= Magnetohydrodynamic momentum equation
{title2=$\rho D_t\mathbf u=-\nabla p+\mathbf J\times\mathbf B+\rho\nu\nabla^2\mathbf u$}
For constant <mass density> $\rho$, <kinematic viscosity> $\nu$ and <magnetic permeability> $\mu_0$, the incompressible momentum equation is
$$
\partial_t\mathbf u+(\mathbf u\cdot\nabla)\mathbf u=-\rho^{-1}\nabla p+(\mu_0\rho)^{-1}(\nabla\times\mathbf B)\times\mathbf B+\nu\nabla^2\mathbf u.
$$
It 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.
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