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Moving boundary problems

  1. Jun 17, 2011 #1

    hunt_mat

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    Does anyone know any good references for a moving boundary problem? I am looking at a cylinder of charge being injected into a fluid, the PDE is:
    [tex]
    -\nabla^{2}\varphi +a\frac{\partial^{2}\varphi}{\partial t^{2}}+b\frac{\partial\varphi}{\partial t}=0
    [/tex]
    I want [itex]\varphi =\varphi_{0}[/itex], a constant on the moving boundary [itex]x=v_{0}t[/itex]
    Can anyone suggest some possible solutions?
     
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  3. Jun 17, 2011 #2

    Hootenanny

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    Let me start by saying that I could be way off here as I haven't worked with moving boundaries before. However, why don't you shift to moving co-ordinates such that [itex]\eta = x - v_0 t[/itex], then your boundary value problem reduces to

    [tex]-\nabla^{2}\varphi(\eta) + v_0\{av_0 - b\}\varphi(\eta) =0\;,[/tex]
    [tex]\varphi(0) = \varphi_0\;.[/tex]
     
  4. Jun 17, 2011 #3

    hunt_mat

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    I tried this method before and I didn't get really far with it. You are still left with the problem of the origin moving away from you and that doesn't really help you much.
     
  5. Jun 17, 2011 #4

    Hootenanny

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    I'm not sure I follow. In 1D (or a symmetric case in [itex]\mathbb{R}^3[/itex] which reduces to 1D), you will be left with a family hyperbolic or trigonometric functions, depending on the sign. The remaining constant can be determined by the initial distribution of the field - I assumed that this is given.
     
    Last edited: Jun 17, 2011
  6. Jun 17, 2011 #5

    hunt_mat

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    Initially, I am interested in a calculating the electric and magnetic fields from a cylinder of charge moving at a speed v_0 from a plane at x=0. The 1D case has been solved and some very nice solutions have been obtained and now my colleague and I are interested in the 2D case. We reduced the problem down to a damped wave equation which I thought was rather nice.

    I am interested in the solution of [itex]\varphi[/itex] outside of the cylinder.
     
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