Slow Viscous Flow of Rarefied Gases through Short Tubes

The semiempirical equation which was shown by Weissberg to be a very close approximation to the rigorous variational calculation of the pressure drop for creeping flow of incompressible viscous fluids through short tubes has been generalized to take into account the slip of a rarefied gas at the tube wall. The generalized equation is $ω=ωs+βa/λ̄$⁠, where a is the tube radius, $λ̄=(μ/P̄)(12πRT/M)12$⁠, and $ω=F/[14πa2ΔP(8RT/πM)12]$⁠. Here, μ and M are the viscosity and molecular weight of the gas, ΔP is the pressure drop P₁−P₂, P̄ is the average pressure ½(P₁+P₂), R is the gas constant per mole, T is the absolute temperature, and F=P̄Q where Q is the volume flow. The coefficient β and the slip parameter w_s depend only on L/a, the length to radius ratio of the tube: $β=(π/8)/[(L/a)+(3π/8)], ωs=(β2/3π)[(128L/a)+(27π2/4)]$⁠. Excellent agreement is noted between values of w calculated from the equation given here and values obtained by Knudsen for long tubes and by Lund for short tubes.