= Constant-flux porous gravity current
{title2=$t_c=\phi K_bQ/U^3$}
Constant actual volume input $Q$ into a <porous gravity current with background flow> gives early symmetric scales $h\sim[(Q/\phi)^2t/a]^{1/3}$ and $x\sim[a(Q/\phi)t^2]^{1/3}$. For $U>0$, drift dominates after $t_c=\phi K_bQ/U^3$. The late interior depth is $Q/U$ and the steady upstream edge is $x_-=-K_bQ/U^2$, with upstream linear profile $h=U(x-x_-)/K_b$. Upstream <volume flux> is zero because background drift balances gravity spreading.
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