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- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This book provides a coherent and easy accessible approach to rarefied gas dynamics. The author addresses young researchers and professionals who look for a consistent introduction as well as scientists and engineers who deal with molecular gas dynamics in their routine work. It is the first monograph which includes advanced topics like oscillatory flows and sound propagation in the rarefied gas flows. The reader is introduced to the main concepts and recent results of rarefied gas dynamics. No prior knowledge of gas dynamics is needed to develop a sound understanding of the underlying principles of thermodynamics, modern analytical and numerical methods of modelling. The author includes different theoretical and computational methodologies like model kinetic equations, the discrete velocity method, and Monte Carlo methods. It is beneficial to have all of them in one place, since these methods often have different domains of applicability. Most results are given without using mathematical derivations. Readers who want to study this field deeper can choose from a long list of references. Audience: Researchers, engineers and professionals who deal with rarefied gas dynamics in their routine work and need quickly to learn main concepts and results of this field. Advanced and PhD students in physics, mathematics, mechanical engineering and chemical engineering.
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Seitenzahl: 438
Veröffentlichungsjahr: 2015
Ähnliche
Table of Contents
Related Titles
Title Page
Copyright
Preface
List of Symbols
List of Acronyms
Chapter 1: Molecular Description
1.1 Mechanics of Continuous Media and Its Restriction
1.2 Macroscopic State Variables
1.3 Dilute Gas
1.4 Intermolecular Potential
1.5 Deflection Angle
1.6 Differential Cross Section
1.7 Total Cross Section
1.8 Equivalent Free Path
1.9 Rarefaction Parameter and Knudsen Number
Exercises
Chapter 2: Velocity Distribution Function
2.1 Definition of Distribution Function
2.2 Moments of Distribution Function
2.3 Entropy and Its Flow Vector
2.4 Global Maxwellian
2.5 Local Maxwellian
Exercises
Chapter 3: Boltzmann Equation
3.1 Assumptions to Derive the Boltzmann Equation
3.2 General Form of the Boltzmann Equation
3.3 Conservation Laws
3.4 Entropy Production due to Intermolecular Collisions
3.5 Intermolecular Collisions Frequency
Exercises
Chapter 4: Gas–Surface Interaction
4.1 General form of Boundary Condition for Impermeable Surface
4.2 Diffuse–Specular Kernel
4.3 Cercignani–Lampis Kernel
4.4 Accommodation Coefficients
4.5 General form of Boundary Condition for Permeable Surface
4.6 Entropy Production due to Gas–Surface Interaction
Exercises
Chapter 5: Linear Theory
5.1 Small Perturbation of Equilibrium
5.2 Linearization Near Global Maxwellian
5.3 Linearization Near Local Maxwellian
5.4 Properties of the Linearized Collision Operator
5.5 Linearization of Boundary Condition
5.6 Series Expansion
5.7 Reciprocal Relations
Exercises
Chapter 6: Transport Coefficients
6.1 Constitutive Equations
6.2 Viscosity
6.3 Thermal Conductivity
6.4 Numerical Results
Exercises
Chapter 7: Model Equations
7.1 BGK Equation
7.2 S-Model
7.3 Ellipsoidal Model
7.4 Dimensionless Form of Model Equations
Exercises
Chapter 8: Direct Simulation Monte Carlo Method
8.1 Main Ideas
8.2 Generation of Specific Distribution Function
8.3 Simulation of Gas–Surface Interaction
8.4 Intermolecular Interaction
8.5 Calculation of Post-Collision Velocities
8.6 Calculation of Macroscopic Quantities
8.7 Statistical Scatter
Exercises
Chapter 9: Discrete Velocity Method
9.1 Main Ideas
9.2 Velocity Discretization
9.3 Iterative Procedure
9.4 Finite Difference Schemes
Exercises
Chapter 10: Velocity Slip and Temperature Jump Phenomena
10.1 General Remarks
10.2 Viscous Velocity Slip
10.3 Thermal Velocity Slip
10.4 Reciprocal Relation
10.5 Temperature Jump
Exercises
Chapter 11: One-Dimensional Planar Flows
11.1 Planar Couette Flow
11.2 Planar Heat Transfer
11.3 Planar Poiseuille and Thermal Creep Flows
Exercises
Chapter 12: One-Dimensional Axisymmetrical Flows
12.1 Cylindrical Couette Flow
12.2 Heat Transfer between Two Cylinders
12.3 Cylindrical Poiseuille and Thermal Creep Flows
Exercises
Chapter 13: Two-Dimensional Planar Flows
13.1 Flows Through a Long Rectangular Channel
13.2 Flows Through Slits and Short Channels
13.3 End Correction for Channel
Exercises
Chapter 14: Two-Dimensional Axisymmetrical Flows
14.1 Flows Through Orifices and Short Tubes
14.2 End Correction for Tube
14.3 Transient Flow Through a Tube
Exercises
Chapter 15: Flows Through Long Pipes Under Arbitrary Pressure and Temperature Drops
15.1 Stationary Flows
15.2 Pipes with Variable Cross Section
15.3 Transient Flows
Exercises
Chapter 16: Acoustics in Rarefied Gases
16.1 General Remarks
16.2 Oscillatory Couette Flow
16.3 Longitudinal Waves
Exercises
Appendix A: Constants and Mathematical Expressions
A.1 Physical Constants
A.2 Vectors and Tensors
A.3 Nabla Operator
A.4 Kronecker Delta and Dirac Delta Function
A.5 Some Integrals
A.6 Taylor Series
A.7 Some Functions
A.8 Gauss–Ostrogradsky's Theorem
A.9 Complex Numbers
Appendix B: Files and Listings
B.1 Files with Nodes and Weights of Gauss Quadrature
B.2 Files for Planar Couette Flow
B.3 Files for Planar Heat Transfer
B.4 Files for Planar Poiseuille and Creep Flows
B.5 Files for Cylindrical Couette Flows
B.6 Files for Cylindrical Heat Transfer
B.7 Files for Axi-Symmetric Poiseuille and Creep Flows
B.8 Files for Poiseuille and Creep Flows Through Channel
B.9 Files for Oscillating Couette Flow
References
Index
End User License Agreement
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Guide
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: Molecular Description
Figure 1.1 Potential versus intermolecular distance .
Figure 1.2 Scheme of binary collision.
Figure 1.3 Deflection angle versus impact parameter for LJ potential.
Chapter 4: Gas–Surface Interaction
Figure 4.1 Scheme of gas–surface interaction.
Chapter 5: Linear Theory
Figure 5.1 Scheme of surfaces and .
Chapter 9: Discrete Velocity Method
Figure 9.1 One-sided scheme along characteristic.
Figure 9.2 Polar coordinates.
Chapter 10: Velocity Slip and Temperature Jump Phenomena
Figure 10.1 Coordinates for the calculation of slip and jump coefficients.
Figure 10.2 Velocity deviation (a) and velocity itself (b) versus coordinate in viscous slip problem.
Figure 10.3 Dimensionless heat flow versus coordinate in viscous slip problem.
Figure 10.4 Dimensionless velocity versus coordinate in thermal slip problem.
Figure 10.5 Dimensionless heat flow versus coordinate in thermal slip problem.
Figure 10.6 Deviations of gas temperature from reference temperature (a) and from surface temperature (b) versus coordinate in temperature jump problem.
Chapter 11: One-Dimensional Planar Flows
Figure 11.1 Scheme and coordinates of planar Couette flow.
Figure 11.2 Dimensionless velocity versus coordinate in planar Couette flow.
Figure 11.3 Scheme for heat transfer between two plates.
Figure 11.4 Temperature deviation versus coordinate in planar heat transfer.
Figure 11.5 Scheme and coordinates of planar Poiseuille and creep flows.
Figure 11.6 Perturbations and in the middle point () versus velocity .
Figure 11.7 Coefficients , , and for gas flow between two plates versus rarefaction parameter . Comparison of data based on S-model with those obtained in Ref. [94] applying the BE with HS and LJ potentials for helium (He) and xenon (Xe).
Figure 11.8 Coefficients , , and for gas flow between two plates versus rarefaction parameter . Comparison of data based on diffuse scattering with those obtained in Ref. [124] applying the CL scattering kernel.
Figure 11.9 Dimensionless velocities and for gas flow between plates versus coordinate : solid lines–kinetic equation, dashed line–slip solution (11.86).
Figure 11.10 Dimensionless heat flows and for gas flow between plates versus coordinate : solid lines–kinetic equation, dashed line–hydrodynamic solution (11.90).
Chapter 12: One-Dimensional Axisymmetrical Flows
Figure 12.1 Scheme of cylindrical Couette flow.
Figure 12.2 Scheme to obtain (12.19).
Figure 12.3 Perturbations versus angle at the middle point between cylinders for and : (a) without split, (b) with split.
Figure 12.4 Scheme to calculate (see Eq. (12.32)).
Figure 12.5 Dimensionless velocity in cylindrical Couette flow versus radial coordinate at .
Figure 12.6 Scheme of heat transfer between two cylinders.
Figure 12.7 Dimensionless heat flow between two cylinders versus rarefaction for several values of radii ratio . Numerical data are reported in Ref. [35]. Reproduced from Ref. [136] with permission.
Figure 12.8 Dimensionless heat flow between two cylinders versus rarefaction for = 65: curves–theoretical results [35] based on CL kernel (4.17); symbols–experimental data [37]. Reproduced from Ref. [35] with permission.
Figure 12.9 Temperature deviation between two cylinders at .
Figure 12.10 Scheme of cylindrical Poiseuille and creep flows.
Figure 12.11 Scheme to obtain (12.102).
Figure 12.12 Poiseuille coefficient for gas flow through tube versus rarefaction . Comparison of data based on S-model with those obtained in Ref. [144] applying the BE with HS potential.
Figure 12.13 Coefficients and for gas flow through tube versus rarefaction . Comparison of data based on diffuse scattering with those obtained in Ref. [34] applying the CL scattering kernel.
Figure 12.14 Dimensionless velocities and for gas flow through tube versus coordinate : solid lines – kinetic equation, dashed line – slip solution (12.85).
Figure 12.15 Dimensionless heat flows and for gas flow through tube versus coordinate : solid lines – kinetic equation, dashed line – hydrodynamic solution (11.90).
Chapter 13: Two-Dimensional Planar Flows
Figure 13.1 Scheme for gas flow through a channel of rectangular cross section.
Figure 13.2 Poiseuille (a) and thermal creep (b) coefficients for rectangular channel versus rarefaction parameter and aspect ratio : solid line – [148, 151]; dashed line – Eqs. (13.11) and (12.86); point-dashed line – Eqs. (13.25) and (13.26). Reproduced from Ref. [136] with permission.
Figure 13.3 Scheme and computational domain for gas flow through slit and short channel.
Figure 13.4 Poiseuille coefficient for gas flow through short channel versus rarefaction : solid line – numerical solution [166]; point-dashed line – free-molecular solution, Eqs. (13.25) and (13.26).
Figure 13.5 Dimensionless flow rate for gas flow through a slit (a) and short channel (b) versus rarefaction : – Ref. [175]; and 0.99 – Ref. [178].
Figure 13.6 Comparison between linear [161] and nonlinear [178] theories for slit flow ().
Figure 13.7 Distributions of density , temperature , and bulk velocity along the slit axis at . Reproduced from Ref. [175] with permission.
Figure 13.8 Scheme of end correction for channel.
Figure 13.9 Pressure distribution along the symmetry axis at the inlet part.
Figure 13.10 Density deviation in the end-correction problem versus longitudinal coordinate at . Reproduced from Ref. [187] with permission.
Figure 13.11 Comparison of the end-correction formula for channel (13.81) supported by the data of Table 13.9 (solid lines) with numerical values (circles) of from Table 13.4.
Chapter 14: Two-Dimensional Axisymmetrical Flows
Figure 14.1 Scheme for flow through a short tube.
Figure 14.2 Poiseuille coefficient for gas flow through short tube versus rarefaction , Refs [189, 190]: solid lines – kinetic equation, point-dashed lines – free-molecular limit.
Figure 14.3 Dimensionless flow rate for gas flow through orifice (a) and short tube (b) versus [194, 195, 197].
Figure 14.4 Comparison between linear Eq. (14.23) and nonlinear [194, 195] theories for orifice flow ().
Figure 14.5 Distributions of density , temperature , and bulk velocity along the orifice axis at . Reproduced from Ref. [195] with permission.
Figure 14.6 Scheme of end correction for tube.
Figure 14.7 Comparison of end-correction formula for tube (13.81) supported by data in Table 14.5 (solid lines) with numerical values (circles) of from Ref. [189].
Figure 14.8 Flow rates through short tube versus time : up – , down – , left – , right – ; solid lines – inlet, dashed lines – outlet, pointed lines – stationary flow, Table 14.4. Reproduced from Ref. [201] with permission.
Figure 14.9 Distributions of the density , temperature , and longitudinal bulk velocity along tube axis at , , and . Reproduced from Ref. [201] with permission.
Chapter 15: Flows Through Long Pipes Under Arbitrary Pressure and Temperature Drops
Figure 15.1 Scheme of stationary flow through long pipe.
Figure 15.2 Pressure distribution along circular tube: - (a) , reproduced from Ref. [136] with permission; -(b) .
Figure 15.3 Scheme of transient flow through long pipe.
Figure 15.4 Pressures and for transient flow through long tube versus dimensionless time . Reproduced from Ref. [207] with permission.
Figure 15.5 Dimensionless flow rate given by Eq. (15.37) for transient flow through long tube versus dimensionless time : solid line – inlet , dashed line – outlet . Reproduced from Ref. [207] with permission.
Chapter 16: Acoustics in Rarefied Gases
Figure 16.1 Scheme of oscillating Couette flow.
Figure 16.2 Perturbation at , , and : solid line – nonsplitting scheme; dashed line – splitting scheme.
Figure 16.3 Amplitudes of bulk velocity and shear stress at the surface () for oscillating Couette flow versus dimensionless distance . Reproduced from Ref. [215] with permission.
Figure 16.4 Scheme of acoustic wave generation.
Figure 16.5 Amplitudes of pressure tensors and at receptor () versus : pointed line – Eqs. (16.113) and (16.114); point-dashed line – Eq. (16.76) for .
Figure 16.6 Phases of pressure tensors and at receptor () versus : pointed line – Eqs. (16.113) and (16.114); point-dashed line – Eq. (16.76) for .
List of Tables
Chapter 1: Molecular Description
Table 1.1 Parameters and for LJ and AI potentials
Table 1.2 Parameters of
ab initio
potential given by (1.18)
Chapter 4: Gas–Surface Interaction
Table 4.1 Accommodation coefficient extracted from the experimental data [32] applying the diffuse–specular kernel (4.15): a–atomically clean silver, b–atomically clean titanium, c–titanium covered by oxygen
Table 4.2 Accommodation coefficients and extracted in Refs [33–35] from the experimental data [36, 37] applying the CL kernel (4.17): surface is typically technical
Chapter 6: Transport Coefficients
Table 6.1 Dimensionless viscosity and thermal conductivity based on the LJ potential versus reduced temperature , Ref. [21]
Table 6.2 Viscosity and thermal conductivity based on the AI potential versus temperature , Refs [59–61]
Chapter 7: Model Equations
Table 7.1 Constants , , and of model collision operator
Chapter 10: Velocity Slip and Temperature Jump Phenomena
Table 10.1 Heat flow rate based on the S-model and CL kernel, Ref. [80]
Table 10.2 Viscous slip coefficient based on different equations and potentials. Diffuse gas–surface interaction
Table 10.3 Viscous slip coefficient based on the S-model and CL kernel, Refs [33, 85]
Table 10.4 Experimental values of viscous slip coefficient and the corresponding values of tangential momentum accommodation coefficient
Table 10.5 Thermal slip coefficient based on different equations and potentials. Diffuse gas–surface interaction
Table 10.6 Thermal slip coefficient based on the S-model and CL kernel, Refs [33, 85]
Table 10.7 Thermal slip coefficient extracted from experimental data
Table 10.8 Temperature jump coefficients based on different equations and potentials. Diffuse gas–surface interaction
Table 10.9 Temperature jump coefficients based on the CL kernel (4.17), Ref. [33]
Chapter 11: One-Dimensional Planar Flows
Table 11.1 Dimensionless shear stress in planar Couette flow versus rarefaction obtained by various methods: BGK by the scheme (11.27), BE based on HS in Ref. [91], and DSMC based on AI for argon in Ref. [75]
Table 11.2 Dimensionless heat flow between two plates versus rarefaction obtained from the S-model by the scheme (11.67) and that obtained by the DSMC method based on AI for argon in Ref. [76]
Table 11.3 Numerical scheme parameters and coefficients , , , for gas flow between plates versus rarefaction based on the code given in Section B.4.
Chapter 12: One-Dimensional Axisymmetrical Flows
Table 12.1 Numerical scheme parameters and shear stress at for cylindrical Couette flow versus rarefaction and radii ratio based on the code given in Section B.5.
Table 12.2 Numerical scheme parameters and dimensionless heat flow at and versus rarefaction for cylindrical heat transfer based on the code given in Section B.6.
Table 12.3 Numerical scheme parameters and coefficients , , , and for gas flow through a circular tube versus based on the code given in Section B.7.
Chapter 13: Two-Dimensional Planar Flows
Table 13.1 Coefficients , , and versus aspect ratio
Table 13.2 Numerical scheme parameters and coefficients , , , and at for gas flow through rectangular channel versus rarefaction based on the program listing in Section B.8.
Table 13.3 Coefficients and for gas flow through slit () versus rarefaction , Ref. [164]
Table 13.4 Poiseuille coefficient for gas flow through short channel versus length-to-height ratio and rarefaction : Refs [167, 168]
Table 13.5 Thermal creep coefficient for gas flow through short channel versus length-to-height ratio and rarefaction : and 5 – Refs [166, 170], and 100 – Ref. [167]
Table 13.6 Dimensionless flow rate for gas flow through slit () versus pressure ratio and rarefaction : – DSMC, Ref. [175]; and 0.99 – BGK, Ref. [178]
Table 13.7 Dimensionless flow rate for gas flow through short channel versus length-to-height ratio , pressure ratio , and rarefaction
Table 13.8 Coefficients for interpolating formula (13.79)
Table 13.9 Poiseuille coefficient , end-correction , and domain sizes , for channel versus rarefaction , Ref. [187]
Chapter 14: Two-Dimensional Axisymmetrical Flows
Table 14.1 Dimensionless flow rate for gas flow through short tube versus , Ref. [189]
Table 14.2 Dimensionless flow rate for gas flow through tube of finite length versus rarefaction , Ref. [190]
Table 14.3 Dimensionless flow rate for gas flow through orifice versus pressure ratio and rarefaction based on the AI potential for gas argon at K, [195]
Table 14.4 Dimensionless flow rate for gas flow through short tube versus pressure ratio and rarefaction obtained by the DSMC method based on HS molecular model, Refs [196, 197]
Table 14.5 Poiseuille coefficient , end correction , and domain sizes , for tube versus rarefaction , Ref. [198]
Table 14.6 Approximate time needed to establish steady flow in tube. Reproduced from Ref. [201] with permission
Chapter 15: Flows Through Long Pipes Under Arbitrary Pressure and Temperature Drops
Table 15.1 Dimensionless flow rate for tube calculated by (15.10) with (12.119).
Table 15.2 Dimensionless flow rate for tube calculated by (15.17) with (12.119) and (12.120), .
Table 15.3 Dimensionless flow rate for the tube of variable radius calculated by (12.119) and (12.120), , constant, Ref. [206].
Table 15.4 Dimensionless time needed to establish the equilibrium in tube versus , Ref. [207].
Chapter 16: Acoustics in Rarefied Gases
Table 16.1 Numerical scheme parameters, amplitudes , , and phases , at oscillating plate () for oscillating Couette flow based on the code given in Section B.9.
Table 16.2 Coefficient and obtained as a solution of Eqs. (16.80)–(16.83) for and , Ref. [218].
Table 16.3 Amplitudes of pressure tensors and at the plates ( and ) for longitudinal wave versus rarefaction and oscillation speed parameter , Ref. [218].
Table 16.4 Phases of pressure tensor and for longitudinal wave at the plates ( and ) versus rarefaction and oscillation speed parameter (), Ref. [218].
Appendix A: Constants and Mathematical Expressions
Table A.1 Fundamental physical constants
Table A.2 Standard atomic weights, [227]
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Felix Sharipov
Rarefied Gas Dynamics
Fundamentals for Research and Practice
Author
Prof. Dr. Felix Sharipov
Departameto de Física
Universidade Federal do Paraná
CaixaPostal 19044
81531-990, Curitiba
Brazil
All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.
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Print ISBN: 978-3-527-41326-3
ePDF ISBN: 978-3-527-68507-3
ePub ISBN: 978-3-527-68553-0
Mobi ISBN: 978-3-527-68554-7
oBook ISBN: 978-3-527-68552-3
Cover Design Adam Design, Weinheim, Germany
Preface
During lectures, seminars, and mini-courses given by me in universities, research institutes, and schools, I was frequently asked to suggest a book for beginners in order to learn quickly fundamentals and main results of rarefied gas dynamics. During last decades, the interest to this area drastically increased due to a necessity to model gas flows in many technologies related to rarefied gas flows. For instance, many technological processes take place in vacuum chambers under low pressure conditions where the continuous medium mechanics is not valid anymore. A further optimization of such processes requires more detailed information about gas flows in complex geometrical configurations. Another example of application of rarefied gas dynamics is the rapid miniaturization of electronic and mechanical equipment, which led to the necessity to take into account the gas rarefaction. In fact, the molecular mean free path became close to characteristic sizes of the miniaturized equipment even under the atmospheric pressure. Thus, the number of researchers and engineers dealing with rarefied gases drastically increased. In spite of many excellent books in this area, it was not so easy to suggest one of them that would describe the fundamentals of rarefied gas flows in concise and easily acceptable form. The present textbook intends to fill this lacuna. It is addressed to students, researchers, and engineers who wish to learn fundamentals and main results of rarefied gas dynamics and then to apply this knowledge to their practice.
In the first part of the present book, the main concepts related to velocity distribution function, Boltzmann equation, gas–surface interaction are given in an easily acceptable form. The main techniques to model rarefied gases such as discrete velocity method and direct simulation Monte Carlo method are described. Most of the results are given without hard mathematical derivations, but many references are suggested to those readers who want to study this field deeper. In the second part of the book, the classical problems of fluid dynamics, namely, Couette flow, heat transfer between solid surfaces, flows through various kinds of pipes, wave propagation, and so on, are solved analytically and numerically. Both linear and nonlinear transport phenomena are considered. For the sake of simplicity, only a single monatomic gas is considered, but some recommendations about the applicability of these results to polyatomic gases and gaseous mixtures are given. The book draws more attention to deterministic approaches such as the discrete velocity method. Indeed, in many technological processes, the Mach number is so small that the probabilistic methods widely used in aerothermodynamics become time consuming because of the statistical scatter. In situations when the Mach number is extremely small, the deterministic methods based on numerical solution of the kinetic equation become unique tools for a modeling of rarefied gas flows.
Most of the numerical solutions given in this book are provided by numerical codes based on the discrete velocity method with recommended input data, allowing the readers to obtain new results which are not reported in papers. The codes are neither optimized nor parallelized, but they require modest computational effort and can be run in an ordinary ultrabook. The readers can modify the code and solve new more complicated problems. Each chapter ends by exercises helping the readers to understand better the chapter matter and to apply it to some practical situations.
It is hoped that the manner to describe the deterministic method will enable many students, researchers, and engineers to learn easily the main concepts and results of rarefied gas dynamics. In the future, this knowledge will make easier the study of other books and papers in this field.
The manuscript of this book has been used in a course that I teach at the Post-Graduation in Physics of Federal University of Paraná. I wish to thank my students and colleagues for comments on the present manuscript.
Curitiba, BrazilApril, 2015
Felix Sharipov
List of Symbols
characteristic size
linearized scattering operator, Eq. (5.51)
coefficients in model equations, Eqs. (7.44), (7.45)
discretized scattering operator, Eq. (9.10)
linearized scattering operators in power expansion, Eq. (5.80)
amplitude of
, Eq. (16.5)
impact parameter, Fig. 1.2
cut-off impact parameter
tensor in ellipsoidal model, Eq. (7.32)
dimensionless molecular velocity, Eq. (5.2)
specific heat per particle at constant pressure, Eq. (6.4)
magnitude in polar coordinates, Eq. (9.18)
coefficients of finite difference scheme, Eqs. (9.37), (9.47)
coefficients of finite difference scheme, Eq. (9.76)
potential zero distance, Eq. (1.19)
specific internal energy, Eqs. (1.13), (2.24)
internal energy, Eq. (1.12)
kinetic energy of relative motion, Eq. (1.22)
energy flow rate, Eqs. (11.75), (12.78), (13.3),
velocity distribution function, Eq. (2.1)
set of functions, Eq. (9.1)
Maxwellian, Eq. (2.37)
global Maxwellian, Eq. (5.1)
reference Maxwellian, Eq. (5.28)
surface Maxwellian, impermeable surface, Eq. (5.56)
surface Maxwellian, permeable surface, Eq. (5.58)
cumulative function, Eq. (8.3)
representation of model particles, Eq. (8.26)
bulk source term, Eqs. (5.34)
dimensionless bulk source term, Eqs. (7.49)
relative velocity, Eq. (1.20)
average relative speed, Eq. (3.34)
maximum relative speed
center mass velocity, Eq. (8.30)
dimensionless flow rate, Eqs. (15.3), (15.20)
Poiseuille coefficient for short channel and tube, Eqs. (13.56), (14.16)
Poiseuille coefficient for infinite channel and tube, Eqs. (11.81), (12.80), (13.5)
Poiseuille coefficient for short channel and tube, Eqs. (13.60), (14.16)
thermal creep coefficient for short channel, Eq. (13.56)
thermal creep coefficient for infinite channel and tube, Eqs. (11.81), (12.80), (13.6)
thermal creep coefficient for short channel, Eq. (13.60)
perturbation function, Eq. (5.6)
split part of perturbation function, Eq. (11.118)
set of perturbation functions, Eq. (9.6)
reference perturbation, Eq. (5.30)
surface source term, Eq. (5.59)
term of model equation, Eq. (7.48)
coefficient, Eq. (13.11)
imaginary unit
unit tensor, Eq. (A.9)
Bessel function, Eq. (A.27)
function, Eq. (A.29)
imaginary part of complex number, Eq. (A.39)
flow vector of energy, Eq. (2.20)
thermodynamic flux, Eq. (5.81)
flow vector of mass, Eq. (2.19)
flow vector of particles, Eq. (2.18)
flow vector of entropy, Eq. (2.36)
flux of property
, Eq. (2.17)
wave number, Eq. (16.1)
Boltzmann constant, Table A.1
attenuation coefficient, Eq. (16.4)
Knudsen number, Eq. (1.33)
equivalent free path, Eq. (1.32)
length-to-height ratio,
Figure 13.3
length-to-radius ratio,
Figure 14.1
dimensionless distance, Eq. (16.13)
linearized collision operator, Eq. (5.21)
linearized BGK collision operator, Eq. (7.8)
linearized ellipsoidal collision operator, Eq. (7.34)
linearized S-model collision operator, Eq. (7.19)
mass of one particle, Eq. (1.1)
mass of gas
atomic weight, Eq. (1.2)
momentum flux tensor, Eq. (2.21)
mass flow rate, Eqs. (11.75), (12.78), (13.2), (13.39), (14.5)
number density, Eq. (1.5)
reference number density, Eq. (5.28)
number of particles
Avogadro number, Table A.1
number of collisions in cell, Eq. (8.24)
number of model particles
number of model collisions in cell, Eq. (8.27)
number of nodes for
variable
number of nodes for velocity
number of nodes for
variable
Loschmidt number, Table A.1
pressure, Eq. (1.7)
pressure tensor, Eq. (2.28)
Prandtl number, Eq. (6.4)
heat flow vector, Eq. (2.32)
mechanocaloric coefficient for infinite channel and tube, Eqs. (11.82), (12.81), (13.7)
heat flow coefficient for infinite channel and tube, Eqs. (11.82), (12.81), (13.8)
dimensionless heat flow vector, Eq. (5.16)
collision integral, Eq. (3.5)
BGK collision integral, Eq. (7.1)
ellipsoidal model of collision integral, Eq. (7.31)
S-model collision integral, Eq. (7.17)
heat flow rate due to velocity gradient, Eq. (10.15)
position vector
intermolecular distance
radial coordinate
scattering kernel, Eq. (4.4)
Cercignani–Lampis scattering kernel, Eq. (4.17)
diffuse scattering kernel, Eq. (4.10)
diffuse–specular scattering kernel, Eq. (4.15)
molar gas constant, Table A.1
random number
