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

2010 ◽  
Vol 133 (2) ◽  
Author(s):  
Toshiyuki Doi
Keyword(s):  
Energy Transfer ◽  
Boltzmann Equation ◽  
Rarefied Gas ◽  
Numerical Data ◽  
Time Dependent ◽  
Sound Waves ◽  
Low Frequencies ◽  
Resonance Effects ◽  
Linearized Boltzmann Equation ◽  
Wide Range

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.

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

2014 ◽  
Vol 136 (8) ◽  
Author(s):  
Toshiyuki Doi
Keyword(s):  
Boltzmann Equation ◽  
Mass Flow Rate ◽  
Knudsen Number ◽  
Mass Flow ◽  
Flow Rate ◽  
Poiseuille Flow ◽  
Rarefied Gas ◽  
Thermal Transpiration ◽  
Wide Range ◽  
Accommodation Coefficients

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.

scholarly journals Oscillatory rarefied gas flow inside rectangular cavities

2014 ◽  
Vol 748 ◽  
pp. 350-367 ◽  
Author(s):  
Lei Wu ◽  
Jason M. Reese ◽  
Yonghao Zhang
Keyword(s):  
Aspect Ratio ◽  
Gas Flow ◽  
Rarefied Gas ◽  
Scaling Law ◽  
Frequency Range ◽  
Damping Force ◽  
Driven Cavity ◽  
Linearized Boltzmann Equation ◽  
Wide Range ◽  
Oscillation Frequencies

AbstractTwo-dimensional oscillatory lid-driven cavity flow of a rarefied gas at arbitrary oscillation frequency is investigated using the linearized Boltzmann equation. An analytical solution at high oscillation frequencies is obtained, and detailed numerical results for a wide range of gas rarefaction are presented. The influence of both the aspect ratio of the cavity and the oscillating frequency on the damping force exerted on the moving lid is studied. Surprisingly, it is found that, over a certain frequency range, the damping is smaller than that in an oscillatory Couette flow. This reduction in damping is due to the anti-resonance of the rarefied gas. A scaling law between the anti-resonant frequency and the aspect ratio is established, which would enable the control of the damping through choosing an appropriate cavity geometry.

Solution of the Boltzmann equation for electrons in external electric fields

1970 ◽  
Vol 4 (1) ◽  
pp. 145-159 ◽  
Author(s):  
L. Stenflo
Keyword(s):  
Wave Propagation ◽  
Boltzmann Equation ◽  
Electric Fields ◽  
Transverse Wave ◽  
Analytical Solutions ◽  
Density Fluctuation ◽  
Time Dependent ◽  
Exact Analytical Solutions ◽  
External Electric Fields ◽  
Linearized Boltzmann Equation

A plasma in a time-dependent external electric field is considered. Collisions are described by means of Fokker—Planck and BGK models. Longitudinal and transverse wave propagation is treated by means of exact analytical solutions of the linearized Boltzmann equation. The results generalize previous calculations on dispersion relations and density fluctuation spectra.

CONSERVATION LAWS AND EXACT SOLUTIONS OF THE BOLTZMANN EQUATION

1989 ◽  
Vol 03 (03) ◽  
pp. 215-223 ◽  
Author(s):  
DANIEL C. MATTIS ◽  
ANTHONY M. SZPILKA ◽  
HUA CHEN
Keyword(s):  
Exact Solution ◽  
Boltzmann Equation ◽  
Joule Heating ◽  
Time Dependent ◽  
Lorentz Model ◽  
Time Approximation ◽  
Ohm’S Law ◽  
Relaxation Time Approximation ◽  
Wide Range ◽  
Ohm's Law

The distribution function f which satisfies the time-dependent Boltzmann equation (BE) for a Lorentz model with perfectly elastic random scatterers is proved nonnegative, and is computed exactly when backscattering dominates. Joule heating and Ohm’s law are recovered, although f has no steady-state limit, contrary to the relaxation-time approximation. (The conventional approximation to the time-independent BE also yields Ohm’s law but not the Joule heating and, worse, it unphysically predicts f<0.) The exact solution is compared with various effective-temperature approximations, and is shown to remain very nearly unchanged over a wide range of times even in the presence of a small amount of inelastic scattering.

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