We examine a time-dependent, three-dimensional rotation of magnetic ellipsoidal particles in a two-dimensional, simple shear flow and a uniform magnetic field. We consider that the particles have paramagnetic and ferromagnetic properties, and we compare their rotational dynamics due to the strengths and directions of the applied uniform magnetic field. We determine the critical magnetic field strength that can pin the particles' rotations. Above the critical field strength, the particles' stable steady angles were determined. In a weak magnetic regime (below the critical field strength), a paramagnetic particle's polar angle will oscillate toward the magnetic field plane while its azimuthal angle will execute periodic rotations. A ferromagnetic particle's rotation depends on its initial angles and the magnetic field strength and direction. Even when it is exposed to a critical magnetic field strength, its rotational dynamics will either be pinned in or out of the magnetic field plane. In a weak magnetic regime, a ferromagnetic particle will either execute out-of-plane rotations or will oscillate toward the magnetic field plane and perform periodic rotations. For both particles, we analytically determine the peaks and troughs of their oscillations and study their time-dependent rotations through analytical and numerical analyses.


Mathematics and Statistics

Second Department

Mechanical and Aerospace Engineering

Research Center/Lab(s)

Center for Research in Energy and Environment (CREE)

Keywords and Phrases

Ferromagnetic Materials; Ferromagnetism; Magnetic Bubbles; Magnetic Fields; Paramagnetism; Shear Flow, Critical Field Strength; Critical Magnetic Field; Ferromagnetic Particles; Ferromagnetic Properties; Magnetic Ellipsoidal Particles; Magnetic Field Strengths; Three-Dimensional Rotation; Uniform Magnetic Fields, Rotation

International Standard Serial Number (ISSN)

1070-6631; 1089-7666

Document Type

Article - Journal

Document Version

Final Version

File Type





© 2019 The Authors, All rights reserved.

Publication Date

01 Oct 2019