An Efficient and Robust Method for Simulating Two-Phase Gel Dynamics

Abstract

Abstract. We develop a computational method for simulating models of gel dynamics where the gel is described by two phases, a networked polymer and a fluid solvent. The models consist of transport equations for the two phases, two coupled momentum equations, and a volume-averaged incompressibility constraint, which we discretize with finite differences/volumes. The momentum and incompressibility equations present the greatest numerical challenges since i) they involve partial derivatives with variable coefficients that can vary quite significantly throughout the domain (when the phases separate), and ii) their approximate solution requires the "inversion" of a large linear system of equations. For solving this system, we propose a box-type multigrid method to be used as a preconditioner for the generalized minimum residual (GMRES) method. Through numerical experiments of a model problem, which exhibits phase separation, we show that the computational cost of the method scales nearly linearly with the number of unknowns, and performs consistently well over a wide range of parameters. For solving the transport equation, we use a conservative finite volume method for which we derive stability bounds.Key words. multiphase flow, mixture theory, multigrid, Vanka relaxation, coupled relaxation, box relaxation, Krylov, GMRES, preconditioning AMS subject classifications. 65M55, 65F10, 65M06, 92E20, 76T991. Introduction. Polymer gels are used in a variety of industrial applications and appear naturally in many biological systems. Gels are made of two materials: a polymer network and a solvent. By weight and volume gels are mostly solvent, but the rheology of the gel can range from a very viscous fluid to an elastic solid. In addition to viscoelastic stresses, gels exhibit chemical stresses which result in swelling and deswelling behavior.A commonly used model for the mechanics of gels is the two-phase flow (or twofluid) model [3,9,11,12,13,18,33]. Each phase, network and solvent, is treated as a continuum and moves according to its own velocity field. Each region in space is composed of a mixture of the two phases that is described by the volume fractions of the phases. The equations of motion for the gel consist of two coupled momentum equations and two continuity equations. These models of gels are based on mixture theory [14,15], which has been used for many applications and is becoming increasingly popular in models of biological materials such as tissue [21], tumors [7, 23], cytoplasm [3, 11, 12, 18], and biofilms [2,8,9]. While much work has gone in to advancing these models, the same is not true for the development of efficient numerical methods for simulating them. Since repeated simulation of models is crucial for validating their usefulness, the need for fast and robust computational methods is essential.In this study, an efficient and robust computational methodology for simulating gel dynamics is proposed for a model problem that shares several characteristics of many other models in the literature [3,9,11,12...

A high-resolution finite-difference method for simulating two-fluid, viscoelastic gel dynamics

“…A full analysis of how the viscosity effects the choice of stable time-step for this method applied to the Navier-Stokes equations is given in [30]. We conclude by noting that in general, it is possible to have osmotic potentials such that Ψ (θ n ) < 0, which results in a destabilizing force that promotes de-mixing or phase separation [9,35,36]. Our method also works for these cases, however the time-step restriction (51) should be modified so that the absolute value of Ψ (θ n ) is used.…”

“…This method is then used as the preconditioner for the generalized minimum residual (GMRES) method [28]. The presence of inertia in the system (26) makes it better conditioned than the viscous dominated system from [9], and the iterative method converges in very few iterations. In the numerical examples presented in Section 4, the maximum number of iterations required by the method was 8, with the most common numbers being 3 and 4.…”

“…The method we use for solving (26) is based on the preconditioned Krylov subspace method first introduced for gels in [9] and developed further in [10,11]. The gel model considered in these studies was for two immiscible viscous-dominated fluids and did not contain inertial effects.…”

“…The preconditioner for the resulting linear system was based on a multigrid procedure with box-relaxation (or symmetric coupled Gauss-Seidel smoothing) [25,26,27]. We generalized the box-type smoother from [9] for (26) and combined it with a multigrid V-cycle. This method is then used as the preconditioner for the generalized minimum residual (GMRES) method [28].…”

“…There are similarities of this method with previously developed techniques for treating single-phase viscoelastic fluids [8,13,14]. We use a second order finite-difference discretization of the momentum and incompressibility equations and adapt our iterative method from [9] to handle nonzero Reynolds number flows. This iterative method uses a Krylov subspace method together with a multigrid preconditioner for solving this coupled set of equations without splitting.…”