Numerical solutions of an accurate diffuse interface model of the incompressible resistive MHD free surface flow

Jul 16, 2026

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This paper develops a new incompressible resistive magnetohydrodynamic (MHD) free-surface flow model using a thermodynamically consistent diffuse interface description of the moving interface. Matched asymptotic analysis proves formal convergence to the corresponding sharp-interface model. A fully decoupled linear finite element method is constructed to preserve the discrete divergence-free constraint of the magnetic field. Numerical studies then validate reliability via magnetic damping effects on bubble dynamics and provide quantitative comparisons against inductionless and sharp-interface ALE formulations.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages21

Reading Time53 min

Introduction

Mathematical model

Numerical method

Convergence and stability analysis

Numerical experiments

arXiv :2607 .07025v1 [math .NA] 8 Jul 2026 Numerical solutions of an accurate diﬀuse interface model of the incompressible resistive MHD free surface ﬂow Maojun Li, Jiancheng Wang, Zeyu Xia ∗, Liwei Xu School of Mathematical Sciences, University of Electronic Science and Technology of China, Sichuan, 611731, P.R. China Abstract In this paper, we derive a new model to simulate the incompressible resistive magnetohydrodynamic (MHD) free surface ﬂow. A thermodynamically consistent diﬀuse interface method is adopted to characterize the moving interface in the modeling process. The formal convergence of the proposed MHD free surface ﬂow model to the sharp interface model is established via a matched asymptotic argument, and the model can be solved without the need for sophisticated free surface capturing schemes. We design a fully decoupled linear ﬁnite element scheme that preserves the divergence-free constraint of the magnetic ﬁeld at a discrete level. The reliability and robustness of the proposed model and algorithm are validated through numerical investigations of the magnetic damping eﬀect on bubble dynamics. In particular, we provide a quantitative numerical comparison of the present results with those obtained from an inductionless MHD model and a sharp interface arbitrary Lagrangian–Eulerian model. Keywords: magnetohydrodynamics, free surface ﬂow, ﬁnite element method, asymptotic convergence 2020 MSC: 76W05, 65M60, 76T10 1. Introduction When a magnetic ﬁeld is applied to an electrically conducting and non-magnetic ﬂuid (e.g. , liquid metals, plasmas, and strong electrolytes), the induced magnetohydrodynamic (MHD) eﬀect can signiﬁcantly change the ﬂuid dynamics via the Lorentz force, resulting in fundamentally distinct ﬂow behaviors. Given the ubiquitous presence of such a scenario in industrial processes, numerous studies have been carried out to investigate the underlying ﬂow mechanism [7] and develop structure-preserving numerical schemes [16, 17 , 19 , 23] . Although MHD ﬂows exhibit extremely complex dynamics in engineering applications, the frequent occurrence of free surfaces further complicates the ﬂuid dynamics. Representative examples include magnetic stirring and damping in the continuous casting and reﬁning of metals [7, 11], MHD instabilities of free surfacesin Hall–Héroult aluminum reduction cells [9, 15], liquid metal batteries [15, 30], and plasma-facing components of tokamaks [21, 24] . In this work, we consider incompressible resistive MHD free surface ﬂows in a ﬁxed and bounded domain Ω ⊂ R3 , which contains two immiscible ﬂuids occupying two time-dependent open subdomains Ω± (t), respectively. From a macroscopic perspective, the free surface is a manifold of codimension one [12], and the two subdomains Ω± (t) are separated by a sharp interface Γ(t) = Ω+ (t) ∩ Ω − (t) that should not be in contact with the boundary ∂Ω . Consequently, the ﬂow dynamics can be governed by the following resistive MHD equations in each subdomain Ω± (t) with the jump conditions on the interface Γ(t), ρ± (∂tu + u · ∇u) = ∇ · 􀀀2η±D (u) − pI􀀁 + J × B + ρ±g in Ω± (t), (1a) ∇ · u = 0 in Ω± (t), (1b) ∂tB + ∇ × E = 0 in Ω± (t), (1c) ∇ × B = µ0 J in Ω± (t), (1d) J = σ± (E + u × B) in Ω± (t), (1e) ∇ · B = 0 , ∇ · J = 0 in Ω± (t), (1f) ∗ Corresponding author Email addresses: [limj@uestc.edu.cn](limj@uestc.edu.cn) (Maojun Li), [202311110403@std.uestc.edu.cn](202311110403@std.uestc.edu.cn) (Jiancheng Wang), [zeyuxia@uestc.edu.cn](zeyuxia@uestc.edu.cn) (Zeyu Xia), [xul@uestc.edu.cn](xul@uestc.edu.cn) (Liwei Xu) [[2ηD(u) − pI]]nΓ = −λκnΓ[[u]] = 0 [[B]] = 0 nΓ × [[E]] = 0 [[J]] · nΓ = 0 VΓ = u · nΓ on Γ(t), on Γ(t), on Γ(t), on Γ(t), on Γ(t), on Γ(t) . (1g) (1h) (1i) (1j) (1k) (1l) In the above system, the unknowns are the ﬂuid velocity u, the pressure p, the magnetic ﬁeld B, the electric ﬁeld E, and the current density J. In addition, D (u) = ~~1~~2 􀀀 ∇u + (∇u)⊤ 􀀁 and I st

FAQs

What physical phenomenon does the paper model?

The work models incompressible resistive MHD free-surface flows, where an applied magnetic field couples electromagnetic effects to fluid motion through the Lorentz force, and a moving free surface separates two immiscible fluids.

How is the moving interface handled in the proposed model?

The model uses a thermodynamically consistent diffuse interface method to represent the evolving interface, rather than tracking a sharp interface directly during computation.

What numerical scheme is proposed, and what key constraint does it preserve?

A fully decoupled linear finite element scheme is designed to preserve the divergence-free constraint of the magnetic field at the discrete level.

Jul 16, 2026

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