Bubble of air in resistor follows $ r^{3}$ law? I used a program to simulate the resistance of a resistor when a small bubble (compared to size of the resistor) of air of radius $r$ was placed inside the resistor, as $r$ increased.
I plotted the resistance of the resistor versus radius $r$ and it seems to follow perfectly a 3rd degree polynomial. I was wondering if this law has been proven or whether it is easy to derive or whether it is wrong?
 A: A small bubble will reduce the cross sectional area of the resistor for a short distance. The "effective area" will be $A_0 - \pi r^2$, and this reduced area will be effective for a length "that scales with $r$".

The result is a change in resistance that, for small $r$, will look like (thanks Sammy Gerbil for pointing out an error in an earlier version... )
$$\Delta R \propto \frac{r}{A_0 - \pi r^2}$$
When $\pi r^2 << A_0$, we can rewrite this as
$$\begin{align}\Delta R &\propto \frac{r}{A_0}\frac{1}{1 - \frac{\pi r^2}{A_0}}\\
&\approx \frac{r}{A_0}\left(1 + \frac{\pi r^2}{A_0}\right)\
\end{align}$$
So it seems to me there would be both a linear, and a cubic, term in the change in resistance. Is that the kind of polynomial you are seeing?
A: This may be relevant and should be a comment but too long (also do not know how to include a figure in a comment): If you look at a sphere of radius $r$ and resistivity $\rho$ determine the resistance between the points A and B ( see the figure) as follows:
The resistance of the annulus of thickness $dl$ is $dR = \frac{\rho dl}{\pi L^2}$, where $L^2 + (l-r)^2 = r^2$. This gives $$dR = \frac{\rho}{\pi}\frac{dl}{r^2 - (l-r)^2} = \frac{\rho}{\pi}\frac{dl}{2lr - l^2}$$ so I was expecting the total resistance would be given as the sum of the separate resistances of the shells $R =  \frac{\rho}{\pi} \int^{2r}_0\frac{dl}{2lr - l^2} = \frac{\rho}{\pi r} \int^2_0 \frac{dx}{2x - x^2}$. Unfortunately, this appears to diverge, not sure why. 
A: If you mutliply all lengths of a resistor by $L$, the resistance goes as $\frac{R_0}{L}$. So consider a cubical resistor centered on the bubble which scales, in parallel with a O shaped resistor with area $A_0-L^2$. This you can use to find the asymptotics.
