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Research Papers: Fundamental Issues and Canonical Flows

Plane Thermal Transpiration of a Rarefied Gas Between Two Walls of Maxwell-Type Boundaries With Different Accommodation Coefficients

[+] Author and Article Information
Toshiyuki Doi

Assistant Professor
Department of Applied Mathematics and Physics,
Graduate School of Engineering,
Tottori University,
Tottori 680-8552, Japan
e-mail: doi@damp.tottori-u.ac.jp

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received September 30, 2013; final manuscript received January 30, 2014; published online May 19, 2014. Assoc. Editor: Ali Beskok.

J. Fluids Eng 136(8), 081203 (May 19, 2014) (9 pages) Paper No: FE-13-1582; doi: 10.1115/1.4026926 History: Received September 30, 2013; Revised January 30, 2014

Plane thermal transpiration of a rarefied gas between two walls of Maxwell-type boundaries with different accommodation coefficients is studied based on the linearized Boltzmann equation for a hard-sphere molecular gas. The Boltzmann equation is solved numerically using a finite difference method, in which the collision integral is evaluated by the numerical kernel method. The detailed numerical data, including the mass and heat flow rates of the gas, are provided over a wide range of the Knudsen number and the entire range of the accommodation coefficients. Unlike in the plane Poiseuille flow, the dependence of the mass flow rate on the accommodation coefficients shows different characteristics depending on the Knudsen number. When the Knudsen number is relatively large, the mass flow rate of the gas increases monotonically with the decrease in either of the accommodation coefficients like in Poiseuille flow. When the Knudsen number is small, in contrast, the mass flow rate does not vary monotonically but exhibits a minimum with the decrease in either of the accommodation coefficients. The mechanism of this phenomenon is discussed based on the flow field of the gas.

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Figures

Grahic Jump Location
Fig. 1

Velocity profile uT(x2) (Eq. (15)) for α2=1 and α1=0,0.1,0.5, and 1. (a) Kn =0.1, (b) Kn =1, and (c) Kn =10

Grahic Jump Location
Fig. 2

Profile QT(x2) (Eq. (18)) of the heat flow for α2=1 and α1=0,0.1,0.5, and 1. (a) Kn =0.1, (b) Kn =1, and (c) Kn =10

Grahic Jump Location
Fig. 3

Mass flow rate coefficient mT (Eq. (21)) as a function of α1 and α2. (a) Kn =0.1, (b) Kn =1, and (c) Kn =10. The case α1=α2=0 represents the limiting solution for α1=α2=+0 (Eq. (31)). Note the relations in Eqs. (25) and (26).

Grahic Jump Location
Fig. 4

Mass flow rate coefficient mT (Eq. (21)) as a function of α1 for α2=0,0.3, and 1 (Kn =0.1). Open circle (○): numerical solution, cross (×): limiting solution for α1=α2=+0 (Eq. (31)), and the solid line (——): asymptotic solution for small Knudsen numbers (Eq. (37)).

Grahic Jump Location
Fig. 5

Mass flow rate coefficient mT (Eq. (21)) as a function of Kn for (α1,α2)=(0.1,1), and (0.5,1). Open symbols (○,Δ): present results and closed symbols (•, ▲): Ref. [26]; circles (○,•): (α1,α2)=(0.5,1) and triangles (Δ, ▲): (α1,α2)=(0.1,1). The solid line (——): asymptotic solution for small Knudsen numbers (Eq. (37)). Note the difference between the ordinates.

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