Define the rate of strain tensor $e_{i j}$ in terms of the velocity components $u_{i}$.

Write down the relation between $e_{i j}$, the pressure $p$ and the stress tensor $\sigma_{i j}$ in an incompressible Newtonian fluid of viscosity $\mu$.

Prove that $2 \mu e_{i j} e_{i j}$ is the local rate of dissipation per unit volume in the fluid.

Incompressible fluid of density $\rho$ and viscosity $\mu$ occupies the semi-infinite domain $y>0$ above a rigid plane boundary $y=0$ that oscillates with velocity $(V \cos \omega t, 0,0)$, where $V$ and $\omega$ are constants. The fluid is at rest at $y=\infty$. Determine the velocity field produced by the boundary motion after any transients have decayed.

Evaluate the time-averaged rate of dissipation in the fluid, per unit area of boundary.