Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 345 / 1 / c

Let the fixed current volume per unit channel width beA suitable high-Reynolds number deep-ambient gravity-current front condition iswhere the order-one Froude number $\mathrm{Fr}$ records the selected front closure. In the dilute limit the mixture's reduced gravity iswith ambient and carrier-fluid density ${\rho }_a$.

The well-mixed particle volume of species $i$ is $V_0{\varphi }_i$. Its deposition rate through the base of length $L$ is $L{\hat{W}}_i{\varphi }_i$, where ${\hat{W}}_i<0$. The resulting gravity-current box model is thereforeIt conserves fluid volume while suspended particle volume, and therefore the driving buoyancy, decreases by deposition.

When ${\hat{W}}_1={\hat{W}}_2=\hat{W}$, both concentrations acquire the same decay factor. The total reduced gravity consequently obeysEliminating time with the front equation givesIntegration from the initial state yieldswhereThe particle-laden gravity current reaches its runout length of a gravity current when $g^{\prime }$ vanishes. Since $\hat{W}<0$,Thus $C=5\mathrm{Fr}$ for the stated front condition; the common normalization $\mathrm{Fr}=1$ gives $C=5$.

Because $\hat{W}<0$, species 1 now settles more slowly and species 2 more rapidly than in the equal-speed case. The supplied inequality says that species 2 provides the larger initial particle-density contribution to the reduced gravity; with exact Boussinesq density contrasts, the corresponding condition is ${\tilde{\varphi }}_1({\rho }_1-{\rho }_a)<{\tilde{\varphi }}_2({\rho }_2-{\rho }_a)$. The dominant buoyancy contribution is therefore removed earlier, so $g^{\prime }(t)$ and the gravity-current front condition fall below their equal-settling values at first order in $ϵ$. The runout length decreases. The slower loss of the weaker species-1 contribution only partly compensates for this effect.