= Hydraulic control of a zero-PV rotating channel
{title2=$Q_c=\frac bg[\sqrt{gh_1+(fb)^2/16}-fb/4]^3$}
= Zero-PV channel control
{synonym}
At a hydraulic control, the specific-energy function of a <zero-potential-vorticity rotating channel flow> is stationary with respect to wall depth at fixed discharge and width. Writing $v=u_1+fb/2$ gives $v(v+fb/2)=gh_1$ and $Q_c=bv^3/g$, hence the displayed discharge. The fully wetted positive-flow solution requires $fb<\sqrt{2gh_1}$; at equality the other wall dries and the first wall velocity vanishes. Beyond that limit the assumed two-wet-wall profile must be replaced.
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