Magnetic Fluid Dynamics in Engineering#
Magnetic Fluid Dynamics (MFD) in engineering involves the study of the behavior of ferrofluids or magnetizable fluids under the influence of magnetic fields. These fluids are colloidal suspensions of ferromagnetic or superparamagnetic nanoparticles in a carrier liquid. The behavior of such fluids is influenced by both fluid dynamics and magnetic interactions. Below are the mathematical models and example test cases for Magnetic Fluid Dynamics in engineering:
Mathematical Models#
MHD Continuity Equation: $\( \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0 \)\( where \)\rho\( is the fluid density, and \)\mathbf{v}$ is the fluid velocity.
MHD Momentum Equation: $\( \rho \frac{D\mathbf{v}}{Dt} = -\nabla p + \rho \mathbf{g} + \nabla \cdot \boldsymbol{\tau} + \mathbf{J} \times \mathbf{B} + \nu \nabla^2 \mathbf{v} \)\( incorporating the viscous term (\)\nu \nabla^2 \mathbf{v}$) for fluid viscosity.
Magnetic Induction Equation: $\( \frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B} - \eta \nabla \times \mathbf{B}) \)\( where \)\mathbf{B}\( is the magnetic field, and \)\eta$ is the magnetic diffusivity.
Magnetic Force Equation: $\( \mathbf{F}_{\text{magnetic}} = \nabla (\mathbf{m} \cdot \mathbf{B}) \)\( where \)\mathbf{m}$ is the magnetic moment of the ferrofluid.