$(35$ points$)$ Centrifugal deformation of an air$-$liquid interface. Consider an open cylindrical bucket of radius $R,$ initially filled with an incompressible Newtonian liquid to height $h_0.$ The ambient air pressure above the liquid is $P_0.$ The bucket is rotated at constant angular velocity, $.$ At steady state, the liquid is in rigid body rotation, and the centripetal force on the fluid results in deformation of the liquid$-$air interface, which can be described by a radially$-$varying liquid height, $h(r).$

For this problem, you may assume that the bucket and fluid are very deep such that the effect of the bottom wall on the flow is negligible.

$($a$)(15pts)$ Assuming that the flow is unidirectional in the $-$direction, derive an expression for the velocity field. $($Hint: use the $-$component of the Navier$-$Stokes equation$)$

$($b$)(10$ pts$)$ Assuming that the surface tension effects are negligible $($i$.$e$.,$ low interfacial energy and small deformation of boundary$),$ derive an expression for the pressure in the fluid in terms of the local fluid height, $h(r).$

$($c$)(10pts)$ Assuming that the total volume of fluid is conserved, determine $h(r).$