2.3 Electric Potential Method

The second approach for the current density is to solve the electric potential equation and calculate the current density using Ohm's law. In general, the electric field $\vec{E}$ can be expressed as:

where $\varphi$ and $\vec{A}$ are the scalar potential and the vector potential, respectively. For a static field and assuming $\vec{b}<<\vec{B_0}$, Ohm's law given in Equation  2.2-2 can be written as:

For sufficiently conducting media, the principle of conservation of electric charge gives:

The electric potential equation is thus given by:

The boundary condition for the electric potential $\varphi$ is given by:

for an insulating boundary, where $\vec{n}$ is the unit vector normal to the boundary, and

for a conducting boundary, where $\varphi_0$ is the specified potential at the boundary. The current density can then be calculated from Equation  2.3-2.

With the knowledge of the induced electric current, the MHD coupling is achieved by introducing additional source terms to the fluid momentum equation and energy equation. For the fluid momentum equation, the additional source term is the Lorentz force given by:

which has units of N/m $^3$ in the SI system. For the energy equation, the additional source term is the Joule heating rate given by:

which has units of W/m $^3$.

For charged particles in an electromagnetic field, the Lorentz force acting on the particle is given by:

where $\it {q}$ is the particle charge density (Coulomb/m $^3$) and $\nu_p$ is the particle velocity. The force $\vec{F}_p$ has units of N/m $^3$.

For multiphase flows, assuming that the electric surface current at the interface between phases can be ignored, the electric conductivity for the mixture is given by:

where $\sigma_i$ and $\nu_i$ are respectively the electric conductivity and volume fraction of phase $\it {i}$. $\sigma_m$ is used in solving the induction equations.