Hartmann number

Remarks

It is a dimensionless number: $\sigma B_0^2$ has the units of a force per volume per speed, as does $\mu / L^2$, so $\mathrm{Ha}^2 = \sigma B_0^2 L^2 / \mu$ has no units. Example, Hartmann flow: a fluid driven by a constant pressure gradient between the planes $y = \pm L$, in a transverse field $B_0 \mathbf{e}_y$. In the limit of small magnetic Reynolds number, its velocity profile is proportional to $\cosh(\mathrm{Ha}) - \cosh(\mathrm{Ha} \, y / L)$. For large $\mathrm{Ha}$ this is flat in the middle and drops to $0$ in layers of thickness about $L / \mathrm{Ha}$ near the walls. For $\mathrm{Ha} \to 0$ the profile becomes the parabola of plane Poiseuille flow. Non-example: $\sigma B_0^2 L / (\rho U)$ is the Stuart number, or interaction parameter, which compares the Lorentz force with inertia, not with viscosity; it equals $\mathrm{Ha}^2 / \mathrm{Re}$.