= Cubic-input similarity for a draining gravity current
{title2=$h,l\propto t,\quad x_N\propto t^2$}
For a one-sided <draining gravity current> with injected volume per unit width $V_0(t/\tau)^3$, define
$$
C=\left(\frac{\nu V_0^2}{g\tau^6}\right)^{1/5},\quad
D=\left(\frac{gV_0^3}{\nu\tau^9}\right)^{1/5},\quad
\xi=\frac{x}{Dt^2},\quad h=CtH(\xi),\quad l=CtL(\xi).
$$
The <similarity solution> satisfies
$$
H-2\xi H'-\frac13(H^3H')'=-K(1+H/L),\qquad
\phi(L-2\xi L')=K(1+H/L),
$$
where $K=k[g^3\tau^3/(\nu^3V_0)]^{2/5}$. The inlet condition is $-H(0)^3H'(0)=9$, the integral of $H+\phi L$ is one, and both depths vanish at the advancing front. The endpoint $\xi_N$ gives $x_N=\xi_NDt^2$ and depends on $K$ and $\phi$.
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