Air at flows steadily through a long plane microchannel of half-width

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Thermal, Statistical & Complex Systems · Kinetic theory and transport in gases

Fluid Dynamics, Ch The stress tensor completed: constitutive law and fluid properties

Problem

Air at

20^{\circ}\mathrm{C}

flows steadily through a long plane microchannel of half-width

h=2.0\,\mu\mathrm{m}

, driven by a pressure gradient. The mean absolute pressure in the channel is

0.50\,\mathrm{atm}

, and the pressure drop along it is small compared with this. At

20^{\circ}\mathrm{C}

and

1\,\mathrm{atm}

the mean free path of air is

68\,\mathrm{nm}

. Take the slip length equal to the mean free path. ((a)) Find the mean free path at

0.50\,\mathrm{atm}

and

20^{\circ}\mathrm{C}

, and the Knudsen number

Kn=\ell_{\rm mfp}/h

. Which regime is this? ((b)) By what factor does slip raise the flow rate above the no-slip prediction for the same pressure gradient? ((c)) At this pressure, what is the smallest half-width for which the slip correction to the flow rate stays below

5\%

? ((d)) Keeping

h=2.0\,\mu\mathrm{m}

, below what absolute pressure does

Kn

pass

10^{-1}

, where the slip condition alone stops being enough?

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