Mobile-volume peak of an exponentially forced porous current ($t_{\max }=\tau \log (1/s)/(1-s)$)

For residual fraction $0<s<1$, differentiation of $V=Q\tau (e^{-st/\tau }-e^{-t/\tau })/(1-s)$ gives its unique positive maximum at the displayed time. The maximum volume is $Q\tau s^{s/(1-s)}$. When $s=0$ the mobile volume approaches $Q\tau$ monotonically, with no finite maximum. The limiting $s\to 1$ expression is $V=Qt{e}^{-t/\tau }$ with peak at $t=\tau$.