Hartmann flow rate with normal-field walls ($Q=\frac{2G{L}^3}{\nu }\frac{H\coth H-1}{H^2}$)

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.