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Let $S>0$ denote volume loss per unit downstream length and take $Q_0>0$. Mass balance gives $Q_x=-S$, soAssume fluid is removed carrying its local axial velocity, with no additional axial force on the retained flow. Under this inviscid seepage approximation, the horizontal momentum equation and its Bernoulli head are unchanged. On the flat bed, hydraulic control with prescribed seepage therefore usesThe calculation applies on a reach with positive remaining discharge. Constant loss cannot be extended indefinitely past $x=Q_0/S$ while preserving this positive-flow model.

At a regular control, $F=1$ and the same differentiated energy condition requires $q_x=0$, where $q=(Q_0-Sx)/b(x)$. Differentiating givesWrite $D=\sqrt{Q_0^2+S^2{L}^2}$. The two algebraic roots are $(Q_0\pm D)/S$. At the plus root $Q=-D<0$, so it is outside the stipulated positive-flow branch. The admissible control candidate isIt lies upstream of the geometric throat. At this point $Q(x_c)=D$ andIt is a maximum of positive $q$: differentiating the numerator of $q_x$ at $x_c$ gives $2S{x}_c-2Q_0=-2D<0$. ConsequentlyA control exists only if the reach includes this point and boundary conditions supply the critical conserved head $\mathcal{B}=3h_c/2$ and a suitable transcritical connection. Merely prescribing $Q_0$ and $S$ does not force that head. At $S=0$ the limiting location is zero and the usual throat depth is recovered.