How do I solve for w and p in an incompressible flow using Euler's equation?

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SUMMARY

The discussion focuses on solving for the pressure \( p \) and the vertical velocity component \( w \) in an incompressible flow using Euler's equation. The user successfully derived the pressure equation \( p(z) = p_0 - \rho g z \), where \( \rho \) is the constant density and \( g \) is the acceleration due to gravity. The challenge remains in determining the complete solution for \( w \) and confirming the presence of one free parameter in the solution. Further guidance on applying Euler's equation is suggested to advance the solution.

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I'm stumped on a HW question that I just can't seem to proceed on.

Homework Statement



An incompressible ( rho = constant ) flow in 2 dimensions [x = (x,z)], with F = (0,-g), satisfies Euler's equation. For this flow, the velocity is u = (u0,w(x)), where u0 is a constant, with w = 0 on x = 0 and p = p0 on z = 0. Find the solution for w and p, and show that it contains one free parameter.

The Attempt at a Solution



I've managed to get p(z) = p0 - rho*g*z, though I don't really know where to go from there or if I've even done it right to begin with.

Any guidance would be much appreciated, thankyou.
 
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Perhaps you should start by actually writing down Euler's equation and trying to solve it.
 
That's what I did do and ended up with that p(z) I stated.

Anyhow, I've managed to make headway now. Thanks for your help.
 

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