Chauvi ere and Lozinski as deterministic Fokker-Planck (FP) equations. The first option gives rise to stochastic numerical methods [6], [7], which hav

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1 Simulation of dilute polymer solutions using a Fokker-Planck equation Cédric Chauvi ere a and Alexei Lozinski b;λ a Division of Applied Mathematics, Brown University, Box F, Providence, RI 9, USA. b FSTI - ISE - LMF, Ecole Polytechnique Fédérale de Lausanne, CH 5 Lausanne, Switzerland. Abstract The main goal of this work is to rehabilitate the use of the Fokker-Planck equation for simulating complex flows of non-newtonian fluids which do not possess closed-form constitutive equations. Anumerical method based on the Fokker-Planck equation is applied to the simulation of a dilute solution of polymer modeled by FENE dumbbells. The proposed method is tested for both homogeneous planar extensional flow and the flow around a cylinder in a channel. Introduction Non-Newtonian fluids which have a memory of past deformation are called viscoelastic fluids and are present in a wide range of applications e.g. multigrade oils, food processing, biological fluids such as blood. In this paper we shall be concerned with the numerical simulation of dilute polymer solutions which may be regarded as being a low concentration of polymer chains in a Newtonian solvent. Thus, we can assume that the polymer chains do not interact with each other. The solvent is assumed to be incompressible and isothermal and we restrict ourselves to the case of inertialess flows (zero Reynolds number). Therefore, the mass and momentum equations have the form of the Stokes equations in which the extra-stress fi, i.e. the polymeric part of the Cauchy stress tensor, is given as a source term: = rp + s r u + r fi ; () r u =; () where u, p and s denote, respectively, the velocity, pressure and the viscosity of the Newtonian solvent. The extra-stress usually comes from a supplementary equation (closed-form constitutive equation) or a mesoscopic model. Over the past twenty years, numerous numerical methods have been designed for improving the discretization of constitutive equations for viscoelastic flows. It would be too lengthy to enumerate them here and the interested reader is referred to the excellent review in [9]. Mesoscopic models are generally regarded as being (potentially) more realistic than their closedform counterparts but their numerical simulation requires much more work. Mathematically, these models can be written in two equivalent forms (see [7]): either as stochastic differential equations or Λ corresponding author, alexei.lozinski@epfl.ch

2 Chauvi ere and Lozinski as deterministic Fokker-Planck (FP) equations. The first option gives rise to stochastic numerical methods [6], [7], which have become very popular during the last years. In contrast, simulations of dilute polymer solutions through the use of an FP equation are very rare in the literature and have been limited, so far, to very simple flows. Since the pioneering work of Warner [] in 97, followed thirteen years later by Fan [4] for the simulation of simple shear flows for FENE (finitely extensible non-linear elastic) dumbbells, no attempt has been made to use the FP equation for the simulation of more complex flows using this model. It is the purpose of this paper to show that using an FP equation may be a good alternative to stochastic computations for the simulations of dilute polymer solutions, even when complex flows are considered. The paper is organized as follows. In the next section, we recall the equations used to model dilute polymer liquids (the FENE model) and in Section 3 we describe how these equations may be discretized. The computation of the polymeric extra-stress from the solution of our FP equation is then detailed in Section 4. In Section 5 we illustrate our method with two numerical examples: a simple planar extensional flow and the flow past a cylinder in a channel. Finally, in Section 6 we draw some conclusions about the potential of the FP equation to simulate complex viscoelastic flows. Modeling dilute polymer solutions The idea for the modeling of dilute polymeric solutions is to replace the polymer chains by dumbbells which consist of two beads connected by a spring. A dumbbell whose centre of mass is at x at time t is fully described by a configuration end-to-end vector q(t; x). Let ψ(t; x; q) be the probability density function (pdf) of the random process q(t; x). In other words ψ(t; x; q)dq denotes the probability that a dumbbell at position x and at time t is to be found with the configuration vector in the box [q; q + dq]. The pdf satisfies the Fokker-Planck equation, which can be written (see [], [7]) as Dψ Dt +div q (»q F(q))ψ = qψ; (3) where D=Dt is the material derivative,» is the transposed velocity gradient tensor (» ij i =@x j ), is the relaxation time of the fluid and F(q) is the spring force, which takes the following form for FENE dumbbells: q F(q) = jqj b : (4) In the relation above, the parameter b controls the maximum extensibility of the dumbbells which cannot exceed p b. The computation of the velocity field with ()-() is decoupled from the computation of the pdf, so in equation (3), we can assume that the velocity gradient isknown. Since the vector q lies in a disc of radius p b, it is natural to represent it in polar coordinates, i.e q = r cos ; q = r sin ; with r [; p b] and [; ß]: (5) Introducing two functions b ( ) and b ( ) of the angle as follows b ( ) =» cos +» +» b ( ) =» sin +» +» sin ; (6) cos +»» ; (7)

3 Simulation of dilute polymer solutions 3 a detailed expression of equation (3) may be written as Dψ Dt = rb b + br b r + b (b r ) ψ ψ : (8) Equation (8) is completed with the initial condition at time t =, which corresponds to the equilibrium solution (» = ) and takes the following form: ψ eq = b + ßb In the next section, we will show how equation (8) may be discretized. 3 Discretization of the equations b= r : (9) In order to satisfy the boundary conditions at r = and r = p b, we shall use instead of ψ(t; x;r; ) a new unknown ff(t; x; ; ) defined by s ψ(t; x;r; )= ff(t; x; ; ); () where the relation between r and [ ; ] is r = b + and s is a fixed, strictly positive real number. To prevent the spring force (4) from becoming infinite for r = p b,we should have ψ =at r = p b (or = ) and this is achieved by introducing the multiplier (( )=) s into (). We have found experimentally that our numerical method has the best stability properties under s =. We have used this value of s to obtain the results of Section 5. Note that the pdf should be symmetric in q, thatexplains why we have chosen the new variable to be a function of r. In order to decouple the computation in the physical space Ω from the computation in the configuration space, we use the following time-splitting for equation (8) b ff i+= ff i = L FP (ff i+= ); t () ff i+ ff i+= +(u i+ r)ff i+ =; t () where L FP refers to the Fokker-Planck operator appearing on the right-hand side of (8) after the change of variable (). Thus we have reduced the task of solving a four-dimensional problem for ff(t; x; ; ) at a given time t to solving two-dimensional problems for ff i+= ( ; ) and ff i+ (x) respectively. Each step of the splitting ()-() requires a specific type of discretization to account for the type of the equation and the space in which the solution is sought. For equation (), we search for an approximate solution ff( ; ) of the form ff( ; ) = XNR NF i= X j= ff ijh i ( ) cos(j )+ XNR NF i= X j= ff ijh i ( ) sin(j ); (3) where fh i ( )g»i»nr are Lagrange interpolating polynomials based on the Gauss-Legendre points i (see the book of Canuto et al. [] for more details). Note that the set f i g is chosen so that it does not include the points = and = since the boundary conditions there are already

4 4 Chauvi ere and Lozinski taken into account by (). Only the Fourier modes of even order are kept in (3) because of the symmetry of ff( ; ). Inserting (3) into (), we form the product of () with a test function fh k ( ) cos(l )g»k»n R»l»NF or fh k( ) sin(l )g»k»n R and integrate over configuration space. The integrals with respect to are evaluated using Gauss-Lobatto quadrature rule whereas the integrals with»l»nf respect to can be computed analytically. The resulting linear system for the unknown coefficients ffij and ffij is LU decomposed and a back substitution gives the numerical solution of equation (). We note that this spectral method conserves the integral of the pdf over the configuration space provided s in () is an integer from to N R. An SUPG element-by-element spectral element method with constant upwinding factor, as detailed by Chauvi ere and Owens [3], is used to discretize the hyperbolic equation () in Ω. Periodic boundary conditions are applied at inflow andoutflow for the flow around a cylinder in a channel. The Stokes equations ()-() are discretized using a spectral element method, as described in the paper by Owens and Phillips [8], for example. 4 Computation of the extra-stress Having computed the pdf at x at time t, the extra-stress which will serve as a source term for the Stokes equations ()-() is given by what is known as the Kramers expression [7]. For the FENE model and a two dimensional problem it is fi (x;t) = p b + ( I+ < q Ω F(q) >) (4) b where the symbol Ω denotes the tensor product of two vectors and p is the polymeric viscosity of the fluid. The brackets < > in (4) denote the statistical average over configuration space, i.e. < q Ω F(q) >= Z D(; p b) q Ω F(q)ψ(t; x; q)dq: (5) Using (4),(5) and (), the evaluation of (4) gives the following expressions for the components of the extra-stress! fi xx = p b + XNR ψ + ßb! i ( + i ) s i (ff i + ff b 6 i) ; (6) fi xy = p fi yy = p b + ßb b 6 b + where! i are the quadrature weights. 5 Numerical results b XNR i= ψ + ßb 6 5. Planar extensional flow i= s! i ( + i ) i ff i; (7) XNR i= s! i ( + i ) i (ff i ff i)! ; (8) As a first test, we consider a homogeneous planar extensional flow for which the velocity field is given by (u x ;u y )=( "x; "y), and " is the extension rate. One can find an analytical expression of the pdf

5 Simulation of dilute polymer solutions 5 for the steady-state solution of (3) in this case (see []). It has the form b= ψ = C jqj exp( "(q b q )) (9) where C is some normalization constant. We report the results of two computations for different values of the extension rate (" = and " = 5), the other parameters being p = = and b =. Table gives the computed values of the xx-component of the extra-stress and the relative error for different resolutions in the case " =. We can see that the error is very small even for low resolutions. At this extension rate, the solution is very smooth (see Figure, which represents the pdf for the computational mesh (N F ;N R )=(6; 3)) and only afewpoints are needed for our spectral method to properly capture the solution. When the extension rate is increased up to " = 5, getting accurate numerical results requires much more refined meshes, as is shown in Table. This is because the exact solution at high extension rates possesses huge gradients which are very localized. Capturing such a solution requires highly refined meshes. Figure shows the numerical oscillatory solution for a low level of discretization (N F ;N R )=(6; 3) and Figure 3 shows how the solution can be properly captured when the mesh is refined up to (N F ;N R )=(; 4): In the next section we will apply the same numerical method to a complex flow. 5. Flow in complex geometry The problem of planar viscoelastic flow around a cylinder (see Figure 4) is used to test our method for more complex geometries. This benchmark problem is rheologically rich since it offers both shear flows (near solid walls) and extensional flows in the wake of the cylinder. We choose the aspect ratio Λ=R=H ==, where H is the width of the channel and R is the radius of the cylinder. We set the ratio s =( s + p ) equal to.59, and b equal to. The time steps are chosen equal to t =:5. The flow domain is divided into 6 spectral elements and polynomials of degree N =6are used in the two spatial direction for the representation of the variables (with exception of the pressure). The pressure basis polynomials are chosen to be of degree less in each direction so as to ensure that the discrete problem is well posed (see []). The iterations are stopped when the following convergence criterion is fulfilled for all collocation points x Ω : ku(x;t i+ ) u(x;t i )k» 4 : () t For this problem, we may define the Deborah number as De = U R ; () where U is the average velocity at entry or exit. For the FENE model, this benchmark problem has only been investigated with stochastic simulations of Monte-Carlo type [5]. We fix the Deborah number equal to.8, obtained by taking ( ; U;R)=(:8; ; ) and we increase the resolution of the configuration space from (N F ;N R )=(4; 9) up to (N F ;N R )=(6; 3). The convergence of our numerical scheme with mesh refinement is then established by plotting the component fi xx of the extra-stress on the axis of symmetry (jxj > ) and on the cylinder surface (jxj < ). This is shown in Figure 5, where for the two finest meshes the results match very well. We also show the contours of fi xx over the whole physical domain in Figure 6, to demonstrate the smoothness of the plot. Our results are obtained on a personal computer with the CPU Pentium III at 8MHz. The CPU time per time step was.3, 6.4 and 4.5 seconds for the meshes mentioned at Figure 5. time steps were needed to reach the steady state.

6 6 Chauvi ere and Lozinski 6 Conclusion In this paper, we have shown that computing complex viscoelastic flows using an FP equation is now possible. The key ingredient to make such a simulation possible is the time-splitting which reduces the computation of a four-dimensional problem to the computation of two-dimensional problems. The advantage of this approach over stochastic simulations is that the resulting solution is not polluted by statistical noise. The method introduced in this paper is very general and can also be applied to models describing concentrated polymer solutions such as reptation models []. Acknowledgment. The authors would like to thank Robert Owens for helpful discussions and for proofreading the manuscript. The work of the second author has been supported by the Swiss National Science Foundation, grant number References [] Bird RB, Armstrong RC, Hassager O. Dynamics of polymeric liquids, vol., Kinetic theory, nd ed. New York, Chichester, Brisbane, Toronto, Singapore: Wiley-Interscience, 987. [] Canuto C, Hussaini MY, Quarteroni A, Zang TA. Spectral Methods in Fluid Dynamics. Berlin, Heidelberg: Springer Verlag, 988. [3] Chauvi ere C, Owens RG. A new spectral element method for the reliable computation of viscoelastic flow. Comput. Methods Appl. Mech. Engrg. ; 9: [4] Fan XJ. Viscosity, first normal-stress coefficient, and molecular stretching in dilute polymer solutions. J. Non-Newtonian Fluid Mech. 985; 7:5-44. [5] van Heel APG, Hulsen MA, van den Brule BHAA. On the selection of parameters in the FENE-P model. J. Non-Newtonian Fluid Mech. 998; 75:53-7. [6] Hulsen MA, van Heel APG, van den Brule BHAA. Simulation of viscoelastic flows using Brownian configuration fields. J. Non-Newtonian Fluid Mech. 997; 7:79-. [7] Öttinger HC. Stochastic Processes in Polymeric Fluids. Berlin: Springer Verlag, 996. [8] Owens RG, Phillips TN. Steady viscoelastic flow past a sphere using spectral elements. Int. J. Numer. Meth. Engrg. 996; 39: [9] Owens RG, Phillips TN. Computational Rheology. Singapore: Imperial College Press/World Scientific,. [] Warner HR. Kinetic theory and rheology of dilute suspensions of finitely extensible dumbbells. Ind. Eng. Chem. Fundam. 97; :

7 Simulation of dilute polymer solutions 7 List of Figures Plot of the pdf ψ for " = and (N F ;N R )=(6; 3): Plot of the pdf ψ for " = 5 and (N F ;N R )=(6; 3): Plot of the pdf ψ for " = 5 and (N F ;N R )=(; 4): Cylinder radius R placed symmetrically in a D channel of half width H Profile of the component fi xx on the cylinder surface and in the wake of the cylinder for three levels of discretization Contour of fi xx at De =:8 for FENE model List of Tables Numerical results for planar extensional flow at " = (the exact value of fi xx is ) Numerical results for planar extensional flow at " = 5 (the exact value of fi xx is 4: )

8 8 Chauvi ere and Lozinski Figure : Plot of the pdf ψ for " = and (N F ;N R )=(6; 3): Figure : Plot of the pdf ψ for " = 5 and (N F ;N R )=(6; 3):

9 Simulation of dilute polymer solutions Figure 3: Plot of the pdf ψ for " =5and (N F ;N R ) = (; 4): H R Figure 4: Cylinder radius R placed symmetrically in a D channel of half width H.

10 Chauvi ere and Lozinski 9 8 (N F,N R )=(6,3) (N F,N R )=(5,) (N F,N R )=(4,9) 7 Txx x Figure 5: Profile of the component fi xx on the cylinder surface and in the wake of the cylinder for three levels of discretization. Figure 6: Contour of fi xx at De =:8 for FENE model.

11 Simulation of dilute polymer solutions N R N F Computed Relative error 5 8: : : : : : : : : :6 8: :5 Table : Numerical results for planar extensional flow at " = (the exact value of fi xx is ). N R N F Computed Relative error 5 diverging : :35 : : 3 5 4: : : : 7 Table : Numerical results for planar extensional flow at " = 5 (the exact value of fi xx is 4: )

Chapter 1 Introduction

Chapter 1 Introduction This thesis is concerned with the behaviour of polymers in flow. Both polymers in solutions and polymer melts will be discussed. The field of research that studies the flow behaviour