[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83174-en":3,"doc-seo-83174-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83174,2336464648746,"Skyler","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Numerical solutions of an accurate diffuse interface model of the incompressible resistive MHD free surface flow","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.","arXiv :2607 .07025v1 [math .NA] 8 Jul 2026  \nNumerical solutions of an accurate diffuse interface model of the incompressible resistive MHD free surface flow  \nMaojun Li, Jiancheng Wang, Zeyu Xia ∗, Liwei Xu  \nSchool of Mathematical Sciences, University of Electronic Science and Technology of China, Sichuan, 611731, P.R. China  \nAbstract  \nIn 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.  \nKeywords: magnetohydrodynamics, free surface flow, finite element method, asymptotic convergence  \n2020 MSC: 76W05, 65M60, 76T10  \n1. Introduction  \nWhen 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] .  \nIn 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),  \nρ± (∂tu + u · ∇u) = ∇ · 􀀀2η±D (u) − pI􀀁 + J × B + ρ±g in Ω± (t), (1a)  \n∇ · u = 0 in Ω± (t), (1b)  \n∂tB + ∇ × E = 0 in Ω± (t), (1c)  \n∇ × B = µ0 J in Ω± (t), (1d)  \nJ = σ± (E + u × B) in Ω± (t), (1e)  \n∇ · B = 0 , ∇ · J = 0 in Ω± (t), (1f)  \n∗ Corresponding author  \nEmail addresses: [limj@uestc.edu.cn](limj@uestc.edu.cn) (Maojun Li), [202311110403@std.uestc.edu.cn](202311110403@std.uestc.edu.cn) (Jiancheng Wang),  \n[zeyuxia@uestc.edu.cn](zeyuxia@uestc.edu.cn) (Zeyu Xia), [xul@uestc.edu.cn](xul@uestc.edu.cn) (Liwei Xu)  \n[[2ηD(u) − pI]]nΓ = −λκnΓ[[u]] = 0  \n[[B]] = 0 nΓ × [[E]] = 0 [[J]] · nΓ = 0  \nVΓ = u · nΓ  \non Γ(t),  \non Γ(t),  \non Γ(t),  \non Γ(t),  \non Γ(t),  \non Γ(t) .  \n(1g)  \n(1h)  \n(1i)  \n(1j)  \n(1k)  \n(1l)  \nIn 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","cbCaijzAWONrkWZT","https://ap.wps.com/l/cbCaijzAWONrkWZT","pdf",1739842,3,1,21,"English","en",105,"# Introduction\n# Mathematical model\n# Numerical method\n# Convergence and stability analysis\n# Numerical experiments","[{\"question\":\"What physical phenomenon does the paper model?\",\"answer\":\"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.\"},{\"question\":\"How is the moving interface handled in the proposed model?\",\"answer\":\"The model uses a thermodynamically consistent diffuse interface method to represent the evolving interface, rather than tracking a sharp interface directly during computation.\"},{\"question\":\"What numerical scheme is proposed, and what key constraint does it preserve?\",\"answer\":\"A fully decoupled linear finite element scheme is designed to preserve the divergence-free constraint of the magnetic field at the discrete 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physical phenomenon does the paper model?","Question",{"text":75,"@type":76},"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.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the moving interface handled in the proposed model?",{"text":80,"@type":76},"The model uses a thermodynamically consistent diffuse interface method to represent the evolving interface, rather than tracking a sharp interface directly during computation.",{"name":82,"@type":73,"acceptedAnswer":83},"What numerical scheme is proposed, and what key constraint does it preserve?",{"text":84,"@type":76},"A fully decoupled linear finite element scheme is designed to preserve the divergence-free constraint of the magnetic field at the discrete 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