a material element, S(t), of Γ(t), D ρ dx =, (3.3) Dt S(t) where S(t) = {(x, f(x, t)) : a(t) < x < b(t)} for an x-segment. Changing variables, we have D Dt b(t) a(t) ρ(x, t) + (f x (x, t)) dx =. (3.3) Then bringing the derivative inside the integral results in = b(t) a(t) ρ t (x, t) + (f x (x, t)) + ρ(x, t) + (fx (x, t)) f x(x, t)f xt (x, t) dx +b (t)ρ(b(t), t) + (f x (b(t), t)) a (t)ρ(a(t), t) + (f x (a(t), t)). (3.33) 3

59 Using equation (3.) and noting that a (t) = u(a(t), t) and b (t) = u(b(t), t), = = = b(t) a(t) b(t) ρ t + f x + ρf x(v x u x f x uf xx ) + f x + (ρu + f x) x dx (3.34) ρ t + f x + ρf x(v x u x f x uf xx ) a(t) + f x +ρ x u + fx + ρf xf xx u + ρu x + f + f x dx (3.35) x b(t) ρ t + f x + ρf xv x ρu x fx a(t) + f x +ρ x u + fx + ρu x + f x dx. (3.36) This holds for any interfacial element, S(t), so ρ t + f x + ρf xv x ρu x fx + ρ x u + f + f x + ρu x + f x =. (3.37) x Now manipulating, = ρ t + ρf xv x ρu x f x + f x + ρ x u + ρu x (3.38) = ρ t + ρf xv x ρu x f x + ρu x + ρu x f x + f x + uρ x (3.39) = ρ t + uρ x + ρf xv x + ρu x + f x, (3.4) as desired. Then, we can add in the diffusion using the same derivation as above. We can also similarly derive the equation for ρ for a y-segment, starting with the conservation of the surfactants along an arbitrary interface element. 3

60 3. Timestepping We will first introduce the timestepping assuming a given velocity field, u = (u, v), that can be evaluated at the segments (or at any point in domain, Ω), as needed. Therefore, the system we will be solving includes the advection of the four segments and solving for the surfactant concentration on the segments, f t + uf x = v (3.4) ( ) ρ t + uρ x + ρ (u x + v x f x ) = ρ x, (3.4) + f x P e Γ + f x + f x for an x-segment and, x g t + vg y = u (3.43) ( ) ρ t + vρ y + ρ (v y + u y g y ) = ρ + g y P e Γ + g y + g y, (3.44) y for a y-segment. Also, we must keep track of the connectivity between segments and interpolate the location of the interface and the surfactant concentration in overlapping regions of the segments. Strang splitting is used to step forward one timestep. The splitting method is based on the methods shown in [6]. The steps used in the timstepping are, Step A y 3

61 (x-segments) f t + uf x = v (3.45) ρ t =, (3.46) and Step B (x-segments) f t = (3.47) ( ) ρ t + uρ x + ρ (u x + v x f x ) = ρ x (3.48) + f x P e Γ + f x + f x Step A and Step B, above, are just shown for x-segments. They are similar for y- segments, where equation 3.45 in Step A is replaced by equation 3.43, and equation 3.48 in Step B is replaced by equation During each timestep, we must also evaluate the velocity field on the interface and do a reconstruction of the interfaces and of the interfacial surfactant concentration, as stated above. The scheme to step one timestep, t, forward from t to t + t is, Step A (timestep t ) Evaluate the velocity on the interface x Step B (timestep t) Step A (timestep t ) 33

62 Reconstruction of the interface and ρ on the interface Evaluate the velocity on the interface (3.49) For each step here, we loop over the four segments. Also, it is important to note that for each step we use the newest values possible. For example, when evaluating ρ in Step B, we use the resulting f and g from the previous Step A in those equations. The velocity is handled as presented in this scheme because it will allow us to take just one Navier-Stokes step per timestep when we include solving for the velocity field in the system. Then, we can interpolate the velocity to the location of the interface, as needed. Step B in scheme 3.49 is solved using Strang splitting again. The steps used to solve the equations for ρ are, Step B(A) (x-segments) f t = (3.5) ρ t + uρ x =, (3.5) Step B(B) 34

63 (x-segments) f t = (3.5) ρ t + ρ (u x + v x f x ) + f x =, (3.53) and Step B(C) (x-segments) f t = (3.54) ( ) ρ t = ρ x. (3.55) P e Γ + f x + f x Once again, the steps for the x-segments are just shown, and the steps for the y-segments are similar. Step B(A) updates the advection term of equation 3.48, Step B(B) updates the source term in the equation, and Step B(C) updates the diffusion term. taken, In scheme 3.49 when Step B is advanced by t, the following substeps are x Step B(A) (timestep t ) Step B(B) (timestep t ) Step B(C) (timestep t) Step B(B) (timestep t ) 35

64 Step B(A) (timestep t ). (3.56) Once again, for each step, we loop over the four segments. Also, at each step we are using the newest values of ρ, f, and g available. Strang splitting is used to give second order accuracy [39]. In Chapter 7 of [6], there is a discussion of different time splitting methods and analysis giving the order of accuracy is shown for certain cases. Here, we are using splitting in a dimensional manner, so that we solve for only one variable at a time. Also, we are using splitting when solving for ρ, allowing us to handle the different parts of the equation (the advection term, the diffusion term, and the source term) using different methods. LeVeque states that the analysis he presents is applicable to both of these cases. In [6], two different splitting methods are presented, so called Godunov and Strang splitting. Godunov splitting is when we solve the equations by consecutively taking one t timestep of each substep, to complete a full timestep of size t.. Strang splitting is what we do above when we use splitting. Strang splitting has a symmetry in the steps taken. The idea is to first take a t timestep of the first step of the equation and then a t timestep of the second step, and then a t timestep of the first step again. We use this symmetric idea but generalize it to allow us to split more than once. This type of splitting was presented by Strang in [39], where he studies a linear hyperbolic problem being solved in two dimensions using the Lax-Wendroff finite difference method. Godunov splitting gives only first order accuracy in time while Strang split- 36

65 ting gives second order accuracy. LeVeque in [6] presents the analysis to show the order of convergence in time for these two splitting methods. He studies a one dimensional variable where splitting is used to handle the different parts of the equation. In this example analysis, the assumption is made that the linear operators do not depend explicitly on time for simplification. Strang also shows this result when using Strang splitting for his model problem and also discusses stability in [39]. In the next few sections we discuss how we discretize Step A and Step B from above. In these sections, we assume that we are stepping from time, t n, to time, t n+, and that t n+ t n = dt in each step. The timestep, dt, is t or t as stated in the schemes above. When we state that we have specific values at t n, we mean the newest values available for a variable at that step. So for example in Step B, f n and ρ n are the f and ρ we have solved for in the previous Step A. Also, in these discretizations, x and y, are the gridsizes of the x and y-segments, respectively. 3.3 Discretization of the Interface Advection Equations Here, we discretize Step A, equations 3.45 and 3.46, for both x and y-segments, (x-segments) f t + uf x = v (3.57) ρ t =, (3.58) 37

66 and (y-segments) g t + vg y = u (3.59) ρ t =. (3.6) Once again following [44], the second order Lax-Wendroff scheme is used to discretize the equations for f and g. Assuming we have f, ρ, u, and v at time, t n, for an x-segment, we find f (and therefore u and v) at time t n+ where t n+ t n = dt using, f n+ j = f n+ j + dt ( [ ]) u n j (Do u n j )(D o fj n ) + u n j D + D fj n D o vj n (ṽn + dt j vj n ũn j u n ) j D o fj n, (3.6) dt dt where D o w n j = wn j+ w n j x (3.6) D + D w n j = wn j+ w n j + w n j x (3.63) f n+ j = f n j + dt(v n j u n j D o f n j ), (3.64) (w can be f, u, or v) and ũ n j and ṽ n j are velocities evaluated at the temporary surface position f n+ j. Solving for ρ just gives ρ n+ = ρ n. Similarly for the y-segments, we assume that we have g, u, and v at time, t n, 38

67 and solve for g (and therefore u and v) at time t n+, where t n+ t n = dt using, g n+ j = g n+ j + dt ( [ ]) v n j (Do vj n )(D o gj n ) + vj n D + D gj n D o u n j (ũn j u n ) j + dt dt ṽn j v n j dt D o g n j, (3.65) where g n+ j = g n j + dt(u n j v n j D o g n j ). (3.66) In these equations, D o w n j and D + D w n j are defined as in equations 3.6 and 3.63 with x replaced by y for the y-segments (w can be g, u, or v), and ũ n j and ṽ n j are velocities calculated at the temporary surface position g n+ j. Once again for ρ, ρ n+ = ρ n. The equations for f and g are solved using quadratic (second order) extrapolation boundary conditions. So if a segment has N points we set the boundary condition using the following, w N+ = 3w N 3w N + w N (3.67) w = 3w 3w + w 3, (3.68) where w is either f or g. Also, if we need values of u and v at the boundaries we compute them using the same extrapolation. 39

68 3.4 Discretization of the Surfactant Concentration Equations To solve the equations for the surfactant concentration on the interface, ρ, (x-segments) and (y-segments) f t = (3.69) ( ) ρ t + uρ x + ρ (u x + v x f x ) = ρ x, (3.7) + f x P e Γ + f x + f x x g t = (3.7) ( ) ρ t + vρ y + ρ (v y + u y g y ) = ρ + g y P e Γ + g y + g y, (3.7) y we use splitting as shown above in the timestepping. So here we will discuss how to discretize the three different steps, B(A), B(B) and B(C). Step B(A), the advection of ρ, equations 3.5 and 3.5, are discretized in the same manner as the advection of the interface is discretized, using the second order Lax-Wendroff scheme. For x-segments, assuming we have ρ n, f n, u n and v n, we can evaluate ρ n+, f n+, u n+, v n+, where t n+ t n = dt, using y ρ n+ j = ρ n j dtu n j D o ρ n j + dt un j (D o u n j D o ρ n j + u n j D + D ρ n j ) dt u n+ j u n j D o ρ n j, dt (3.73) 4

69 where D o w n j and D + D w n j are defined in equations 3.6 and 3.63 (w can be u or ρ). This simplifies, since the equation for f n+ gives f n+ = f n and therefore u n+ = u n, to ρ n+ j = ρ n j dtu n j D o ρ n j + dt un j (D o u n j D o ρ n j + u n j D + D ρ n j ). (3.74) Similarly for y-segments, ρ n+ j = ρ n j dtv n j D o ρ n j + dt vn j (D o v n j D o ρ n j + v n j D + D ρ n j ), (3.75) since g n+ = g n and v n+ = v n. We treat the boundary conditions here as we did above for the advection of the interface, using quadratic extrapolation boundary conditions for ρ. For Step B(B), the source terms which handle the surfactant concentration as the interface stretches, equations 3.5 and 3.53, we solve the ordinary differential equation exactly, but still approximate u x, v x, and f x and v y, u y and g y. So then the discretization is, where t n+ t n = dt, ρ n+ j = ρ n j e dt Dou n j +Dovn j Dofn j +(Dof j n), (3.76) for an x-segment and ρ n+ j = ρ n j e dt Dov j n +Doun j Dogn j +(Dog j n), (3.77) for a y-segment. Again here we have ρ n, f n, g n, u n, and v n and are solving for ρ n+ (since f n+ = f n or g n+ = g n and therefore u n+ = u n and v n+ = v n ). We 4

70 use quadratic extrapolation boundary conditions for f or g, u, and v. Finally, we treat the diffusion Step B(C), implicitly on the interface using ideas of domain decomposition with the segments. We use the multiplicative Schwarz method as discussed in Section.4. of [47] and in Section 6.. of []. The idea is to solve ρ t = P e Γ sρ on each segment with boundary conditions from the neighboring segments in an iterative manner. Figure 3.3 helps to explain the scheme. The top figure shows all four segments combined to form Γ on which we want to solve the diffusion equation. The middle figure shows segments and, and the bottom figure shows segments 3 and 4. Note that these figures have much coarser grids and much fewer points than when actually conducting simulations. We only solve the diffusion equation for the surfactant concentation on the black points shown in the figure. We then interpolate the solution at the blue and red points from the neighboring segments (since these points are always in the overlapping region between segments). First, we solve ρ t = P e Γ sρ on segments and with Dirichlet boundary conditions at the blue points. We take as an initial guess this boundary condition to be ρ n j at the blue points. Then, we solve for ρ on segments 3 and 4 with Dirichlet boundary conditions at the blue points. The values of ρ at the blue points on segments 3 and 4 are interpolated from the ρ just solved for on segments and. We then again solve for ρ on segments and using Dirichlet boundary conditions, using interpolated values for ρ at the blue points from the newest values of ρ on segments 3 and 4. We continue in this manner until ρ converges on the entire interface. Whenever interpolation is used in this method, we use cubic 4

71 Figure 3.3: This figure explains the domain decompsition scheme used on the segments to solve for the diffusion of the surfactants. The top figure shows all the segments forming Γ on which we want to solve for ρ. The middle figure shows segments and and the bottom figure shows segments 3 and 4. The black points are where we solve the system for ρ on each segment, and the blue points are where we apply the Dirichlet boundary conditions in the iterative scheme. 43

72 interpolation. The maximum of the relative error is used to decide when to stop iterating. The tolerance is set to. For all the given velocity field cases shown in Section 3.6, the scheme needed less than ten iterations to converge. The convergence analysis is presented in [47]. Demmel, [], summarizes some results pertaining to our scheme. He states that if solving on two domains with an overlap region and solving the systems on the domains exactly, as in our case, the number of iterations is independent of gridsize. This is the behavior we see for the different simulations run on varying gridsizes in Section 3.6. Now we will discuss how we solve for ρ on each segment. We need to discretize equations 3.54 and 3.55 on each x-segment and the corresponding equations on each y-segment. We use a Crank-Nicolson scheme for time integration to step from time, t n, to time, t n+, where as before t n+ t n = dt, ρ n+ j = ρ n j + P e Γ s n j + P e Γ s n+ j ( dt x ( dt x s n j+ s n+ j+ In this equation, when solving on an x-segment, ) (ρ n j+ ρ n j ) (ρ n s n j ρ n j ) j ) (ρ n+ j ρ n+ j ).(3.78) (ρ n+ j+ ρn+ j ) s n+ j s n j+ s n j = = ( ( f n j+ fj n + x ( ( f n j+ fj n + x ) ) ) ) (3.79). (3.8) 44

73 These terms, s n j and s n, are similarly defined for a y-segment with f replaced by j+ g and x replaced by y. Since, f t = and g t = (and therefore f n+ j = fj n and g n+ j = gj n ), s n+ j = s n j, s n+ = s n, and s n+ = s n. j+ j+ j j This leads to a tridiagonal system being solved for each segment, B C A B C A N B N C N A N B N ρ n+ ρ n+ ρ n+ N = Ẽ E E N Ẽ N, (3.8) where A j = dt P e Γ x s n j B j = + dt P e Γ x s n j C j = dt P e Γ x s n j E j = ρ n j + P e Γ s n j s n j ( s n j+ dt x ( s n j s n j+ ) + s n j+ (3.8) (3.83) (3.84) ) (ρ n j+ ρ n j ) (ρ n s n j ρ n j ). (3.85) j (Note that for a y-segment, each x would be replaced by a y.) In this system, 45

74 Ẽ and ẼN are just E and E N plus the Dirichlet boundary conditions set at the blue points discussed in the Schwarz method above. 3.5 Reconstruction In Section 3., we include a reconstruction step for the location of the segments and the interfacial surfactant concentration. These steps are included in the description of the SPM in [44]. Also, many more details of the implementation of the SPM, as well as more results are presented in [43]. These references discuss the update of just the location of the interface and not how to handle a function which is defined on Γ such as ρ. Remember in our situation, the number of segments is fixed to four. The methods shown in these references include the addition and removal of segments. In the reconstruction, first the extrema closest to the end of each segment is found. We know the endpoints of each segment will be modified depending on the neighboring segments extrema. The overlapping region is kept as large as possible for each segment, while making sure it is still a function of the appropriate axis. Next, we remove points from a segment or add points to a segment so that it is defined up to its neighboring segments extrema. We add and remove points by keeping track of where the segment is defined on the axis (we just note the endpoints on the independent axis and modify these in this reconstruction). When we add points, we also need to add values of f and g and ρ. We do this by interpolating on the neighboring segment and evaluating the value at the point we have added on the segment. This is possible because points added are always in 46

75 an overlapping region between two segments. Also, at this time we replace the two points at each end of the segment (that were there before we added or removed any points) from the neighboring segment. We do this because we have been using extrapolating boundary conditions to solve the equations for f, g, and ρ. Finally, we blend the overlapping regions between two segments, since the segments and surfactants on the segments were updated independently but are considered to be representing the same section of the interface, Γ. We do this by interpolating values of f or g and ρ from the neighboring segment to a point (x j, f j ) on a x-segment or to a point (g j, y j ) on a y-segment. We denote the interpolated value, fj or g j and ρ j. We set the new values as a weighted average between the old values already on the segment and the interpolated values from the neighbor, f new j = f j + θ(m j )( f j f j ) (3.86) g new j = g j + θ(m j )( g j g j ) (3.87) ρ new j = ρ j + θ(m j )( ρ j ρ j ), (3.88) where m j is the slope of f or g. This is done to give the largest weight to the segment representation with the smallest slope. We choose, θ(m) = β β ( ) m < β m β m β β m > β. (3.89) 47

76 We set β =, as it is considered the default value in [44]. Anytime we interpolate from the neighboring segment in the reconstruction, we use cubic interpolation. The blending completes the reconstruction of both the interface and the surfactant concentration on the interface and we are ready to begin the next timestep. 3.6 Order of Convergence In this section, we show that with a given velocity field the numerical scheme described in this chapter is second order. The results are shown with three different given velocity fields, varying P e Γ, and varying initial concentrations of surfactant, ρ o. In this study, only the equations to advect the interface and the equations for ρ are being solved. We nondimensionalize as described in Section., with ρ nondimensionalized by ρ. We have the exact solution of the interface in all cases and the exact solution of ρ when P e Γ =. When P e Γ =, we take the error for ρ on a grid with size h to be the difference, ρ h ρ h. We compute the L and L norm of the error for all cases. We have found that the area of the drop and total surfactant along the interface have a convergence rate of two. Here, we plot the relative error in the area and in the total surfactant versus time for one gridsize. All simulations here were run until time, t = 3. 48

77 The first given velocity field tested was an oscillating ellipse, u = cos(πt) cos(y.5) sin(x.5) (3.9) v = cos(πt) sin(y.5) cos(x.5). (3.9) The initial interface was set to be a circle centered at (.5,.5) with radius.35. With this given velocity field we tested two initial conditions of the surfactant concentration, ρ o =.5 and ρ o = (sin(θ) + )/3, θ = arctan(y.5, x.5), and two Peclet numbers, P e Γ = and P e Γ =. In Figure 3.4, the advection of the interface is shown, which remains the same for all four cases, and in Figure 3.5, we show the nondimensionalized surfactant concentration for the case with ρ o =.5 and P e Γ =. In Figures 3.6, 3.7, and 3.8, we see that the convergence of the interface and the surfactant concentration is second order. In these cases, t =.75 x. Also, the relative error in the area inside the drop and the relative error in the total surfactants along Γ for each case is shown when x = y =.5. In all the convergence plots in this section the green line is a reference line with slope. 49

78 Γ at t= Γ at t= Γ at t= Γ at t= Γ at t= Γ at t=3 Figure 3.4: The interface at varying times with the oscillating ellipse given velocity field defined by equations 3.9 and

79 ρ at t= ρ at t= ρ at t= ρ at t= ρ at t= ρ at t=3 Figure 3.5: The surfactant concentration along the interface at varying times with the corresponding Γ shown in Figure 3.4. Initially, the surfactant concentration was set to be a constant along the interface, ρ o =.5. The surfactants are plotted versus the arclength of the interface. The stars in Figure 3.4 correspond to when the arclength here is. 5

80 Error x = y A( t ) A( ) A( ) 6 x time Figure 3.6: The error in the location of the interface for the oscillating ellipse case, equations 3.9 and 3.9. The green line in all the convergence plots in this section is a reference line with slope. The plot on the right shows the relative error in the area within the drop versus time when x = y =.5. In this plot, A(t), is the area in the drop at time t, measured numerically. 5

81 Error x = y ρ ( t ) ρ ( ) ρ ( ) 6 x time Error x = y ρ ( t ) ρ ( ) ρ ( ) 6 x time Figure 3.7: The error in ρ when ρ o =.5. The plots on the right show the relative error in the total amount of surfactant versus time. Here, ρ(t) is the total amount of surfactant along Γ at time t, computed numerically. The top plots correspond to P e Γ =, and the bottom plots correspond to P e Γ =. 53

82 Error x = y ρ ( t ) ρ ( ) ρ ( ) 6 x time Error x = y ρ ( t ) ρ ( ) ρ ( ) 6 x time Figure 3.8: The error for ρ when ρ o = (sin(θ) + )/3, θ = arctan(y.5, x.5). The plots on the right show the relative error in the total surfactants versus time. The top plots correspond to P e Γ = and the bottom plots correspond to P e Γ =. 54

83 Next, we tested a rigid body rotation, u = π (x.5) + (y.5) 3 sin φ (3.9) v = π (x.5) + (y.5) 3 cos φ, (3.93) where φ = arctan(y.5, x.5). The initial interface was set to be a circle centered at (.5,.) with radius.7. We tested the same four cases as for the oscillating ellipse given velocity field. In Figures 3.9 and 3., the interface location is shown and the surfactant concentration for the case with ρ o = (sin(θ) + )/3, θ = arctan(y., x.5), and P e Γ = is shown. In Figures 3., 3., and 3.3, we see that the convergence of the interface location and the surfactant concentration is second order as in the previous case. In these cases, t =.5 x. For the cases with ρ o =.5, we show no error plots for ρ because the error was of machine precision. Also, we plot the relative error in the area inside the drop and the relative error in the total surfactants along Γ for each case when x = y =.5. Note that for the cases where ρ o =.5, we see no difference in the error in the total surfactants for the varying P e Γ numbers. This is expected since ρ stays constant throughout the rigid body rotation in all of these cases. 55

84 Γ at t= Γ at t= Γ at t= Γ at t= Γ at t= Γ at t=3 Figure 3.9: The interface at varying times when advected with the rigid body rotation velocity field, given by equations 3.9 and

85 ρ at t=.5.5 ρ at t= ρ at t= ρ at t= ρ at t= ρ at t=3 Figure 3.: The surfactant concentration at varying times with the corresponding Γ shown in Figure 3.9. Initially, the surfactants were set to be ρ o = (sin(θ) + )/3, where θ = arctan(y., x.5). The surfactants are plotted versus the arclength of the interface. The stars in Figure 3.9 correspond to when the arclength here is. 57

86 Error x = y A( t ) A( ) A( ) 4 x 4 3 time Figure 3.: The error in the location of the interface for the rigid body rotation case, equations 3.9 and The plot on the right shows the relative error in the area inside the drop versus time when x = y =.5. ρ ( t ) ρ ( ) ρ ( ).5 x 4 3 time ρ ( t ) ρ ( ) ρ ( ).5 x 4 3 time Figure 3.: The relative error in the total surfactants versus time when ρ o =.5. The left plot corresponds to P e Γ = and the right plot corresponds to P e Γ =. 58

87 Error x = y ρ ( t ) ρ ( ) ρ ( ) 4 x 4 3 time Error x = y ρ ( t ) ρ ( ) ρ ( ).5 x 4 3 time Figure 3.3: The error in ρ when ρ o = (sin(θ)+)/3, θ = arctan(y., x.). The plots on the right show the relative error in the total surfactants versus time. The top plots correspond to P e Γ = and the bottom plots correspond to P e Γ =. 59

88 The last case we tested was a rotating velocity field, u = sin (πx) sin(πy) (3.94) v = sin (πx) sin(πx). (3.95) The initial interface was set to be a circle centered at (.3,.3) with radius.. We run the velocity field forward until time t =.3 and then reverse it, and continue in this periodic manner until time, t = 3. We tested the same four cases as for the previous two given velocity fields. In Figures 3.4 and 3.5, the interface location and the surfactant concentration is shown for the case with ρ o =.5 and P e Γ =. In Figures 3.6, 3.7, and 3.8, we see that the convergence of the interface location and the surfactant concentration is second order as with the previous velocity fields. In these cases, t =.75 x. Also, we plot the relative error in the area within the drop and the relative error in the total surfactants for each case when x = y =.5. Note, that in the cases where P e Γ =, the rate of convergence (measured numerically) is higher than second order. With the previous two velocity fields, Figures 3.7 and 3.8, and Figure 3.3, we did not note a higher rate of convergence when P e Γ =, as we do here. 6

89 Γ at t= Γ at t= Γ at t= Γ at t= Γ at t= Γ at t=.6 Figure 3.4: The interface at varying times with the rotating velocity field defined in equations 3.94 and We only show from time, t =, to time, t =.6, because the deformation of the drop is just repeated until time, t = 3. 6

90 ρ at t=.5.5 ρ at t= ρ at t= ρ at t= ρ at t= ρ at t=.6 Figure 3.5: The surfactant concentration at varying times with the corresponding Γ shown in Figure 3.4. Initially the surfactant concentration was set to a constant along the interface, ρ o =.5. The surfactants are plotted versus the arclength of the interface. The stars in Figure 3.4 correspond to when the arclength here is. 6

91 Error x = y A( t ) A( ) A( ) 4 x 4 3 time Figure 3.6: The error in the interface location for the rotating velocity field, equations 3.94 and The plot on the right shows the relative error in the area within the drop versus time when x = y =.5. 63

92 Error 3 3 x = y ρ ( t ) ρ ( ) ρ ( ).5 x 3 3 time Error 4 3 x = y ρ ( t ) ρ ( ) ρ ( ).5 x 3 3 time Figure 3.7: The error in ρ when ρ o =.5. The plots on the right show the relative error in the total surfactants versus time. The top plots correspond to P e Γ = and the bottom plots correspond to P e Γ =. 64

93 Error 4 3 x = y ρ ( t ) ρ ( ) ρ ( ) x 3 3 time Error 4 3 x = y ρ ( t ) ρ ( ) ρ ( ).5 x 3 3 time Figure 3.8: The error in ρ when ρ o = (sin(θ) + )/3, θ = arctan(y.3, x.3). The plots on the right show the relative error in the total surfactants versus time. The top plots correspond to P e Γ = and the bottom plots correspond to P e Γ =. 65

94 3.7 Discretization of the Navier-Stokes Equations We solve the Navier-Stokes equations, u t + (u )u + p u = f in Ω (3.96) Re u = in Ω. (3.97) using a projection scheme. The idea of the scheme is to first solve equation 3.96 for the velocity field, while neglecting the pressure term. Then this velocity field is projected to fulfill the divergence free condition [9]. We use the code found in [36] as a structure for the solver. The numerical method in [36] is based on a scheme presented in [4]. The force from the interface is denoted f in equation We discuss how to compute f in Section First, we present the timestepping used to solve the Navier-Stokes equations and then the spatial discretization. We use Einstein notation in this section to simplify and make clearer the steps taken to solve these equations. In this notation, u = (u, v) = (u, u ), x = (x, y) = (x, x ), and f = (f, f ) and i =, and j =, Timestepping We step forward from t n to t n+, where t n+ t n = dt. An intermediate velocity, u i, the pressure, p n+, and the new divergent free velocity field, u n+, are solved i 66

95 for, u i u n+ i u n i dt p n+ x i u n i dt = x j (u n i u n j ) + f n j u i + Re x j ( ) u i x j (3.98) = (3.99) dt x i = (u n i u n j ) + fj n pn+ + ( ) u i. (3.) x j x i Re x j x j This scheme is implemented using four steps. First, the nonlinear terms and force are treated by solving for an intermediate velocity, u i, u i u n i dt = x j (u n i u n j ) + f n j. (3.) Then the viscosity is treated implicitly, solving for u i, u i u i dt = Re x j ( ) u i. (3.) x j Combining these two steps is the same as solving equation The next step is the pressure correction, p n+ x i = dt u i x i. (3.3) This correction enforces the divergence free condition. This can be seen by subtracting equation 3.98 from equation 3., u n+ i u i dt = pn+ x i, (3.4) 67

96 and then taking the divergence, ( u n+ ) i u i x i dt = x i ( ) p n+, (3.5) x i and enforcing the divergence free condition for u n+, un+ i x i =, resulting in equation 3.3. Finally, the new divergent free velocity field, u n+ i, is solved for using equation 3.4, u n+ i = u i dt pn+ x i. (3.6) The timestepping here is just first order. This is because the time splitting is Godunov splitting as presented in [6] and as we are basically just using forward and backward Euler in the steps. The viscous terms are solved implicitly, which allows us to take timesteps on the order of h x and h y, where h x and h y are the spatial resolutions of the grid in the x and y direction Spatial Discretization The spatial discretization uses a staggered grid, as shown in Figure 3.9. The pressure points are in the center of a grid cell and the velocity points are at the edges of the cell. (Also, note f is defined in the same location as u and f is defined in the same location as u.) The spatial discretization of the nonlinear terms is a weighted average between a central scheme and an upwind scheme. The weighted average uses the constant, 68

97 (u ) k,l+ (u ) k,l p k,l (u ) k+,l (u ) k,l Figure 3.9: The staggered grid on which the Navier-Stokes equations are solved. γ, ( ( maxk,l (u ) k,l γ = min. dt max, max ) ) k,l (u ) k,l,. (3.7) h x h y The nonlinear terms, (u i u j ) = x j (u ) x + (u u ) x (u u ) x + (u ) x (3.8) are computed so that the first term is located in the same location as (u ) k,l and the second term is located in the same location as (u ) k,l. The central discretization 69

98 of these terms at a point (k, l) is, (u (u ) ) (u k+ ) =,l k,l x h x (3.9) (u u ) = (u ) k,l+ (u ) k,l+ (u ) k,l (u ) k,l x h y (3.) (u u ) = (u ) k+,l (u ) k+,l (u ) k,l (u ) k,l x h x (3.) (u (u ) ) (u k,l+ = ) k,l x h y, (3.) where (u i ) k+,l = (u i) k+,l + (u i ) k,l (u i ) k,l+ = (u i) k,l+ + (u i ) k,l (3.3). (3.4) The location of these values is shown in Figure 3.. Linearly combining the centrally discretized terms with the upwind scheme 7

99 gives the following discretization at a point (k, l), (u ) x = h x [( (u u ) x = [( h y ( (u u ) x = [( h x (u ) γ (u k+ ),l k+,l (ũ ) k+ ( )] (u ) γ (u k ),l k,l (ũ ) k,l (u ) k,l+ (u ) k,l,l ) (u ) k,l+ γ (u ) k,l+ (ũ ) k,l+ )] (u ) k,l γ (u ) k,l (ũ ) k,l ) (ũ ) k+,l ) (3.5) (3.6) (u ) k+,l (u ) k+,l γ (u ) k+,l ( )] (u ) k,l (u ) k,l γ (u ) k,l (ũ ) k,l (3.7) (u ) = ) [((u ) γ (u k,l+ x h ) k,l+ (ũ ) y k,l+ ( )] (u ) γ (u k,l ) k,l (ũ ) k,l. (3.8) In the upwind scheme, the following is used, (ũ i ) k+,l = (u i) k+,l (u i ) k,l (ũ i ) k,l+ = (u i) k,l+ (u i ) k,l (3.9). (3.) These terms are located at the same location as their corresponding average term (equations 3.3 and 3.4). For example, (ũ i ) k+,l is located where (u i ) k+,l is. We can rewrite the terms in the linear combinations to see how the upwinding goes 7

100 into effect, ( (u ) γ (u k+ ),l k+,l (ũ ) k+ ( (u ) k+,l ( γ)(u ) k+,l + ( + γ)(u ) ) k,l ( (u ) k+,l ( + γ)(u ) k+,l + ( γ)(u ) ) k,l ( (u ) k,l+,l ) (u ) k,l+ γ (u ) k,l+ (ũ ) k,l+ (u ) k,l+ ( ( + γ)(u ) k,l + ( γ)(u ) k,l+ ) = (3.) (u ) k+,l (u ) k+,l < ) = (3.) (u ) k,l+ ( (u ) k,l+ ( γ)(u ) k,l + ( + γ)(u ) ) k,l+ (u ) k,l+ < ( ) (u ) k+,l (u ) k+,l γ (u ) k+,l (ũ ) k+,l = (3.3) ( (u ) k+,l ( + γ)(u ) k,l + ( γ)(u ) ) k+,l (u ) k+,l ( (u ) k+,l ( γ)(u ) k,l + ( + γ)(u ) ) k+,l (u ) k+,l < ( ) (u ) γ (u k,l+ ) k,l+ (ũ ) k,l+ = (3.4) ( ( + γ)(u ) k,l + ( γ)(u ) ) k,l+ (u ) k,l+ (u ) k,l+ ( ( γ)(u ) k,l + ( + γ)(u ) k,l+ ) (u ) k,l+ (u ) k,l+ (The terms not rewritten here are very similar to the terms that are shown here.) The central scheme here is second order, but the upwind scheme is only first order. Therefore, the weighted average is between first and second order. The other terms are much simpler. The viscosity terms are discretized in the <. 7

101 (u ) k,l+/ (u ) k /,l+ (u ) k,l+ (u ) k+,l+/ (u ) k+/,l+ (u ) k,l p k,l (u ) k+,l (u ) k+/,l (u ) k,l+/ (u ) k,l / (u ) k,l (u ) k+,l / (u ) k /,l (u ) k+/,l Figure 3.: The staggered grid with all the terms used in the spatial discretization of the nonlinear terms. 73

102 following manner, x j ( ) ui x j = (u i) k,l (u i ) k,l + (u i ) k+,l h x + (u i ) k,l (u i ) k,l + (u i ) k,l+ h y, (3.5) where the location of each term is at (u i ) k,l. For the pressure correction equation, equation 3.3, each of the following terms is defined where p k,l is located, p x i = p k,l p k,l + p k+,l h x + p k,l p k,l + p k,l+ h y (3.6) u i = (u ) k+,l (u ) k,l + (u ) k,l+ (u ) k,l. (3.7) x i h x h y Finally, for the update to u n+ i, equation 3.6, located, p x i is defined where (u i ) k,l is p = p k+,l p k,l x h x (3.8) p = p k,l+ p k,l. x h y (3.9) All of the discretizations for the terms not including the nonlinear terms are second order since we use central schemes. For both u and p, the boundary conditions are set to be periodic in the x- direction. In the y-direction, Dirichlet boundary conditions are set for u and Neumann boundary conditions are set for p. The Dirichlet conditions are set to be shear flow boundary conditions that will be described in Chapter 4. The boundary 74

103 conditions are set in this manner for each step presented in the timestepping. 3.8 Coupling the Navier-Stokes Equations with the Interface and Surfactants 3.8. Timestepping In Section 3., we discussed how we step forward from time t to t + t assuming a given velocity field. When we add in the solving of the fluid flow, the Strang splitting scheme to step one timestep is now, Navier-Stokes step (timestep t ) Evaluate the velocity on the interface Step A (timestep t ) Evaluate the velocity on the interface Step B (timestep t) Step A (timestep t ) Reconstruction of the interface and ρ on the interface Navier-Stokes step (timestep t ). (3.3) 75

104 Rather then solving the Navier-Stokes step twice consecutively over two timesteps, we just solve it once. So then the code is initialized by stepping the Navier-Stokes solver t, and then the number of timesteps required are completed, Step A (timestep t ) Evaluate the velocity on the interface Step B (timestep t) Step A (timestep t ) Reconstruction of the interface and ρ on the interface Navier-Stokes step (timestep t). Evaluate the velocity on the interface. (3.3) For the final timestep, the Navier-Stokes solver is advanced t, instead of t, as is required to get the velocity field at the final time. The Navier-Stokes step included in the scheme is, f t = (3.3) g t = (3.33) ρ t = (3.34) u t + (u )u + p Re u = ReCa δ (Γ, (σκˆn + sσ), x) (3.35) u =, (3.36) 76

105 where f, g, ρ, u, and p are solved for. Here, if we are stepping from t n to t n+, f n+ = f n, g n+ = g n, and ρ n+ = ρ n, so really u n+ and p n+ are just solved for as described in Section 3.7. Remember, the newest values available are always used. In Section 3.7, the Navier-Stokes equations were solved assuming a force, f, on the right hand side of the momentum equation Surface Tension Forces Now, we discuss how this force is computed from the interface. Assuming, we are stepping from t n to t n+, we have the interface at time, t n, and the surfactants on the interface at this time, ρ n. Then f n is computed where, f = ReCa δ (Γ, (σκˆn + sσ), x). (3.37) To be able to solve for the force on Γ, the surface tension on each segment is needed. We use the nonlinear equation of state for σ and nondimensionalize by σ c and ρ, as presented in Section., σ = + E log( ρ). (3.38) In this equation, E is the elasticity number. The surface gradient of the surface tension, s σ, is also needed, s σ = ˆτ σ s = ˆτ E ρ ρ s. (3.39) 77

106 To compute s σ, ρ s also needed, and ˆτ are needed, and to compute σκˆn on Γ, ˆn and κ are ρ s = ρ (3.4) x + f ( x ) f, x (3.4) + f x + f x ˆτ = ˆn = ( ) f x, + f x + f x (3.4) κ = f xx, (3.43) ( + fx) 3/ for x-segments and ρ s = ρ (3.44) y + g y ( ) g y, (3.45) + g y + g y ˆτ = ˆn = ( ) g, y + g y + g y (3.46) κ = g yy, (3.47) ( + gy) 3/ for y-segments. (Sometimes the values are the negative of the ones presented here depending on how the independent axis for a segment is defined.) This completes the evaluation of F(X(s)), F(X(s)) = ReCa (σκˆn + sσ), (3.48) 78

107 on the interface. Now, we discuss how to compute the singular force on the Navier- Stokes staggered grid following [45], f = δ(γ, F(X(s)), x), (3.49) where δ is defined to have support on Γ such that, δ(γ, g(x(s)), x)z(x) dx = g(x(s))z(x(s)) ds. (3.5) Ω Γ The delta function is regularized to spread the force to the velocity grid following the idea of Peskin s Immersed Boundary Method [33]. We approximate δ(γ, F(X(s)), x) with, δ ɛ (Γ, F(X(s)), x) = δ ɛx (x X(s))δ ɛy (y Y (s))f(x(s)) ds, (3.5) Γ where δ ɛi (x) = ( ) x φ, (3.5) h i h i and i = x, y. In this definition of the delta functions, the epsilons refer to the width of support of the delta function. We choose φ to be the piecewise cubic function in [45], φ(x) = x x + x 3 x x + 6 x 6 x 3 < x. (3.53) In this case, the width of the delta function, ɛ x or ɛ y, is h x or h y. 79

108 Tornberg and Engquist in [45] show that the error of approximating δ(γ, g(x(s)), x) with δ ɛ (Γ, g(x(s)), x), as measured by, E = h xh y δ ɛ (Γ, g, x k,l )z(x k,l ) (k,l) Γ g(x(s))z(x(s))ds, (3.54) is O(h q ), where h = max(h x, h y ) and q is the moment order of the one dimensional delta function approximation, equation 3.5. A one dimensional delta function approximation is said to be of moment order q (δ ɛ Q q ) if, h k δ ɛ (x k X) =, (3.55) and h k δ ɛ (x k X)(x k X) p = p =,,..., q. (3.56) In our case, δ ɛx and δ ɛy given by equation 3.5, defined using the piecewise cubic function, have moment order 4 and therefore the error E, defined by equation 3.54, is of O(h 4 ). For a linear partial differential equation with such a singular source, one can further show that error away from the singularities will be O(h q ). At the location of the singularity however, one is to expect a first order error in maximum norm. We also use the delta function regularization to interpolate the velocity field from the Navier-Stokes grid to points on the interface, u(x(s)) = u(x)δ ɛx (x X(s))δ ɛy (y Y (s)) dx. (3.57) Ω 8

109 One of the benefits of interpolating in this manner is that total power in the domain is equal to the power from the interface as shown in Section 5 of [33], = = = Ω Ω Γ Γ u(x) f(x) dx (3.58) ( ) u(x) F(X(s))δ ɛx (x X(s))δ ɛy (y Y (s)) ds dx (3.59) Γ ( ) F(X(s)) u(x)δ ɛx (x X(s))δ ɛy (y Y (s)) dx ds (3.6) Ω F(X(s)) u(x(s)) ds. (3.6) 3.9 Order of Convergence with the Navier-Stokes Solver A convergence study when we include the Navier-Stokes equation in the system to solve for the fluid flow is presented. Since results of drop deformations in shear flow will be shown, the convergence study is conducted on drops in shear flow as well. In these shear flow simulations, the characteristic length, L, is the diameter of the drop initially and the characteristic velocity, U, is the terminal speed of the top and bottom walls of the domain. In Chapter 4, we explain how the boundary conditions are set to create the shear flow, usually the speed of the walls is ramped up from rest to a constant terminal speed. Here, just one case is shown, where the Reynolds number is, the capillary number is.4, the elasticity number is. and the Peclet number is, with and without surfactants. When surfactants are included, we choose to use the 8

110 nonlinear equation of state and to nondimensionalize by σ c and ρ and to set the initial surfactant concentration to ρ o =.4. These two cases are run until time t =.5 in a domain. We study the convergence of the interface location, surfactant concentration (when included), the velocity field, and the pressure. Note that the area inside the drop and the total amount of surfactant on the drop converges. Here, we show the relative error in these values versus time for the different grids used. First, we study each case when only space refinement is used, so the Navier-Stokes grid is set to h x = h y = 5, 5, and 45 and the segments respectively have the following gridsizes, x = y =,, and.9. In all of these simulations, t = Then, space and time are both refined. In this case h x, h y, x, and y are set as before and t =.9 x. We measure the error numerically as the difference in the value on the grid and the the value on the next coarsest grid. The error is computed using both the L norm and the L norm. In all the plots, the dotted lines are the maximum norm, the solid lines are the L norm, the green line is a reference line with slope, and the red line is a reference line with slope. In Figure 3., the case with surfactants is shown. The drop contour is plotted at the initial and final times. Also, the surfactant concentration on the interface is shown at the initial and final times. Then in Figure 3., the velocity inside and outside the drop is shown at the final time. The velocity at the initial time was set to zero. Also, the pressure at the initial and final times is plotted. Note the jump in the pressure which is discussed below. 8

111 ρ s ρ s Figure 3.: The top two plots show the drop contour at the initial and final times when surfactants are included. The bottom two plots show the surfactant concentration on the interface at the same times. The stars on the interface correspond to zero arclength in the surfactant concentration plots. 83

112 Figure 3.: The top plots show the velocity field inside and outside the drop at the final time, t =.5. For the plot showing the velocity field inside the drop, the maximum velocity is.34. The bottom plots show the pressure at the initial and final times. 84

113 In Figure 3.3, we show convergence plots for the interface location, velocity field, and pressure when we refine in space, for the case with no surfactants. For the velocity field plots throughout this section, the black lines are the error in u, the horizontal component of the velocity, and the blue lines are the error in the vertical component, v. Also, we plot how the relative error in the area inside the drop changes with time for the three resolutions. There is a little over first order convergence in the interface location, close to second order convergence for the velocity field in the L norm but below first in the maximum norm, and the pressure converges at above first order as well. This is what is expected with spatial refinement as the Navier-Stokes solver is spatially discretized with a mix of first and second order schemes. The same plots are shown in Figure 3.4 but now the refinement is both in space and time. The convergence for the interface location is still a little over first order. The order of convergence for the velocity field has dropped to first order when measuring error using the L norm and is still below first in the maximum norm, and the order for the pressure has dropped below first in both norms. We expected the order of convergence to drop to first when we refine both time and space, since the timestepping used for the Navier-Stokes solver is only first order. The analytic solution for the pressure has a jump discontinuity across the interface. Here, this jump is regularized over a few grid points as seen in Figure 3.. The order of convergence in the pressure is affected by this numerical treatment. The treatment of this discontinuity is a well-known difficulty in interface tracking methods, as will be discussed at the end of this section. 85

114 x 4 Error in Γ 4 A( t ) A( ) A( ) 6 3 x = y 3 time Error in (u,v) Error in p x = y x = y Figure 3.3: Convergence plots for the interface, velocity field, and pressure, when only refining in space. In all the convergence plots in this section, the dotted black lines are the error measured using the maximum norm, the black solid lines are the error measured using the L norm, the green line is a reference line with slope, and the red line is a reference line with slope. Also, the relative error in the area inside the drop versus time is plotted. A(t) is the area inside the drop at time t measured numerically. The black line is the finest grid and the red is the coarsest grid. 86

115 4 x 4 Error in Γ 4 A( t ) A( ) A( ) x = y 3 time Error in (u,v) Error in p x = y x = y Figure 3.4: Convergence plots for the interface, velocity field, and pressure with refinement in both space and time. Also, the relative error in the area inside the drop versus time is shown. The black line is the finest grid and the red is the coarsest grid. 87

116 In Figure 3.5, the convergence plots when we just refine in space for the case with surfactants are shown. Here, the behavior is the same as with the clean drop. Additionally, ρ has a convergence rate below first order. The lower rates of convergence for ρ may have to do with the treatment of the pressure and the velocity fields right around the interface. This is included in the discussion below, when ideas of how to improve the pressure discretization are presented. Finally, in Figure 3.6, the convergence plots when we refine both space and time and include surfactants are shown. We note that we see the same differences between this case and just refining in space as in the case of clean drops. Once again the rate of convergence for ρ is below first order. 88

117 Error in Γ 4 Error in ρ x = y 6 3 x = y.5 x 4.5 x 4 A( t ) A( ) A( ).5 ρ ( t ) ρ ( ) ρ ( ).5 3 time 3 time Error in (u,v) Error in p h x = h y h x = h y Figure 3.5: Convergence plots for the interface, surfactants, velocity field, and pressure, when only refining in space. In this case, surfactants are included. The relative error in the area inside the drop versus time and in the total amount of surfactant versus time is shown. When including the relative error in the total surfactants on Γ, ρ(t) is the total amount of surfactant along Γ at time, t, computed numerically. 89

118 Error in Γ 4 Error in ρ x = y 6 3 x = y 4 x 4 4 x 5 A( t ) A( ) A( ) 3 ρ ( t ) ρ ( ) ρ ( ) 3 3 time 3 time Error in (u,v) Error in p h x = h y h x = h y Figure 3.6: Convergence plots for the interface, surfactants, velocity field, and pressure with refinement in both space and time. Surfactants are included here. The relative error in both the area inside the drop versus time and the total surfactants versus time is shown. 9

119 Above, we mentioned attempting to find a better way to handle the pressure. If the discontinuity in the pressure can be treated better, then this should improve the quality of the velocity field near the interface and therefore also improve the quality of the interface and of ρ. We may presently be seeing greater error in ρ since the derivatives of the velocities at the interface and derivatives of the interface are used to compute ρ. An attempt was made to implement and use the Immersed Interface Method [5], explicitly enforcing the jump conditions at the interface instead of making use of a reqularized delta function to spread the forces onto the grid. Without implicit (in time) treatment of the interface, this method has a strict stability limit on the timestep size, as commented on in [5] and [48]. This made it unfeasible to use for our purposes. The development of an implicit interface treatment for the SPM will be a direction of future research. We also implemented a finite element Navier-Stokes solver before the present finite difference solver described in Section 3.7. We implemented two different schemes in DOLFIN, a finite element package used for solving differential equations [8]. The first scheme was a fractional step θ-scheme based on ideas presented in [6, 4, 3] and the second scheme used a projection approach [3]. The coupling between these finite element solvers and the interface method caused spurious currents to develop in the velocity field creating oscillations in the interface. The methods implemented did not include any special treatment of the advection term in the Navier-Stokes equations and also did not include any numerical diffusion; either of these could possibly have reduced these spurious currents and the resulting oscillations. 9

120 Initially, a finite element solver was desired because in variational form the capillary force, σκˆn, can be treated using the Laplace-Beltrami operator as shown in [7]. This would prevent having to compute the curvature which might also be causing some of the larger errors in the pressure. The curvature at times is oscillatory, even with the present Navier-Stokes solver, as seen in Section This is due to taking two derivatives of the interface shape to compute the curvature. We plan on continuing the development of a finite element solver which would allow us to avoid computing the curvature explicitly. 9

121 Chapter 4 Results All the results presented in this chapter are drop deformations in shear flow. The values used in this chapter will be the nondimensionalized ones introduced in Section.. As stated before, in simulations with shear flow, the characteristic length, L, is the diameter of the drop initially and the characteristic velocity, U, is the terminal speed of the top and bottom walls of the domain. Here, ρ is nondimesionalized by ρ and σ by σ c, and the nonlinear form of the equation of state for σ is used. The shear flow is created by setting Dirichlet boundary conditions for the velocity on the top and bottom walls of the domain. The nondimensional velocity of the top wall is ramped up from (, ) to (, ) by time t =. The velocity of the top wall is set to u = ( ( cos(πt)), ) before time, t =, and to u = (, ) after time, t =. The bottom wall is set to have negative the velocity of the top wall, going from (, ) to (, ). The flow is initialized with a quiescent velocity field. In Section 4.., this method of setting up the shear flow is compared with 93

122 another method, where the speed of the walls are not ramped up but the top wall is set to have velocity (, ) and the bottom wall is set to have velocity (, ). We state very clearly when this second method is used. In this case, the flow is initialized as a linear shear flow and this is not a solution to the shear flow problem with a drop. In both cases periodic boundary conditions are used on the left and right walls of the domain. The interface is initially a circle centered in the middle of the domain with radius.5. Also, in all cases, the elasticity number is fixed, E =., and the Peclet number is fixed, P e Γ =. Convergence tests for drop deformations in shear flow using our numerical method have been shown in Section 3.9. Based on the varying gridsizes shown there, we fix the grids used for all the simulations in this chapter. The grids for the velocity field and pressure are set so that h x = h y =.5 and for the segments so that x = y = 9. We choose these grids to allow the ratio of x to h x and y to h y to be.5, since it is natural and computationally affordable to have a higher resolution for the segments, but these grids cannot be taken too fine or we introduce too small scales along the interface. The ratio of.5 is a good compromise and is similar to ratios chosen by others when using different methods which take the same issues into consideration [33]. The timestep is set to t =.. The time to run one of the simulations with a 8 domain on a personal computer with a. GHz processor and GB of RAM to time, t =, is about ten minutes. 94

123 When we present the results, we measure the drop deformation, D = L d B d L d + B d, (4.) where L d is the half-length of the drop and B d is the half-breadth. In discussions of drop deformations in shear flow, it is important to note that depending on the other parameters there is a critical capillary number under which the drop reaches a steady state and above which the drop continues to stretch. Here, examples of drops in flows with capillary numbers above and below the critical number are shown. 4. Clean Interfaces First, results with no surfactants on the interface are shown. 4.. Varying Domain Lengths Since periodic boundary conditions in the streamwise direction are set, there is an effect of the neighboring drops on the deformation of the drop if the domain is too short. We show a drop deforming in a shear flow with Reynolds number, Re =, and capillary number, Ca =.4. At these parameter values with no coverage the drop reaches steady state. The length of the domain is varied, Lx =, 4, 6, 8,, and the height of the domain is kept fixed, Ly =. The deformation of the drop, at steady state, varies significantly from Lx = to Lx = 4, then not very significantly to Lx = 6, and visually not at all to 95

124 Figure 4.: Steady state contours of the drop with varying lengths. The red drop is when Lx = and the black drop is when Lx = 4, 6, 8, and. Lx = 8 and. This is seen in the contours of the drop in Figure 4. and in the deformation plots in Figure 4.. Also, by studying the velocity fields at the periodic boundary, Figure 4.3, we can see if the neighboring drops are having an effect on the drop, i.e. if the profile deviates from a linear shear flow. From these results, it was determined to run the rest of the simulations on 8 domains. 96

125 D t D t Figure 4.: Deformation plots for clean drops with varying domain lengths. The red line is when Lx =, the green line is when Lx = 4, the blue line is when Lx = 6, and the black line is when Lx = 8 and..5.5 y y.5.5 u v Figure 4.3: The velocity fields at the periodic boundary. The plot on the left shows the horizontal component of the velocity, and the plot on the right shows the vertical component of the velocity. The vertical component of the velocity is not solved for at the periodic boundary due to the staggered grid of the Navier- Stokes solver. We plot the average of the vertical component at points to the right and at points to the left of the periodic boundary, when we show the velocity field at this boundary. Again, the red dots are when Lx =, the green dots are when Lx = 4, the blue when Lx = 6, and the black when Lx = 8 and. 97

126 4.. Two Different Initial Conditions In the introduction of this section, we discussed two different ways in which the shear velocity field is set. Either the top and bottom walls are ramped up and the initial condition for the velocity field is set to, u = (, ), or the shear flow is set initially and the drop is put into this flow. In the second case the top and bottom walls have speed and the initial condition is u = ( + y, ). In Figure 4.4, we show the deformation of a drop with the two different initial conditions, with the Reynolds number set to and the capillary number set to.4. The two cases have different transient behavior but eventually result in the same steady state. In the case where the flow is initialized with a linear velocity profile, there are larger transients and larger drop deformations. This behavior is expected since the initial flow in this case is not a solution to the problem. In Figure 4.5, the drop contours at time t = and at steady state are shown. These two different initial conditions and some results are discussed in detail in [4]. They compare their results to what was found earlier in [37] where the shear flow was only initialized with a linear velocity profile. 98

127 D t Figure 4.4: Deformation plots for clean drops with the two different initial conditions. The red line shows the deformation when initialized with a linear velocity field and the black line shows when initialized with a quiescent velocity field Figure 4.5: Drop contours with the two different initial conditions. The plot on the left is at time, t =, and the plot on the right shows that both cases reach the same steady state and gives the steady state contour. Again, the red is the contour when initialized with a linear velocity field and the black shows when initialized with a quiescent velocity field. 99

128 4..3 Varying Reynolds Numbers We study the clean interface with varying Reynold numbers, Re =.,,, 5, 5, and, with a fixed capillary number of.4. Here, as stated above, the results are simulated on a 8 domain and the velocity field is initialized with a quiescent flow while the speed of the top and bottom walls of the domain is ramped up. For Re =.,, and 5, the capillary number, Ca =.4 is below the critical capillary number and the drops reach steady state. For Re = 5 and the drops continue to stretch and deform. In Figure 4.6, the deformation plots for the cases which reach steady state and the corresponding steady state contours are shown. In Figure 4.7, the contour for Re = 5 and is shown at time t = 5. In Figure 4.8, the velocity at the periodic boundary is plotted for all cases. When the drop stretches more there is a greater effect from the neighboring drops, even though that effect is still very minor. In Figure 4.9, the velocity fields inside and outside the drop for the cases with Reynolds numbers and at time, t = 5, is plotted. The steady state case has velocity vectors that are tangent to the interface, as expected.

129 D t Figure 4.6: Deformation plots and steady state contours for clean drops with Reynolds numbers,.,,, and 5, which are black, blue, green, and red, respectively.

130 .5.5 Figure 4.7: The contours for the drops that do not reach steady state at time t = 5. The case with Re = 5 is magenta and the one with Re = is cyan..5.5 y y.5.5 u 5 5 v x 3 Figure 4.8: The velocity fields at the periodic boundary. The plot on the left shows the horizontal component of the velocity, and the plot on the right shows the vertical component of the velocity. Again, the colors correspond to the different Reynolds numbers,. is black, is blue, is green, 5 is red, 5 is magenta, and is cyan.

131 Figure 4.9: The velocity field inside and outside the drop at time t = 5. The top plots are the case with Reynolds number and the drop is at steady state. For the plot with the velocity shown inside the drop, the maximum velocity is.36. The bottom case is with Reynolds number, and the maximum velocity in the interior of the drop is.89. 3

132 4..4 Varying Capillary Numbers Now we study clean drops with varying capillary numbers, Ca =.5,.,.4,.7 and, with a fixed Reynolds number, Re =. For the drops with capillary numbers.5,. and.4, the drop reaches steady state and for Ca =.7 and the drop continues to deform, so the critical capillary number is between.4 and.7. In Figure 4., the deformation plots for the cases which reach steady state and the corresponding steady state contours are shown. In Figure 4., the contours for Ca =.4 and.7 are shown at time t = 5. In Figure 4., the velocity at the periodic boundary is plotted for all cases. For the smallest capillary number there is a timestep constraint to maintain stability in the system, so t =. in this case. 4

133 .4.3 D t Figure 4.: Deformation plot and steady state contours for the clean drops with capillary numbers.5,., and.4, which are black, blue, and green, respectively. 5

134 .5.5 Figure 4.: The contours for the drops that do not reach steady state at time t = 5. The capillary numbers are.7, which is magenta, and, which is cyan..5.5 y y.5.5 u 4 4 v x 4 Figure 4.: The velocity fields at the periodic boundary. The plot on the left shows the horizontal component of the velocity, and the plot on the right shows the vertical component of the velocity. Again, the colors correspond to the different capillary numbers,.5 is black,. is blue,.4 is green,.7 is magenta, and is cyan. 6

135 4..5 Comparison to Previous Work We compared some of the clean drop results to the work in [46] and [4]. Tornberg et al. use a front-tracking method to represent the interface and use a finite element Navier-Stokes solver described in [7]. To compare with the results presented in [4], simulations were run on a domain and gridsizes of h x = h y =.5 and x = y = 9 with t =. were used. We are only able to compare results when the viscosity ratio parameter λ =, meaning the viscosity inside and outside the drop are equal. We begin by looking at varying Reynolds numbers for fixed capillary numbers. In Figure 4.3, Ca =. and.4 and the Reynolds number is varied, Re =,, 5,. Note that all these drops reach steady state and do not deform as much as above when we have varying Reynolds numbers on a much longer domain. This is an effect of having periodic drops so close to one another. These drop contours look like the ones in [4] for these cases. Also, Tornberg shows some results on a slightly longer domain, 3. The same results are shown in Figure 4.4. In this longer domain, the simulation with Ca =.4 does not reach steady state and the contour is shown at t = 8. Previously, two methods to initialize the velocity field were discussed and how this was presented in [46] and [4]. We see the same behavior as in these previous works, that when the drop is placed in a linear velocity field initially, it overshoots and then settles back to steady state. We were able to replicate the deformation plots showing this behavior in [4], which are very similar to the deformation plots shown in section

136 The figures in [4], which we compare our results to, are Figures 7.4, 7., and 7. for varying Reynolds numbers and are Figures 7.6 and 7.7 for the comparison between initial conditions. The comparison with this previous work verifies even further the method implemented and used here to track the interface. 8

137 Figure 4.3: On the left, the Ca =. and on the right, the Ca =.4. In each plot, a range of Reynolds numbers, Re =,, 5, and, which are black, blue, green, and red, respectively, is shown Figure 4.4: Here, the Reynolds number is fixed to, while the Ca =. and.4, and is black and red respectively. The steady state drop is shown for Ca =. but the drop with Ca =.4 does not reach steady state and is shown at time, t = 8. 9

138 4. Contaminated Interfaces We add in surfactants to the interfaces in this section. The elasticity number is kept fixed at E =. and the Peclet number is kept fixed at P e Γ =. We continue to work on a 8 domain with quiescent initial conditions for the velocity field. We have chosen to nondimensionalize by σ c and ρ (as discussed in Section.), where σ c is the surface tension of a clean drop and ρ is the maximum surfactant concentration possible on the drop. This choice allows us to more easily understand and interpret the effects of varying amounts of initial surfactant on Γ. 4.. Varying Initial Surfactant Concentrations Numerical experiments were conducted to see the effect of varying initial surfactant concentrations on the interface, ρ o. In these simulations, the Reynolds number and capillary number are fixed, Re = and Ca =.4. With more surfactants on the interface, the drops deform more. This is seen in Figure 4.5 where the contours and deformation plots for a range of ρ o values are plotted. In Figure 4.6, we see that having surfactants on the interface does not change how much of an effect the periodic drops have on one another. In Figure 4.7, the velocity fields inside and outside a drop with no surfactants and a drop with ρ o =.4 are shown. The behavior of the velocity fields do not differ much from one another, beyond the fact that the drop with surfactants is more deformed. In Figure 4.8, ρ ρ o is plotted versus arclength and the corresponding surface tension difference, σ σ o, is also shown. The initial surface tension, σ o, varies as ρ o varies since σ o = σ(ρ o ). This figure allows us to understand how the deformation

139 of the drop advects the surfactants and how this effects the surface tension through the equation of state. Note that all values plotted here are nondimensional. Also, here surfactants for ρ o = are plotted and they just remain zero while having no effect on the drop deformation and fluid velocity. It is important to notice that Γ (ρ ρ o) does not have to be equal to zero for the total amount of surfactant to be conserved, since the length of Γ is changing. In this figure, the Marangoni stress, σ, is also shown. Plots and a discussion of the the capillary stress, σκ, are s included in Section The capillary stress and the Marangoni stress have competing effects on the drop. As the capillary stresses increase, the drop deforms more. As the Marangoni stresses increase, the forces act to prevent the drop from deforming. In this section, we have shown the behavior of the drop and surfactants for varying initial surfactant concentrations, ρ o. Note that as ρ o increases, both the capillary and Marangoni stresses increase. This means that as the surfactant concentration increases, there is a competing effect from the two stresses. The capillary stress dominates and therefore we see that as ρ o increases the drop deforms more. The Marangoni stress effect results in pushing surfactants away from the tips. This is why ρ o can be set to large values, for example.975 (the highest value used in these tests), without σ quickly becoming negative at points along the interface, as it would if ρ values were allowed to peak at values even closer to, as shown in Section.3.

140 D t Figure 4.5: Deformation plot and steady state contours for drops with varying amounts of surfactants. The initial surfactant concentration is set to ρ o =,.,.3,.4,.5,.7,.9, and.975 and is black, blue, green, red, magenta, cyan, yellow, and orange respectively.

141 .5.5 y y.5.5 u 4 4 v x 4 Figure 4.6: The velocity fields at the periodic boundary. The plot on the left shows the horizontal component of the velocity, and the plot on the right shows the vertical component of the velocity. Again, the colors show the velocities for varying initial surfactant concentrations, as in Figure

142 Figure 4.7: We show the velocity field inside and outside the steady state drop. The top plots are the case with ρ o =. For the plot with the velocity shown inside the drop, the maximum velocity is.36. The bottom case is with ρ o =.4, and here, the maximum velocity in the interior of the drop is.9. 4

143 .5 ρ ρ o σ σ o s s Marangoni stress s Figure 4.8: The plot on the top shows how the surfactants have progressed in time, ρ ρ o versus arclength, for different initial surfactant concentration values on the drop. The plot in the middle is the surface tension difference, σ σ o, plotted versus arclength. And the plot on the bottom is the Marangoni stress versus arclength. The colors are the same as above, ρ o = is black, ρ o =. is blue, ρ o =.3 is green, ρ o =.4 is red, ρ o =.5 is magenta, ρ o =.7 is cyan, ρ o =.9 is yellow, and ρ o =.975 is orange. The stars on the contours in Figure 4.5 correspond to arclength here. 5

144 4.. Varying Reynolds Numbers Now, the initial surfactant concentration is set to ρ o =.4 and the capillary number is set to.4 and the Reynolds number is varied. The behavior is similar to what was seen for the clean case. In Figure 4.9, the contours and deformation plots for the cases that reach steady state are shown. Figure 4. has contour plots of the cases, Re = 5 and, which do not reach steady state. When surfactants are included, we use a finer grid for the cases, Re = 5 and. The grids used are h x = h y =.5, x = y =, and t =

145 D t Figure 4.9: Deformation plot and steady state contours for the drops with Reynolds numbers,.,,, and 5, which are black, blue, green, and red respectively. Here, surfactants are included with initial surfactant concentration, ρ o =.4. 7

146 .5.5 Figure 4.: The contours for the drops that do not reach steady state at time t = 5. The Reynolds numbers are 5 for the magenta drop and for the cyan drop..5.5 y y.5.5 u 5 5 v x 3 Figure 4.: The velocity fields at the periodic boundary. Again, the colors correspond to the varying Reynolds numbers,. is black, is blue, is green, 5 is red, 5 is magenta, and is cyan. 8

147 4..3 Varying Capillary Numbers Finally, drops in shear flow with Re = and ρ o =.4 are shown while the capillary number is varied. Simulations were run with Ca =.5,.,.4,.7, and. As shown in Figure 4., when the capillary number is.5,., and.4, the drops reach steady state. When the capillary number is.7 and, they continue to deform, as happened with the clean drop, as is shown in Figure 4.3. We see in Figure 4.4 that the varying capillary numbers (even with surfactants) does not have much of an effect on how the periodic drops interact with one another. Finer grids are used for the cases Ca =.5,.7, and.9, h x = h y =.5, x = y =, and t =.5. For Ca =.5, the timestep is decreased as in 8 the clean case, t =.5. 9

148 .4.3 D t Figure 4.: Deformation plots and steady state contours for the drops with capillary numbers.5, which is black,., which is blue, and.4, which is green.

149 .5.5 Figure 4.3: The contours for the drops that do not reach steady state at time t = 5. The capillary numbers are.7 for the magenta drop and for the cyan drop..5.5 y y.5.5 u 4 4 v x 4 Figure 4.4: The velocity fields at the periodic boundary. Again, the colors correspond to the different capillary numbers,.5 is black,. is blue,.4 is green,.7 is magenta, and is cyan.

150 4..4 Capillary Stress We have mentioned previously how one area of improvement in the method would be the computation of the curvature, κ. The curvature, κ, is a very sensitive quantity to compute numerically, since it requires approximating second derivatives of the interface shape. This means that any small perturbation of the interface will be magnified substantially. Here, the capillary stress, σκ, is shown for the different cases presented in this chapter. In Figure 4.5, the capillary stress is shown for varying ρ o values. The corresponding results for these cases were presented in Section 4... As ρ o increases, there is a trend for the oscillations to increase. In Figure 4.6, one of these cases, ρ o =.4, is shown at varying times. The drop is approaching steady state at time, t = 6.5, and has reached steady state by time, t =. There is a trend for the oscillations in the stress to get worse as the drop stays at steady state longer.

151 capillary stress capillary stress capillary stress capillary stress 6 4 ρ o = 4 s 6 4 ρ o =.3 4 s s 6 4 ρ o =.5 ρ o =.9 4 s capillary stress capillary stress capillary stress capillary stress 6 4 ρ o =. 4 s 6 4 ρ o =.4 4 s 6 4 ρ o =.7 4 s 6 4 ρ o = s Figure 4.5: The capillary stress versus arclength is plotted for varying initial surfactant concentrations. These stresses go along with the results presented in Section

152 ρ o =.4, t = ρ o =.4, t = 6 6 capillary stress s capillary stress s ρ o =.4, t = 5 ρ o =.4, t = 6 6 capillary stress s capillary stress s ρ o =.4, t = 5 ρ o =.4, t = capillary stress s capillary stress s Figure 4.6: The capillary stress for coverage, ρ o =.4, at varying times. 4

153 In Figure 4.7, the capillary stress for varying Reynolds numbers is shown. The corresponding results for these cases were presented in Section 4... Note that for larger Reynolds numbers the oscillations in the curvature and therefore in the capillary stress are worse. Also, the oscillations seem to dominate in the tip regions for all cases. Finally, in Figure 4.8, the capillary stress for varying capillary numbers is shown. The corresponding results for these cases were presented in Section Here, the oscillations decrease significantly as the capillary number increases. In this section, we have shown how the quality of the curvature and capillary stress vary with different parameters. The quality of the curvature needs to be closely watched as it is computed by taking two derivatives of the interface. We discussed in Section 3.9, the implementation of a finite element solver for the Navier-Stokes equations, which would enable us to evaluate the force from the drop on the fluid, without having to explicitly evaluate the curvature. Another possible way to improve the quality of the curvature would be to filter out high frequencies. There is however a delicate balance in such a treatment in order to obtain smoothness in the computed curvature without damping out real variations that should be present. 5

154 6 Re =. 6 Re = capillary stress 4 capillary stress s 3 4 s 6 Re = Re = 5 capillary stress 4 capillary stress s 3 4 s 5 Re = 5 5 Re = capillary stress 5 capillary stress s s Figure 4.7: The capillary stress versus arclength for varying Reynolds numbers. These stresses go along with the results presented in Section

155 6 Ca =.5 6 Ca =. capillary stress capillary stress s Ca = s 5 capillary stress capillary stress s 5 Ca = Ca = s capillary stress s Figure 4.8: The capillary stress versus arclength for varying capillary numbers. These stresses go along with the results presented in Section

156 4..5 Comparison with Previous Work When including surfactants, we compared our work with that of Xu, Li, Lowengrub, and Zhao [49]. They use a level set method to represent the interface, developed by [3], and they use the Immersed Interface Method to solve the Stokes equations. We are not able to quantitatively compare to the work presented there since we are solving the Navier-Stokes equations and the boundary conditions are different. We have periodic boundary conditions in the x-direction and they use Dirichlet boundary conditions. We ran cases like the one they show for one drop, with small Reynolds numbers, Re = and.. We see comparable behavior to what is shown in Figures a and a. Also, we see very similar behavior for the surfactant concentration, surface tension, Marangoni and capillary stresses as shown in Figures b and c. In these cases, we do not have the problem with oscillations in the curvature, κ, that we had above. This is most likely because the drop is deforming so much and is continuing to deform in these cases and because the Reynolds number is small and the capillary number is set to.7. 8

157 Part II Soluble Surfactants 9

158 Chapter 5 Mathematical Model 5. Equations In the second part of this thesis, the surfactants are soluble in the bulk fluid around the interface. The surfactant concentration in the bulk fluid, C, is tracked separately, with a boundary condition term at the interface, Γ. This boundary condition handles the exchange of surfactant to and from the bulk fluid. Also, a source term is added to the equation for the interfacial surfactant concentration, ρ, equation.7, from Chapter, which handles the exchange of surfactant to and from the interface. Figure 5. shows a schematic of the full problem. The region outside of Γ(t), which is evolving in time, is denoted Ω (t), and the entire domain, as before, Ω. Here, we will solve for the surfactant concentration outside the drop in Ω and only allow an exchange with that side of the drop. The red specks in Ω in the figure represent the surfactants in the bulk fluid and the red on Γ represents the 3

159 Ω (t) Γ(t) Figure 5.: This figure shows the problem configuration. In this part of the thesis, the surfactants are soluble in the bulk fluid. The red represents the surfactants in the bulk and on the interface. surfactants on the interface. The incompressible Navier-Stokes equations continue to be solved for the fluid velocity, u, and pressure, p, with the force from the interface included on the right hand side as a singular source term, ρ Ω ( u t ) + (u )u + p µ u = δ (Γ, (σκˆn s σ), x) in Ω (5.) u = in Ω, (5.) where ρ Ω and µ are the density and the viscosity of the fluid, respectively. In the force term, κ is the curvature along Γ, ˆn is the unit normal vector of Γ pointing into the drop, σ is the surface tension coefficient, and s is the surface gradient 3

160 operator. The interface, Γ, between the two fluids will continue to be represented and advected using the Segment Projection Method as presented in Section 3.. The same equation is solved for the surfactant concentration along the interface, as was derived in Chapter, but now a source term that allows surfactants to be absorbed and desorbed to and from Γ is included, Dρ Dt + ρ( s u) = D Γ sρ + k a C Γ (ρ ρ) k d ρ, (5.3) where k a is the adsorption coefficient, k d is the desorption coefficient, and ρ is the maximum surfactant concentration on Γ. The bulk surfactant concentration at the interface, C Γ, is evaluated from the bulk surfactant concentration, C. The surfactant concentration in the bulk fluid is modeled with an advection diffusion equation with a mixed Neumann-Dirichlet boundary condition at Γ(t), C t + u C = D C in Ω (t) (5.4) D C n = k ac(ρ ρ) k d ρ at Γ(t). (5.5) Note that the domain on which C is solved, Ω (t), is time dependent. In [7], a discussion of soluble surfactants and their absorption and desorption to and from the interface, and their equations can be found. C is the concentration of surfactant in the bulk fluid and is measured in amount of surfactant per area. Equation 5.4 is valid in the bulk outside of the drop, Ω (t), and u in this equation is the velocity of the bulk fluid. This is in difference to the 3

161 interfacial surfactant concentration, ρ, which is measured in amount of surfactant per unit length, and for which equation 5.3 is solved on the interface Γ, given the velocity field u at Γ. The surface tension coefficient, which in part determines the force the interface applies on the fluid, still only depends on the surfactant concentration on the interface. The linear equation relating ρ and σ is given by equation.5, and the nonlinear relation is given by equation.6. This relation closes the system, which is quite complex. The surface tension forces are determined by the interface location and shape together with the concentration of surfactant on the interface, which is in turn affected by the concentration of surfactant in the bulk. These forces affect the fluid velocity, which in turn alters the interface shape and location and the surfactant concentration. 5. Nondimensionalization The system presented here is nondimensionalized, just as the system with insoluble surfactants was in Section.. The characteristic length is L and the characteristic velocity is U, which are once again defined in the results. As stated in Section., we choose the characteristic surfactant concentration on Γ to be ρ, and the characteristic surface tension coefficient to be the surface tension of a clean bubble, σ c. We make this choice over using the equilibrium concentration of surfactant and the corresponding surface tension coefficient because with the extra source term in the equation for ρ, ρ is no longer constant over time. Γ Finally, the characteristic surfactant concentration in the bulk is C, the max- 33

162 imum surfactant concentration in the bulk. If the surfactant concentration in the bulk fluid is above some critical value, then the surfactant molecules form micelles, large groups of molecules where the hydrophobic tails bundle together. Above this critical value, adding surfactants does not effect the surface tension since the added molecules just form more micelles [34]. Also, these molecules are not free to be absorbed by the interface. Therefore, choosing C to nondimensionalize C is a physical choice based on the behavior of surfactant molecules, and should be set to this critical value. The equations for the fluid flow are, as before, u t + (u )u + p Re u = ReCa δ (Γ, (σκˆn sσ), x) in Ω (5.6) u = in Ω. (5.7) with the linear equation of state for σ, σ = Eρ, (5.8) and the nonlinear equation of state, σ = + E log( ρ). (5.9) Then, for the interfacial surfactant concentration, the nondimensional equation is, Dρ Dt + ρ( s u) = ( P e sρ + α C Γ ( ρ) ) Γ La ρ, (5.) 34

163 and for the bulk surfactant concentration, the nondimensional equation is, C t + u C = P e C in Ω (5.) C ( n = αkp e C( ρ) ) La ρ at Γ. (5.) The nondimensional numbers, as before, are the Reynolds number, capillary number, interfacial Peclet number, and elasticity number, respectively, Re = ρ ΩUL µ, Ca = Uµ σ c, P e Γ = LU D Γ, E = RT ρ σ c. (5.3) The additional nondimensional numbers are the bulk Peclet number, Langmuir number, Hatta number, and the nondimensional adsorption length, respectively, P e = LU D, La = C k a, α = Lk ac, K = ρ, (5.4) k d U LC which are discussed in []. 35

164 Chapter 6 Numerical Method The idea behind the numerical method presented in this thesis is to have regular grids on which the equations for the variables, Γ, ρ, C, u, and p can be solved. In Chapter 3, this is done for the interface and the interfacial surfactant concentration using the Segment Projection Method. In this section, we will continue with this idea, and show that the equation for C can be solved on a regular two dimensional grid, with a special treatment of the boundary condition at Γ. 6. Timestepping Once again, as in Section 3., the timestepping is introduced with a given velocity field u = (u, v). The system to be solved now includes the equation for the bulk surfactant concentration and the exchange of surfactants between the interface and the bulk. The entire system includes the advection of the segments and solving for 36

165 the surfactant concentration on the segments, f t + uf x = v (6.) ( ) ρ t + uρ x + ρ (u ( x + v x f x ) = D Γ ρ x + α C Γ ( ρ) ) + f x + f x + f x La ρ, x (6.) for an x-segment and, g t + vg y = u (6.3) ( ) ρ t + vρ y + ρ (v ( y + u y g y ) = D + g Γ ρ y + g y + g y + α C Γ ( ρ) ) y La ρ, y (6.4) for a y-segment, and solving for the bulk surfactant concentration, C t + uc x + vc y = P e (C xx + C yy ) in Ω (6.5) C ( n = αkp e C( ρ) ) La ρ at Γ. (6.6) As in Section 3., when the system just included the surfactants on Γ, Strang splitting is used to step forward one timestep. Strang splitting is used because it gives second order convergence in time, as discussed in Section 3.. There are numerous different steps that will need to be included in the splitting. Step A and Step B from Section 3. are the same except now the extra source term, modeling the transfer of surfactants to and from Γ in the equations for ρ as shown in equations 6. and 6.4, is included. In these steps, as Γ and ρ are updated, C is 37

166 kept fixed. A step to solve for the bulk surfactant concentration in Ω must also be included, Step C f t = (6.7) g t = (6.8) ρ t = (6.9) C t + uc x + vc y = P e (C xx + C yy ), (6.) with the boundary condition at Γ, C n = αkp e ( C( ρ) La ρ). Once again, in each timestep, we need to include the evaluation of the velocity at Γ and the reconstruction of the interface and the surfactants along the interface as presented in Section 3.5. Also, the velocity field on the grid used for C must be evaluated in each timestep. Also, an evaluation of C at Γ, C Γ, to be used in the equations for ρ, is included. Since the domain, Ω (t) is changing in time, at times there are points in the domain where C must be solved for but previous values of the bulk surfactant concentration do not exist. In these situations, C at points which were previously in Ω and the boundary condition on Γ are used to interpolate values of C at points which are newly in Ω. Points in Ω (t) are tracked by evaluating an indicator function, which is if the point is in the domain and if it is not. Also, note that there may be some points that were in the domain at the previous timestep but 38

167 are no longer in the domain. is Then, the timestepping scheme to step forward t from time, t n, to time, t n+, Step A (timestep t ) Evaluate the velocity on the interface Store the indicator function and evaluate the new one Interpolate C to points that are newly in the domain given by Γ Evaluate C on the interface Step B (timestep t ) Step C (timestep t) Step B (timestep t ) Step A (timestep t ) Reconstruction of the interface and ρ on the interface Evaluate the velocity on the interface Evaluate the velocity on the grid for C. (6.) For all the steps here which involve the interface, we loop over the four segments, as was done when there were only insoluble surfactants. Also, as before, the newest values possible for each step are used. 39

168 Splitting is used to solve Step B as discussed in Section 3.. Note, that now Step B is solved with a timestep of t, so the timesteps now change to, Step B(A) (timestep t 4 ) Step B(B) (timestep t 4 ) Step B(C) (timestep t ) Step B(B) (timestep t 4 ) Step B(A) (timestep t 4 ), (6.) where Step B(A) and Step B(C) are defined as before in Section 3., while keeping C fixed. Step B(B) now includes the new transfer term, Step B(B) (x-segments) f t = (6.3) ρ t + ρ (u ( x + v x f x ) = α C Γ ( ρ) ) + f x La ρ (6.4) C t =, (6.5) and 4

169 (y-segments) g t = (6.6) ρ t + ρ (v ( y + u y g y ) = α C + g Γ ( ρ) ) y La ρ (6.7) C t =. (6.8) Also, Strang splitting is used to solve Step C in Ω. In each substep, f t = g t = ρ t =, and hence f, g, and ρ are fixed. The splitting separates between the advection and the diffusion terms, Step C(A) C t + uc x + vc y =, (6.9) and Step C(B) C t = P e (C xx + C yy ), (6.) and in both steps, the mixed Neumann Dirichlet boundary condition is applied at Γ. Then, when Step C is advanced by t, really the following substeps are taken, Step C(A) (timestep t ) 4

170 Step C(B) (timestep t) Step C(A) (timestep t ). (6.) In Sections 3.3 and 3.4 the discretizations of Step A, Step B(A), and Step B(C) are presented. Note that in Step A and B, the velocities used have been evaluated at Γ, and in Step C, the velocities used have been evaluated at the gridpoints for C. In the next few sections, we discuss how we discretize Step C and treat the boundary condition, evaluate C on Γ, interpolate C when needed, and discretize the additional terms handling the transfer of surfactant from and to the interface in Step B. In these discussions, we are stepping forward from time, t n, to time, t n+, where t n+ t n = dt. The timestep, dt, is as defined in the schemes in this section, t, t t, or. Note that when we state that a variable is at time, 4 tn, we mean the newest value of the variable available. Also, in these discretizations, the gridsizes for the x and y-segments are x and y, as before in Part I, and the gridsizes for C are x c and y c. 4

171 6. Discretization of the Interfacial Surfactant Concentration Equations To solve for ρ while allowing an exchange of surfactant with the bulk fluid, f t = (6.) ( ) ρ t + uρ x + ρ (u ( x + v x f x ) = D Γ ρ x + α C Γ ( ρ) ) + f x + f x + f x La ρ, x (6.3) is solved on x-segments and, g t = (6.4) ( ) ρ t + vρ y + ρ (v ( y + u y g y ) = D + g Γ ρ y + g y + g y + α C Γ ( ρ) ) y La ρ. y (6.5) is solved on y-segments. The time splitting used to solve for ρ is discussed in Section 6.. Step B(A) and Step B(C) are the same as in the insoluble case and the discretization for these steps is presented in Section 3.4. Here, the discretization of Step B(B), equations 6.4 and 6.7, is discussed. These equations are solved exactly. In these equations, u x, v x, f x, u y, v y, and g y, and C Γ are still approximated. We step from time, t n to t n+, where t n+ t n = dt. The values, ρ n, f n, g n, u n, v n and C n, are known and f n+ = f n, g n+ = g n, so then u n+ = u n and v n+ = v n and C n+ = C n. So we just are solving for ρ n+ in 43

172 this step, where, ( ρ n+ j = ρ n j Rn j Mj n ) e M n j dt + Rn j M n j (6.6) M n j = D ou n j + D o v n j D o f n j + (D o f n j ) + αc n Γ + α La (6.7) R n j = αc n Γ, (6.8) for an x-segment and M n j = D ov n j + D o u n j D o g n j + (D o g n j ) + αc n Γ + α La (6.9) R n j = αc n Γ, (6.3) for a y-segment. In these equations, D o w n j = wn j+ w n j x, (6.3) for an x-segment and for a y-segment, x is just replaced by y. If for any j point, M n j =, then in that case, ρ n+ j = ρ n j + R n j dt. (6.3) In these discretizations, C n Γ represents the numerical evaluation of Cn at the interface. The evaluation of this value is discussed in Section

173 6.3 Discretization of the Bulk Surfactant Concentration Equations Now, the discretization of Step C, the step that updates the bulk surfactant concentration, equation 6. is presented. In Section 6.4, we discuss how we discretize the boundary condition at Γ. We solve for C in all of Ω, even though C is valid only in Ω. The indicator function, introduced in Section 6. is used to keep track of where C is valid. Therefore, in this section, how to solve Step C in all of Ω, without the boundary condition at Γ, is presented. To step Step C(A), equation 6.9, forward dt from time, t n, to time, t n+, a Lax-Wendroff scheme is used, C n+ ij = Cij n dt(u n ijd o,x Cij n + vijd n o,y Cij) n dt ( D+,t u n ijd o,x Cij n + D +,t vijd n o,y Cij) n + dt ( ( u n ij Do,x u n ijd o,x Cij n + u n ijd +,x D,x Cij)) n + dt ( ( v n ij Do,y vijd n o,y Cij n + vijd n +,y D,y Cij)) n + dt ( u n ij D o,x vijc n ij n + vijd n o,y u n ijcij) n + dt ( ) u n ij vijd n o,x D o,y Cij n. (6.33) 45

174 In this equation, D o,x w n ij = wn i+j w n i j x c (6.34) D o,y wij n = wn ij+ wij n y c (6.35) D +,t wij n = wn+ ij wij n dt (6.36) D +,x D,x w n ij = wn i+j w n ij + w n i j x c D +,y D,y w n ij = wn ij+ w n ij + w n ij y c (6.37) (6.38) D o,x D o,y w n ij = wn i+j+ w n i+j w n i j+ + w n i j 4 x c y c, (6.39) where w can be C, u, or v. Note that in this step, u n+ = u n and v n+ = v n, since the velocity is not updated in Step C. Also, f n+ = f n, g n+ = g n, and ρ n+ = ρ n. Then, equation 6.33, simplifies to, C n+ ij = Cij n dt(u n ijd o,x Cij n + vijd n o,y Cij) n + dt ( ( u n ij Do,x u n ijd o,x Cij n + u n ijd +,x D,x Cij)) n + dt ( ( v n ij Do,y vijd n o,y Cij n + vijd n +,y D,y Cij)) n + dt ( u n ij D o,x vijc n ij n + vijd n o,y u n ijcij) n + dt ( ) u n ij vijd n o,x D o,y Cij n. (6.4) To step Step C(B), equation 6., forward from time, t n, to time, t n+, a Crank- 46

175 Nicolson scheme is used, as was done for the diffusion of ρ, C n+ ij = Cij n + dt P e + dt P e ( C n i+j C n ij + C n i j x c ( C n+ i+j Cn+ ij + C n+ i j x c + Cn ij+ Cij n + Cij n ) yc + Cn+ ij+ ) Cn+ ij + C n+ ij yc.(6.4) This leads to the following system being solved at each timestep, A B B A B B A B B A B B A B A B C n+ C n+ C n+ Ny = E n E n E n Ny, (6.4) where A = D xy Dx Dx Dx D xy Dx Dx D xy Dx Dx Dx D xy, (6.43) 47

176 B = C n+ j = Dy Dy Dy C n+ C n+ C n+ Nx, Ej n = E n E n E n Nx, (6.44), (6.45) and D x = dt, D P e x y = dt, D c P e yc xy = + D x + D y, (6.46) Eij n = Cij n + D x ( C n i+j Cij n + Ci j) n D y ( ) + C n ij+ Cij n + Cij n. (6.47) This system is solved over the entire domain, Ω. In Section 6.4, we will add in the boundary condition for Γ into this system. Note, once again, f n+ = f n, g n+ = g n, and ρ n+ = ρ n in this step. 48

177 6.4 An Embedded Boundary Method for the Soluble Surfactants An embedded boundary method at Γ, presented by Kreiss, Petersson, and Yström in [], is used to handle three different situations involving the coupling of the bulk surfactant concentration and the location of the interface,. to enforce the boundary condition for C at Γ. to interpolate C to new points in the domain 3. to evaluate C at the interface. For each situation, C needs to be evaluated at some point x, at which we do not already have a value of the bulk surfactant concentration. The main steps of this algorithm are the following:. Find a point on the interface, x Γ, such that the line which connects x and x Γ is normal to Γ. (If x already lies on Γ then x Γ = x.). From x Γ, use the normal direction (which connects x Γ and x), to find three points along this line in Ω (t), that cuts either vertical or horizontal gridlines. Denote these three points as x i, x ii, and x iii. 3. The values, C i, C ii, and C iii at the points x i, x ii, and x iii are in general not known. We use quartic interpolation to interpolate from grid values to these points. 49

178 4. Use the values C i, C ii, and C iii, and the boundary condition, C ( n = αkp e C( ρ) ) La ρ. (6.48) to do a one dimensional extrapolation/interpolation along the normal line defined in (). This defines the value of C at the evaluation point, x. (6.49) The details in the algorithm differ for the three situations for which we use this embedded boundary method. In each situation x is defined differently. In the third situation the formulas for the evaluation of x in step (4) are different from the other two situations Enforcing the Boundary Condition In Step C, we are trying to solve C n+ on Ω n+, defined using Γ n+ (Γ n updated by t), as presented in Section 6.. When updating Step C(A), a nine point stencil is used, and when updating Step C(B), a five point stencil is used, as shown in Figure 6.. If any point in the stencil lies outside Ω n+, inside the drop, then C must be evaluated at that point using the embedded boundary method described above. This point is x in algorithm We show examples of the two different stencils with points lying inside the drop in Figure 6.. Using this method allows us to enforce the boundary condition at Γ and decouple the solving of C in Ω n+ from inside the drop, as desired. 5

179 Figure 6.: The two stencils used in the update of C. The stencil on the left, the five point stencil, is used for the diffusion step, Step C(B), and the stencil on the right, the nine point stencil, is used for the advection step, Step C(A). Ω n+ Γ n+/ Figure 6.: Two examples when points in Ω n+ have points in their stencil which lie inside the drop, the red points. When this happens an embedded boundary method is used to evaluate a value of C at the point inside the drop. The red points are what is denoted as x in algorithm

180 The point we need to extrapolate C to is x, the red points in Figure 6.3. First, we must find x Γ, the point on Γ following step () of the algorithm. The magenta points in Figure 6.3 are x Γ. Depending on the direction of the normal, we then find three points, x i, x ii, and x iii, which either cut the x-axis or the y-axis. This is step () of the algorithm. Both directions are shown in Figure 6.3. These points are the green points shown in the figure. First, the value of C i, C ii, and C iii are interpolated at these three points, step (3) of the algorithm. Quartic interpolation is used to do this, ( C k = C db 6 dx + db dx ) db dx ( 3 C 3 db dx 5 db dx + ) db 3 + dx ( 3 C 3 3 db dx + db dx ) db 3 + dx ( 3 db C 4 3 dx db dx + ) db 3, (6.5) 6 dx 3 where k = i, ii, or iii. C, C, C 3 and C 4 are the values of C at the four points we are interpolating to C k from. These points are the blue points in Figure 6.3. Note that the interpolation can be in either the x or y-direction. When the interpolation is in the x-direction, dx = x c and when the interpolation is in the y-direction, dx = y c. The distance between the point we are interpolating to, x k, and the point where C is located is denoted db in the equation above. At times, there are not four points in Ω n+ where x k lies between the first and fourth points to interpolate C k. When this happens, skewed interpolation and/or lower orders of interpolation are used. 5

181 The last step of the algorithm evaluates the value of C at x, C. We fit a cubic polynomial for C through x, x i, x ii, and x iii. To be able to solve for the coefficients of this polynomial, four conditions are needed. Three of those conditions are the values of C, C i, C ii, and C iii, at x i, x ii, and x iii. The fourth condition is the boundary condition, equation 6.48, at x Γ. Then we can evaluate the polynomial at x, where C = K g i C i g ii C ii g iii C iii g o, (6.5) ( g o = K ξ Γ 6 ξ + ξ Γ ξ ) ( ξγ ξ 3 6 ξ + ξ Γ ξ ) ξγ (6.5) ξ ( 3 g i = K 3 ξ Γ ξ 5 ξγ ξ + ) ( ξγ ξ 3 ξ 5ξ Γ ξ + 3 ) ξγ (6.53) ξ ( 3 g ii = K 3 ξ Γ ξ + ξ Γ ξ ) ( ξγ ξ 3 ξ + 4ξ Γ ξ 3 ) ξγ (6.54) ξ ( 3 ξ Γ g iii = K 3 ξ ξγ ξ + ) ( ξγ ξ 3 3 ξ ξ Γ ξ + ) ξγ, (6.55) ξ 3 and K = αkp e( ρ) and K = αkp e La ρ. In the equations for g, ξ = x i x = x i x ii = x ii x iii and ξ Γ = x x Γ. We interpolate ρ to evaluate ρ at x. This extrapolation to C is fourth order, and the interpolation to get C i, C ii, and C iii is also fourth order. This is done to such high order to allow us to compute second derivatives of C to second order. If when evaluating C n+, we need C at time, t n, we just evaluate using C n, ρ n, and Γ n+. But in the Crank-Nicolson scheme when solving for the diffusion of C, we also need some values that lie inside the drop at time, t n+. Then, 53

182 Ω n+ Γ n+/ Figure 6.3: This figure explains how we extrapolate C at the red points using values of C in Ω n+, Γ, and ρ. the same algorithm described above is used and C n+ is computed as a linear combination of C n+ at the twelve points in Ω n+, the blue points in Figure 6.3, plus a constant. Then when solving this system, equation 6.4, we just replace the row corresponding to solving the Crank-Nicolson scheme for C n+, equation 6.4, with this linear combination of the twelve blue points. Here, we have described how we implement the boundary condition for C at Γ, as Γ evolves in time, while still solving C on a regular grid and using standard finite difference schemes to solve for C. 54

183 6.4. Interpolation of C to New Points in the Domain We use the same algorithm to interpolate C when it is needed. In Section 6., this idea was introduced. The interpolation of C is explained using Figure 6.4. We define Ω n to be the region outside the drop defined by Γ n, when we have taken one step of Step A with a timestep of t when stepping forward from time, tn, to time, t n. When we are stepping forward t, from time, t n, to time, t n+, we have C n on Ω n, and need to solve for C n+ on Ω n+. In the figure, this region where C is being solved, Ω n+, is defined by the solid blue Γ. There may be some points, like the red points shown in the figure, that were not in Ω n but are in Ω n+. For these points, values of C n must be interpolated, using points where C n is already defined, the location of Γ n+, and ρ n. Also, note that there may be some points that were in Ω n, but are no longer in Ω n+, the green points shown in the figure. 55

184 Γ n+/ Ω n+ Γ n / Figure 6.4: This figure shows when and why we have to interpolate values of C n. The red points are points where C n does not exist but where C n+ is evaluated. The red points are denoted x in the algorithm for the embedded boundary method. 56

185 Ω n+ Γ n+/ Figure 6.5: This figure shows the same method as shown in Figure 6.3, except now we are interpolating C n to the red points, which are in Ω n+. We do this interpolation of C at x, the red points in the figure, using exactly the same steps as for the extrapolation when we needed values of C at points that lied outside the domain, interior to the drop. Figure 6.5, shows the same scheme implemented for this interpolation. The only difference is that ξ Γ in equations 6.5 to 6.55 is now defined to be ξ Γ = x x Γ. 57

186 6.4.3 Evaluating C at the Interface We also use the same method to evaluate C on the interface, C Γ, which is needed to update ρ. The scheme to evaluate C Γ is shown in Figure 6.6. Since this is needed in the equations for ρ, C Γ is evaluated at (x j, f j ) or (g j, y j ), the discretization of Γ. The point on Γ where we are evaluating C Γ is denoted x in the algorithm for the embedded boundary method and the value of C Γ at this point is denoted C. We follow steps (), (), and (3) of algorithm 6.49 explained in detail when enforcing the boundary condition for C at Γ. The last step of the algorithm evaluates the value of C at x, C. We continue to fit a cubic polynomial for C through x, x i, x ii, and x iii. To be able to solve for the coefficients of this polynomial we need four conditions. These conditions are the same as before. Three of those conditions are the values of C, C i, C ii, and C iii, at x i, x ii, and x iii. The fourth condition is the boundary condition, equation 6.48, at x Γ = x. The fact that x Γ = x leads to a different formula for C, C = K ξ Γ g i C i g ii C ii g iii C iii g o, (6.56) 58

187 where g o = K ξ Γ 3ξ Γ + 6ξξ Γ + ξ (ξ Γ + ξ)(ξ Γ + ξ) (6.57) g i = (ξ Γ + ξ)(ξ Γ + ξ) ξ (6.58) g ii = ξ Γ (ξ Γ + ξ) ξ (ξ Γ + ξ) g iii = ξ Γ (ξ Γ + ξ) ξ (ξ Γ + ξ) (6.59) (6.6) and K = αkp e( ρ) and K = αkp e La ρ. In these equations for g, ξ = x i x ii = x ii x iii and ξ Γ = x i x. Note that here, we do not have to interpolate ρ as x is a point in the discretization of Γ and therefore ρ is already defined at x. Whenever C is evaluated at the interface, it is done explicitly with the most recently computed C. 59

188 Ω n+ Γ n+/ Figure 6.6: This figure also shows the same embedded boundary method described in this section, but now the situation is slightly different, since C is being evaluated at the red points on Γ. 6

189 This completes the description of the discretization of the full problem. We have presented in this thesis a numerical method to solve a two phase flow problem with insoluble and soluble surfactants. In the next few sections, we once again present a convergence study with a given velocity field and then when coupled with the Navier-Stokes equations. 6.5 Order of Convergence In this section, we show that the method developed in this thesis for soluble surfactants is second order with a given velocity field. We have already shown in Section 3.6, that the method is second order for clean drops and for drops with insoluble surfactants. The convergence results are presented for two different velocity fields and varying bulk Peclet numbers P e. In these tests, three equations are being coupled, the equations to advect the interface, the equations for ρ, the surfactants along the interface, and the equation for the bulk surfactant concentration, C. The nondimensionalization is as described above in Section 5., using ρ to nondimensionalize ρ. The exact solutions are used to compute the error when we have the exact solutions. In the situations where we do not, the error for a value on a grid with size h is evaluated to be the difference between that value and the value on the grid at size h. In these results, the error is evaluated using the L norm and the L norm. The area of the drop continues to converge as was the case with insoluble surfactants. The relative error in the area of the drop versus time is shown here for the finest grids. 6

190 In all cases here, the initial surfactant concentration along Γ is set to ρ o =.4. To satisfy the boundary condition for C at Γ, the bulk surfactant concentration is initially set to a constant C o = La =, α = and K =.. La ρ o ρ o. The parameters are set such that P e Γ =, The first given velocity field tested was a constant velocity field, (u, v) = (, ). The initial interface is a circle with radius.3 centered at (.5, ). We study this case with just one Peclet number, P e =. This case was run until time, t =. In Figure 6.7, the initial and final positions of the interface and the initial and final surfactant concentration along the interface are shown. In Figure 6.8, the bulk surfactant concentration, C, as well as the bulk surfactant concentration at Γ, C Γ is shown at the initial and final times of the simulations. In this case the surfactant concentration along the interface and in the bulk stays constant as the drop moves. In Figure 6.9, the convergence of the interface location is shown to be second order and the convergence of the surfactant concentration along the interface and in the bulk is shown to be much greater than second order. Also, the relative error in the area within the drop versus time is plotted for x = y = 4. In all of the convergence plots in this section, the green and red lines are reference lines with slope and, respectively. Also, the timestep is always set to be, t =.9 x =.9 y for all the convergence results presented in this section. 6

191 ρ s ρ s Figure 6.7: The top two plots show the drop contour at the initial and final times when advected using (u, v) = (, ). The bottom two plots show the surfactant concentration on the interface versus arclength at the same times. The stars on the interface correspond to zero arclength for the surfactant concentration plots. 63

192 C Γ C Γ s s Figure 6.8: The top plots show the bulk surfactant concentration at the initial and final times. The bottom two plots show the bulk surfactant concentration at Γ at the same times. 64

193 .5 x 5 Error in Γ A( t ) A( ) A( ) x = y.5 time Error in ρ 5 Error in C 5 3 x = y x c = y c Figure 6.9: Error plots for the interface and surfactants. The relative error in the area inside the drop versus time for x = y = is also shown. 4 65

194 The second velocity field tested was a given shear flow, (u, v) = ( + y, ). (This is different then the shear flows presented in the results as the velocity is set for the entire domain and the Navier-Stokes equations are not solved.) The initial interface is a circle with radius.5 centered at (, ). Convergence results are presented for P e =,, and. This case was run until time, t =.5. In Figures 6. and 6., the interface location, the surfactant concentration along the interface, the bulk surfactant concentration, and the bulk surfactant concentration at the interface at the initial and final times are shown. In Figure 6., the convergence for the interface location is shown to be second order in both norms. Also, the relative error in the area is plotted versus time for x = y =. In Figure 6.3, the convergence for ρ and C is plotted 448 for the three cases, P e =, and. The convergence rate for the surfactant concentration on the interface is second order in all cases and in both norms. For C, the convergence rate is very close to second order in the L norm, and a little below second order in the max norm for P e = and. In the L norm the convergence of C is sometimes below second order (between first and second) because of grid effects. When evaluating C at gridpoints using the embedded boundary method, the error can be larger if the evaluation point lies very close to the interface. 66

195 ρ s ρ s Figure 6.: The top two plots show the drop contour at the initial and final times when advected using, (u, v) = ( + y, ). The bottom two plots show the surfactant concentration on the interface at the same times. The stars on the interface correspond to zero arclength for the surfactant concentration plots. 67

196 C Γ C Γ s s Figure 6.: The top plots show the bulk surfactant concentration at the initial and final times. The bottom two plots show the bulk surfactant concentration at Γ at the same times. 68