On Scaling of Diffuse–Interface Modelsଁ

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Chemical Engineering Science 61 (2006) 2364–2378 www.elsevier.com/locate/ces

On scaling of diffuse–interface modelsଁ

V.V. Khatavkar, P.D.Anderson∗, H.E.H. Meijer

Materials Technology, Dutch Polymer Institute, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Received 30 March 2005; received in revised form 8 September 2005; accepted 14 October 2005

Abstract Application of diffuse–interface models (DIM) yields a system of partial differential equations (PDEs) that generally requires a numerical solution. In the analyses of multiphase flows with DIM usually an artificial enlargement of the interface thickness is required for numerical reasons. Replacing the real interface thickness with a numerically acceptable one, while keeping the surface tension constant, can be justified based on the analysis of the equilibrium planar interface, but demands a change in the local part of the free energy. In a non-equilibrium situation, where the interface position and shape evolve with time, we need to know how to change the mobility in order to still model the same physical problem. Here we approach this question by studying the mixing of two immiscible fluids in a lid-driven cavity flow where the interface between the two fluids is stretched roughly linearly with time, before break-up events start. Scaling based on heuristics, where the mobility is taken inversely proportional to the interface thickness, was found to give fairly well results over the period of linear interface stretching for the range of Péclet numbers and viscosity ratios considered when the capillary number is O(10). None of the scalings studied was, however, able to capture the break-up events accurately. ᭧ 2005 Elsevier Ltd. All rights reserved.

Keywords: Diffuse–interface model; Fluid mechanics; Interface; Mixing; Modelling; Scaling

1. Introduction thickness, an idea first introduced by van der Waals (1893). Various thermodynamic variables change continuously over Multiphase flow problems and moving boundary problems this interfacial region and the thickness of this interfacial region are more frequently being solved using a diffuse–interface is closely related to the finite range of molecular interactions. model. Examples are mixing and rheology of two immiscible Thermodynamically, the finite interaction range is represented fluids including interfacial tension (Chella and Viñals, 1996; by making the free energy of the system to depend not only Keestra et al., 2003), topological transitions i.e., breakup and on the local composition but also on the composition of the coalescence of drops (Lowengrub and Truskinovsky, 1998; immediate environment. By using a mean-field approximation, Jacqmin, 1999; Verschueren, 1999; Lee et al. 2002a,b), con- the non-local effect in the free energy of the system can be rep- tact line dynamics (Seppecher, 1996; Jacqmin, 2000; Chen resented by the dependence on the local composition gradients et al., 2000) and thermocapillary flow (Jasnow and Viñals, rather than on the non-local composition (Cahn and Hilliard, 1996; Verschueren, 1999). For reviews see Anderson et al. 1958). This free energy of the system determines both the inter- (1998) and Naumann and He (2001). face thickness and surface/interfacial tension that now appears In the diffuse–interface approach, the interface between two as a distributed stress over the interfacial region. The conse- immiscible fluids/phases is considered to have a small but finite quence is that the original moving boundary problem is replaced by a set of partial differential equations with solutions that are continuous throughout the system, but have large gradients in ଁ Expanded version of a talk presented at Interdisciplinary Workshop on the interfacial region. The position and shape of the interface Diffuse Interface Models (Lyon, France, January 2004). ∗ Corresponding author. Tel.: +31 40 247 4823; fax: +31 40 244 7355. is a part of the solution of this set of PDEs and hence topolog- E-mail address: [email protected] (P.D. Anderson) ical transitions such as break-up and coalescence of domains URL: http://www.mate.tue.nl/mate/showemp.php/19. appear as an interplay between convection and diffusion and

V.V. Khatavkar et al. / Chemical Engineering Science 61 (2006) 2364–2378 2365 are handled in a natural way without requiring any interface The study of mixing of two immiscible fluids in a lid driven ‘surgery’. In addition, DIM allows the use of fixed grids with a cavity is interesting for this problem since the interfacial stretch consequent ease of numerical implementation, especially if an (defined as a ratio of the interface perimeter S at any time instant extension to three dimensions is required, e.g. using high-order to the interface perimeter S0 at time equals zero) is large (about spectral elements. 15) and increases (roughly linearly) with time until the structure For the success of the method when applied to multiphase becomes fine enough for break-up and coalescence events to flow problems, it is necessary to be able to capture the interfacial become dominant. So, the various scalings can be compared in region, which has a width of about a few nanometers, appro- two situations: one where the interface is only being stretched priately. Usually the characteristic length scale of the problem and the other where topological transitions become important. of interest is in the range of microns to millimeters. This intro- Previous numerical work on mixing of two fluids with differ- duces a dimensionless group into the problem: the Cahn number ent viscosities in a lid driven cavity flow includes the analyses (Ch) which is defined as the ratio of interface thickness to this of Bigg and Middleman (1974) and Chakravarthy and Ottino characteristic length scale. Typical values of the Cahn number (1996). Both studies use the marker-and-cell (MAC) method are O(10−5) or less. To capture this thin interface numerically, to track the evolution of the interface between fluids, however, sufficient spatial resolution is required in the interfacial region neglect surface/interfacial tension forces. Bigg and Middleman otherwise the computational scheme may run into numerical (1974) performed calculations for relatively small values of the stability problems. On the other hand, a proper resolution of the interfacial stretch, while in Chakravarthy and Ottino (1996) in- interfacial region (on the fixed grid that is generally employed) terfacial stretch reaches values as high as 60. We will extend leads to the need of such a fine mesh within the whole domain the work done by Chella and Viñals (1996) by studying the ef- that the method becomes prohibitively expensive, both in terms fect of viscosity ratio on interfacial stretch following Anderson of computer memory and computational time, notwithstanding (1999) who found that the interfacial tension tends to decrease the present computer resources. This problem of efficiently interfacial stretching, in accordance with the results of Chella capturing the real interface may be dealt with either by using and Viñals (1996). adaptive remeshing (Barosan et al., 2005) or by focusing on small length-scale problems for e.g. phase separation of block 2. Model equations copolymer systems (Zvelindovsky et al., 2000) or the rheology of immiscible polymer blends (Roths et al., 2002; Keestra et al., The diffuse–interface model considered here uses a specific 2003). Alternatively, to retain the simplicity and ease of coding form of the Helmholtz free energy function based on the ap- accompanied with a fixed grid, the real interface may be re- proach of Cahn and Hilliard (1958) placed with a numerically acceptable thick interface (Jacqmin, ∇ = + 1 |∇ |2 1999, 2000; Verschueren, 1999; Jamet et al., 2001) and this f(c, c) f0(c) 2 c , (1) solution is considered in this paper. Several questions arise as where c is the mass fraction of one of the components, f is the a consequence of this replacement, the first is whether we still 0 homogeneous part of the specific free energy and is the gra- can describe the same interfacial tension? Based on the analysis dient energy parameter. The homogeneous part f is here, for of an equilibrium planar interface the answer to this question 0 simplicity but without further restrictions approximated by the appears to be yes (Verschueren, 1999; Jamet et al., 2001). The so-called ‘c4’ approximation (Gunton et al., 1983) also known second question is which parameters in the model should be as the Landau–Ginzburg free energy: changed? Justification of replacing the real interface thickness with a numerical interface thickness is based on an analysis = 1 4 − 1 2 f0(c) 4 c 2 c , (2) of an equilibrium case which does not account for the non- equilibrium situation. So, the parameters that describe the evo- where and are both positive constants for an isothermal lution of the system from non-equilibrium to equilibrium condi- system below its critical temperature. Combining Eqs. (1) and tions need to be changed. One of those parameters in the model (2), f can now be written as, is the mobility, contained in the Péclet number Pe. The third f = 1 c4 − 1 c2 + 1 |∇c|2. (3) question is how Pe should be changed? At least three different 4 2 2 scalings were proposed in the literature. Requiring that in the The chemical potential is defined as variational derivative with limit of Ch tending to zero, the solutions of the sharp interface respect to concentration of the specific Helmholtz free energy limit should be recovered from the diffuse–interface models, given by Eq. (3) and reads, Starovoitov (1994) and Lowengrub and Truskinovsky (1998) ∝ f found Pe 1/Ch. Using the same argument, Jacqmin (1999) = = c3 − c − ∇2c. (4) showed that the mobility M scales with the interface thickness c ∝ a = = like M with 1 a<2; for a 1, Pe constant and for For a planar interface (with z as the direction normal to the = ∝ a 2, Pe 1/Ch. Considering the limit of finite interface thick- interface) at equilibrium ( = 0 in Eq. (4)) the corresponding ∝ nesses, Verschueren (1999) proposed Pe Ch. Apart from the concentration profile is found to be: Pe ∝ 1/Ch and Pe ∝ Ch scalings, in this paper, we will investi- ∝ 2 gate Pe Ch, which follows if the interface thickness, instead z c(z) = cB tanh √ , (5) of the domain size, is used as a characteristic length scale. 2 ARTICLE IN PRESS

2366 V.V. Khatavkar et al. / Chemical Engineering Science 61 (2006) 2364–2378 =± where√cB / are the equilibrium bulk solutions and 3. System definition = / is the interface thickness. The interfacial tension is the excess free energy per unit surface area due to the Mixing of two immiscible fluids initially stratified within a inhomogeneity in c in the interfacial region (Rowlinson and square, two-dimensional cavity of size L as shown in Fig. 1 is Widom, 1989) considered. The interface (defined by the position of contour c=0) is initially flat and parallel to the direction of shear. Shear ∞ 2 dc is applied by moving the bottom wall while the remaining walls = dz. (6) −∞ dz are at rest. The velocity of the moving wall is prescribed by Using the equilibrium concentration profile integrating the x 2 x 2 vx = 1 − ,vy = 0, at y = 0. (13) above equation yields: L L √ This function equals the one in Chella and Viñals (1996).We 2 2 c2 = B . (7) have used this unphysical velocity profile for simplicity as it 3 avoids discontinuity in the tangential velocity at the corners and therefore lowers the computational cost. More importantly, for In general we obtain the main question of scaling of DIM addressed here, this devi- c2 ation from the real problem of spatially uniform wall velocity ∝ B . (8) is less likely to be significant while still offering a benchmark to compare our results with. We note, however, that the numer- For mass conservation of the individual components, the mass ical scheme employed here can, without restriction, be applied fraction c should satisfy the local balance equation: to the real problem of spatially uniform wall velocity. To investigate the effect of viscosity ratio on stretch, two jc + v · ∇c = ∇ · M∇, (9) possible arrangements can be used: the more viscous fluid is jt initially in contact with the moving wall or it is not. where M is the mobility. In Eq. (9), known as the Cahn–Hilliard 3.1. Non-dimensionalized governing equations equation, the diffusional flux is assumed to be proportional to the gradient of the chemical potential. This equation was orig- The governing equations are non-dimensionalized using the inally used to describe the initial stages of spinodal decompo- dimensionless variables: sition. Finally, the equations for flow (neglecting external body 2 2 forces) are obtained by coupling the momentum and total mass ∗ c ∗ ∗ ∗ f c = , ∇ = L∇, = ,f= ; balance equations with the DIM equations, which yields a mod- 2 cB cB cB ified Navier–Stokes equation (Lowengrub and Truskinovsky, ∗ v ∗ tV ∗ pL ∗ v = ,t= ,p= , = . 1998; Verschueren, 1999): V L V 1 1 jv + (v · ∇v) =−∇g + ∇ · [(∇v) + (∇v)T] The characteristic velocity scale V is the maximum value of vx jt in Eq. (13). + ∇c, (10) For the characteristic length scale either the domain length or the interface thickness can be used. Here, we opted for the = + where g is the Gibbs free energy (g f p/ ), and are domain length since for our scaling study the interface thickness the density and shear viscosity, respectively, that can depend on will be varied and, also, because the imposed velocity varies c, p is the pressure and v is the velocity. Here, we consider the over this distance. density-matched case (where for simplicity will be set equal The system of equations reads, (after dropping the asterisks, to one hereafter) which gives for the equation of continuity, and assuming creeping flow) ∇ · v = 0. (11) jc 1 + v · ∇c = ∇2, (14) jt Pe For the viscosity we use a linear relationship, = c3 − c − C2∇2c, (15) + − h = c 1 − c 1 1 2 . (12) = 1 4 − 1 2 + 1 2|∇ |2 2 2 f 4 c 2 c 2 Ch c , (16) ∗ T This expression is based on the assumption that the components 0 =−∇p + ∇ · [(∇v) + (∇v) ] 1 of the binary mixture form an ideal mixture. In non-ideal cases, + (∇c − ∇f), (17) the dependence of the viscosity on the concentration is known CaCh to be complex (Qunfang and Yu-Chun, 1999). However, in the ∇ · v = 0, (18) present case where the components mix only over a small length + − scale of the interfacial region, the use of this simple linear ∗ = = c 1 − 1 c 1 . (19) relationship Eq. (12) is valid. 1 2 2 ARTICLE IN PRESS

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bulk solubility, which is proportional to interface thickness, also changes. Keeping results unchanged in non-equilibrium situations requires understanding on how to scale the mobility or equivalently the Péclet number, with the interface thickness. We compare three scalings viz. Pe ∝ 1/Ch, Pe ∝ Ch and ∝ 2 = Pe Ch. We chose Ch 0.01 as the (arbitrary) base case (the same as used by Chella and Viñals (1996)) and increase Ch by a factor of 4 to Ch = 0.04 to compare various scalings. The resulting values of Pe for various scalings are given by Pe = (ChPe)base case/Ch, Pe = Ch(Pe/Ch)base case and Pe = 2 2 Ch(P e/Ch)base case. We compare various scalings for different values of the Péclet number (105 and 6250) for the base case, the capillary number (500, 50, 5 and 0.5) and the viscosity ratio (1 and 10).

5. Numerical method

Instead of solving Eqs. (14)–(19) for c, , v and p in a coupled manner, we decouple the set into a flow problem Eqs. (16)–(19) and a concentration problem Eqs. (14)–(15). Fig. 1. Schematic representation of mixing of two immiscible fluids in a The flow problem is solved using a primitive variable i.e., lid-driven cavity flow. the velocity–pressure formulation and discretized by a standard Galerkin finite element method. These discretized equations The dimensionless groups that appear are: Péclet number Pe, written in matrix form read, capillary number Ca, Cahn number Ch and viscosity ratio , S LT v Mf and are defined as v = v , (22) L 0 p 0 VL2 V Pe = , Ca = 1 ,C= , = 1 h . (20) where v is the discretized velocity, p is the discretized pressure, M cB L 2 Sv is the stiffness matrix containing the viscous terms, L is the A no mass flux condition gives the boundary conditions for matrix due to divergence operation, LT is the matrix for the concentration and chemical potential: gradient operation, M is the mass matrix, and fv is the right hand side containing the 1 (∇c − ∇f)term. ∇c · n = 0, ∇ · n = 0, (21) CaCh The discretized set (22) of linear algebraic equations is solved where n is the unit normal vector to the boundary. using an integrated method with an iterative (conjugate gradients square) solver with incomplete Cholesky decomposition 4. Scaling as preconditioner. In the integrated method (Segal, 1995), both velocity and pressure are used as unknowns i.e., degrees of The application of the diffuse–interface approach to larger freedom. Due to the absence of pressure in the continuity equa- scale systems, like dispersive mixing, introduces two distinct tion, a zero block appears in the main diagonal of the matrix. It length scales: the length scale of the drop diameter, initially is therefore, possible that the first pivot during the elimination typically 1 mm, and that of the interface thickness (), which process for the ILU preconditioner is zero. In order to ensure −5 is about a few nanometers, yielding Ch of O(10 ) or less. In that this does not happen, unknowns are renumbered per level, these cases, it is usually not possible to numerically capture and also globally, so that first velocities and then the pressure the physical value of the interface thickness and, hence, the unknowns are used during the matrix assembling. Taylor–Hood real interface thickness is replaced by a numerically acceptable quadrilateral elements, with continuous pressure, that employ one. Upon artificial enlargement of the interface thickness, it a biquadratic approximation for velocity and a bilinear approx- is necessary to ascertain that we are still describing the system imation for pressure are used. with the same effective surface tension and diffusion. From Two second-order differential Eqs. (14) and (15) that consti- the analysis√ of an equilibrium planar interface we know that tute a concentration problem are solved in a coupled way. For = ∝ 2 / and cB / . Increasing while keeping and the temporal discretization of Eq. (14) a first-order Euler im- cB (equilibrium bulk composition) constant requires that / plicit scheme is employed so that the discretized time deriva- should remain unchanged. Using this information, together tive reads: (cn − cn−1)/t where t is the time step size, and with the definition of interface thickness, we find that the the superscript n represents the current time level. The non- product should be constant. So, when we change the in- linear c3 term in Eq. (15) is linearized by a standard Picard it- n 2 n terface thickness keeping the surface tension the same, we eration which yields (ci−1) ci where subscript i represents the change the homogeneous part of the free energy. Note that the ith Picard iteration at time level n. A spectral element method ARTICLE IN PRESS

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(Patera, 1984; Timmermans et al., 1994) is used for spatial The scheme to advance in time is as follows: 0 discretization of the set of equations. In this method, similar Step 1: Initialize c0 to be the locally equilibrated concentra- to finite element method, the computational domain is divided tion profile. into Nel non-overlapping sub-domains e and a spectral ap- Step 2: Compute f, and . proximation is applied on each element. The basis functions, , Step 3: Solve the system (22) for velocity (v1) with terms that are used for spatial discretization, are high-order Lagrange containing concentration treated explicitly. interpolation polynomials through Gauss–Lobatto integration Step 4: Solve the system (23) iteratively for concentration points defined per element. (c1) and chemical potential (1). Iterations are required due to n = n−1 The full discretized set of linearized equations, written in the non-linear term. Iteration is started with ci−1 c and | n − n | matrix form reads: stopped when max ci ci−1 c. ⎡ ⎤ Step 5: Update the time and repeat steps 2–4. t + n−1 n n−1 The numerical method described above is implemented in M tN S ci Mc ⎣ Pe ⎦ = 0 , (23) the finite element package SEPRAN (Segal, 1995). n 0 [1 − (cn )2]M − C2SM i i−1 h 5.1. Validation

n where ci is the discretized concentration at the ith Picard iter- To validate the solution of both the flow problem and the n ation at time step n, i is the discretized chemical potential at concentration problem, the case of Ca=∞ is considered. In this n−1 the ith Picard iteration at time step n, c0 is the discretized case, both problems are independent and in Eq. (17) the term concentration at time step n − 1, M is the mass matrix, N is the 1/CaCh(∇c − ∇f) is equal to zero. Also, a semi-analytical convection matrix, S is the diffusion matrix. This set is solved expression for the velocity field in the cavity in the absence using the above mentioned iterative solver. of the interfacial tension, that satisfies the boundary condition

Fig. 2. Validation of the numerical solution of the velocity ﬁeld obtained using mesh: 90 × 90 elements, by comparison with the semi-analytical solution. ARTICLE IN PRESS

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Fig. 3. Inﬂuence of different spectral order approximation for concentration and chemical potential on interface length stretch. Parameters used in computations: Pe = 6250, = 1 and mesh: 72 × 72 elements.

(13), is given by Chella and Ottino (1985). A numerical solution ing. Five values of the viscosity ratio viz. 0.1, 1, 2.5, 5 and 10 for velocity field obtained using a mesh that consists of 90×90 are used. Unless indicated otherwise, all the results are obtained elements is compared with the semi-analytical solution, see in using t = 0.01 and a mesh of 90 × 90 elements. Results are Fig. 2. The difference between the numerical and the analytical presented in two forms, a plot of the interface profile (contour solution is small, of O(10−2). The convergence criteria used c = 0) at different time instants and a plot of interfacial stretch | k − k−1| for the iterative solver was max1 j N vj vj v with vs time. −8 the tolerance v equal to 10 . Here, j is the node number, N denotes the total number of nodes and k is the iteration number. All the results obtained in this paper use this same termination 6.1. Scaling results criteria for the iterative solver. Using the analytical expression for the velocity field, the In order to make the comparison of various scalings for Pe, concentration problem is solved with different spectral order when the real thin interface is replaced by a numerically thick approximations viz. 2, 3 and 4 for concentration and chem- interface, numerical simulations were performed choosing = ical potential. Other parameters used in the simulation are Ch 0.01 as the base case value that represents the thin inter- Pe = 6250, t = 0.00625,C = 0.01 and a 72 × 72 elements face in this study. For the test cases, the interface thickness is h = mesh. The convergence criteria for the Picard iteration, which increased by a factor of four to give Ch 0.04. The intention here is to see which, if any, of the three scaling strategies gives is same for all simulations, is based on max |(c )n − 1 j N j i = (c )n | with set equal to 10−6. Here, i is the iteration results for Ch 0.04 that are similar to those obtained with j i−1 c c = number of Picard iteration and n represents nth time step. From the base case i.e., with Ch 0.01. First, results of simulations = 5 = = the converged solution, the interface stretch is calculated by where Pe 10 , Ca 500 and 1 were used for the base computing the length of the contour c = 0 using a user-written case (these values of Pe, Ca, and Ch were chosen based on subroutine in MatlabTM. Fig. 3 compares the interface stretch the results reported by Chella and Viñals, 1996) are shown in obtained for different spectral order approximations for c and Fig. 4. The interface profile obtained from the base case (top . Since the difference between the results of spectral order of row in Fig. 4) is similar to that shown in Fig. 1 of Chella and = 3 2 and 4 is small, a spectral order of 2 is used in the rest of Viñals (1996) which was obtained using Pe 10 and capil- simulations to speed up computations. lary number (C in their notation) of 0.1. This correspondence between results obtained using seemingly different values of Pe (105 and 103) and Ca (500 and 0.1) is not surprising: since 6. Results and discussion in Chella and Viñals (1996), the interface thickness is used as characteristic length scale for non-dimensionalization of Simulation results are presented in this section, first for scal- the model equations, while we have used the domain length ing and then for the effect of viscosity ratio on interface stretch- as characteristic length scale. Taking this difference into ARTICLE IN PRESS

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Fig. 4. Comparison of interface proﬁle (c = 0) after various time instants obtained from various scalings with the base case for Pe = 105, Ca = 500, = 1, t = 0.01 and mesh: 72 × 72 elements.

Fig. 5. Comparison of interface stretch obtained from various scalings with the base case for Pe = 105, Ca = 500, = 1, t = 0.01 and mesh: 72 × 72 elements.

consideration we find Pe = Pe/Ch where Pe is the Péclet shown in Fig. 5, where the interface stretch is plotted against number defined with the interface thickness as the length time. At this point in time, the structure has become fine scale. Also, their capillary number C = 1/CaCh. This yields enough so that further stretching results in breakup of domains Pe = 105and Ca = 1000. We used Ca = 500 and, in fact, if (defined as splitting of domains having concentration values we compare the results of Ca =∞with Ca = 500, we see no of ±1 and hence can be monitored by following c = 0 contour difference. From Fig. 4 we learn that of all the scalings tested, level), as can be seen from the interface profile for base case at the interface profile obtained by quadratic scaling most closely t = 20. Such topological events have a dominant effect on the resembles that of the base case, upto a dimensionless time of interface stretching which none of the scaling strategies stud- 15. Upto this time, interface stretching is (roughly) linear, as ied is able to capture. Mesh size and time step dependency of ARTICLE IN PRESS

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Fig. 6. Mesh dependency of interface stretch for Pe = 105, Ca = 500, = 1 and t = 0.01.

Fig. 7. Time step dependency of interface stretch for Pe = 105, Ca = 500, = 1 and mesh 72 × 72 elements. the results was checked by increasing the number of elements diffusion via a decrease in Pe from 105 to 6250, increases the to 90 in each direction and by increasing the time step to 0.05 resistance to interfacial deformation. Moreover, the topological in two separate runs. Results of this exercise are summarised transitions become noticeable at early times, with the result that in Figs. 6 and 7 which clearly demonstrates that the results are the time period over which interface stretching is linear reduces mesh size and time step independent. to t ≈ 10. Next, we lower the value of Pe used for base case from 105 It is interesting and also relevant to see how the scalings work to 6250. It is apparent from Figs. 8 and 9 that an increase in when inter-facial tension has some signiﬁcance. This happens ARTICLE IN PRESS

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Fig. 8. Comparison of interface proﬁle (c = 0) at different time instants obtained from quadratic scaling with the base case for Pe = 6250, Ca = 500, = 1, t = 0.01 and mesh: 90 × 90 elements.

Fig. 9. Comparison of interface stretch obtained from quadratic scalings with the base case for Pe =6250, Ca=500, =1, t =0.01 and mesh: 90×90 elements.

Fig. 10. Comparison of interface proﬁle (c = 0) at various time instants obtained from quadratic scaling with the base case for Pe = 6250, Ca = 50, = 1, t = 0.01 and mesh: 90 × 90 elements. ARTICLE IN PRESS

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Fig. 11. Comparison of interface stretch obtained from quadratic scaling with the base case for Pe =6250, Ca=50, =1, t =0.01 and mesh: 90×90 elements.

Fig. 12. Comparison of interface profile (c = 0) at various time instants obtained from quadratic scaling with the base case for Pe = 6250, Ca = 5, = 1, t = 0.01 and mesh: 90 × 90 elements. when Ca is O(1). As an intermediate step, first we chose to deviate totally from the base case, see Figs. 14 and 15. Finally, use Ca = 50. The difference in the results of the base case for we investigated how the scalings work when the viscosities of Ca = 500 and Ca = 50 is small, suggesting that Ca should be the two fluids are unequal. For this, we considered a case where lowered further. Unfortunately, the quadratic scaling now gives the viscosity ratio between the two fluids is 10 and the more pronounced differences, compare Figs. 10 and 11. Nevertheless, viscous fluid is initially in contact with the moving wall. Here the time at which the linear stretching stops, and topological too, the quadratic scaling strategy gives the best results as can transitions become important is predicted fairly well. Next, Ca be seen from Figs. 16 and 17. is lowered by a factor of ten to a value of 5. In this case, the quality of match between the quadratic scaling interface profile and that corresponding to the base case is good only upto t = 5 6.2. Effect of viscosity ratio on stretch and deteriorates thereafter as can be seen in Figs. 12 and 13. Inspite of the poor match of the interface profiles, the time To study the influence of the viscosity ratio on stretching, period over which stretching is roughly linear and the value of an initial arrangement where the more viscous fluid is, except stretch at the end of linear stretching fairly agrees. With a further for = 0.1, in contact with the moving wall was chosen. The lowering of Ca to 0.5, the quadratic scaling gives results that following viscosity ratios are explored: =0.1, 1, 2.5, 5 and 10. ARTICLE IN PRESS

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Fig. 13. Comparison of interface stretch obtained from quadratic scaling with the base case for Pe =6250, Ca=5, =1, t =0.01 and mesh: 90 ×90 elements.

Fig. 14. Comparison of interface proﬁle (c = 0) at various time instants obtained from quadratic scaling with the base case for Pe = 6250, Ca = 0.5, = 1, t = 0.01 and mesh: 120 × 120 elements.

Fig. 18 compares the mixing profile at various time steps where stretch values for = 2.5 and 5 are (roughly) equal (although the more viscous fluid is represented with a white colour and the underlining profiles are different, see Fig. 18) and more the less viscous fluid with a black colour. Clearly, the interface than the corresponding values for = 10 and 1. The maximum profile depends on the viscosity ratio and increasing it from 1 stretch value and the time at which it is attained increases with to 10 the leading edge of the less viscous fluid that shears into viscosity ratio. the more viscous fluid becomes increasingly blunt. In fact for In contrast to 1, for = 0.1 the interface stretching is = 10, a round blob of less viscous fluid can be seen at all initially quite slow until the more viscous fluid comes in contact times. with the moving wall. This happens at t ≈ 7.5. After this the As a consequence of this change in the interface profile, stretching is rapid for a brief time period till t ≈ 10 and then stretching is a non-monotonic function of viscosity ratio, see falls again. The interface stretch attained is quite less than that Fig. 19. During the time period from t=5tot ≈ 16 the interface for 1. ARTICLE IN PRESS

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Fig. 15. Comparison of interface stretch obtained from quadratic scaling with the base case for Pe = 6250, Ca = 0.5, = 1, t = 0.01 and mesh: 120 × 120 elements.

Fig. 16. Comparison of interface proﬁle (c = 0) after various time instants obtained from various scalings with the base case for Pe = 105, Ca = 500, = 10, t = 0.01 and mesh: 90 × 90. ARTICLE IN PRESS

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Fig. 17. Comparison of interface stretch obtained from various scalings with the base case for Pe =105, Ca=500, =10, t =0.01 and mesh: 90×90 elements.

5 Fig. 18. Interface proﬁle for different viscosity ratios. Parameters used in the computations: Pe =10 , Ca=500, Ch =0.01, t =0.01 and mesh: 90×90 elements. ARTICLE IN PRESS

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5 Fig. 19. Effect of viscosity ratio on interface stretch. Parameters used in the computations: Pe =10 , Ca=500, Ch =0.01, t =0.01 and mesh: 90×90 elements.

7. Conclusion References

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