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"