Viscosity of Argon at Temperatures > 2,000 K from Measured Shock Thickness

Abstract-
Mott-Smith's approximate theory of plane 1D shock structure (Phys. Rev., 82, 885-92, 1951; Phys. Rev., 5, 1325-36, 1962) suggests, for any intermolecular potential, the average number of collisions undergone by a molecule as it cross the shock quickly approaches a limit as the Mach number increases. We check this with DSMC calculations and show that it can be used to estimate the gas viscosity at high temperatures from measurements of shock thickness.

We consider a monatomic gas (g = 5/3) for five different collision models and hence five different viscosity laws m = m(T). The collision models are: the variable hard sphere, s µ 1/g^2u, with three values of u; the generalized hard sphere; and the Maitland-Smith potential. For shock Mach numbers M₁ ≥ 4.48, all these collision models predict a shock thickness D = 11.0l_s, where l_s is a suitably defined `shock length scale', with a scatter » 2.5%. This shock length depends on the upstream flow speed, downstream density and a collision cross-section derived from the viscosity of the gas at a temperature T_g, characteristic of the collisions at relative speed g = u₁ - u₂ between upstream and downstream molecules.

Using D = 11l_s and the experimental measurements of shock thickness in argon given by Alsmeyer (J. Fluid Mech. 74, 498-513, 1976), we estimate the viscosity of argon at high values of T_g. These estimated values agree with the viscosity of argon recommended by the CRC Handbook of Chemistry and Physics (2001) at T » 1,500 K. For T ≥ 2,000 K, for which there appears to be no reliable direct measurements of viscosity, our estimated values lie between the extrapolated values recommended by the CRC Handbook and those predicted by the simple power law m = m_ref (T/T_ref)^0.72, with T_ref = 300 K and m_ref = 2.283×10^-5 Pa.s. Taking the error in the experimental measurements of D as the scatter in the results of Alsmeyer (± 2%), we estimate the uncertainty in the viscosity deduced from the shock thickness measurements as less than ± 5%. To this accuracy, our results agree with the power law predictions and disagree with the CRC Handbook values, for T ≥ 3,000 K.

Reference
M. N. Macrossan and C. R. Lilley, Physics of Fluids v 15 (11), Nov 2003

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