Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 332 / 1

Write the given pressure-dependent ratio of permeability of a porous medium to sponge thickness aswhere $K$ is constant. The fluid pressure drop across the sponge is the hydrostatic pressure of the current,because the air pressures above and below are equal. Therefore Darcy's law gives the downward volume flux per unit horizontal area asThus the sponge drainage obeys a power law in the current thickness, as described by pressure-dependent Darcy drainage.

Let $z$ measure height above the sponge. Under the long-wave approximation, hydrostatic pressure gives $p_x=\rho g{H}_x$, and the horizontal balance for viscous fluid flow isThe no-slip boundary condition $u(0)=0$ and stress-free boundary condition $u_z(H)=0$ giveIntegrating this lubrication theory profile gives the horizontal volume flux per unit span

Local mass conservation includes the downward loss found in part i:Consequently the required nonlinear partial differential equation isIf the imposed inlet flux is $Q_0{t}^{\alpha }$ and the moving front is $x=x_N(t)$, sufficient boundary conditions areFor a current released onto a dry substrate one also takes the initial condition $H(x,0)=0$ away from the source. The front position is part of this moving-boundary problem.

Seek a similarity solution in the dimensionally completed formwhere the fixed scales absorb $A,D$, and $Q_0$. Its inlet volume flux scales asso the prescribed source requiresThe time derivative, horizontal lubrication theory term, and pressure-dependent Darcy drainage term scale respectively asEquating the three exponents givesSolving these linear equations together with the inlet condition yields the similarity exponents of a draining gravity currentEquivalently,and, up to constant dimensional scales,

For $\alpha =2$, part iii gives $a=1$, $b=2$, and $\beta =1$. PutSubstitution into the nonlinear partial differential equation gives the ordinary differential equationwith boundary conditions

To find the leading edge, set $y={\eta }_N-\eta$ and suppose $F\sim C{y}^p$. The two singular terms in the ordinary differential equation have orders $y^{4p-2}$ and $y^{p-1}$. Their exponents agree only when $p=1/3$. Their leading coefficients then satisfywhereas $F$ and the constant drainage $D$ are lower-order terms. HenceThe draining gravity current therefore has a one-third-power leading edge and its flux vanishes there.