Viscous Gravity Current — Thin Film Equation

Lubrication theory: h_t + ∂_x(h³ ∂_x h) = 0

Fluid Parameters

Dynamics

Front position x_f = 0.00
x_f ~ t^(1/3) self-similar
Volume V = 0.00
Max height h_max = 0.00
t = 0.00
Physics: A viscous fluid spreading under gravity on a flat surface obeys the thin film equation (lubrication approximation): ∂h/∂t = ∂/∂x[h³/(3μ)·(ρg ∂h/∂x − σ ∂³h/∂x³)]. For a constant-volume gravity current, the Barenblatt self-similar solution gives: front position x_f ~ (g/μ)^(1/5)·(V³t)^(1/5) = t^{1/5} (2D) or t^{1/8} (axisymmetric). The height profile is parabolic h(x,t) ~ (x_f²−x²). Surface tension introduces a capillary regularization and can prevent or trigger fingering instabilities.