An incompressible fluid $($density $$ and viscosity $)$ fills the gap between two long, concentric cylinders, as shown. The inner cylinder has radius $R_1$ and rotates at a constant angular velocity of ${}_1$ clockwise. The outer cylinder has inner radius $R_2$ and rotates at a constant angular velocity of ${}_2$ counterclockwise. Assume that the flow is laminar and steady. Neglect gravitational forces.

$($a$)$ Sketch the velocity profile in the gap, $v_{}(r),$ and identify important features.

$($b$)$ Derive an expression for the velocity profile, $v_{}(r),$ in terms of the variables $R_1,R_2,{}_1,{}_2,,,$ and $r.$