Chapter 8, Problem 45P

The classic Poiseuille flow (Eq. 8.12), 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, slip will occur at the walls, and the flow rate and velocity will be increased for a given pressure gradient. In Eq. 8.11, c₁ will still be zero, but c₂ must satisfy the slip condition u = l ∂u/∂r at r = R. Derive the velocity profile and flow rate of gas flow in a micro- or nanotube which has such a slip velocity on the wall. Calculate the flow rate when R = 10 μm, μ = 1.84 × 10⁻⁵ N · s/m², the mean free path l = 68 nm, and −∂p/∂x = 1.0×10⁶ Pa/m.