Explain what is meant by a Stokes flow and show that, in such a flow, in the absence of body forces, $\partial \sigma_{i j} / \partial x_{j}=0$, where $\sigma_{i j}$ is the stress tensor.

State and prove the reciprocal theorem for Stokes flow.

When a rigid sphere of radius $a$ translates with velocity $\mathbf{U}$ through unbounded fluid at rest at infinity, it may be shown that the traction per unit area, $\boldsymbol{\sigma} \cdot \mathbf{n}$, exerted by the sphere on the fluid, has the uniform value $3 \mu \mathbf{U} / 2 a$ over the sphere surface. Find the drag on the sphere.

Suppose that the same sphere is free of external forces and is placed with its centre at the origin in an unbounded Stokes flow given in the absence of the sphere as $\mathbf{u}_{s}(\mathbf{x})$. By applying the reciprocal theorem to the perturbation to the flow generated by the presence of the sphere, and assuming this to tend to zero sufficiently rapidly at infinity, show that the instantaneous velocity of the centre of the sphere is

$$\mathbf{V}=\frac{1}{4 \pi a^{2}} \int_{r=a} \mathbf{u}_{s}(\mathbf{x}) d S$$

Part II