Hartmann flow

Hartmann flow is a fully developed viscous conducting-fluid flow driven along a planar channel while a uniform magnetic field crosses the walls. Induction and the Lorentz force density couple its velocity and tangential induced field. For no-slip, normal-field walls separated by $2L$, the velocity is a hyperbolic cosine profile that approaches plane Poiseuille flow at zero imposed field and develops a uniform core with Hartmann layers at large Hartmann number.

For a streamwise pressure gradient $-\rho G$ and walls at $z=\pm L$ with zero tangential magnetic field, the volumetric flow rate per unit span isIts zero-field value is $2G{L}^3/(3\nu )$; it decreases strictly for $H>0$ and is asymptotic to $2G{L}^3/(\nu H)$. The partial-fraction expansion of the hyperbolic cotangent proves monotonicity directly.

A Hartmann layer is the thin viscous-magnetic boundary layer next to a wall crossed by the applied magnetic field. For channel half-width $L$ and large Hartmann number $H$, its thickness is $\delta \sim L/H=\sqrt{{\mu }_0\rho \nu \eta }/|B_0|$. It restores the no-slip boundary condition and the specified magnetic wall condition outside the nearly uniform Hartmann flow core.

The Hartmann number measures the strength of magnetic coupling relative to viscous and resistive effects in a channel:Here $L$ is the channel half-width, $\nu$ the kinematic viscosity, $\eta$ the magnetic diffusivity and ${\sigma }_c$ the electrical conductivity. In Hartmann flow, large $H$ produces thin magnetic-viscous wall layers and a flatter, slower core for fixed pressure gradient.