Efficient Fully Discrete Finite Element Scheme for the Ferrohydrodynamic Rosensweig Model and Simulations of Ferrofluid Rotational Flow Problems
Abstract
This article focuses on numerical approximations of the ferrohydrodynamic (FHD) Rosensweig model of ferrofluids, which comprises the Navier-Stokes equations, the angular momentum equations, the magnetization equations, and the magnetostatic equation, as well as the simulations of the ferrofluid rotational flow problems. The model's strong coupling feature, encompassing both linear and nonlinear aspects, poses a significant challenge in the development of efficient numerical algorithms, particularly for fully decoupled-type schemes with unconditional energy stability. A significant new challenge in the Rosensweig model, as opposed to the relatively well-studied Shliomis FHD model, is the coupling with the extra angular momentum equations, particularly seen in the linear coupling between the angular velocity of the particle spinning and the flow velocity, demanding innovative numerical strategies to effectively address this complexity. By introducing a new nonlocal auxiliary variable and constructing an ordinary differential equation with a special structure for it, we can simplify the complex coupling terms in the modified but equivalent governing system via explicit discretization. This novel method, along with the ZEC (zero energy contribution) decoupling method, reconstruction of the magnetostatic equation, the spatial finite element method, and the second-order projection method for hydrodynamics, allows us to obtain a fully discrete scheme that is unconditionally energy stable, fully decoupled, linear, and second-order accurate in time. Only a few independent linear elliptic/parabolic problems with constant coefficients need to be solved at each time step. The unconditional energy stability and well-posedness of the scheme are also established. In addition, simulations, including 2D/3D spin-up and annular flows, are implemented to verify the stability and accuracy of the scheme, with the numerical results exhibiting good agreement with experimental and physical analyses.
Recommended Citation
G. D. Zhang et al., "Efficient Fully Discrete Finite Element Scheme for the Ferrohydrodynamic Rosensweig Model and Simulations of Ferrofluid Rotational Flow Problems," SIAM Journal on Scientific Computing, vol. 47, no. 1, pp. B28 - B58, Society for Industrial and Applied Mathematics, Jan 2025.
The definitive version is available at https://doi.org/10.1137/24M1640914
Department(s)
Mathematics and Statistics
Keywords and Phrases
decoupled; ferrohydrodynamics; finite element; magnetic field; second-order; unconditional energy stability
International Standard Serial Number (ISSN)
1095-7197; 1064-8275
Document Type
Article - Journal
Document Version
Citation
File Type
text
Language(s)
English
Rights
© 2025 Society for Industrial and Applied Mathematics, All rights reserved.
Publication Date
01 Jan 2025
