Draining gravity current

A draining gravity current loses fluid through its substrate while spreading horizontally. For a two-dimensional viscous current of thickness

H (x, t)

H_{t} = A (H^{3} H_{x})_{x} - D H^{1 - β}, A = \frac{ρ g}{3 μ},

p = ρ g H

and

k / h \propto p^{- β}

Similarity exponents of a draining gravity current

If a draining gravity current is supplied with volume flux proportional to

t^{α}

H = t^{a} F (x / t^{b}), a = \frac{2 α + 1}{5}, b = \frac{3 α + 4}{5},

provided

β = \frac{1}{a} = \frac{5}{2 α + 1} .

These relations follow by balancing the time derivative, horizontal lubrication theory flux, vertical drainage, and imposed inlet flux.

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