Past exam of the flows, fluctuations and complexity course of the University of Oxford / 2016 / 2 / b

Let the cylinder radius be $a$, with $h/a\ll 1$ and slowly varying thickness. Arc length is $s=a\theta$, with increasing $\theta$ following the upward-moving surface at the right-hand side. Gravity has tangential component $-g\cos \theta$. In the leading thin-film balance, normal viscous gradients dominate tangential ones, curvature corrections are small, and the longitudinal hydrostatic-pressure gradient is smaller by the aspect ratio. Neglect capillary-pressure gradients and imposed gas shear. The tangential Stokes balance is thenThese assumptions also require that inertia and curvature effects from rotation remain small on the thin-film scale; an arbitrary high-speed coating need not obey this equation.

No-slip at the solid wall and negligible shear stress at the free surface giveTwice integrating the constant normal forcing determines the two integration constants:The profile is most strongly retarded by gravity on the ascending side. The surface velocity is $U-g{h}^2\cos \theta /(2\nu )$, not generally equal to the wall speed.