xanthium
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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.
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.