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Stokes flow

  1. Apr 20, 2015 #1
    1. The problem statement, all variables and given/known data
    Let a spherical object move through a fluid in R3. For slow velocities, assume Stokes’ equations apply. Take the point of view that the object is stationary and the fluid streams by. The setup for the boundary value problem is as follows: given U = (U, 0, 0), U constant, find u and p such that Stokes’ equation holds in the region exterior to a sphere of radius R, u = 0 on the boundary of the sphere and u = U at infinity.
    The solution to this problem (in spherical coordinates centered in the object) is called Stokes’ Flow:
    stokes.png
    where p0 is constant and n = r/r .
    (a) Verify this solution.
    (b) Show that the drag is 6πRνU and there is no lift.

    If someone can help it would be great.

    2. Relevant equations


    3. The attempt at a solution
    a)
    I started using the stokes equations but couldn't get there.
     
  2. jcsd
  3. Apr 20, 2015 #2
    What coordinate system did you use, and what form of the stokes equations did you use? How can we help if you don't show us what you have done so far?

    Chet
     
  4. Apr 20, 2015 #3
    Using spherical coordinates (that is what asked in the problem I guess) and these equations
    stokes2.png
     
  5. Apr 20, 2015 #4
    I'm not sure about the n (U.n) term, it stays just U/r2?

    Thanks,
    Mark
     
  6. Apr 20, 2015 #5
    OK. Express these equations in spherical coordinates, of course without the longitudinal dependence because of symmetry.

    Chet
     
  7. Apr 20, 2015 #6
    To prove the divergence equal to zero, only the term in r exists, so we have
    stokes3.png
    I solved that but it's not 0, I guess the problem is in the n (U.n) term...
     
  8. Apr 20, 2015 #7
    This is not correct for a couple of reasons. First of all, you have written the equation for the divergence in cylindrical coordinates, rather than spherical coordinates. Secondly, you have omitted the derivative with respect to the latitudinal coordinate.

    Also, please write down for me the spherical components of the fluid velocity vector in the radial direction and latitudinal direction.

    Chet
     
  9. Apr 20, 2015 #8
    Yes, actually when I solved it I used the equation for spherical and not that one, I put it wrong. (it's with both r squared).
    But I only used the radial direction in the fluid velocity... how do we have a latitudinal component if n and U only have terms in r?
     
  10. Apr 20, 2015 #9
    .
     
    Last edited: Apr 20, 2015
  11. Apr 20, 2015 #10
    The components of U they gave are for cartesian coordinates. I guess they should have mentioned that. You need to convert this to spherical coordinates.

    Chet
     
  12. Apr 20, 2015 #11
    So I rewrite U = (Ucosθ, Usinθ, 0), my vr will be something like -3/4*R*Ucosθ(1/r+1/r3) - 1/4R3*Ucosθ(1/r3-3/r5)+Ucosθ ?
     
  13. Apr 20, 2015 #12
    No. It looks like you have several algebra errors in there. Try again, or show more details please.

    Chet
     
  14. Apr 20, 2015 #13
    Hum... how would stay n(U.n)? cos(θ)U/r? probably not...
     
  15. Apr 20, 2015 #14
    No. n is the unit vector in the radial direction. So n(U.n)=Ucosθn.

    Chet
     
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