Doctoral Dissertations

Keywords and Phrases

Cahn–Hilliard Phase-Field Model; Ferrohydrodynamics; Finite Element Method; Mixed Continuous–Discontinuous Galerkin Method; Shliomis Magnetization Model; Two-Phase Ferrofluid Flow

Abstract

"Ferrofluids are magnetic nanoparticle suspensions whose motion couples surface tension, flow field, magnetostatics, and magnetization dynamics. This dissertation develops, analyzes, and validates an energy-stable finite element method for a two-phase ferrofluid model that couples the Cahn-Hilliard equations with the full Shliomis model of single-phase ferrofluids, retaining its damping torque term, magnetic torque term, and magnetic stress term.

The spatial discretization is a mixed continuous Galerkin (CG) and discontinuous Galerkin (DG) formulation. It uses continuous ��2 elements for the phase field, chemical potential, velocity, and magnetostatic potential, discontinuous ��2 elements for the magnetization, and discontinuous ��1 elements for the pressure. The temporal discretization is semi-implicit and semi-explicit with stabilization. The coupled nonlinear system is solved by a block-Gauss-Seidel Picard iteration.

The target model is proved to satisfy a continuous energy law. The fully discrete scheme is proved to satisfy the corresponding discrete free-energy law. Stability is shown to be unconditional in terms of the mesh size and time step size. The proof relies on two independent cancellation mechanisms. The first one is the cancellation between the magnetic torque term and the magnetic stress term. The second one is the cancellation between the Kelvin-force term and the magnetization-transport term.

The method is implemented in both sequential and parallel adaptive C++ codes, by using deal.II for the finite element data structures, p4est for the distributed mesh adaptation, and Trilinos for the sparse linear algebra. Verification uses manufactured-solution tests and discrete diagnostics. Numerical studies include diamond-droplet relaxation, field-driven elongation, parameter sensitivity, and Lotus-type spike formation under a non-uniform magnetic field"-- Abstract, p. iii

Advisor(s)

He, Xiaoming

Committee Member(s)

Tomas, Ignacio
Singler, John R.
Hong, Qingguo
Zhang, Yanzhi

Department(s)

Mathematics and Statistics

Degree Name

Ph. D. in Mathematics

Publisher

Missouri University of Science and Technology

Publication Date

2026

Pagination

xi, 126 pages

Note about bibliography

Includes_bibliographical_references_(pages 117-122)

Rights

© 2026 Mahdi Gharehbaygloo , All Rights Reserved

Document Type

Dissertation - Open Access

File Type

text

Language

English

Thesis Number

T 12616

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Included in

Mathematics Commons

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