= Hartmann flow rate with normal-field walls
{c}
{title2=$Q=\frac{2GL^3}{\nu}\frac{H\coth H-1}{H^2}$}
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 is
$$
Q(H)=\frac{2GL^3}{\nu}\frac{H\coth H-1}{H^2}.
$$
Its zero-field value is $2GL^3/(3\nu)$; it decreases strictly for $H>0$ and is asymptotic to $2GL^3/(\nu H)$. The <partial-fraction expansion of the hyperbolic cotangent> proves monotonicity directly.
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