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Laplace equation in rectengualar channel (Fluid mechanics)

  1. Jun 5, 2013 #1
    1. The problem statement, all variables and given/known data
    Estimate the speed a potential flow in gravity field would have in direction [itex] y [/itex] in rectangle channel with depth [itex] h [/iteh] and length [itex] l [/itex]. The fluid is incompressible and on the surface [itex] z = 0 [/itex] we have boundary condition [itex] \dfrac{\partial^2 \phi}{t^2} + g\dfrac{\partial \phi}{\partial z} = 0 [/itex]

    2. Relevant equations
    [itex] \nabla^2 \phi = 0, \vec v = \nabla \phi(x,y,z,t), [/itex]
    The free surface is described by [itex] \zeta(x,y,t) = \dfrac{1}{g} \dfrac{\partial \phi}{\partial t}[/itex] with the
    ansatz [itex] \phi = Z(z) e^{-\omega t}e^{i(k_1x + k_2y)} [/itex] which is for channel with depth [itex] h [/itex] and infinity length.

    3. The attempt at a solution

    We put the ansatz in Laplace and obtain [itex] Z^{\prime \prime} - \underbrace{(k_1^2 + k_2^2)}_{k}Z = 0[/itex]

    The solution is with boundary condition [itex] \nabla \phi (-h) = 0 \Rightarrow \dfrac{d }{d z } Z(-h) = 0 [/itex]

    We obtain a solution [itex] \phi( \vec r, t ) = \dfrac{ig}{\omega}A \dfrac{ch(k(z+h))}{ch kh} e^{i(k_1x + k_2y - \omega t)} [/itex]

    My problem is how to express the boundary limits on the walls at [itex] (0,-h) \cup (l,-h) [/itex] expanding in the y direction.
  2. jcsd
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