Similarity exponents of a draining gravity current
= Similarity exponents of a draining gravity current
If a <draining gravity current> is supplied with <volume flux> proportional to $t^\alpha$, a first-kind <similarity solution> has
$$
H=t^aF(x/t^b),
\qquad
a=\frac{2\alpha+1}{5},
\qquad
b=\frac{3\alpha+4}{5},
$$
provided
$$
\beta=\frac1a=\frac5{2\alpha+1}.
$$
These relations follow by balancing the time derivative, horizontal <lubrication theory> flux, vertical drainage, and imposed inlet flux.