Numerical analysis of the Poiseuille flow and the thermal transpiration of a rarefied gas through a pipe with a rectangular cross section based on the linearized Boltzmann equation for a hard sphere molecular gas

2010 ◽  
Vol 28 (4) ◽  
pp. 603-612 ◽  
Author(s):  
Toshiyuki Doi
Keyword(s):  
Boltzmann Equation ◽  
Numerical Analysis ◽  
Cross Section ◽  
Poiseuille Flow ◽  
Hard Sphere ◽  
Rarefied Gas ◽  
Rectangular Cross Section ◽  
Molecular Gas ◽  
Thermal Transpiration ◽  
Linearized Boltzmann Equation

scholarly journals Poiseuille and thermal transpiration flows of a highly rarefied gas: over-concentration in the velocity distribution function

2011 ◽  
Vol 669 ◽  
pp. 242-259 ◽  
Author(s):  
SHIGERU TAKATA ◽  
HITOSHI FUNAGANE
Keyword(s):  
Mass Flow ◽  
Hard Sphere ◽  
Rarefied Gas ◽  
Velocity Distribution Function ◽  
Molecular Gas ◽  
Approximation Procedure ◽  
Molecular Models ◽  
Iterative Approximation ◽  
Thermal Transpiration ◽  
Linearized Boltzmann Equation

Poiseuille and thermal transpiration flows of a highly rarefied gas are investigated on the basis of the linearized Boltzmann equation, with a special interest in the over-concentration of molecules on velocities parallel to the walls. An iterative approximation procedure with an explicit error estimate is presented, by which the structure of the over-concentration is clarified. A numerical computation on the basis of the procedure is performed for a hard-sphere molecular gas to construct a database that promptly gives the induced net mass flow for an arbitrary value of large Knudsen numbers. An asymptotic formula of the net mass flow is also presented for molecular models belonging to Grad's hard potential. Finally, the resemblance of the profiles between the heat flow of the Poiseuille flow and the flow velocity of the thermal transpiration is pointed out. The reason is also given.

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.

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