Consider flow of an incompressible fluid of uniform density $\rho$ and dynamic viscosity $\mu$. Show that the rate of viscous dissipation per unit volume is given by

$$\Phi=2 \mu e_{i j} e_{i j},$$

where $e_{i j}$ is the strain rate.

Determine expressions for $e_{i j}$ and $\Phi$ when the flow is irrotational with velocity potential $\phi$. Hence determine the rate of viscous dissipation, averaged over a wave period $2 \pi / \omega$, for an irrotational two-dimensional surface wave of wavenumber $k$ and small amplitude $a \ll k^{-1}$ in a fluid of very small viscosity $\mu \ll \rho \omega / k^{2}$ and great depth $H \gg 1 / k$.

[You may use without derivation that in deep water a linearised wave with surface displacement $\eta=a \cos (k x-w t)$ has velocity potential $\phi=-(\omega a / k) e^{-k z} \sin (k x-\omega t)$.]

Calculate the depth-integrated time-averaged kinetic energy per wavelength. Assuming that the average potential energy is equal to the average kinetic energy, show that the total wave energy decreases to leading order like $e^{-\gamma t}$, where

$$\gamma=4 \mu k^{2} / \rho$$