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.