The classic Poiseuille flow is for no-slip conditions at the walls. If the fluid is a gas, and when the mean free path, l (the average distance a molecule travels before collision with another molecule), is comparable to the length-scale L of the flow (micro- or nano scale flows), slip will occur at the walls, and the flow rate and velocity will be increased for a given pressure gradient. In the equation for flow inside a cylinder: u= r^2/4μ (∂p/∂x)+ c1/μ lnr+ C2

c1 will still be zero, but c2 must satisfy the slip boundary condition u = l∂u/∂r at r =R.

Required:
a. Derive the velocity and flow rate of gas flow in such a micro- or nano tube which has such a slip velocity at the wall.
b. Calculate the flowrate when R=10μm, μ=1.84 X 10^-5 Ns/m^2, the mean free path 68 nm, and opla1.0 X 10^6 Pa/m.