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
<br /> \frac{d\sigma}{d\Omega} = \frac{a^2}{4}<br />
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
<br /> \textbf{p}_s + \textbf{p}_i = \textbf{p}_s&#039; + \textbf{p}_i&#039;<br />
where the s and the i denote respectively the momentum of the sphere and the ith molecule. The prime denotes the momentum after collision. (I assumed that molecules do not collide simultaneously.)

The change in speed of the sphere is
<br /> dv = \frac{|\textbf{p}_s&#039; - \textbf{p}_s|}{M}<br />
From accelaration dv/dx if the x direction is chosen along the movement of the sphere we can find the resistance force
<br /> F = M\frac{dv}{dx}<br />
Now I need to evaluate either \textbf{p}_s&#039; - \textbf{p}_s or \textbf{p}_i - \textbf{p}_i&#039;.

Anyone can help?