The energy equation for the motion of a viscous, incompressible fluid states that

$$\frac{d}{d t} \int_{V} \frac{1}{2} \rho u^{2} d V+\int_{S} \frac{1}{2} \rho u^{2} u_{i} n_{i} d S=\int_{S} u_{i} \sigma_{i j} n_{j} d S-2 \mu \int_{V} e_{i j} e_{i j} d V$$

Interpret each term in this equation and explain the meaning of the symbols used.

Consider steady rectilinear flow in a (not necessarily circular) pipe having rigid stationary walls. Deduce a relation between the viscous dissipation per unit length of the pipe, the pressure gradient $G$, and the volume flux $Q$.

Starting from the Navier-Stokes equations, calculate the velocity field for steady rectilinear flow in a circular pipe of radius $a$. Using the relationship derived above, or otherwise, find the viscous dissipation per unit length of this flow in terms of $G$.

[Hint: In cylindrical polar coordinates,

$$\nabla^{2} w(r)=\frac{1}{r} \frac{d}{d r}\left(r \frac{d w}{d r}\right)$$