Momentum correction for lateral mass addition ($u{u}_x=-g(h+H{)}_x+(Q_x/A)(u_a-u)$)

In a sectionally uniform flow, a volume source $Q_x$ per downstream length brings axial velocity $u_a$. Momentum conservation gives the displayed source correction, with $A$ the wetted area. The head derivative is $[h+H+u^2/(2g){]}_x=Q_x(u_a-u)/(gA)$. Removing fluid at the local velocity preserves the head in this approximation. Adding fluid at rest does not: its acceleration reduces that head. The volume source alone cannot specify the momentum exchange without a velocity assumption.