Cubic-input similarity for a draining gravity current ($h,l\propto t,x_N\propto t^2$)

For a one-sided draining gravity current with injected volume per unit width $V_0(t/\tau {)}^3$, defineThe similarity solution satisfieswhere $K=k[g^3{\tau }^3/({\nu }^3{V}_0){]}^{2/5}$. The inlet condition is $-H(0{)}^3{H}^{\prime }(0)=9$, the integral of $H+\varphi L$ is one, and both depths vanish at the advancing front. The endpoint ${\xi }_N$ gives $x_N={\xi }_{N}D{t}^2$ and depends on $K$ and $\varphi$.