Magnetohydrodynamic momentum equation ($\rho D_t\mathbf{u}=-\nabla p+\mathbf{J}\times \mathbf{B}+\rho \nu {\nabla }^2\mathbf{u}$)

For constant mass density $\rho$, kinematic viscosity $\nu$ and magnetic permeability ${\mu }_0$, the incompressible momentum equation isIt is the Navier-Stokes equation with Lorentz force density. The magnetic force decomposes into magnetic tension and a magnetic pressure gradient. A uniform streamwise pressure gradient need not make ordinary pressure independent of transverse coordinates.