Solution to Navier-Stokes Equation for dynamic boundary

[xanthium]

I'm looking to get a full solution to the Navier-Stokes equation to describe fluid flow through a pipe with moving surfaces.

For now I am just concerned with a two dimensional system. The upper and lower boundaries are parallel to the x-axis. The surfaces of the boundaries move sinusoidally according to:
Vb(x)=v0*cos(k0*x)

Eliminating several terms from the Navier-Stokes equations, I think the only relevant terms that I need to solve are in the following two equations:

nu*grad^2*v(x,y)+grad*p(x,y)=0

grad*v(x,y)=0

A possible solution that I am trying to test is:
Vx(x,y)=Vx0*e^(i*k0*x)*cos(ky*y)
Vy(x,y)=Vy0*e^(i*k0*x)*sin(ky*y)
P(x,y)=P0*e^(i*k0*x)*e^(i*q*y)

Where Vx0,ky,Vy0,P0,q are constants to be determined. It is clear from the boundary conditions (the no-slip condition in particular) that Vx0=V0. Other than that, I am not sure how to get the other constants or even if this solution works completely.

Any help or suggestions would be very much appreciated.

 

[Andy Resnick]

I'm looking to get a full solution to the Navier-Stokes equation to describe fluid flow through a pipe with moving surfaces.

For now I am just concerned with a two dimensional system. The upper and lower boundaries are parallel to the x-axis. The surfaces of the boundaries move sinusoidally according to:
Vb(x)=v0*cos(k0*x)

Eliminating several terms from the Navier-Stokes equations, I think the only relevant terms that I need to solve are in the following two equations:

nu*grad^2*v(x,y)+grad*p(x,y)=0

grad*v(x,y)=0

A possible solution that I am trying to test is:
Vx(x,y)=Vx0*e^(i*k0*x)*cos(ky*y)
Vy(x,y)=Vy0*e^(i*k0*x)*sin(ky*y)
P(x,y)=P0*e^(i*k0*x)*e^(i*q*y)

Where Vx0,ky,Vy0,P0,q are constants to be determined. It is clear from the boundary conditions (the no-slip condition in particular) that Vx0=V0. Other than that, I am not sure how to get the other constants or even if this solution works completely.

Any help or suggestions would be very much appreciated.

I'm a little confused... are the walls moving in their plane (back and forth) or out of plane (up and down)?

The first problem is (IIRC) solved- look up "Stokes' first problem" and "Stokes' second problem"

 

[charng]

I understand your desire for a complete solution to the Navier-Stokes equation for fluid flow through a pipe with moving surfaces. However, it is important to note that the Navier-Stokes equation is a highly complex partial differential equation that describes the motion of fluids and is notoriously difficult to solve analytically. In fact, there is no general analytical solution for the Navier-Stokes equation, and even for simple boundary conditions, solutions can only be found for certain simplified cases.

That being said, your proposed solution is a valid approach and it is possible that it could work for your specific system. However, it would require further analysis and verification through numerical simulations or experimental data. Additionally, the constants you have chosen may need to be adjusted to match your specific boundary conditions and physical properties of the fluid.

I would suggest consulting with a fluid dynamics expert or utilizing numerical methods such as computational fluid dynamics to obtain a more accurate and reliable solution for your system. This will also allow you to explore a wider range of boundary conditions and parameters to better understand the behavior of the fluid flow in your system.

Overall, the Navier-Stokes equation is a powerful tool for understanding fluid dynamics, but it is important to recognize its limitations and the need for further analysis and validation in order to obtain accurate solutions.