## Mathematics and Statistics Faculty Research & Creative Works

### Three-Dimensional Rotation of Paramagnetic and Ferromagnetic Prolate Spheroids in Simple Shear and Uniform Magnetic Field

#### Abstract

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.

#### Department(s)

Mathematics and Statistics

#### Second Department

Mechanical and Aerospace Engineering

#### Research Center/Lab(s)

Center for Research in Energy and Environment (CREE)

#### Second Research Center/Lab

Center for High Performance Computing Research

#### 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

Final Version

text

English

01 Oct 2019

COinS