A flow of helium is generated in a circular microtube by axial pressure and temperature gradients. The microtube has a length l 5 5 mm and a uniform cross-section with a diameter 2r2 5 10 μm. In a section where the flow may be considered as locally fully developed, the gradients are dp/dz 5 25 Pa/μm and dT/dz 5 0.5 K/μm, for a pressure p 5 100 kPa and a temperature T 5 400 K.

Assume a tangential momentum accommodation coefficient σ 5 0.9.

i. Write the velocity distribution as a function of the Knudsen number in a dimensionless form: uz (r, Kn).

ii. Calculate the part of the velocity increase due to slip at the wall.

iii. Calculate the part of the velocity increase due to thermal creep.