A viscous fluid is contained in a channel between rigid planes $y=-h$ and $y=h$. The fluid in the upper region $\sigma<y<h$ (with $-h<\sigma<h$ ) has dynamic viscosity $\mu_{-}$ while the fluid in the lower region $-h<y<\sigma$ has dynamic viscosity $\mu_{+}>\mu_{-}$. The plane at $y=h$ moves with velocity $U_{-}$and the plane at $y=-h$ moves with velocity $U_{+}$, both in the $x$ direction. You may ignore the effect of gravity.

(a) Find the steady, unidirectional solution of the Navier-Stokes equations in which the interface between the two fluids remains at $y=\sigma$.

(b) Using the solution from part (a):

(i) calculate the stress exerted by the fluids on the two boundaries;

(ii) calculate the total viscous dissipation rate in the fluids;

(iii) demonstrate that the rate of working by boundaries balances the viscous dissipation rate in the fluids.

(c) Consider the situation where $U_{+}+U_{-}=0$. Defining the volume flux in the upper region as $Q_{-}$and the volume flux in the lower region as $Q_{+}$, show that their ratio is independent of $\sigma$ and satisfies

$$\frac{Q_{-}}{Q_{+}}=-\frac{\mu_{-}}{\mu_{+}}$$