= Solution
In the same idealized momentum model, write $S=-|S|$. The discharge now increases downstream. The admissible maximum of $Q/b$ moves to $x_c>0$, beyond the geometric throat, because added discharge initially offsets the widening channel. The formula $x_c=-SL^2/(Q_0+D)$ remains valid, and the critical depth depends on $S^2$ through $D$. The positive-flow reach is now restricted upstream by $Q_0+|S|x>0$, rather than downstream by exhaustion of the discharge. The same critical-head and boundary-connection conditions are still needed.
Actual injection introduces a further physical issue: newly added fluid need not have the stream's axial <velocity>. Let its axial <velocity> be $u_a$ and let $A=bh$. The <momentum correction for lateral mass addition> is
$$
u u_x=-g(h+H)_x+\frac{Q_x}{A}(u_a-u),\qquad
\mathcal B_x=\frac{Q_x(u_a-u)}{gA}.
$$
Removal at the local <velocity>, or ideal co-moving addition, gives the conserved-head analysis above. Addition at rest instead accelerates new fluid and decreases the head. It cannot be modeled solely by changing the sign of $S$ in a constant-head calculation.
For example, flat-bed addition at rest gives $(1-F^2)h_x=F^2h[b_x/b-2Q_x/Q]$. Its regular-control compatibility is $b_x/b=2Q_x/Q$, rather than the co-moving condition $b_x/b=Q_x/Q$; in the parabolic channel it gives the different candidate $x_c=-SL^2/Q_0$. Thus \b[the sign reversal moves the ideal control downstream, but its quantitative prediction depends on the momentum carried by the injected water]. The source specifies the volume rate only, so an injection-velocity assumption must accompany any definite constant-energy answer.
Back to article page
