A particle of arbitrary shape and volume $4 \pi a^{3} / 3$ moves at velocity $\mathbf{U}(t)$ through an unbounded incompressible fluid of density $\rho$ and viscosity $\mu$. The Reynolds number of the flow is very small so that the inertia of the fluid can be neglected. Show that the particle experiences a force $\mathbf{F}(t)$ due to the surface stresses given by

$$F_{i}(t)=-\mu a A_{i j} U_{j}(t)$$

where $A_{i j}$ is a dimensionless second-rank tensor determined solely by the shape and orientation of the particle. State the reason why $A_{i j}$ must be positive definite.

Show further that, if the particle has the same reflectional symmetries as a cube, then

$$A_{i j}=\lambda \delta_{i j}$$

Let $b$ be the radius of the smallest sphere that contains the particle (still assuming cubic symmetry). By considering the Stokes flow associated with this sphere, suitably extended, and using the minimum dissipation theorem (which should be stated carefully), show that

$$\lambda \leqslant 6 \pi b / a .$$

[You may assume the expression for the Stokes drag on a sphere.]