 Solution

First neglect inertia relative to the $O(ϵ)$ pressure correction, requiring $F\ll 1$ for smooth fields on these scales. The positive velocity isMultiplying by area gives the discharge $Q=A^{5/4}-(ϵ/4)A^{3/4}{A}_x+O({ϵ}^2)$. HenceThe leading equation is a nonlinear kinematic wave equation. The pressure correction has positive diffusion coefficient for $A>0$ and smooths steep gradients. Dropped terms are $O({ϵ}^2)$ and inertial terms $O(ϵF)$ on the assumed scales; sufficiently thin fronts require a separate balance rather than an unrestricted Taylor expansion.