The liquid flow is incompressible, steady state and effect of mass transfer and voltization of liquid can be neglected.

I am trying to start from vector form of Navier-stokes equation

\rho D/Dt =-grad(P)+\rho +\mu (laplacian)^2

and continuity equation

div(\rho )=0

But shall i set the pressure gradient zero? Suppose, i am talking about a half liquid-filled closed long cylinder rotating on its own axis, everything same as earth. I am trying to find out laminar flow patterns, and no-slip condition on cylinder wall in contact with liquid holds.

for the static cylinder with quiescent liquid level, pressure gradient exists at inner cylinder wall submerged in liquid. and although the flow is incompressible, some region of the cross section should contain no fluid and hence have zero liquid density. if a definite volume fraction of the cylinder is filled, then How would I incorporate the effect in differential equation?