 Particle-laden current with suction

A steady well-mixed dilute particle-laden gravity current above an absorbing porous bed loses carrier fluid at speed $W$ and particles at total downward speed $W+V$, where $V$ is their relative settling velocity magnitude. Volume conservation and the particle budget give $Q=Q_0-Wx$ and $\varphi =(Q/Q_0{)}^{V/W}$ for $W>0$. The momentum conservation equation is $(Q^2/h+g_0^{\prime }\varphi h^2/2{)}^{\prime }=-Wu$ if the withdrawn fluid carries its local horizontal velocity. Consequently $u{u}^{\prime }+g_0^{\prime }\varphi h^{\prime }+g_0^{\prime }h{\varphi }^{\prime }/2=0$. At $W=0$, $\varphi =e^{-Vx/Q_0}$ and the momentum flux is constant.