Decoupled Energy Stable Schemes for Phase-Field Models of Two-Phase Complex Fluids

Abstract

Abstract. We consider in this paper numerical approximations of phase-field models for twophase complex fluids. We first reformulate the phase-field models derived from an energetic variational formulation into a form which is suitable for numerical approximation and establish their energy laws. Then, we construct two classes, stabilized and convex-splitting, of decoupled time discretization schemes for the coupled nonlinear systems. These schemes are unconditionally energy stable and lead to decoupled elliptic equations to solve at each time step. Furthermore, these elliptic equations are linear for the stabilized version. Stability analysis and ample numerical simulations are presented.Key words. phase-field, liquid crystals, multiphase flows, Navier-Stokes, Allen-Cahn, CahnHilliard, stability AMS subject classifications. 65M12, 65M70, 65Z05 DOI. 10.1137/130921593 1. Introduction. Complex fluids have great practical utility since the microstructure can be manipulated to produce useful mechanical, optical, or thermal properties [4,23]. Numerical studies of multiphase complex fluids is a rich area for research with important applications that can be addressed. Some numerical difficulties are (i) the moving interfaces between various components; (ii) open issues regarding wellposedness of many multiphase models leading to confusion as to whether numerical pathologies are due to the model or the methods; and most important, (iii) nonlinear coupling (hydrodynamics, interface, and microstructure) makes it very difficult to design easy-to-implement, energy stable numerical schemes.For the first two issues above, the diffuse-interface/phase-field models, whose origin can be traced back to [38] and [46], have been proved efficient with much success. A particular advantage of the phase-field approach is that models can often be derived from an energy-based variational formalism (energetic variational approaches), leading to well-posed nonlinear coupled systems that satisfy thermodynamics-consistent energy dissipation laws. This makes it possible to carry out mathematical analysis and design numerical schemes which satisfy a corresponding discrete energy dissipation law. In recent years, the phase-field method has become one of the major tools to study a variety of interfacial phenomena (cf. [17,2,36,19,32,49] and recent review papers [44,22] and the references therein).For phase-field models which satisfy a dissipative energy law, it is especially desirable to design numerical schemes that preserve the energy dissipation law at the

Efficient energy stable numerical schemes for a phase field moving contact line model

“…(3.9). The introduction of an explicit velocity u n * in (3.28) follows a similar approach used in Minjeaud [25] (see also Shen and Yang [37]). Comparing to the replacement of ũ n+1 with u n , the replacement of ũ n+1 with u n * in Eq.…”

“…A particular advantage of the phase-field approach is that the derived models are usually well-posed nonlinear partial differential equations that satisfy thermodynamics-consistent energy dissipation laws, which makes it possible to carry out mathematical analysis and further design numerical schemes which satisfy corresponding discrete energy dissipation laws. Thus phase field models recently have been the subject of many theoretical and numerical investigations (cf., for instance, [5,8,[12][13][14]19,21,24,31,33,[35][36][37][38]43]). …”

Linear unconditional energy‐stable splitting schemes for a phase‐field model for nematic–isotropic flows with anchoring effects

“…In this work, we study a modification of the model presented in where we take into account the anchoring effects and an arbitrary interpolation function can be considered to localize the nematic region (in practice, we consider a fifth‐order polynomial). We derive new linear splitting schemes for nematic–isotropic mixtures, taking into account viscous, mixing, nematic, and anchoring effects, which allows us to split the computation of the three pairs of unknowns ( v , p ) (velocity–pressure), ( c , μ ) (phase field–chemical potential), and ( d , w )(director vector–equilibrium) in three different steps.…”

Decoupled, Energy Stable Schemes for Phase-Field Models of Two-Phase Incompressible Flows

“…The techniques developed in this paper can be used to construct efficient numerical schemes in other situations. For example, we have recently extended the approach in this paper to a phase-field model for two-phase complex fluids with matching density [35].…”