Navier-Stokes Problem: Solving for Pressure Gradient in a 2D Rectangular Cavity

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The discussion revolves around solving the Navier-Stokes equations for a 2D rectangular cavity filled with liquid, where the top plate moves at a velocity V0. The flow is approximated as one-dimensional due to the dimensions L and H, leading to a simplified pressure gradient equation. Participants agree on the necessity of an integral constraint to ensure net flow rate is zero, and they explore the implications of including vertical velocity components. The conversation highlights the importance of pressure gradients in driving flow and discusses the setup of equations under various assumptions, including low Reynolds numbers. Overall, the problem is deemed solvable with the right approach to pressure gradients and boundary conditions.
  • #31
I'm having second thoughts on your equation for ##\omega##. Shouldn't it be a function of x times a function of y, not the sum of such functions?

cosh cos, cosh sin, sinh cos, sinh sin
 
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  • #32
Chestermiller said:
I'm having second thoughts on your equation for ##\omega##. Shouldn't it be a function of x times a function of y, not the sum of such functions?

cosh cos, cosh sin, sinh cos, sinh sin

Shoot, you're totally right. Then ##\nabla \cdot \psi = \omega## is not separable. I can't think of a way to analytically solve. How would you do this numerically? Finite difference ##\nabla^4 \psi=0##?
 
  • #33
joshmccraney said:
Shoot, you're totally right. Then ##\nabla \cdot \psi = \omega## is not separable. I can't think of a way to analytically solve. How would you do this numerically? Finite difference ##\nabla^4 \psi=0##?
No. Solve the equations involving both omega and psi. The only tricky part is numerically specifying the boundary conditions on omega. But I know how to do that.
 
  • #34
Yea, how do you do that?
 

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