Solid-body rotation in a rotating tank

The velocity components have to satisfy the Navier-Stokes equations, satisfy the conditions of incompressibility, and verify the following boundary conditions  

One of the unknowns to be determined is the position h r) of the free surface relative to the bottom of the tank. 

In our problem, the gravitational acceleration vector is oriented downward along Oz, that is, fr= fe= and fz = -pg. With [17.5], the Navier-Stokes equations reduce to  

Because of the symmetry of revolution, the pressure does not depend on angle 6. The velocity field verifies the condition of incompressibility. The only solution to [17.9] that verifies the boundary conditions [17.6] and [17.7] is  

The pressure field is determined by solving the Navier-Stokes equations for the radial and axial components. We obtain  

---