The figure below shows two long concentric cylinders with a viscous fluid filling the annular gap between them. The outer radius of the inner cylinder is $\backslash ($ r$\_\{1\}\backslash ),$ and the inner radius of the outer cylinder is $\backslash ($ r$\_\{2\}\backslash ).$ The viscous flow occurs between $\backslash ($ r$\_\{1\}\backslash )$ and $\backslash ($ r$\_\{2\}\backslash ).$ The inner cylinder moves axially with a velocity $\backslash ($ u$=$U$\_\{0\}\backslash ).$ The pressure is uniform throughout the fluid in all directions.

a$.$ Derive the velocity profile for the fluid in the gap.

b$.$ Make a rough hand sketch of the velocity profiles in the gap.