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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?

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    $\begingroup$ How exactly do you want to eliminate $u$ from the equation? $\endgroup$ – rafa11111 Dec 11 '18 at 11:47
  • $\begingroup$ 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 $\endgroup$ – Yuriy S Dec 11 '18 at 11:56

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