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Research Papers: Micro/Nanoscale Heat Transfer

Numerical Analysis of the Time-Dependent Energy and Momentum Transfers in a Rarefied Gas Between Two Parallel Planes Based on the Linearized Boltzmann Equation

[+] Author and Article Information
Toshiyuki Doi

Department of Applied Mathematics and Physics, Tottori University, Tottori 680-8552, Japandoi@damp.tottori-u.ac.jp

J. Heat Transfer 133(2), 022404 (Nov 03, 2010) (9 pages) doi:10.1115/1.4002441 History: Received March 27, 2010; Revised July 22, 2010; Published November 03, 2010; Online November 03, 2010

Periodic time-dependent behavior of a rarefied gas between two parallel planes caused by an oscillatory heating of one plane is numerically studied based on the linearized Boltzmann equation. Detailed numerical data of the energy transfer from the heated plane to the unheated plane and the forces of the gas acting on the boundaries are provided for a wide range of the gas rarefaction degree and the oscillation frequency. The flow is characterized by a coupling of heat conduction and sound waves caused by repetitive expansion and contraction of the gas. For a small gas rarefaction degree, the energy transfer is mainly conducted by sound waves, except for very low frequencies, and is strongly affected by the resonance of the waves. For a large gas rarefaction degree, the resonance effects become insignificant and the energy transferred to the unheated plane decreases nearly monotonically as the frequency increases. The force of the gas acting on the heated boundary shows a remarkable minimum with respect to the frequency even in the free molecular limit.

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Figures

Grahic Jump Location
Figure 5

Amplitude |Q| of the energy flow at the two walls as a function of ω/ω0. ((a) and (b)) Kn=0.01, ((c) and (d)) Kn=0.1, and ((e) and (f)) Kn=10; ((a), (c), and (e)) |Q| at x1=0 and ((b), (d), and (f)) |Q| at x1=1. Open circle (○): hard-sphere gas, solid line (——): BKW model, dotted line (- - -) in (a) and (b): Navier–Stokes solution (Eq. 31) for Kn=0.01, dotted line (- - -) with the letters HC in (a) and (b): solution of heat conduction equation (Eq. 40) for Kn=0.01, dotted line (- - -) in (e) and (f): free molecular solution for Kn=∞ (Eq. 45), dot-dashed line (– - –): limiting value of |Q(0)| as ω→∞ (Eq. 47), and vertical dashed lines (– –): ω/ω0=1,2,…,4.

Grahic Jump Location
Figure 4

Amplitude |P| of the normal stress at the two walls as a function of ω/ω0. ((a) and (b)) Kn=0.01, ((c) and (d)) Kn=0.1, and ((e) and (f)) Kn=10; ((a), (c), and (e)) |P| at x1=0 and ((b), (d), and (f)) |P| at x1=1. Open circle (○): hard-sphere gas, solid line (——): BKW model, dotted line (- - -) in (a) and (b): Navier–Stokes solution (Eq. 30) for Kn=0.01, dotted line (- - -) in (e) and (f): free molecular solution for Kn=∞ (Eq. 44), dot-dashed line (– - –): limiting value of |P(0)| as ω→∞ (Eq. 47), and vertical dashed lines (– –): ω/ω0=1,2,…,4.

Grahic Jump Location
Figure 3

The profile of the macroscopic variables (III) Kn=10. (G) ω/ω0=1.541, (H) 2.580, and (I) 4.533. See the caption of Fig. 1.

Grahic Jump Location
Figure 2

The profile of the macroscopic variables (II) Kn=0.1. (D) ω/ω0=1.165, (E) 1.953, and (F) 3.347. See the caption of Fig. 1.

Grahic Jump Location
Figure 1

The profile of the macroscopic variables (I) Kn=0.01. (A) ω/ω0=1.032, (B) 1.939, and (C) 3.068. (a) The amplitude U of the flow velocity, (b) normal stress P, (c) temperature T, and (d) heat flow Q (see Eqs. 14,15,16,17). Solid line (——): present result (hard-sphere gas), dotted line (- - -): time-independent solution (ω=0)(16), and cross (×): solution in the limit ω→∞ (Eq. 47).

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