Asteady full-developed couette flow in the x-direction

Question

Problem-

A steady, full-developed Couette flow in the x-direction (fully-developed means that the velocity does not evolve as a function of distance of x). The velocity profile for the Couette flow is given by v = (ay, 0,0) where a is a constant. All properties have zero gradients in the z-direction (i.e. no dependence of velocity components or pressure on the z-direction). You may assume that the fluid is a Newtonian fluid with constant viscosity.

1. Verify if the fluid is incompressible.
2. Determine the pressure gradient field 
3. Solve for the pressure field p(x,y,z)

V_P = ∂p/∂x, ∂p/∂y, ∂p/∂z

Additional information-

This problem relates to Mechanical Engineering and it is about couette, a fluid flowing in one direction. The velocity of the flow is given. Questions about whether the fluid is incompressible, the pressure gradient field and the pressure field for p(x, y, z).