scholarly journals Inversion of the transverse force on a spinning sphere moving in a rarefied gas

2021 ◽  
Vol 933 ◽  
Author(s):  
Satoshi Taguchi ◽  
Tetsuro Tsuji
Keyword(s):  
Boltzmann Equation ◽  
Knudsen Number ◽  
Rarefied Gas ◽  
Mean Free Path ◽  
Transverse Force ◽  
Surface Velocity ◽  
Translational Velocity ◽  
The Boltzmann Equation ◽  
Wide Range ◽  
The Continuum

The flow around a spinning sphere moving in a rarefied gas is considered in the following situation: (i) the translational velocity of the sphere is small (i.e. the Mach number is small); (ii) the Knudsen number, the ratio of the molecular mean free path to the sphere radius, is of the order of unity (the case with small Knudsen numbers is also discussed); and (iii) the ratio between the equatorial surface velocity and the translational velocity of the sphere is of the order of unity. The behaviour of the gas, particularly the transverse force acting on the sphere, is investigated through an asymptotic analysis of the Boltzmann equation for small Mach numbers. It is shown that the transverse force is expressed as $\boldsymbol{F}_L = {\rm \pi}\rho a^3 (\boldsymbol{\varOmega} \times \boldsymbol{v}) \bar{h}_L$ , where $\rho$ is the density of the surrounding gas, a is the radius of the sphere, $\boldsymbol {\varOmega }$ is its angular velocity, $\boldsymbol {v}$ is its velocity and $\bar {h}_L$ is a numerical factor that depends on the Knudsen number. Then, $\bar {h}_L$ is obtained numerically based on the Bhatnagar–Gross–Krook model of the Boltzmann equation for a wide range of Knudsen number. It is shown that $\bar {h}_L$ varies with the Knudsen number monotonically from 1 (the continuum limit) to $-\tfrac {2}{3}$ (the free molecular limit), vanishing at an intermediate Knudsen number. The present analysis is intended to clarify the transition of the transverse force, which is previously known to have different signs in the continuum and the free molecular limits.

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.

Transient motion of a confined rarefied gas due to wall heating or cooling

1993 ◽  
Vol 248 ◽  
pp. 219-235 ◽  
Author(s):  
Dean C. Wadsworth ◽  
Daniel A. Erwin ◽  
E. Phillip Muntz
Keyword(s):  
Knudsen Number ◽  
Stokes Equations ◽  
Rarefied Gas ◽  
Mean Free Path ◽  
Direct Simulation ◽  
Transient Motion ◽  
Wall Heating ◽  
Gas Response ◽  
The Continuum ◽  
Rarefaction Effects

The transient motion that arises in a confined rarefied gas as a container wall is rapidly heated or cooled is simulated numerically. The Knudsen number based on nominal gas density and characteristic container dimension is varied from near-continuum to highly rarefied conditions. Solutions are generated with the direct simulation Monte Carlo method. Comparisons are made with finite-difference solutions of the Navier–Stokes equations, the limiting free-molecular values, and (continuum) results based on a small perturbation analysis. The wall heating and cooling scenarios considered induce relatively large acoustic disturbances in the gas, with characteristic flow speeds on the order of 20% of the local sound speed. Steady-state conditions are reached after on the order of 5 to 10 acoustic time units, here based on the initial speed of sound in the gas and the container dimension. As rarefaction increases, the initial gas response time is decreased. For the case of a rapid increase in wall temperature, transient rarefaction effects near the wall greatly alter gas response compared to the continuum predictions, even at relatively small nominal Knudsen number. For wall cooling, the continuum solution agrees well with direct simulation at that same Knudsen number. A local Knudsen number, based on the mean free path and the scale length of the temperature gradient, is found to be a more suitable indicator of transient rarefaction effects.

THE CONTRIBUTION OF THE BHATNAGAR–GROSS–KROOK MODEL TO THE DEVELOPMENT OF RAREFIED GAS DYNAMICS IN THE EARLY YEARS OF THE SPACE AGE

2013 ◽  
Vol 25 (01) ◽  
pp. 1340025
Author(s):  
RODDAM NARASIMHA
Keyword(s):  
Boltzmann Equation ◽  
Gas Dynamics ◽  
Rarefied Gas ◽  
Rarefied Gas Dynamics ◽  
Early Years ◽  
Plane Shock ◽  
Bgk Model ◽  
The Boltzmann Equation ◽  
Nonequilibrium Flows ◽  
Space Age

The advent of the space age in 1957 was accompanied by a sudden surge of interest in rarefied gas dynamics (RGD). The well-known difficulties associated with solving the Boltzmann equation that governs RGD made progress slow but the Bhatnagar–Gross–Krook (BGK) model, proposed three years before Sputnik, turned out to have been an uncannily timely, attractive and fruitful option, both for gaining insights into the Boltzmann equation and for estimating various technologically useful flow parameters. This paper gives a view of how BGK contributed to the growth of RGD during the first decade of the space age. Early efforts intended to probe the limits of the BGK model showed that, in and near both the continuum Euler limit and the collisionless Knudsen limit, BGK could provide useful answers. Attempts were therefore made to tackle more ambitious nonlinear nonequilibrium problems. The most challenging of these was the structure of a plane shock wave. The first exact numerical solutions of the BGK equation for the shock appeared during 1962 to 1964, and yielded deep insights into the character of transitional nonequilibrium flows that had resisted all attempts at solution through the Boltzmann equation. In particular, a BGK weak shock was found to be amenable to an asymptotic analysis. The results highlighted the importance of accounting separately for fast-molecule dynamics, most strikingly manifested as tails in the distribution function, both in velocity and in physical space — tails are strange versions or combinations of collisionless and collision-generated flows. However, by the mid-1960s Monte-Carlo methods of solving the full Boltzmann equation were getting to be mature and reliable and interest in the BGK waned in the following years. Interestingly, it has seen a minor revival in recent years as a tool for developing more effective algorithms in continuum computational fluid dynamics, but the insights derived from the BGK for strongly nonequilibrium flows should be of lasting value.

scholarly journals Non-equilibrium dynamics of dense gas under tight confinement

2016 ◽  
Vol 794 ◽  
pp. 252-266 ◽  
Author(s):  
Lei Wu ◽  
Haihu Liu ◽  
Jason M. Reese ◽  
Yonghao Zhang
Keyword(s):  
Boltzmann Equation ◽  
Flow Regime ◽  
Mass Flow Rate ◽  
Knudsen Number ◽  
Mass Flow ◽  
Flow Rate ◽  
Transitional Flow ◽  
Molecular Flow ◽  
Parallel Plates ◽  
The Boltzmann Equation

The force-driven Poiseuille flow of dense gases between two parallel plates is investigated through the numerical solution of the generalized Enskog equation for two-dimensional hard discs. We focus on the competing effects of the mean free path ${\it\lambda}$, the channel width $L$ and the disc diameter ${\it\sigma}$. For elastic collisions between hard discs, the normalized mass flow rate in the hydrodynamic limit increases with $L/{\it\sigma}$ for a fixed Knudsen number (defined as $Kn={\it\lambda}/L$), but is always smaller than that predicted by the Boltzmann equation. Also, for a fixed $L/{\it\sigma}$, the mass flow rate in the hydrodynamic flow regime is not a monotonically decreasing function of $Kn$ but has a maximum when the solid fraction is approximately 0.3. Under ultra-tight confinement, the famous Knudsen minimum disappears, and the mass flow rate increases with $Kn$, and is larger than that predicted by the Boltzmann equation in the free-molecular flow regime; for a fixed $Kn$, the smaller $L/{\it\sigma}$ is, the larger the mass flow rate. In the transitional flow regime, however, the variation of the mass flow rate with $L/{\it\sigma}$ is not monotonic for a fixed $Kn$: the minimum mass flow rate occurs at $L/{\it\sigma}\approx 2{-}3$. For inelastic collisions, the energy dissipation between the hard discs always enhances the mass flow rate. Anomalous slip velocity is also found, which decreases with increasing Knudsen number. The mechanism for these exotic behaviours is analysed.

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