Possible Solution for a Moving Cylinder of Charge Injected into a Fluid

[hunt_mat]

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:
$$-\nabla^{2}\varphi +a\frac{\partial^{2}\varphi}{\partial t^{2}}+b\frac{\partial\varphi}{\partial t}=0$$
I want $\varphi =\varphi_{0}$, a constant on the moving boundary $x=v_{0}t$
Can anyone suggest some possible solutions?

 

[Hootenanny]

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 $\eta = x - v_0 t$, then your boundary value problem reduces to

$$-\nabla^{2}\varphi(\eta) + v_0\{av_0 - b\}\varphi(\eta) =0\;,$$
$$\varphi(0) = \varphi_0\;.$$

 

[hunt_mat]

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.

 

[Hootenanny]

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.

I'm not sure I follow. In 1D (or a symmetric case in $\mathbb{R}^3$ 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.

 

[hunt_mat]

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 $\varphi$ outside of the cylinder.