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Research Papers: Flows in Complex Systems

Poiseuille Flow and Thermal Transpiration of a Rarefied Gas Between Parallel Plates: Effect of Nonuniform Surface Properties of the Plates in the Transverse Direction

[+] 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 October 17, 2014; final manuscript received April 24, 2015; published online June 15, 2015. Assoc. Editor: Prashanta Dutta.

J. Fluids Eng 137(10), 101103 (Oct 01, 2015) (9 pages) Paper No: FE-14-1601; doi: 10.1115/1.4030490 History: Received October 17, 2014; Revised April 24, 2015; Online June 15, 2015

Poiseuille flow and thermal transpiration of a rarefied gas between parallel plates with nonuniform surface properties in the transverse direction are studied based on kinetic theory. We considered a simplified model in which one wall is a diffuse reflection boundary and the other wall is a Maxwell-type boundary on which the accommodation coefficient varies periodically and smoothly in the transverse direction. The spatially two-dimensional (2D) problem in the cross section is studied numerically based on the linearized Bhatnagar–Gross–Krook–Welander (BGKW) model of the Boltzmann equation. The flow behavior, i.e., the macroscopic flow velocity and the mass flow rate of the gas as well as the velocity distribution function, is studied over a wide range of the mean free path of the gas and the parameters of the distribution of the accommodation coefficient. The mass flow rate of the gas is approximated by a simple formula consisting of the data of the spatially one-dimensional (1D) problems. When the mean free path is large, the distribution function assumes a wavy variation in the molecular velocity space due to the effect of a nonuniform surface property of the plate.

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Figures

Grahic Jump Location
Fig. 1

Schematic view of the channel. (a) Configuration of the plates. On the bottom plate, the gray part has a different surface property from that of the white part. (b) View from the X3 direction and the distribution αb(X1) of the accommodation coefficient of the plate at X2 = 0. The numerical analysis is conducted in the shaded rectangular domain D.

Grahic Jump Location
Fig. 2

Solution of a highly rarefied gas. (a) Schematic of the solution of a highly rarefied gas. The surface areas denoted by α1 and α2 are called the α1 part and α2 part, respectively. The shaded region in the gas represents the paths of the molecules arriving at B from α2 part. (b) Macroscopic flow velocity, uP in Eq. (18), of the nearly free molecular solution (25) and (26) of Poiseuille flow (Kn = 100, α1 = 0.5, α2 = 1, A = 1, C = 0.25, and b/L = 0.1). Thick line in 0.5 < x2 < 1: Eq. (27).

Grahic Jump Location
Fig. 3

Distribution of the macroscopic flow velocity, uP in Eq. (18), of Poiseuille flow in the cross section (α1 = 0.5, α2 = 1, b/L = 0.1, and C = 0.5). (a) and (b) Kn = 0.1 and (c) and (d) Kn = 10; (a) and (c) A = 0.2 and (b) and (d) A = 5. 3D plot: present result, dashed line (- - -): 1D solution with the uniform accommodation coefficient α1 (=0.5) on the plate x2 = 0, dashed–dotted line (– - –): 1D solution with the uniform accommodation coefficient α2 (=1) on the plate x2 = 0, and thick line: transition interval CA − 0.1 < x1 < CA + 0.1. Figures 3(b) and 3(d) are shrunk in the x1 direction by the factor of 2. Not all of the computational grid points are used to draw these figures.

Grahic Jump Location
Fig. 4

Distribution of the macroscopic flow velocity, uT in Eq. (18), of thermal transpiration in the cross section (α1 = 0.5, α2 = 1, b/L = 0.1, and C = 0.5). See the caption of Fig. 3. In Figs. 4(a) and 4(b), the dashed line on x1 = 0 is hidden by the 3D plot of the present result. Figures 4(b) and 4(d) are shrunk in the x1 direction by the factor of two.

Grahic Jump Location
Fig. 5

Mass flow rate coefficient mP (Eq. (20)) of Poiseuille flow as a function of C I: α1 = 0.5, α2 = 1, and b/L = 0.1. △: A = 0.2, ▽: A = 0.5, ○: A = 1, ◇: A = 2, and □: A = 5. Dashed line (- - -): Eq. (31). For every C, the markers represent the numerical results for Kn = 0.1, 10, 0.3, 3, and 1 from the top, respectively. Closed circle (•): numerical solution of the linearized Boltzmann equation for a hard-sphere molecular gas, the computation of which is conducted for the Knudsen number Kn/1.270042 so as to make the BGKW and hard-sphere gases acquire the same viscosity [12].

Grahic Jump Location
Fig. 6

Mass flow rate coefficient mT (Eq. (20)) of thermal transpiration as a function of C (α1 = 0.5, α2 = 1, and b/L = 0.1). Dashed line (- - -): Eq. (32). For every C, the markers represent the numerical results for Kn = 10, 3, 1, 0.3, and 0.1 from the top, respectively (see Fig. 5 caption).

Grahic Jump Location
Fig. 7

Mass flow rate coefficient mP (Eq. (20)) of Poiseuille flow as a function of C II: α1 = 0.25, α2 = 0.5, and b/L = 0.1. Dashed line (- - -): Eq. (31). For every C, the markers represent the numerical results for Kn = 0.1, 10, and 1 from the top, respectively (see Fig. 5 caption).

Grahic Jump Location
Fig. 8

Perturbed velocity distribution function ΦP of Poiseuille flow at (x1, x2) = (A/2, 0.75) as a function of the molecular velocity ζ1 and ζ2 (A = 0.2, C = 0.5, α1 = 0.5, α2 = 1, and b/L = 0.1: (a) Kn = 0.1 and (b) Kn = 10

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