Derive the relation between the stress tensor $\sigma_{i j}$ and the rate-of-strain tensor $e_{i j}$ in an incompressible Newtonian fluid, using the result that there is a linear dependence between the components of $\sigma_{i j}$ and those of $e_{i j}$ that is the same in all frames. Write down the boundary conditions that hold at an interface between two viscous fluids.

Viscous fluid is contained in a channel between the rigid planes $y=-a$ and $y=a$. The fluid in $y<0$ has dynamic viscosity $\mu_{-}$, while that in $y>0$ has dynamic viscosity $\mu_{+}$. Gravity may be neglected. The fluids move through the channel in the $x$-direction under the influence of a pressure gradient applied at the ends of the channel. It may be assumed that the velocity has no $z$-components, and all quantities are independent of $z$.

Find a steady solution of the Navier-Stokes equation in which the interface between the two fluids remains at $y=0$, the fluid velocity is everywhere independent of $x$, and the pressure gradient is uniform. Use it to calculate the following:

(a) the viscous tangential stress at $y=-a$ and at $y=a$; and

(b) the ratio of the volume fluxes of the two different fluids.

Comment on the limits of each of the results in (a) and (b) as $\mu_{+} / \mu_{-} \rightarrow 1$, and as $\mu_{+} / \mu_{-} \rightarrow \infty$