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

In summary, the conversation discusses a homework problem involving an incompressible flow in 2 dimensions, with given velocity and pressure conditions. The solution for w and p is found, containing one free parameter. The individual seeking help initially struggles but eventually makes progress with the assistance of someone else.
  • #1
Yalldoor
4
0
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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  • #2
Perhaps you should start by actually writing down Euler's equation and trying to solve it.
 
  • #3
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.
 

1. What is the Euler Equation for fluids?

The Euler Equation is a fundamental equation in fluid mechanics that describes the motion of an inviscid, incompressible fluid. It is based on the laws of conservation of mass, momentum, and energy.

2. How is the Euler Equation derived?

The Euler Equation is derived from the Navier-Stokes equations by neglecting the effects of viscosity and assuming that the fluid is incompressible. This simplification allows for a more basic understanding of fluid dynamics.

3. What is the significance of the Euler Equation in fluid mechanics?

The Euler Equation is significant because it provides a simplified model of fluid flow that can be used to analyze and predict the behavior of fluids in various situations. It is also the basis for more complex equations and models used in fluid dynamics.

4. Can the Euler Equation be used to model all types of fluids?

No, the Euler Equation is only applicable to inviscid, incompressible fluids. Real fluids, such as air and water, have some level of viscosity and compressibility, so the Euler Equation is not an accurate representation for these types of fluids.

5. How does the Euler Equation relate to Bernoulli's principle?

The Euler Equation can be used to derive Bernoulli's principle, which states that in a steady flow of an inviscid, incompressible fluid, the total energy of the fluid remains constant. Bernoulli's principle is a direct consequence of the Euler Equation and is commonly used to explain the behavior of fluids in various applications.

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