# How to rewrite this differential equation

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?

• How exactly do you want to eliminate $u$ from the equation? – rafa11111 Dec 11 '18 at 11:47
• If $u$ is a scalar field then your second equation means that its gradient is $0$ so the respective term in the first equation vanishes – Yuriy S Dec 11 '18 at 11:56