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Solution to Navier-Stokes Equation for dynamic boundary

  1. Oct 20, 2009 #1
    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:

    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:



    A possible solution that I am trying to test is:

    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.
  2. jcsd
  3. Oct 25, 2009 #2

    Andy Resnick

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    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"
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