Couette Flow with Variable Viscosity Consider the laminar, steady flow of a viscous fluid between infinite, parallel plates. The top plate moves at a velocity U$0$ that is small enough that dissipation can be neglected, and is held at a constant temperature of Th $=10$C$.$ The plates are separated by a gap of h $=10$ cm$.$ A heat flux of qw $=240$ W$/$m$2$ is applied to the bottom plate

a$)$ Write differential equations for the velocity and temperature profiles, without plugging in numbers. You will need to start from the compressible version of the equations, since you must consider variability of the fluid properties $($specifically $$ and k$)$ with temperature.

$($b$)$ For water, the thermal conductivity can be approximated as nearly constant over the resulting temperature range in this problem, with a value of k $=0\mathrm{.}6$ W$/$m$$K$.$ Compute the temperature profile $($by hand$),$ and plot the resulting velocity profile u$($y$)$ U$0$ using numerical tools. You will need to either find an empirical relation or tabulated data for $($T$)$ for water.