Past exam of the theoretical mechanics course of the University of Oxford / 2010 / A / 3

Introduce $\mathcal{R}=\rho UL/\mu$. Balancing the streamwise pressure gradient $\Pi /L$ with cross-layer viscosity $\mu U/({\delta }^2{L}^2)$ givesAfter division by that viscous scale, the streamwise momentum equation isThe transverse equation, divided by its pressure-gradient scale, isContinuity scales exactly to ${\hat{u}}_{\hat{x}}+{\hat{w}}_{\hat{z}}=0$. With $\delta \ll 1$, ${\delta }^2\mathcal{R}\ll 1$ and $S=O(1)$, the leading equations are

No slip and impermeability give dimensional $u=-U$, $w=0$ on the upward-moving wall, so $\boxed{\hat{u}=-1,\hat{w}=0}$ there. For the interface normal, $h_x=\delta {\hat{h}}_{\hat{x}}$, and hencein its $x,z$ components. The relevant stresses scale asMultiplying by the inward normal gives the leading tractionIts leading normal component vanishes only if $\hat{p}=0$ at the interface; the tangential component then requires ${\hat{u}}_{\hat{z}}=0$.

The free surface is material. Applying the material derivative to $z-h(x,t)=0$ gives $w=h_t+u{h}_x$, which in the scaled variables becomes

Suppress hats for this calculation. Transverse pressure uniformity and zero surface pressure give $p=0$ throughout the film. Thus $u_{zz}=-S$. Integrating twice, using $u_z(h)=0$ and $u(0)=-1$, givesThe flux is $q={\int }_0^{h}udz=-h+S{h}^3/3$. Integrate continuity across the layer and apply the wall and kinematic conditions to obtain $h_t+q_x=0$. Restoring hats yieldsThe cubic flux contains downward gravity-driven transport and upward wall entrainment.