(i) In a Newtonian fluid, the deviatoric stress tensor is linearly related to the velocity gradient so that the total stress tensor is

$$\sigma_{i j}=-p \delta_{i j}+A_{i j k l} \frac{\partial u_{k}}{\partial x_{l}}$$

Show that for an incompressible isotropic fluid with a symmetric stress tensor we necessarily have

$$A_{i j k l} \frac{\partial u_{k}}{\partial x_{l}}=2 \mu e_{i j} \text {, }$$

where $\mu$ is a constant which we call the dynamic viscosity and $e_{i j}$ is the symmetric part of $\partial u_{i} / \partial x_{j}$.

(ii) Consider Stokes flow due to the translation of a rigid sphere $S_{a}$ of radius $a$ so that the sphere exerts a force $\mathbf{F}$ on the fluid. At distances much larger than the radius of the sphere, the instantaneous velocity and pressure fields are

$$u_{i}(\mathbf{x})=\frac{1}{8 \mu \pi}\left(\frac{F_{i}}{r}+\frac{F_{m} x_{m} x_{i}}{r^{3}}\right), \quad p(\mathbf{x})=\frac{1}{4 \pi} \frac{F_{m} x_{m}}{r^{3}},$$

where $\mathbf{x}$ is measured with respect to an origin located at the centre of the sphere, and $r=|\mathbf{x}|$.

Consider a sphere $S_{R}$ of radius $R \gg a$ instantaneously concentric with $S_{a}$. By explicitly computing the tractions and integrating them, show that the force $G$ exerted by the fluid located in $r>R$ on $S_{R}$ is constant and independent of $R$, and evaluate it.

(iii) Explain why the Stokes equations in the absence of body forces can be written

$$\frac{\partial \sigma_{i j}}{\partial x_{j}}=0$$

Show that by integrating this equation in the fluid volume located instantaneously between $S_{a}$ and $S_{R}$, you can recover the result in (ii) directly.