 Solution

Let $z$ measure distance normal to the plane, with the wall at $z=0$ and the free surface at $z=h$. Take density $\rho$, dynamic viscosity $\mu$, and gravitational acceleration $g$. In lubrication theory, normal momentum balance gives hydrostatic pressure while inertia and streamwise viscous derivatives are smaller. With negligible surface tension and negligible air shear,Apply the no-slip boundary condition $u=v=0$ at $z=0$, and zero leading tangential stress $u_z=v_z=0$ at $z=h$. The velocity profiles areDepth integration gives the volume flux, with $K=\rho g/(3\mu )$:Combining incompressibility with the wall and surface kinematic conditions gives conservation of mass, $h_t+{\partial }_x{q}_x+{\partial }_y{q}_y=0$. Therefore the two-dimensional version of the gravity-driven thin film on an incline satisfiesDirect downslope gravity supplies the advective term, while gradients of the hydrostatic pressure spread the film along the surface. At $\alpha =0$ this reduces to nonlinear gravitational spreading in a horizontal film.