Mobile-volume peak of an exponentially forced porous current
= Mobile-volume peak of an exponentially forced porous current
{title2=$t_{\max}=\tau\log(1/s)/(1-s)$}
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 $s\to1$ expression is $V=Qt e^{-t/\tau}$ with peak at $t=\tau$.