Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 37A (Fluid Dynamics)

Write $d_{+}=\sigma +h$ and $d_{-}=h-\sigma$ for the lower and upper layer thicknesses. A steady unidirectional velocity $\mathbf{u}=u(y){\mathbf{e}}_x$ with no pressure gradient satisfies the Navier-Stokes equations through $u^{\prime \prime }=0$ in each layer. Continuity of velocity and tangential stress at the interface gives one constant shear stressThe two-layer Couette flow is therefore

The fluid exerts the tangential traction $-\tau {\mathbf{e}}_x$ on the upper boundary and $+\tau {\mathbf{e}}_x$ on the lower boundary. Equivalently, the boundaries exert $+\tau {\mathbf{e}}_x$ and $-\tau {\mathbf{e}}_x$ on the fluids, respectively.

The viscous dissipation per unit boundary area is

The rate at which the upper and lower boundaries do work on the fluid, per unit area, isThe definition of $\tau$ giveswhich is the required mechanical-energy balance.

Suppose $U_{+}=U$ and $U_{-}=-U$. Integrating the two velocity profiles gives the volume fluxes per unit spanHereConsequentlyso all dependence on the interface position cancels from the ratio: