Migration of a droplet in a cylindrical tube in the creeping flow regime

Transcription

1 Migration of a droplet in a cylindrical tube in the creeping flow regime Binita Nath, Gautam Biswas, Amaresh Dalal, and Kirti Chandra Sahu Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati , India Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Sangareddy , Telangana, India The migration of a neutrally buoyant droplet in a tube containing another immiscible liquid is investigated numerically in the creeping flow regime. A fully developed velocity profile is imposed at the inlet of the tube. The interface between the two immiscible fluids is captured using a coupled level-set and volume-of-fluid (CLSVOF) approach. The deformation and breakup dynamics of the droplet are investigated in terms of three dimensionless parameters, namely, the ratio between the radius of the undeformed droplet and the radius of the capillary tube, the viscosity ratio between the dispersed and the continuous phases, and the capillary number that measures the relative importance of the viscous force over the surface tension force. It has been observed that the droplet, while traversing through the tube, either approaches a steady bullet like shape or develops a prominent re-entrant cavity at its rear. Depending on the initial droplet size, there exists a critical capillary number for every flow configuration beyond which the drop fails to maintain a steady shape and breaks into fragments. The deformation and breakup phenomena depend primarily on the droplet size, the viscosity ratio and the capillary number. Special attention has been given to the case where the drop diameter is comparable with the tube diameter. A thorough computational study has been conducted to find the critical capillary number for a range of droplets of varied sizes suspended in systems having di erent viscosity ratios. I. INTRODUCTION The study of droplet dynamics in microchannels or capillary tubes depends on various parameters which are responsible for the topological changes at the interface separating the fluids. Two of the dictating parameters are the viscous and surface tension forces, which are important at small scales. The relative strength of both the forces is expressed by the capillary number (Ca µ s V/, where µ s is the viscosity of the continuous phase, V is the velocity scale and is the interfacial tension). Low values of Ca indicate that the stresses due to interfacial tension are stronger than the viscous stresses. Subjected to such conditions, a migrating drop tends to minimize its surface area by changing to a spherical shape. On the other hand, when Ca is higher, the viscous stresses dominate resulting in large droplet deformation, which in turn leads to evolution of asymmetrical droplet shapes. The motion of a droplet inside a channel/pipe in the Stokes flow limit has been studied theoretically by many researchers in the past decades. Hetsroni et al. [1] studied the motion of a small undeformed droplet traversing along the axis of the tube and obtained an equation for the dimensionless droplet velocity. Brenner [2] and Bungay & Brenner [3] conducted a similar study and obtained an equation for the excess dimensionaless pressure drop. They also observed that depending on the viscosity ratio the overall pressure drop can be either positive or negative in the presence of the drops. Hyman & Skalak [4, 5] investigated the case of equally spaced train of neutrally buoyant concentrically located drops considering both deformed and undeformed droplet shapes. Chan & gtm@iitg.ernet.in Leal [6]; Nadim & Stone [7] studied the drop motion addressing the e ect of the viscosity ratio, ( µ d /µ g ) and the capillary number Ca, where viscosities of dispersed fluid and continuous fluid are µ d and µ s,respectively. Recently, Konda et al. [8] investigated the motion of an air bubble in a converging-diverging channel by conducting two-dimensional numerical simulations. It is found that the bubble undergoes oblate-prolate deformation periodically at the early times, which becomes chaotic at the later times in the inertial regime. This occurs due to path instability as well as the Segré-Silberberg e ect [9]. Sutera and co-workers [10, 11] experimentally studied the motion of solid hemispheres and caps in Newtonian fluids assuming it as a model for motion of erythrocytes in capillaries. Prothero & Burton [12, 13] investigated the motion of a train of gas bubbles in a Newtonian fluid as a model for blood flow through capillaries. However, the experiments relevant to the present work are the studies conducted by Karnis et al. [14], Olbricht [15], Ho & leal [16] and Olbricht & Kung [17]. They have investigated the dynamics of droplets migrating through channels or tubes at low Reynolds numbers. Ho & leal [16] also investigated the e ects of the capillary number up to O(10 2 ) and viscosity ratios on the shape of the drops migrating inside a capillary tube. Olbricht & Kung [17] experimentally studied the motion of drops in a capillary tube for a large range of the capillary numbers, drop sizes and viscosity ratios. They reported the e ects of drop size and viscosity ratio on the critical capillary number, above which the drop breakup occurs. Most numerical investigations on the migration of bubble/drop in Poiseuille flow at low Reynolds number (in the Stokes flow approximation) were carried out using boundary-integral method (see e,g. [18 24]). It is to be noted here that boundary-integral method does not solve the complete Navier-Stokes equations. A few oth-

2 2 ers researchers, on the other hand, solved the Navier- Stokes equations and studied the motion of drops in channels using finite-di erence/finite-volume/front tracking methods [8, 25] by conducting two-dimensional numerical simulations. A neutrally buoyant drop in a fully developed channel flow is capable of crossing streamlines as a consequence of drop deformation and formation of a variety of transient non-spherical shapes, in contrast to non-deformable drops and particles. Hence, such a complex situation demands dynamic simulation and high computational accuracy. Moreover, although several researchers numerically investigated the breakup of a droplet in capillary tubes, it has been observed that most of them terminate their computations as the thread of the outer fluid becomes thinner as the time progresses, and when the numerical method adopted becomes di - cult to resolve the interface further in order to get the accurate physics. Thus, the exact physics of post breakup scenario of the drop is a relatively unexplored field. In the present study, the dynamics of a single droplet migrating inside a tube is investigated in a creeping flow regime. A bespoke coupled level-set and volume-of-fluid approach is used. We thoroughly investigate the deformation and translational motion of an axisymmetric drop in a tube, and present its behaviour subject to varied parametric conditions. The influence of the relevant physical parameters, such as the ratio of droplet radius to tube radius, the capillary number and the viscosity ratio between the two phases are discussed in detail. Since the study of the droplet breakup phenomenon is very important from the practical applications point of view, the present work attempts to capture the critical conditions for the breakup of the droplet by studying the dynamics of single drop in a cylindrical tube. The rest of the paper is organised as follows. In section II, we outline the formulation and discuss the validation of the present numerical solver. The results are presented in section III, and concluding remarks are given in section IV. Fullydeveloped flow Inlet r z z 0 II. FORMULATION a 0 R 0 Outlet FIG. 1. Schematic diagram (not to scale) showing the initial flow configuration of a droplet of radius a o located at (r o,z 0) in a cylindrical tube of radius R 0. A fully-developed flow is imposed at the inlet of the tube. The length L of the tube is taken to be equal to 40R 0. The motion of an initially spherical droplet of radius a o migrating through a cylindrical tube of radius R 0 due to an imposed flow at the inlet of a tube (of radius R 0 and length L = 40R 0, as shown in Fig. 1) is considered. The radius of the droplet in the present study lies in the range of 0.1R 0 to 0.95R 0. The droplet and the fluid inside the tube are immiscible. The fluids are also assumed to be incompressible and Newtonian. The viscosity and density of the dispersed phase (droplet) and continuous phase (fluid inside the cylinder) are (µ d, d ) and (µ s, s ), respectively. The flow is assumed to be axisymmetric. Thus, an axisymmetric cylindrical coordinate system (r, z) is used to model the flow dynamics, where r and z represent the radial and axial coordinates, respectively. Initially, the bubble is located at z = z 0 =2R 0. In the present study, the e ect of acceleration due to gravity, g is neglected. A. Governing equations In order to determine the flow characteristics, we solve the equations of mass and momentum conservation, which are given by r u =0, (1) ( + u ru = rp + r µ( )(ru + ru T ) + apple( )r. (2) Here, u (u, v) and p denote the velocity and pressure fields, respectively; where u and v represent the velocity components in the axial and radial directions, respectively; and µ denote the density and viscosity of the fluid, respectively and t represents time. represents the (constant) interfacial tension. apple = r n is the interfacial curvature, in which n is the outward-pointing unit normal at the interface, i.e. towards the continuous medium. This is derived from level set function as n = r r, (3) where the level set function, is the signed distance function from the interface. is zero at the interface and has positive and negative values for the continuous and dispersed phases, respectively. In Eq. (2), the surface-tension force is included as a body force term using the formulation proposed by Brackbill et al. [26]. Here, is the smoothed void fraction field, which is defined using a Heaviside function [27], H( ), as = H( ) 8 >< 1, if >, h i 1 = 2 >: sin, if apple, 0, if <. (4)

3 3 Here, 2 is the interface thickness over which the fluid properties are interpolated. The present simulations were performed using = 3 z/2, where z is the size of the computational cell. The smoothed density ( ) and viscosity µ( ) are expressed by a Heaviside function as ( ) = s +(1 ) d, (5) µ( ) = µ s +(1 )µ d, (6) respectively. In the CLSVOF formulation, the advection equations for the volume fraction ( ) and the level set function ( ) are given by + u r =0, (7) + u r =0, (8) B. Initial and boundary conditions In order to solve the governing equations, the following initial and boundary conditions are implemented. The fluids are stationary at t = 0, i.e. at the starting of the computations. The fully-developed velocity profiles are imposed at the inlet of the tube (z = 0), which are given by u(r) =2V 1 r 2 R0 2, and v(r) =0, (9) where V is the average velocity of the imposed flow. The no-slip and no penetration conditions (u = v = 0) are imposed at the tube wall, i.e. at r = R 0. The outlet boundary conditions, which are Neumann boundary conditions for the velocity components and fixed pressure condition are applied at the outlet of the tube. They are given by at z @z =0, and p = p 0, (10) C. Numerical method The numerical method used to solve the governing equations is a coupled level-set and volume-of-fluid method [27, 28]. The interface is considered as a piecewise linear segment in each cell. The surface tension force at the interface is accounted using the continuum surface force model [26]. In the CLSVOF method, the level set function [29] is deployed to capture the interface, and volume of the fluid fraction is used to conserve the mass due to the moving interface [30]. The governing equations are discretized using the finite-di erence method on a cylindrical coordinate system. The grid size in the radial and axial directions is considered to be the same, i.e r = z. The MAC algorithm [31] is employed to solve the single set of governing equations on a staggered grid with scalars located at the cell centers and velocity components at the center of the cell faces. The convective and viscous terms are discretized using the second-order ENO method [32] and central di erencing, respectively. In the present work, the time stepping procedure is based on an explicit method to maintain the stability of the solution. During the computations, the time steps are chosen to satisfy Courant-Friedrichs-Lewy (CFL), capillary, viscous and gravitational time conditions [33]. The details of the numerical method used in the present study can be found in Gerlach et al. [28] and Chakraborty et al. [33]. The CLSVOF method has been used by many researchers successfully in solving problems involving the study of dynamics of buoyancy driven gas bubbles motion in an unbounded quiescent liquid, the dynamics of bubble formation from a submerged orifice [33 35], impact of a drop on a liquid surface [36], bubble entrapment [37] and in many such practical problems. D. Non-dimensionalisation The radius of the tube, R 0 and average imposed velocity, V at the inlet of the tube are used as the length and velocity scales, while µ s and s are used as the viscosity and density scales, respectively. The results obtained from the numerical simulations are presented in terms of dimensionless parameters. The aspect ratio, a, which is given by a 0 /R 0, i.e. the ratio of the initial droplet radius to the tube radius. The other dimensionless parameters are the Reynolds number, Re svr0 µ s, the capillary number, Ca µsv, the density ratio, = d s and the viscosity ratio, = µ d µ s. In the present study, the Reynolds number is assumed to be small (Re < 1), i.e. the influence of inertia can be assumed to be very small compared to the viscous e ect. Furthermore, as we study the migration of a neutrally buoyant droplet, we set = 1 in the simulations. Thus, the problem is essentially governed by three dimensionless parameters, namely, a, Ca and. The results are presented in terms of dimensionless time, tv/r 0. E. Fluid properties considered The fluid properties are chosen based on the experimental work of Ho & Leal [16]. In their study, the suspending medium was 95.75% by weight glycerine in water at 25 C. The drops that were suspended comprised of a

4 4 solution of silicone oil (dow corning 200 fluid, a dimethyl silioxane polymer) and carbon tetrachloride. The surface tension was measured to be N/m for the interface between the glycerine and mixture of silicone oil and carbon tetrachloride. Four grades of silicone oil having di erent viscosities but almost equal densities (1251 Kg/m 3 ) were used. In the present computational study, these fluid properties are deployed to validate the code. The ratio of the viscosity of the droplet fluid to that of the suspending fluid has been varied in the range of to The droplet size is varied from a =0.1 to0.95 and Ca is varied from 0.1 to 1. F. Validation The present numerical solver is validated by comparing the results obtained from the present study with the corresponding results reported by the previous experimental and numerical investigations [16, 19, 20, 38]. The qualitative and quantitative comparisons obtained are reported below. First, a grid convergence test is conducted based on which a suitable grid size is chosen for generating the rest of the results presented in this study V d 1.9 N z N r 1000, , , , τ FIG. 2. Comparison of the velocity of the drop, V d for di erent grid-meshes ( , , , ). The parameters are Ca =1.0, =0.1 anda =0.7. N z and N r represent the number of grids in the z and r directions of the computational domain, respectively. the present simulations are presented in Fig. 3(b), which agree with the experimentally obtained droplet shapes of Olbritch & Kung [17]. 1. Grid convergence test (a) (b) The grid convergence test is conducted by simulating the migration of a droplet in a cylindrical tube with aspect ratio, a =0.7. The rest of the dimensionless parameters are Ca =1.0 and =0.1. In Fig. 2, the temporal variations of the dimensionless drop velocity, V d for di erent grid-meshes ( , , , ) are shown. We found that the steady velocity of the droplet obtained using , , , are 2.029, 2.05, and 2.07, respectively. The percentage di erences in the steady velocity of the droplet are 1.07%, 0.33% and 0.67%, when we compare V d obtained using with that obtained using , and , respectively. In view of this, and optimising the computational time and cost without compromising the accuracy of the results, grid-meshes are used to generate the rest of the results presented in this study. The corresponding mesh size of 0.02 is employed in all the simulations. The time step was chosen such that the Courant-Friedrichs-Lewy (CFL) and capillary time step conditions are satisfied [33]. 2. Qualitative comparisons The experimental work of Olbritch & Kung [17] shows the steady shapes of a droplet migrating in a capillary tube for a = 0.95 and = The droplet shapes for Ca =0.05, 0.10 and 0.16 (from [17]) are shown in Fig. 3(a). The corresponding droplet shapes obtained from Ca =0.05 Ca =0.1 Ca =0.16 FIG. 3. Comparison of the steady shapes of a droplet of size a =0.95 migrating in a capillary tube for =0.99 for Ca = 0.05, 0.10 and (a) Experimental results [17], and (b) present study. The present study focuses on capturing the droplet deformation and breakup while migrating in a tube. The droplet breakup phenomenon forming re-entrant jet was investigated by Guido & Preziosi [38], who compared the numerical simulation results of Tsai & Miksis [20] with the experimental results of Olbritch [15]. We perform numerical simulation in the same configuration consid-

5 5 ered by these authors. The droplet shapes obtained from the present numerical simulations (panel a) are compared with those of Tsai & Miksis [20] (panel b). The experimentally obtained droplet shapes from Olbritch [15] are also shown in Fig. 4(c). It can be seen that our results show excellent agreement with those presented by Tsai & Miksis [20] for the time instants =0, 2, 4 and 6. Udell [19], though it di ers from the experimental results of Ho & Leal [16] slightly; however the trend remains same. These results also follow the same trend observed in the theoretical investigation of Hetsroni et al. [1] conducted for a small undeformed droplet. They found the dimensionless droplet velocity, (a) =0 =2 =4 =6 V d = a2 + O(a 2 ). (11) (b) Note that the values of V d obtained using Eq. 11 defer from the present results and the results of Martinez & Udell [19] and Ho & Leal [16], but the trend (V d decreases with ) remains the same. This is because Eq. 11 is strictly valid for small and undeformed droplets. presented in Fig. Figs. 5 (c) V d 1.4 (a) Ho and Leal [16] Martinez and Udell [19] Present 1.3 FIG. 4. Comparison of the evolution of droplet shape: (a) Present study, (b) the results of Tsai & Miksis [20]. (c) The shapes of the droplet obtained by Olbritch [15] in the same configuration. The parameters used are Ca = 1, = 0.1 and a = V d (b) Ho and Leal [16] Martinez and Udell [19] Present 3. Quantitative comparisons In order to gain more confidence on our numerical solver, we perform quantitative comparison with the previous investigations. In Figs. 5(a) and (b), we study the e ect of viscosity ratio on the dimensionless droplet velocity, V d for a =0.726 and 0.914, respectively, and compare with the experimental results of Ho & Leal [16] and the numerical simulation results of Martinez & Udell [19]. It is observed that the present computational results agree well with the computational results of Martinez & FIG. 5. The e ect of viscosity ratio on the droplet velocity, V d for (a) a =0.726 and (b) a = Here, Ca =0.1. The results obtained from the present computation are compared with those presented by Ho & Leal [16] and Martinez & Udell [19].

6 6 III. RESULTS AND DISCUSSION After validating our numerical solver by comparing with the previously reported experimental and numerical results, in this section, we conduct an extensive parametric study by varying the aspect ratio (a = ), the capillary number (Ca = ) and the viscosity ratio ( = ). The study is carried out for small and big droplets, which undergo deformation without and with breakup at later times as they migrate inside the cylindrical tube. A. Dynamics of unbroken droplets First, we investigate the dynamic of small droplets migrating inside the tube. These droplets do not break, but deform into oblate shapes as they move in the axial direction. The degree of this deformation largely depends on Ca. The aspect ratios ranging from a =0.1to0.9 are considered. In this study, we fix the radius of the cylinder and vary the initial droplet radius to vary the aspect ratio, a. Based on the previously reported experimental results, the capillary number is varied from 0.1 to 1, while the viscosity ratio is kept in the range 0.1 to E ect of initial droplet size To understand the e ects of droplet size on the dynamics, we consider droplet of di erent sizes migrating in the cylindrical tube of fixed radius, and thereby varying the aspect ratio, a. The rest of the parameters are Ca =0.1 and = 0.1. For a droplet moving along the axis of the tube, the droplet velocity, V d lies between the maximum (centerline) and the average velocities of the undisturbed flow. The maximum or the centerline velocity is twice that of the average velocity in the Hagen-Poiseuille flow. From our basic understanding, it is obvious that if the droplet (placed at the centerline of the tube) is very small in size, it moves almost with the centerline velocity of the tube. However, as the droplet size increases, the steady velocity attained by the droplet is much less than the centerline velocity, but higher than the average flow velocity in the tube. Figure 6 depicts these phenomena. We can see that for a droplet for a =0.1 (small droplet), the steady dimensionless velocity of the droplet, V d is almost equal to 2 and increasing the droplet size gradually decreases the droplet velocity, which reaches to a plateau for a>0.6. This behaviour is also observed (not shown) for film thickness (the thickness between the droplet and wall) as well. The elongated drops are termed as Taylor drops. Similar findings on droplet velocity and film thickness for long drops were reported in the classical studies of Taylor [39] (experimental) and Bretherton [40] (theoretical). FIG. 6. The e ect of the variation of the aspect ratio, a (i.e initial droplet size) on the droplet velocity, V d. The steady shapes of the droplet for a =0.3, 0.5, 0.7 and 0.9 are inserted in the plot. The rest of the parameters are Ca =0.1 and = E ect of capillary number The influence of the capillary number is investigated for a droplet with aspect ratio, a = 0.7 migrating inside the fluid stream in a tube. The viscosity ratio, is considered to be 0.1. As discussed in the introduction, several researchers in the past have numerically computed the shapes of a droplet suspended in a creeping flow through a capillary tube. In the review of Guido & Preziosi [38], it is reported that with the increase in the value of Ca the droplet becomes elongated in the axial direction. Figure 7 depicts the steady shapes of a droplet for di erent Ca. It can be seen that the droplet elongates when the capillary number is increased. Close inspection of Fig. 7 also reveals that with increasing Ca, the front end of the drop becomes more tapered while the curvature at the trailing end decreases. This phenomenon was also observed in the past by Prothero & Burton [12, 13]. It was also reported [11 14, 16] that the onset of a re-entrant cavity appears for Ca 0.75 in case of a = Recently, Cherukumudi et al. [41] also reported the formation on indentation at the back of a drop for Ca 1. The phenomenon of formation of cavity or indentation can be clearly seen in Fig. 7. For Ca apple 0.7, we see that no such indentation is formed but when Ca is around unity, a prominent indentation is observed at the trailing edge due to the change of the sign of the curvature. We observe that for large values of the capillary number (Ca > 1, for this set of parameters), a droplet does not reach to a steady shape. For Ca > 1, the indentation grows and allows the continuous fluid to penetrate inside the droplet and beyond a critical value of the capillary number the droplet breaks. This will be discussed in the next section, while discussing the dynamics of disintegrating droplets. Figure 7 also illustrates the streamlines inside the

7 to the flow. After this initial phase, the droplet velocity, V d gradually increases as the droplet attains a steady streamlined shape. Once a steady shape is obtained the drop continues to migrate in the axial direction with the same steady velocity (for > 3 approximately, for this set of parameters). One interesting observation noted here is that at high Ca, the drop velocity exceeds the centerline velocity of the fluid, 2V. This was also observed by Lac & Sherwood [43], where it was revealed that an elongated droplet with < 0.5 travels faster than 2V, the maximum velocity of the imposed flow at the inlet of the tube E ect of viscosity ratio FIG. 7. The steady shapes attain by a droplet for a =0.7 migrating in a tube for di erent values of Ca. The streamlines (in the relative reference frames moving with the droplets) are also shown in this figure for di erent values of Ca. The viscosity ratio, = Ca 0.1 V d τ FIG. 8. The temporal variation of droplet velocity, V d as it is migrating inside the tube for di erent values of Ca. The other parameters are a = 0.7 and = 0.1. droplet signifying circulation within the droplet. The motion of the medium outside the droplet exists continuous with the rotational motion inside the droplet. The liquid motion in the core of the droplet can be approximated by a Hill s spherical vortex [42]. If a Hill s vortex flow through a constricted passage, it experiences strain tending to make it oblate with major axis along the axis of symmetry. The temporal variations of dimensionless velocity of the droplet, V d are shown in Fig. 8 for Ca = , 0.7 and 1. The parameter values considered are the same as those used to generate Fig. 7. It is seen that during the initial time ( < 0.5 approximately), a slight decrease in the droplet velocity is observed due to the onset of shape distortion, which increases the droplet resistance In Figs. 9 (a), (b), (c) and (d), the e ect of viscosity ratio is investigated on the dynamics of a droplet (a =0.7) for di erent values of the capillary number. The variations of steady droplet velocity, V d with in Fig. 9 (a) show that increasing viscosity ratio, decreases V d for the range of the capillary numbers considered. It can also be observed that for a fixed value of, increasing Ca increases the steady droplet velocity, V d. In Figs. 9 (b) and (c), the e ect of and Ca is investigated on the droplet deformation by plotting the variations of dimensionless length l d and width, w d of the droplet with Ca for di erent values of. It can be seen that an increase in either Ca or (or both) causes the droplet to deform more in the axial direction, and hence its length (l d )increases, while width (w d ) decreases. Fig. 9 (d) shows the e ect of Ca and on the thickness of the continuous fluid film surrounding the drop. In accordance with the observation from Figs. 9 (b) and (c), it is obvious that the film thickness (f t ) would increase as the droplet gets elongated. Thus, as expected, we see in Fig. 9(d) that with the increase in either Ca or (or both) increases the film thickness f t. The inspection of this figure also reveals that the e ect of viscosity ratio is independent of size of the droplet. Note that length and width of the droplet, and film thickness presented in Fig. 9 are non-dimensionalised with the radius, R 0 of the tube. B. Dynamics of disintegrating droplets The breakup of a droplet into smaller droplets is brought about due to the instabilities. Unless a very high capillary number is considered, a small drop does not break into smaller droplets in a pressure-driven capillary tube. However, if the aspect ratio, a is large, the drop breaks into daughter droplets even at low capillary numbers. The particular the capillary number that causes a drop to break is termed here as critical capillary number. The same terminology was also used in the previous studies [17, 38]. The value of the critical capillary number for a particular drop size is dependent on the viscosity ratio of the system. As observed in Fig. 7, for Ca = 1, a

8 V d 1.68 (a) Ca (c) w d Ca l d (b) Ca f t (d) Ca FIG. 9. (a) Variation of droplet velocity, V d with viscosity ratio, for di erent values of Ca. Variations of dimensionless (b) length of the droplet, l d, (c) width, w d of the droplet, and (d) film thickness, f t with Ca for di erent values of the viscosity ratio,. The aspect ratio, a is fixed at 0.7. =0 =1 =2 =3 =4 =18 clearly visible indent is formed at the trailing edge of the droplet (for a =0.7). Thus, we now study the e ect of a on the splitting of the droplet for Ca = 1. Also we investigate on the e ect of on drop breaking phenomenon for a fixed a and Ca. 1. E ect of droplet size for a system with =0.1 and Ca =1 The deformation of a droplet depends primarily on the size of the drop with respect to the tube radius, a. Tounderstand the importance of the aspect ratio of a droplet on its deformation and breakup, we perform numerical simulations for a =0.7, 0.8 and 0.9 in a flow system with =0.1 and Ca = 1. These results are discussed next. Droplet dynamics for a =0.7 For a = 0.7 (i.e for a relatively smaller droplet), the droplet shapes are plotted for di erent times in Fig. 10. It can be seen that the droplet undergoes large deformation during the initial phase ( <1). It continues to deform till = 3 for this set of parameter values. During this period, the shape of the droplet is adjusted due to the competition between the inertial and viscous forces, which are dominant at the leading and trailing edges, of the droplet respectively. In this case, as the droplet is small in size, the tube wall is su ciently away from the edge of the drop and thus the viscous drag due to the wall e ect is much smaller compared to the inertial force at the leading edge. At = 4, we see that the drop attains a steady bullet like shape and there after it traverses in the tube maintaining the same steady shape. FIG. 10. The shape evolution of a droplet for Ca =1and =0.1. The aspect ratio a =0.7. Droplet dynamics for a =0.8 The dynamics of a droplet for a =0.8 is investigated in Fig. 11 while keeping the rest of the parameters the same as those used for a =0.7. The shape evolution of the droplet for a = 0.8 reveals some interesting phenomena. In this case, the droplet surface is closer to the pipe wall as compared to a =0.7case, which enhances the e ect of the drag force. It is observed that the droplet continues its deformation (unsteady state) for a long time. At an early time ( = 2), the indent becomes very prominent at the trailing edge of the droplet. At t 4, it can be seen that the continuous fluid pushes its way through the drop forming a penetrating jet. Then, the droplet fluid forms a thin layer towards the leading edge and continues thinning while the tail part of the drop joins back, thus leaving a small part of the continuous fluid trapped in between them (see = 6). The results at between 6 and 8 show how the leading part of the drop separates from the main droplet body. At = 7.25, the position of the droplet is nicely captured showing the advancing front daughter droplet with two small satellite droplets followed by the large drop. At = 7.75, an interesting phenomenon occurs, when the two small satellite droplets merge with the large drop. This merger happens due to the phenomenon of Ostwald ripening. From = 8to

9 9 10, the entire drop merges back and proceeds towards attaining a steady shape ( = 15). During the entire period ( =0 15), the drop gradually tends to become more streamlined thus increasing the gap between the drop surface and the wall of the tube. Therefore, the drag force due to the tube wall e ect decreases and inertial force at the leading edge increases gradually during this period. =0 =2 =4 =6 =6.5 =7.25 methods that yield results for very large or very small drops [17] cannot be applied for droplets whose size is comparable to the tube diameter. The shape evolution of a droplet for a =0.9 is shown in Fig. 12. It can be seen that the deformation dynamics is di erent from that observed in the previous case (a =0.8). In this case, the droplet is more elongated (see droplet shape at = 4, 6 and 8). An obvious reason for this elongation is the initial droplet size is bigger than the previous case. Another reason is that due to the close contact of the drop surface with the tube wall, the droplet experiences more drag. The trailing edge of the droplet is more elongated at the early times ( = 1 to 4) owing to the dominance of the drag force in the presence of the wall. After t = 4, the droplet undergoes thinning and then breaks up into daughter droplets and satellite drops ( = 6 to 8) in the same way as that in the earlier case (a = 8). From =8 to 11, the phenomenon of Ostwald ripening is very prominent. From = 11 to 18, the drops eventually coalesce and merge back to form a single drop. It is observed that the situation at = 18 (on the verge of leaving the domain) is quite similar to the situation at = 9 in Fig. 11, hence it can be conjectured that for an extended domain the drop will gradually attain a steady shape. =7.75 =8 =0 =2 =4 =6 =7 =8 =8.5 =9 =8.75 =9 =9.5 =9.75 =10 =10.25 =10.5 =11 =1 =12 =10 =15 =12.5 =13.25 =13.5 =14 =17 =17.75 =18 =18.5 FIG. 11. The shape evolution of a droplet for Ca =1and =0.1. The aspect ratio a =0.8. Droplet dynamics for a =0.9 The aspect ratio is increased to a =0.9 and the dynamics of a droplet is investigated in Fig. 12 while keeping the rest of the parameters the same as those used for a =0.7 and a =0.8. This is an interesting case as the initial droplet radius is neither very small nor very large as compared to the radius of the tube. The main reason for the choice of this aspect ratio is that the asymptotic FIG. 12. The shape evolution of a droplet for Ca =1and =0.1. The aspect ratio a = E ect of viscosity ratio on the dynamics of breaking up droplets From the results presented in Figs. 10, 11 and 12, we learnt that a drop with aspect ratio, a =0.9 elongates

10 10 more and forms several daughter droplets and satellite droplets at the intermediate times before merging back to a single drop. Thus, we consider a droplet of a =0.9 and investigate the e ect of viscosity ratio on droplet migration and breakup. We consider three viscosity ratios in this study, = 10 3,0.58 and 2.04 in Figs. 13, 14 and 15, respectively. Ca = 1 is taken for all the viscosity ratios. Inspection of Figs. 13, 14 and 15 reveals that increasing viscosity ratio, increases the tendency of droplet breakup into smaller droplets. =0 =1 =2 =3 under the influence of surface tension. The capillary force plays a major role and promotes the formation of cylindrical ligaments of various sizes. Subsequent breakup of these ligaments is brought about by the Rayleigh-Plateau instability mechanism culminating in the formation of droplets of smaller sizes. The above mentioned explanation applies well to both low and high viscosity ratios. However, for higher viscosity ratios (say for = 2.04), the shear force on the ligament surfaces becomes relatively high in magnitude. In the face of counteracting interfacial tension, it elongates the ligaments more ( =8) before the ligaments break-up into droplets (Fig. 15). =0 =2 =4 =5 =5 =7 =6 =6.5 =9 =11 =7 =8 =13 =14 =13 =16 =15 =16 FIG. 13. The shape evolution of a droplet for Ca = 1. The aspect ratio a =0.9. =10 3 and =19 =20 The mechanisms of droplet break up and generation of smaller droplets are similar to the mechanisms of atomization that have been extensively researched. The mechanism of atomization typically consists of 3 steps: (1) formation of liquid sheets; (2) the liquid sheets undergo disintegration into drops and ligaments; (3) the ligaments further break up into finer droplets. Theoretical analyses of the atomization process to predict droplet size have yielded models providing good agreements with the experimental data. Fraser et al. [44] and Dombrowski and Johns [45] have described the process of liquid sheet atomization and the explanations have close resemblance to the processes that are observed in Figs. 13, 14 and 15. The di erence is that, in our case, the liquid drop is being disintegrated in a liquid medium. Under this situation, the trailing edges of the deformed drop are gradually transformed into liquid sheets (clearly evident from Figs. 14 and 15). For the disturbances (sinuous or varicose), the sheets are assumed to fragment into pieces of lengths dictated by a dominant wave mode. These fragmented parts roll up FIG. 14. The shape evolution of a droplet for Ca = 1. The aspect ratio a =0.9. C. Critical capillary number, Ca cr = 0.58 and Finally, in order to investigate the e ects of viscosity ratio and the size of the drop on the critical value of Ca that leads to drop breakup, we run a number of computations for di erent systems by varying the viscosity ratio from 10 3 to 2.5. The critical capillary number (Ca) is calculated based on the drop size for each combination of the rest of the governing parameters. Table I depicts the value of the critical capillary number, Ca cr obtained with respect to the aspect ratio (a) for di erent values of viscosity ratio,. From this study, we observe some physics related to drop breakup. When we focus column wise in Table I,

11 11 =0 =5 =8 =12 =14 = Ca cr 3 a =18 =20.5 FIG. 15. The shape evolution of a droplet for Ca = 1. The aspect ratio a =0.9. = 2.04 and TABLE I. The critical capillary number for di erent values of aspect ratio, a and viscosity ratio a it is observed that for each value of, the critical capillary number decreases as the aspect ratio, a (which is equivalent to droplet size) increases. The reason for such behavior can be attributed to the fact that when the droplet is very small, its velocity is influenced more by the centerline velocity of the external flow, due to which it moves fast in the axial direction without experiencing much deformation. Also the drop being far from the wall, experiences a less resistance. Such small drops require high Ca (low surface tension) in order to undergo breakup. However, as the drop size keeps on increasing Ca cr decreases, because the drop approaches the wall and the resistance to its flow increases. Row wise inspection of Table I (i.e. inspection of the variation of Ca cr with for a fixed a) reveals that the value of critical capillary number, Ca cr decreases (for most values of a considered) with the increase in viscosity ratio, for a fixed aspect ratio, a (i.e. drop size). This is because increasing increases the tendency of the droplet to deform more, and hence Ca cr decreases. In Fig. 16. it can be observed that while this phenomenon is true for droplets with a>0.5, the same does not seem to be followed for smaller drops (a <0.5). For smaller drops, an interesting phenomenon occurs. At first Ca cr decreases as usual but this constant decrement pertains only to a certain range of. For >1 (approximately), Ca cr again increases. This behavior is very prominent for a =0.4 as seen in the Table I. This non-monotonic variation of Ca cr is first noticed for =0.5. It is to FIG. 16. Variation of critical capillary number Ca cr with viscosity ratio, for di erent values of a. be noted here that, it is di cult to find Ca cr for >1, because under such a situation (of high capillary number and also high ) the droplet continuously deforms and most portions of the drop behave as a thin film, which continuously tends to elongate without breaking up. In this study, we have restricted our computations upto Ca = 6.52; thus the capillary number required for the breaking of a small drop (a =0.3) beyond = 1 has not been computed. IV. SUMMARY Numerical simulations of a neutrally buoyant droplet migration in a cylindrical tube in the creeping flow regime are conducted using a CLSVOF method. The axisymmetric Navier-Stokes and continuity equations are solved to determine the dynamics of a droplets migrating in a capillary tube with an imposed fully-developed flow at the inlet of the tube. In the context of physiological flows, the parameters (droplet size and capillary diameter) considered are relevant for the motion of red blood cells in arterioles, the motion of artificial capsules in industrial applications related to food and polymer processings, and two phase flows in a porous media, to name a few. Extensive validations of the numerical solver adapted are presented before proceeding with detailed investigation of the present problem. A parametric study is conducted by varying the ratio of droplet radius to the tube radius, capillary number and viscosity ratio between the droplet and the surrounding fluid injected at the inlet of the tube. First, we focus on the dynamics of unbroken droplets (small droplets) and find that for small droplets, increasing the size of the droplet decreases the droplet velocity, owing to increased deformability and the influence of the close proximity with the wall. For very small droplets, the drop velocity approaches the maximum velocity of the surrounding fluid flowing in the tube.

12 12 Then, we investigate the e ect of and Ca on the droplet dynamics by fixing the droplet size to a =0.7. We found that increasing Ca increases the deformation and velocity of the droplet. For Ca 1 and =0.1, an indentation is formed at the rear end of the drop, which is apprehended to grow further or disappear depending on the flow conditions. It is observed that with the increase in viscosity ratio, the drop deforms more and endure more resistance to the flow, thereby decreasing its velocity. We found that for a fixed value of Ca, the drop deformation increases with the increase in, such that the length of the drop increases and the width of the drop decreases, which in turn increases the thickness of the fluid film surrounding the droplet. We then perform numerical investigations to understand the physics involving disintegrating droplets migrating inside the tube. By keeping the viscosity ratio and capillary number fixed 0.1 and 1, respectively, droplet size is varied from to a =0.7 to 0.9. We observe that when the drop size is comparable to the tube diameter (a =0.9), the tendency of the drop to break into smaller droplets increases. For a = 0.8, we observe that the daughter droplets merge back to the main droplet at the later times. The merged droplet then traverses with a steady shape. On the other hand, the drop with a = 0.7, deforms initially and develops an indentation at the trailing edge. This indentation dies down and the drop attains a steady shape after some time without breaking up. As we observe that the droplet with size comparable to the tube diameter (a = 0.9) reveals interesting physics of breakup, we investigate the e ect of viscosity ratio for this droplet. We notice that low viscosity drops get breakup due to a re-entrant jet formed at the rear end, which gradually penetrates carrying the external fluid into the drop. 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