= Solution
The leading lubrication momentum equations and incompressibility condition are
$$
0=-p_x+\rho g\sin\alpha+\mu u_{yy},
\qquad
0=-p_y-\rho g\cos\alpha,
\qquad
u_x+v_y=0.
$$
At the solid plane $y=0$, no slip and no penetration give $u=v=0$. At the free surface $y=h(x,t)$, neglecting surface tension and ambient viscous stress gives
$$
p=p_{\rm atm},
\qquad
u_y=0,
$$
to leading order. The <kinematic boundary condition for a free-surface graph> is
$$
\boxed{v=h_t+u h_x
\quad\text{at }y=h.}
$$
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