Governing Equations for MHD

The governing equations for MHD have two components classical fluid dynamics and electromagnetics. The former includes mass continuity equation and N-S equation. The latter includes Maxwell s equation, current continuity equation, and constitutive equations. For an incompressible electrolyte solution of density, p, and viscosity, p, the continuity and N-S equations are, respectively, described as 

The current flux for various ionic species of electrolyte solutions can be obtained using the ionic flux density, Nj, of species, K, given as 

This equation is known as Nernst-Plank equation. Under steady state conditions  

The potential in the electrolyte solution is governed by the local electroneutrality condition  

The above electroneutrality assumption holds everywhere except in the thin Debye screening layer next to the solid surface. The potential drop across the Debye layer can be significant even though it is only a few nanometers in thickness. It may be noted that ionic mass transport affects the current density, J, which influences the flow field by Lorentz body force. Therefore, one needs to solve simultaneously the full mathematical model consisting of continuity, N-S equation, Nernst-Planck equation, and the local electroneutrality conditions with the appropriate boundary condition. 

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