A spherical bubble of radius a moves with velocity $\mathbf{U}$ through a viscous fluid that is at rest far from the bubble. The pressure and velocity fields outside the bubble are given by

$$p=\mu \frac{a}{r^{3}} \mathbf{U} \cdot \mathbf{x} \quad \text { and } \quad \mathbf{u}=\frac{a}{2 r} \mathbf{U}+\frac{a}{2 r^{3}}(\mathbf{U} \cdot \mathbf{x}) \mathbf{x}$$

respectively, where $\mu$ is the dynamic viscosity of the fluid, $\mathbf{x}$ is the position vector from the centre of the bubble and $r=|\mathbf{x}|$. Using suffix notation, or otherwise, show that these fields satisfy the Stokes equations.

Obtain an expression for the stress tensor for the fluid outside the bubble and show that the velocity field above also satisfies all the appropriate boundary conditions.

Compute the drag force on the bubble.

[Hint: You may use

$$\int_{S} n_{i} n_{j} d S=\frac{4}{3} \pi a^{2} \delta_{i j}$$

where the integral is taken over the surface of a sphere of radius a and $\mathbf{n}$ is the outward unit normal to the surface.]