Thermal transpiration for the linearized Boltzmann equation.(English)Zbl 1107.35004

Summary: The phenomena of thermal transpiration due to the boundary temperature gradient is studied on the level of the linearized Boltzmann equation for the hard-sphere model. We construct such a flow for a highly rarefied gas between two plates and also in a circular pipe. It is shown that the flow velocity parallel to the plates is proportional to the boundary temperature gradient. For a highly rarefied gas, that is, for a sufficiently large Knudsen number $$\kappa$$, and the flow velocity between two plates is of the order of log $$\kappa$$, and the flow velocity in a pipe is of finite order. Our analysis is based on certain pointwise estimates of the solutions of the linearized Boltzmann equation.

MSC:

 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35Q35 PDEs in connection with fluid mechanics 76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
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References:

 [1] ; Handbook of mathematical functions with formulas, graph, and mathematical tables, 1001–1003. Dover, New York, 1972. [2] Bardos, Comm Pure Appl Math 39 pp 323– (1986) [3] ; ; The mathematical theory of dilute gases. Applied Mathematical Sciences, 106. Springer, New York, 1994. · Zbl 0813.76001 [4] Asymptotic theory of the Boltzmann equation. II. Rarefied Gas Dynamics (Proc. 3rd Internat. Sympos., Palais de l’UNESCO, Paris, 1962), Vol. I, 26–59. Academic Press, New York, 1963. [5] Kinetic theory of gases, 327–337. McGraw-Hill, New York, 1938. [6] Knudsen, Ann Phy (Leipzig) 31 pp 205– (1910) [7] Kinetic theory of gases. secs. 83, 84. Dover, New York, 1961. [8] Maxwell, Philos Trans R Soc 170 pp 231– (1879) [9] Niimi, J Phys Soc Jpn 30 pp 572– (1971) [10] Ohwada, Phys Fluids A 1 pp 2042– (1989) [11] Sone, J Phys Soc Jpn 21 pp 1836– (1966) [12] Kinetic theory and fluid dynamics. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston, Boston, 2002. [13] Sone, Phys Fluids 11 pp 1672– (1968) [14] Phys Fluids 13 pp 1651– (1970)
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