ANSYS FLUENT 12.0 Magnetohydrodynamics Module Manual

2.2 Magnetic Induction Method

In the first approach, the magnetic induction equation is derived from Ohm's law and Maxwell's equation. The equation provides the coupling between the flow field and the magnetic field.

In general, Ohm's law that defines the current density is given by:

$\vec{\jmath} = \sigma \vec{E}$

(2.2-1)

where $\sigma$ is the electrical conductivity of the media. For fluid velocity field $\vec{U}$ in a magnetic field $\vec{B}$ , Ohm's law takes the form:

$\vec{\jmath} = \sigma (\vec{E} + \vec{U} \times \vec{B})$

(2.2-2)

From Ohm's law and Maxwell's equation, the induction equation can be derived as:

$\frac{\partial \vec{B}}{\partial t} + (\vec{U} \cdot \nabla)... ...{\mu \sigma} \nabla^2 \vec{B} + (\vec{B} \cdot \nabla) \vec{U}$

(2.2-3)

From the solved magnetic field $\vec{B}$ , the current density $\vec{\jmath}$ can be calculated using Ampere's relation as:

$\vec{\jmath} = \frac{1}{\mu} \nabla \times \vec{B}$

(2.2-4)

Generally, the magnetic field $\vec{B}$ in a MHD problem can be decomposed into the externally imposed field $\vec{B_0}$ and the induced field $\vec{b}$ due to fluid motion. Only the induced field $\vec{b}$ needs to be solved.

From Maxwell's equations, the imposed field $\vec{B_0}$ satisfies the following equation:

$\nabla^2\vec{B_0} - \mu\sigma{'} \frac{\partial \vec{B_0}}{\partial t} = 0$

(2.2-5)

where $\sigma{'}$ is the electrical conductivity of the media in which field $\vec{B_0}$ is generated. Two cases need to be considered.

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