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On Scaling of Diffuse–Interface Modelsଁ ARTICLE IN PRESS 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 0009-2509/$ - see front matter ᭧ 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2005.10.035 ARTICLE IN PRESS 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.
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