# Unsteady-state slip of a gas near an infinite plane with diffusion-mirror reflection of molecules

- 25 Downloads
- 1 Citations

## Abstract

On the basis of a “two-point” approximation for the distribution function, an approximate analytical solution is obtained to the Rayleigh problem describing (for an arbitrary moment of time) the unsteady-state slip of a rarefied gas near a surface with diffusion-mirror reflection of molecules. In the solution of the problem it was postulated that the characteristic value of the macroscopic velocity of the gas is small in comparison with the speed of sound. An approximate analogy is established with the propagation of free vibrations in an electrical line of infinite length. An investigation is made of the limiting transition σ→ 0 (σ is the fraction of mirror-reflected particles) in an exact solution of the Boltzmann problem. It is shown that, with σ ≪ 1, the rate of slip for any given moment of time is determined from the hydrodynamic equations of motion. An investigation is made of the nonsingularity of the limiting transitions (t→∞, σ→ 0; σ→ 0, t→ ∞) with determination of the rate of thermal slip.

## Keywords

Reflection Mathematical Modeling Distribution Function Mechanical Engineer Exact Solution## Preview

Unable to display preview. Download preview PDF.

## Literature cited

- 1.G. G. Stokes, “On the effect of internal friction of fluids on the motion of pendulums,” Trans. Cambridge Phil. Soc.,9, No. 8 (1851). Mathematical and Physics Papers, Cambridge (1901), pp. 1–141.Google Scholar
- 2.Rayleigh, “On the motion of solid bodies through viscous liquids,” Phil. Mag.,21, 697–711 (1911).Google Scholar
- 3.H. T. Yang and L. Lees, Rayleigh's problem at low Mach number according to kinetic theory of gases,” J. Math. Phys.,35, No. 3, 195–235 (1956).Google Scholar
- 4.E. P. Gross and E. A. Jackson, “Kinetic theory of the impulsive motion of an infinite plane,” Phys. Fluids,1, No. 4, 318–328 (1958).Google Scholar
- 5.Yu. A. Koshmarov, “Flow of a rarefied gas around a wall suddenly set into motion,” Inzh. Zh.,3, No. 3, 433–441 (1963); V. P. Shidlovskii, Introduction to the Theory of a Rarefied Gas [in Russian], Izd. Nauka, Moscow (1965).Google Scholar
- 6.H. T. Yang and L. Lees, “The Rayleigh problem for low Reynolds numbers according to the kinetic theory of gases,” in: Gasdynamics of Rarefied Gases [Russian translation], Izd. Inostr. Lit., Moscow (1963), pp. 325–375.Google Scholar
- 7.L. Trilling, “Asymptotic solution of the Boltzmann-Krook equation for the Rayleigh shear flow problem,” Phys. Fluids,7, No. 10, 1681–1691 (1964).Google Scholar
- 8.C. Cercignani and F. Sernagiotto, “Rayleigh's problem at low Mach numbers according to kinetic theory,” in: Rarefied Gas Dynamics, Vol. 1, Academic Press, New York (1965), pp. 332–353.Google Scholar
- 9.M. Epstein, “Linearized Rayleigh's problem with incomplete surface accomodation,” in: Rarefied Gas Dynamics, Academic Press, Vol. 1, New York (1969), pp. 255–265.Google Scholar
- 10.C. Cercignani, Mathematical Methods in Kinetic Theory, Plenum (1969).Google Scholar
- 11.A. V. Lykov, The Theory of Thermal Conductivity [in Russian], Izd. Vysshaya Shkola, Moscow (1967).Google Scholar
- 12.M. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Series, and Products, Academic Press (1966).Google Scholar
- 13.E. P. Gross and E. A. Jackson, “Kinetic models and the linearized Boltzmann equation,” in: Mekhanika [Periodic collection of translations of foreign articles], No. 5 (1960), pp. 65–81.Google Scholar
- 14.M. A. Lavrent'ev and B. V. Shabat, Methods in the Theory of Functions of a Complex Variable [in Russian], Izd. Nauka, Moscow (1973).Google Scholar
- 15.V. A. Ditkin and P. I. Kuznetsov, Handbook on Operational Computation [in Russian], Izd. GITTL, Moscow-Leningrad (1951).Google Scholar
- 16.S. K. Loyalka and J. W. Cipolla, “Thermal creep slip with arbitrary accomodation at the surface,” Phys. Fluids,14, No. 8, 1656–1661 (1971).Google Scholar
- 17.Yu. I. Yalamov, I. N. Ivchenko, and B. V. Deryagin, Gasdynamic calculation of the rate of thermal slip of a gas near a solid surface,” Dokl. Akad. Nauk SSSR,177, No. 1, 74–76 (1967).Google Scholar
- 18.Yu. Yu. Abramov and G. G. Gladush, “Flow of a rarefied gas near an inhomogeneous heated surface,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 2, 20–29 (1970).Google Scholar
- 19.S. K. Loyalka, Approximate method in the kinetic theory,” Phys. Fluids,14, No. 11, 2291–2294 (1971).Google Scholar