Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 73 / 1 / iv

At a fixed location behind the advancing nose, the depth decreases after passage. A fall $-dh>0$ leaves the immobile fluid volume $s\varphi (-dh)$ per unit length, so total local fluid storage changes by $\varphi dh+s\varphi (-dh)=\varphi (1-s)dh$. The thinning-region equation is thereforeThe inlet flux remains $uh(0,t)=Q{e}^{-t/\tau }$. Applying the method of characteristics with thinning-wave speed $u/[\varphi (1-s)]$ gives the residual-trapping attenuation of a porous current:The nose advances into new pore space, which has no residual fluid yet, so its storage factor is still $\varphi$ and its speed is still $u/\varphi$. Consequently $X(t)=ut/\varphi$ and the interior nose trace becomes $h_0{e}^{-st/\tau }$. Using the faster thinning-wave speed for the leading nose would incorrectly erase the attenuation caused by capillary residual trapping. The broken printed reference “(??)” points to the preceding outer profile.