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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 $x_c=-S{L}^2/(Q_0+D)$ 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 isRemoval 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 $(1-F^2)h_x=F^{2}h[b_x/b-2Q_x/Q]$. 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 $x_c=-S{L}^2/Q_0$. Thus 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.