Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 3 / 16A (Fluid Dynamics)

For the parallel flow $\mathbf{u}=(u(y,t),0)$ with no pressure gradient, the Navier-Stokes equations reduce to the diffusion equationSubstitution ofgivesThe boundary velocities require $f(L_0)=1$ and $f(-L_0)=-1$. The problem is antisymmetric under $y\mapsto -y$, so $f$ is odd. Withthe unique odd solution is

Usingin the quotient from part (a), multiplying numerator and denominator by the complex conjugate of the denominator, and taking the real part after multiplication by $e^{i\omega t}$ giveswhereThe denominator follows from

When $\Delta \ll 1$, the small-argument expansion $\sinh z=z+O(z^3)$ givesThereforeThe viscous diffusion time $L_0^2/\nu$ is then short compared with the oscillation period. Viscosity communicates the wall motion across the entire layer almost instantaneously, producing the linear profile of quasi-steady Couette flow.