Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 3 / 7D (Fluid Dynamics)

Take $x$ vertically downward and $y$ outward from the wall, with the film occupying $0<y<h$. The ambient pressure isAt $y=h$, normal stress continuity gives $p=p_a$ and zero ambient shear gives $\mu U^{\prime }(h)=0$; at the wall, no slip gives $U(0)=0$.

For steady parallel flow $u=(U(y),0)$, incompressibility is automatic and Navier–Stokes reduces toSince $p_x={\rho }_{a}g$, integration yields