How can I calculate the frictional force on a moving sphere in an ideal gas?

[gdumont]

Ok, I need to find the frictional force on sphere of radius $a$ and mass $M$ moving with velocity $v$ in an ideal gas at temperature $T$.

If I put myself in the sphere frame, then diffrential cross-section is
$$\frac{d\sigma}{d\Omega} = \frac{a^2}{4}$$
and the total cross-section is $\sigma_{\textrm{tot}}=\pi a^2$. How do I find the frictional force from this? Ellastic collisions between the sphere and the gas particules are assumed.

Any help greatly appreciated.

 

[gdumont]

Ok, here's what I tought:

If the gas has density $\rho$ than the number of molecules in a volume $\sigma_{\textrm{tot}}dx$ is $dN=\pi \rho a^2 dx$. If collisions are ellastic, then
$$\textbf{p}_s + \textbf{p}_i = \textbf{p}_s' + \textbf{p}_i'$$
where the $s$ and the $i$ denote respectively the momentum of the sphere and the $i$th molecule. The prime denotes the momentum after collision. (I assumed that molecules do not collide simultaneously.)

The change in speed of the sphere is
$$dv = \frac{|\textbf{p}_s' - \textbf{p}_s|}{M}$$
From accelaration $dv/dx$ if the $x$ direction is chosen along the movement of the sphere we can find the resistance force
$$F = M\frac{dv}{dx}$$
Now I need to evaluate either $\textbf{p}_s' - \textbf{p}_s$ or $\textbf{p}_i - \textbf{p}_i'$.

Anyone can help?