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Navier Stokes Equation - Flow of waves

  1. Jan 8, 2015 #1
    1. The problem statement, all variables and given/known data

    (a) Show that for an incompressible flow the velocity potential satisfies ##\nabla^2 \phi = 0##. Show further the relation for the potential to be ## \frac{\partial \phi}{\partial t} + \frac{\nabla \phi \dot \nabla \phi}{2} + \frac{p}{\rho} + gz = const.##
    (b)Write out the boundary conditions on velocity and show ##\phi$## is of the form: ## \phi = -\frac{\omega n_0}{k} cos(kx-\omega t) \frac{cosh kz}{sinh kH}## subject to ## \omega^2 = gk \space tanh \space (kH)##
    (c) How does the frequency of the waves vary after the boat is gone?

    29nyz6d.png

    2. Relevant equations


    3. The attempt at a solution

    Part (a)


    For an irrotational flow, ## \nabla \cdot \vec u = 0## so this means that velocity can be written as gradient of some potential ## u = \nabla \phi##.

    Therefore ## \nabla \cdot \vec u = \nabla^2 \phi = 0##.

    Navier stokes equation is given by:
    [tex] \frac{\partial \vec u}{\partial t} + (\vec u \cdot \nabla) \vec u + \frac{1}{\rho} \nabla p + g \vec k = \nu \nabla^2 \vec u [/tex]

    Using the identity and ## \nabla^2 \vec u = 0 ##, ## \nabla \times \vec u \times \vec u = 0## I get the answer.

    Part (b)

    The boundary condition is velocity at ##z=0## is zero (non-slip).

    I don't think this question wants me to simply substitute the given form of ##\phi## to verify it is right. I am expected to derive it.

    Substituting, I get
    [tex] Z \frac{\partial X}{\partial t} + \frac{ (Z\nabla X + X \nabla Z) \cdot (Z\nabla X + X \nabla Z) }{2} + \frac{p}{\rho} + gz = const. [/tex]

    This leads to nowhere. Any idea how to proceed?
     
  2. jcsd
  3. Jan 9, 2015 #2
  4. Jan 13, 2015 #3
    Any help for part (b)??
     
  5. Jan 16, 2015 #4
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