Enhanced Modeling And Efficient Fully Discrete Finite Element Scheme For A Diffuse Interface FHD Model With Different Viscosities And Densities

Abstract

This paper proposes, analyzes, and numerically solves an enhanced model for the two-phase ferro-fluid flows with different viscosities and densities. In addition to the variable parameters for the density and viscosity and a reformulation of the magnetostatic equation, a relative flux term related to the diffusion of the components is also incorporated into the model to ensure a physically consistent and mathematically rigorous system, inspired by Abels et al. (2012) and Shen and Yang (2015). The introduction of this relative flux term plays a pivotal role in the analysis of the energy law and the construction of the numerical scheme. To construct an efficient fully discrete finite element numerical scheme, we introduce an intermediate magnetization for the magnetization equation, add several key stabilization terms due to the explicit treatment of the nonlinear and coupling terms, apply the artificial compressibility approach for the Navier–Stokes equations, and utilize the invariant energy quadratization approach for the Cahn-Hilliard equations. It is proven to be uniquely solvable per time step and unconditionally stable. Meanwhile, the proposed scheme requires no artificial boundary conditions on the pressure. In addition, we employ an adaptive mesh strategy to better capture the diffuse interface in practical numerical simulations. We present several 2D and 3D numerical examples to demonstrate and validate the presented two-phase model and numerical scheme, including the accuracy test, the Spinodal decomposition, the deformation and migration of a ferrofluid droplet in an irregular domain, as well as the Rosensweig instability with large density ratios under the uniformly or nonuniformly applied magnetic field.

Department(s)

Mathematics and Statistics

Comments

Shaanxi Key Science and Technology Innovation Team Project, Grant 2026RS-CXTD-71

Keywords and Phrases

Different densities and viscosities; Stabilization; The artificial compressibility; The invariant energy quadratization method; Two-phase ferrohydrodynamics

International Standard Serial Number (ISSN)

0045-7825

Document Type

Article - Journal

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2026 Elsevier, All rights reserved.

Publication Date

01 Dec 2026

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