In problem $1$ with solve the following,

$($i$)$ Determine the velocity profile and plot the same, when the fluid viscosity varies linearly as a function of temperature in the range defined by the walls" temperature.

$($ii$)$ Determine the temperature profile and plot the same, expressing your result in terms of fluid properties and the temperature and speed of the moving plate.

$($iii$)$ Determine the heat flux at the moving and the stationary plate.

$($iv$)$ Determine the influence of dissipation on the velocity profile and heat transfer between the walls and the fluid. Plot the temperature profile for different Brinkman number to $10.$ And here problem $1$description is as follows : couette flow is generated by the relative motion of two parallel plates. One of the plates is moving on its own ane with the constant velocity U the other one is at rest. The spacing bewtween the two plates is small and is equal to L$($ you can assume the motion between the two plates to be rectangular cordinates woth origin on the plate at rest$)$