Consider a dilute gas whose molecules of mass $m$ have mean velocity of magnitude $\bar{v}$. Suppose that the average velocity in the $x$-direction $u_x$ increases monotonically with $z$, so that $u_x=u_x(z)$ with $\left|u_x\right| \ll \bar{v}$ and all gradients small. There are $n$ molecules per unit volume and their mean free path is $l$ where $l \gg d$ (molecular diameter) and $l \ll L$ (linear dimension of enclosing vessel).
(a) The viscosity $\eta$ is defined as the proportionality constant between the velocity gradient and the stress in the $x$-direction on an imaginary plane whose normal points in the $z$-direction. Find an approximate expression for $\eta$ in terms of the parameters given.
(b) If the scattering of molecules is treated like that of hard spheres, what is the temperature dependence of $\eta$ ? The pressure dependence? Assume a Maxwellian distribution in both cases.
(c) If the molecular scattering cross section $\sigma \propto E_{\mathrm{cm}}^2$, where $E_{\mathrm{cm}}$ is the center-of-mass energy of two colliding particles, what is the temperature dependence of $\eta$ ? Again assume a Maxwellian distribution.
(d) Estimate $\eta$ for air at atmospheric pressure $\left(10 \mathrm{dyn} / \mathrm{cm}^2\right)$ and room temperature. State clearly your assumptions.