Applying boundary conditions on an almost spherical body

[hunt_mat]

I am solving the Laplace equation in 3D:
$$\nabla^{2}V=0$$
I am considering azumuthal symmetry, so using the usual co-ordinates $V=V(r,\theta)$. Now suppose I have two boundary conditions for $[V$, which are:
$$V(R(t)+\varepsilon f(t,\theta),\theta)=1,\quad V\rightarrow 0\quad\textrm{as}\quad r\rightarrow\infty$$
where $\varepsilon\ll1$. The boundary condition does lead itself to separation of variables does it? Would a more general Green function approach be more suitable?

 

[jasonRF]

Since $t$ is a parameter I will ignore it. One classic way to try and find an approximate solution is to use perturbation theory. Your boundary condition can be Taylor expanded, $$
1 = V(R + \epsilon \, f(\theta),\theta) = V(R, \theta) + \epsilon \, f(\theta) V_r(R, \theta) + \frac{1}{2}\left( \epsilon \, f(\theta) \right)^2 V_{rr}(R, \theta) + \ldots $$
where $V_r = \partial V / \partial r$, etc, and look for a solution of the complete boundary value problem of the form,
$$ V(r,\theta) = V^{(0)}(r,\theta) + \epsilon V^{(1)}(r,\theta) + \epsilon^2 V^{(2)}(r,\theta) + \ldots $$
Here $V^{(k)}(r,\theta)$ is simply a function, and the number $k$ tells us what power of $\epsilon$ is in front of it. We plug the expansion for $V(r,\theta)$ into Laplace's equation and collect terms with the same power of $\epsilon$ to find, that for all $k$ we must have $\nabla^2 V^{(k)}(r,\theta)=0$. Likewise, we plug the expansion for $V(r,\theta)$ into the Taylor expanded boundary condition and collect terms with the same power of $\epsilon$. This leads to a series of boundary value problems. The first few boundary conditions are
$$
\begin{eqnarray}
V^{(0)}(R,\theta) & = & 1 \\
V^{(1)}(R,\theta) & = & -f(\theta) V^{(0)}_r(R,\theta)\\
V^{(2)}(R,\theta) & = & -f(\theta) V^{(1)}_r(R,\theta) - \frac{1}{2}f^2(\theta) V^{(0)}_r(R,\theta)\\
\vdots
\end{eqnarray}
$$

That is, you first solve $\nabla^2 V^{(0)} = 0$ with $V^{(0)}(R,\theta) = 1$, then solve $\nabla^2 V^{(1)} = 0$ with $V^{(1)}(R,\theta) = - f(\theta) V^{(0)}_r(R,\theta)$, etc.

You might be able to use separation of variables for each of these. If $\epsilon$ is small enough you probably just need to solve the first few problems, then add up the solutions accordingly.

Jason

 

[jasonRF]

Forgot to address this above.

The boundary condition does lead itself to separation of variables does it?

Unless you are able to define a curvilinear coordinate system such that your boundary is a surface of constant coordinate (such as a sphere in spherical coordinates, ellipse in ellipsoidal coordinates, etc.), then the problem is not separable.