I have the following NS equation and conservation equation:
$$\nu\nabla^2u + v_0\frac{\partial u}{\partial z} - u\cdot\nabla u - \frac{1}{\rho}\nabla p = \frac{\partial u}{\partial t}$$
$$\nabla\cdot u=0$$
Now, I would like to manipulate the above equations to eliminate $x$ and $y$ components of $u$ together with the pressure $p$, such that I obtain the equation: $$\nu\nabla^4u_z + v_0\frac{\partial }{\partial z}\nabla^2u_z = \frac{\partial }{\partial t}\nabla^2u_z$$
Can anyone explain how to get the above final equation?