Lubrication gravity-current flux
= Lubrication gravity-current flux
{title2=$q=-(g/3\nu)h^3h_x$}
For a thin viscous <gravity current> with <hydrostatic pressure> $p=\rho g(h-z)$, a <no-slip boundary condition> at its base and a <stress-free boundary condition> at its top, horizontal momentum balance gives $u=(g/\nu)h_x(z^2/2-hz)$. Integrating across the depth gives the <volume flux> $q=-(g/3\nu)h^3h_x$. Horizontal spreading follows $h_t+q_x=0$ if the substrate is impermeable.