AER1301: Kinetic Theory of Gases Supplementary Notes. Instructor:

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1 1 Course Summary AER1301: Kinetic Theory of Gases Supplementary Notes Instructor: Prof. Clinton P. T. Groth University of Toronto Institute for Aerospace Studies 4925 Dufferin Street, Toronto, Ontario, Canada M3H 5T6 January 8, 2007 The course is intended to provide an introduction to the kinetic theory of gases. Material covered in the course includes discussion of significant length dimensions; different flow regimes, continuum, transition, collision-free; a brief history of gas kinetic theory; equilibrium kinetic theory, the particle distribution function, Maxwell-Boltzmann distribution. collision dynamics, collision frequency and mean free path; elementary transport theory, transport coefficients, mean free path method Boltzmann equation, derivation, Boltzmann H-theorem, collision operators; Generalized transport theory, Maxwell s equations of change, approximate solution techniques, Chapman-Enskog perturbative and Grad series expansion methods, moment closures; derivation of the Euler and Navier-Stokes equations, higher-order closures; free molecular aerodynamics; shock waves. The textbook for the course is Gaskinetic Theory, by Tamas Gombosi [1]. These notes are intended to supplement the material covered in the textbook. groth@utias.utoronto.ca 1

2 2 Course Outline 2.1 Outline of Textbook The material covered in the textbook by Gombosi [1] can be summarized as follows: CHAPTER 1: INTRODUCTION + Chapter 1 provides a brief history of gaskinetic theory and discusses the evolution from hydrostatics, to fluid dynamics hydrodynamics, on through to kinetic theory. + Basic assumptions and limitations of kinetic theory: Molecular Hypothesis: i matter is composed discrete molecules particles; ii all molecules for a given substance are alike and are the smallest quantity of the substance that retains its unique chemical properties. Ideal Perfect Gases: i molecules are point-like structures with no internal structure or internal degrees of freedom monatomic gas assumption; ii molecules only exert forces on one another within the sphere of influence; iii outside the sphere of influence, motion of molecules described by classical mechanics non-relativistic equations of motion. Statistical Nature of Theory: i individual motion of each molecule is not tracked; ii particle phase space velocity distribution function, f or F, describes the probability of or number of particles having a velocity, v, at position x. + Vectors and tensors using vector and Cartesian tensor indexed notation is reviewed. Vector Notation Cartesian Tensor Notation Vector x = x ı + y j + z k x i Inner product scalar x y x i y i Cross product vector x y ǫ ijk x j y k Outer product dyadic, tensor xy = x y x i y j Gradient vector Divergence scalar f B f B i Curl vector B ǫ ijk B k x j Vector Derivative dyadic, tensor B B i x j + Cartesian, cylindrical, and spherical coordinate systems are reviewed and solid angles defined. CHAPTER 2: EQUILIBRIUM KINETIC THEORY + Chapter 2 introduces the concept of a particle phase space velocity distribution function, f and F, and shows how the distribution function relates to macroscopic measured gas dynamic properties e.g.,, u, and T. 2

3 + Maxwell-Boltzmann Distribution Function: F = M = exp 1 3/2 m 2πp/ 2 c 2 = p m 2πθ 3/2 exp 1 2 c 2 θ Maxwell s original derivation of the velocity distribution function under conditions of thermodynamic equilibrium is reviewed. Notion of drifting Maxwellians is discussed with c = v u and where the average bulk velocity is u = m < vf >=< vf > Distribution of Molecular Speeds: Most Probable and Average Speeds: m 3/2 f v = 4πv 2 exp 1 2πkT 2 v m = 2kT m 1/2 v = mv 2 kt 8kT 1/2 πm + Specific heats and equipartition of translational energy for a monatomic gas. - Equilibrium distribution function for a mixture and Dalton s law. - Specific heats for a diatomic gas. CHAPTER 3: BINARY COLLISIONS + Collisional processes are responsible for establishing thermodynamic equilibrium. They are also the microscopic processes governing all macroscopic transport phenomena. Chapter 3 provides a thorough description of binary collision process i.e., the collisions of two particles. + Kinematics of Two-Particle Collisions: center of mass, particle motion in a central force field, angle of deflection, inverse power inter-particle potentials. + Statistical Description of Collisions: collision frequency, mean free time, mean free path, collision cross section. ν = 2nσ v - Collision rates for gas mixtures. - Chemical reactions. τ = 1 ν CHAPTER 4: ELEMENTARY TRANSPORT THEORY λ = vτ = v ν = 1 2nσ + Elementary theories for describing non-equilibrium transport are reviewed in Chapter 4. + Molecular Effusion: escape from a container through a small orifice. Hydrodynamic escape. Kinetic effusion. j kin = 1 4 n v 3

4 + Fluid Dynamic Macroscopic Transport: Diffusion Mass Diffusion: Fick s Law j i = D n Viscous Drag Diffusion of Momentum: Viscosity and Fluid Stresses τ ij = µ [ ui + u j x j Heat Flow Thermal Diffusion: Fourier s Law q i = κ T 2 ] 3 δ u α ij x α + Mean Free Path Method: Assumption that the characteristic length scales are much larger than the mean free path, λ. Provides expressions for the fluid dynamic transport coefficients i.e., diffusion coefficient, D, viscosity, µ, and thermal conductivity, κ. + Flow in a Tube: Knudsen Number: Poiseuille Flow Kn < Slip Flow 0.01 < Kn < 0.1. Free Molecular Flow Kn > CHAPTER 5: THE BOLTZMANN EQUATION Kn = λ l + Chapter 5 provides a derivation of the Boltzmann equation and discusses some of its mathematical properties. + Derivation of the Boltzmann Equation: Underlying Assumptions: i the effective range of the inter-molecular forces are much smaller than the mean free path; and ii molecular chaos colliding particles are uncorrelated and undergo many collisions with other particles before re-colliding. The latter leads to irreversibility in solutions of the Boltzmann equation. + Boltzmann s H-Theorem: F t + v F F i + a i = δf v i δt dh dt 0 + Equilibrium distribution and five summation invariant quantities mass, momentum, energy. + Relaxation Time Approximation for the Collision Operator BGK Model: δf δt = F M τ 4

5 - Relaxation Time Approximation for the Collision Operator multi-species. - Fokker-Planck Approximation for the Collision Operator. CHAPTER 6: GENERALIZED TRANSPORT EQUATIONS + Approximate techniques for constructing general non-equilibrium solutions to the Boltzmann equation invariably give rise to generalized transport equations. Chapter 6 provides a description of Grad-type moment closure methods and the resulting transport equations which arise. The latter must be solved in place of directly solving the Boltzmann equation. It is important to note that moment closure methods provide a means for constructing approximate non-equilibrium solutions to the Boltzmann equation. + Moments of the Boltzmann Equation and Maxwell s Equations of Change: Conserved and non-conserved forms of Maxwell s equations of change. + Grad-Type Moment Closures: 20-Moment Closure, u i, P ij, Q ijk. R ijkl = 1 {P ijp kl } 3 [ijkl] 1 {pδ ij P ij pδ kl P kl } 3 [ijkl] 13-Moment Closure, u i, P ij, q i. Q ijk = 2 5 δ ijq k + δ ik q j + δ jk q i 10-Moment Closure, u i, P ij. 8-Moment Closure, u i, p, q i. Q ijk = 0 P ij = δ ij p Q ijk = 2 5 δ ijq k + δ ik q j + δ jk q i 5-Moment Closure, u i, p: Euler equations and assumption of local thermodynamics equilibrium LTE. P ij = δ ij p Q ijk = 0 + BGK Collision Terms for 13-Moment Closure single species: δp ij δt = 1 τ P ij pδ ij δ δt = δu i δt = δp δt = 0 δτ ij δt = τ ij τ δq i δt = q i τ + Recovery of Navier-Stokes Equations: 20- and 13-moment closures contain Navier- Stokes equations. - Collision Terms for Multi-Species Gases. CHAPTER 7: FREE MOLECULAR AERODYNAMICS 5

6 + For free molecular flows, it is assumed that the mean free path is much larger than the largest characteristic scale of the flow geometry. Under this assumption, the interaction of the gas molecules can be completely neglected and only the interaction of the gas with solid surfaces must be taken into account. Free molecular flow theory is reviewed in Chapter 7. + Flux of Mass, Momentum, and Translational Energy at a Solid Body: Reflection Coefficients. Mass Transfer. Perpendicular Momentum Transfer. Tangential Momentum Transfer. Translational Energy Transfer. + Free Molecular Heat Transfer: Heat transfer between two plates. + Free Molecular Aerodynamic Forces: Pressure and shearing forces. - Free Molecular Heat Transfer to Specific Bodies. Stanton number and thermal recovery factor. - Free Molecular Aerodynamic Forces for Specific Bodies. Lift and drag forces. CHAPTER 8: SHOCK WAVES + The structure of steady one-dimensional planar shocks are studied using fluid dynamic and simple kinetic descriptions in Chapter 8. + Shock-Wave Solutions: Fluid dynamics: Euler equations. Fluid dynamics: Navier-Stokes equations. Kinetic theory: Mott-Smith model. Subjects denoted with a + sign are core material that you will be responsible for in the final exam and subjects denoted with a - sign are additional course material that you will not be examined on. 2.2 Some Additional References A list of additional references dealing with the subject matter covered in this course are given below. Students are encouraged to consult these references to supplement the material covered in the course textbook. Textbook: Gaskinetic Theory, by T. I. Gombosi, Cambridge University Press, An Introduction to Thermodynamics, the Kinetic Theory of Gases, and Statistical Mechanics, by F. W. Sears, Addison-Wesley, Molecular Flow of Gases, by G. N. Paterson, John Wiley and Sons,

7 The Mathematical Theory of Non-Uniform Gases, by S. Chapman and T. G. Cowling, Cambridge University Press, An Introduction to the Kinetic Theory of Gases, by J. H. Jeans, Cambridge University Press, Introduction to Physical Gas Dynamics, by W. G. Vincenti and C. H. Kruger, John Wiley and Sons, Flow Equations for Composite Gases, by J. M. Burgers, Academic Press, Rarefied Gas Dynamics, by M. N. Kogan, Plenum Press, An Introduction to the Theory of the Boltzmann Equation, by S. Harris, Holt, Rinehart, and Winston, Fundamentals of Maxwell s Kinetic Theory of a Simple Monatomic Gas, Treated as a Branch of Rational Mechanics, by C. Truesdell and R. G. Muncaster, Academic Press, Molecular Nature of Aerodynamics, by G. N. Patterson, UTIAS, Extended Thermodynamics, by I. Müller and T. Ruggeri, Springer-Verlag, The Mathematical Theory of Dilute Gases, by C. Cercignani, R. Illner, M. Pulvirenti, Springer-Verlag, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, by G. A. Bird, Oxford Science Publications, Ionospheres: Physics, Plasma Physics, and Chemistry, by R. W. Schunk and A. F. Nagy, Cambridge University Press, Flow Regimes for a Monatomic Gas 3.1 Knudsen Number The Knudsen number, Kn, is a measure of a gas potential to maintain conditions of thermodynamic equilibrium. It is defined as the ratio of the mean free path the average distance traveled by a gas particle between collisions, λ to an appropriate reference length scale, l, characterizing the flow: Kn = λ l. 1 When the mean free path is small compared with the characteristic length scale or dimension of interest i.e., for Kn 1, the gas will undergo a large number of collisions over typical length scales of interest and assumptions of near thermal equilibrium apply. For such flows the continuum hypothesis is valid and conventional fluid dynamic macroscopic descriptions for the fluid behaviour i.e., the Navier-Stokes equations are appropriate. Note that on average gas particles must undergo only about 3 to 4 binary collisions to equilibriate the translational energy modes. When the mean free path becomes comparable to or larger than the characteristic length scale i.e., for Kn 1 and Kn > 1, the gas is unable to maintain conditions of thermal equilibrium and the continuum hypothesis fails. As a consequence, fluid dynamic descriptions based on assumptions of near equilibrium break down. For such flows, a microscopic description for the fluid behaviour is required as may be provided by gas kinetic theory and the Boltzmann equation. 7

8 3.2 Continuum, Slip, Transition, and Free-Molecular Flow Regimes Depending on inter-particle collision rate, and hence the flow Knudsen number, four flow regimes may be identified for a monatomic gas. These four flow regimes and the ranges in the Knudsen number which define each regime are listed below. Continuum Regime Kn 0.01 collision-dominated flow conventional fluid-dynamic equations i.e., the Navier-Stokes Equations are valid Slip-Flow Regime 0.01 < Kn 0.1 fluid dynamic equations can be augmented with slip boundary conditions for the flow velocity and temperature Knudsen layer analyses are generally used to formulate appropriate boundary conditions Transition Regime 0.1 < Kn collisions are less frequent but cannot be neglected very difficult regime to model Free-Molecular Flow Regime Kn > collisionless flow inter-particle collisions negligible, must only consider particle interactions with flow field boundaries Note that values for the Knudsen numbers defining the boundaries between the four regimes given above are not sharp and the values listed are typical values that can be used as guides for determining when non-equilibrium rarefied flow effects are important. In general, the mean free path is related to the fluid viscosity, µ. For hard sphere collisions described in the course textbook, the mean free path is given by λ = 16µ 5 1 2πRT, 2 where, T, and R are the density, temperature, and gas constant. This expression can be used to evaluate the flow Knudsen number given a characteristic length scale. l. Some simple analysis can be used to relate the Knudsen number, Kn, to the Reynolds number Re = ul/µ and Mach number M = u/a, where a = γrt is the sound speed for the gas and γ is the specific heat ratio. For flows for which the Reynolds number is in the range 0 < Re < 100 such that inertial terms are relatively small compared with viscous forces, one can write M = u a = u a Reµ ul = µ aλ Re Re = KnRe, 3 al al 8

9 hypersonic flow transonic flow compressibility negligible subsonic flow supersonic flow M 10 4 Newtonian flow M = viscous, inertialless flow Re = 0 Stokes flow inertia negligible space flight M = 1Re l TRANSITIONAL FLOW M = 0.01Re l FREE-MOLECULAR FLOW micrometerology Kn = 1 boundary M = 1 3 1Re x 4/5 1/2 M = 1Re x TRANSITIONAL FLOW M = 0.01Re x 1/2 CONTINUUM FLOW meteorology incompressible flow M = 0 Kn = 0.01 boundary M = Re x 4/5 aeronautics Re 10 8 Oseen flow viscous fluid laminar boundary layer turbulent boundary layer inviscid flow Re = symmetric flow asymmetric flow Figure 1: Flow Regimes for a Monatomic Gas. or Kn = M Re, 4 where, for this derivation, it has been assumed that µ aλ this assumption can be seen to follow from Eq. 2 for the mean free path given above and the characteristic length l is chosen to be some typical body length. This important result given by Eq. 4 is due to von Karman. For larger values of the Reynolds number in the range 100 < Re < 10 5, inertia effects become important and the flows are typically laminar. If one bases the Knudsen number on the thickness of the laminar boundary layer, δ l 10l/Re 1/2 this expression is valid for a developing flat plate laminar boundary layer, then M = u a = u a Re 1/2 u 10l 1 µ ul aλ = Re 1/2 = 10Kn Re 1/2, 5 a δ 1/2 l Re aδ 1/2 l µre aδ l or Kn = 1 M /2 Re Finally, for Re > 10 5, the flows are typically turbulent. Basing the Knudsen number on the thickness of the turbulent boundary layer, δ t l/3re 1/5 this expression is valid for a developing flat plate turbulent boundary layer, then Re 1/2 or M = u a = u a Re 1/5 u l 1 a 3δ t Re 1/5 = 1 3 Re 1/5 µ ul aδ t µre 1/5 1 aλ Re 4/5 = 1 3 aδ t 3 Kn Re4/5, 7 Kn = 3 M. 8 4/5 Re 9

10 Summarizing, the Knudsen number can be related to the Reynolds and Mach numbers as follows: M 0 < Re < 100, Re 1 M Kn = 10 Re 1/2 100 < Re < 10 5, 9 3 M Re 4/5 Re > The preceding expression for the Knudsen number can be used to explore the boundaries between the flow regimes of a monatomic gas as identified above in M,Re phase space. For Kn = 0.01 and Kn = 1, two graphs of M = MRe corresponding to the continuum-flow/slip-flow and transition-flow/free-molecular-flow boundaries can be determined as shown in Fig. 1 above. The resulting figure clearly illustrates the domains of the continuum, slip, transition, and free-molecular flow regimes for a monatomic over the full range of Mach number and Reynolds number. 3.3 Some Numbers for Perspective For air under conditions of STP standard temperature and pressure, T = 273 K, p = kpa, and µ = kg/m-s, the density is relatively large and the mean free path assuming a hard sphere collision model is λ 0.6 µm. 10 This means for flows with characteristic lengths, l, in the range 0.01 m < l < 1 m the Knudsen number will be very small with Kn < The assumption of thermal equilibrium is certainly valid in these circumstances and non-equilibrium effects would be insignificant. However, given these same flow conditions, if the spatial scale of interest is now on the order l 0.5 µm, then the Knudsen number will be Kn 0.12 and the flow would be expected to lie within the transition regime. This illustrates that even under conditions of STP, non-equilibrium rarefied flow effects are important for micron-scale flows of air i.e., for air flows with length scale on the order of 1 µm. 4 Chapman-Enskog Solution of the BGK Equation The Chapman-Enskog perturbative expansion technique is used to determine a hierarchy of approximate solutions to the BGK kinetic equation. 4.1 BGK Kinetic Equation Assuming that there are no external forces i.e., a i = 0 and adopting a relaxation time approximation for the collision operator of the type first proposed by Bhatnagar et al. [2], the kinetic equation describing the time-evolution of the particle phase space distribution function, F, can be written as F t + v F i = δf δt = F M, 11 τ where M is the Maxwell-Boltzmann distribution function and τ is the characteristic relaxation time for collision processes. The Maxwellian particle phase-space distribution function is given by where θ = p/. M = m 2πp/ 3/2 exp 1 2 c 2 p 10 = m 2πθ 3/2 exp 1 2 c 2 θ, 12

11 4.2 Chapman-Enskog Perturbative Expansion Technique In the Chapman-Enskog perturbative expansion technique [3 5], approximate solutions to a scaled version of the kinetic equation F t + v F i = F M, 13 ǫτ are sought which have the following form: F = M f 0 + ǫf 1 + ǫ 2 f 2 + ǫ 3 f , 14 where ǫ is a scaling parameter introduced for the purposes of the perturbative solution analysis with the understanding that ǫ 1. In general, ǫ Kn. This implies that the relaxation time, τ, is small and that we are interested in perturbative solutions from local thermodynamic equilibrium. The corresponding solution to the unscaled kinetic equation i.e., Eq. 11 given above is then given by F = M f 0 + f 1 + f 2 + f , 15 with the assumptions that f 0 = O1, f 1 = Oǫ, f 2 = Oǫ 2, f 3 = Oǫ 3, etc. 16 Substituting the scaled expansion into the scaled kinetic equation yields M f 0 1 τ [ ] + ǫ f 0 M + v i f 0 M + f1 M t τ [ ] ǫ 2 f 1 M + v i f 1 M + f2 M t τ [ ] ǫ 3 f 2 M + v i f 2 M + f3 M t τ + = For non-trivial solutions, require each term of this expansion for the kinetic equation in powers of ǫ to vanish. At this point, it is very important to notice the distinctions between the Chapman- Enskog and Grad-type moment closure expansion techniques! The Chapman-Enskog approach is formally a perturbative expansion in a small parameter, with each term adding only the next higher-order correction to the solution. The Grad approach is a truncated power series expansion and each term in the expansion can contain solution content of all orders. 4.3 Zeroth-Order Solution: The Euler Equations To zeroth order in the small parameter, ǫ, the solution of the kinetic equation must satisfy f 0 1 M τ = This condition yields and f 0 = 1, 19 F M, 20 11

12 where M = M,u i,p = M,u i,θ. 21 Thus to zeroth-order in ǫ, the particle phase space distribution function is approximated by a local Maxwellian which depends on the local values of, u i, and p or θ. This is the so-called local thermal equilibrium LTE approximation. In this case, the unscaled kinetic equation can be written as or M t M t + v i M = 0, 22 + u i + c i M [ ui t + u j + c j u ] i M = x j c i The latter is the non-conservative form of the kinetic equation expressed in terms of the random particle velocity, c i. Note that the value of the BGK collision operator is zero at this level of approximation. Moment equations describing the transport of, u i, and p or θ can be obtained taking the velocity moments m, mc i, and mc 2 /2 of the approximate kinetic equation. These transport equations can be written as t + u i = 0, 24 u i t + u u i j + 1 p = 0, 25 x j p t + u p i p u i = These moment equations are referred to as the Euler equations and complete the specification of the zeroth-order solution. 4.4 First-Order Solution: The Navier-Stokes Equations The first-order correction, f 1, to the zeroth-order result given above must satisfy M t M + v i + f1 M = τ This condition yields f 1 = τ [ ] M M + v i, 28 M t where the phase space distribution function is now approximated by F M 1 + f Substituting this first-order approximation for the distribution function into the unscaled kinetic equation yields the following approximate kinetic equations: [ ] 1 + f 1 [ ] M + v i 1 + f 1 M = f1 M, 30 t τ 12

13 or [ ] 1 + f 1 M t ] 1 + f 1 M + u i + c i [ x [ i ui t + u j + c j u i x j ] [ ] 1 + f 1 M c i = f1 M. 31 τ For consistency with the zeroth-order solution, it is required that m < f 1 M >= 0, 32 m < v i f 1 M >= 0, m < c i f 1 M >= 0, 33 m 2 < v2 f 1 m M >= 0, 2 < c2 f 1 M >= These consistency conditions follow from the definition of the Maxwellian, M, for which m < M >=, 35 m < v i M >= u i, m < c i M >= 0, 36 m 2 < v2 M >= 3 2 p + 1 m 2 u2, 2 < c2 M >= 3 2 p, 37 m < v i v j M >= u i u j + δ ij p, m < c i c j M >= δ ij p, 38 and the definitions of the velocity moments of any phase space distribution function, for which we require that m < 1 + f 1 M >=, 39 m < v i 1 + f 1 M >= u i, m < c i 1 + f 1 M >= 0, 40 m 2 < v2 1 + f 1 M >= 3 2 p + 1 m 2 u2, 2 < c2 1 + f 1 M >= 3 2 p, 41 in the case that F = M1 + f 1. As with the zeroth-order approximation, if we now take velocity moments m, mc i, and mc 2 /2 of the non-conservative form of the approximate kinetic equation, the moment equations for the first-order solution can be obtained. For the continuity equation one can write < m [ ] 1 + f 1 M > + < m u i + c i [ ] 1 + f 1 M > t x [ i ui < m t + u j + c j u ] i [ ] 1 + f 1 M > x j c i = < m f1 M >, τ which can then be evaluated in stages as follows: t < m1 + f1 M > + u i < m1 + f 1 M > + < mc i 1 + f 1 M > x i ui t + u u i j < m [ ] 1 + f 1 M > x j c i u i [ ] < mc j 1 + f 1 M > x j c i = 1 τ < mf1 M >, 13

14 t + u i u i < m [ x j c i For the momentum equation one can write ] c j 1 + f 1 M t + u u i i + δ ij = 0, x j [ ] < mc α 1 + f 1 M > + < mc α u i + c i [ t > + u i x j < m c j c i 1 + f 1 M >= 0, t + u i = ] 1 + f 1 M < mc α [ ui t + u j + c j u i x j = < mc α f 1 M τ which can then be evaluated in stages as follows: >, > ] [ ] 1 + f 1 M > c i t < mc α1 + f 1 M > + u i < mc α 1 + f 1 M > + < mc α c i 1 + f 1 M > x i ui t + u u i [ ] j < mc α 1 + f 1 M > x j c i u i [ ] < mc α c j 1 + f 1 M > x j c i = 1 τ < mc αf 1 M >, < mc α c i M > + < mc α c i f 1 M > x i ui t + u u i j x j u i x j = 0, < m c i [ c α 1 + f 1 M < m c i [ c α c j 1 + f 1 M δ iα p + < mc α c i f 1 M > +δ iα u i t + u u i j + 1 p + 1 x j x j And finally, for the energy equation one can write < m 2 c2 t [ ] 1 + f 1 M > + < m 2 c2 u i + c i [ ] > < m c α 1 + f 1 M > c i ] > < m [c α c j ] 1 + f 1 M > c i ui t + u u i j = 0, x j m < c i c j f 1 M > ] 1 + f 1 M < m 2 c2 [ ui t + u j + c j u i x j = < m 2 c2f1 M τ 14 >, = > ] [ ] 1 + f 1 M > c i

15 which can then be evaluated in stages as follows: t < m 2 c2 1 + f 1 M > + u i < m 2 c2 1 + f 1 M > + < m 2 c ic f 1 M > ui t + u u i j < m [ ] x j 2 c2 1 + f 1 M > c i u i < m x j 2 c jc 2 [ ] 1 + f 1 M > c i = 1 τ < m 2 c2 f 1 M >, 3 p 2 t u p i + < m 2 c ic 2 M > + < m 2 c ic 2 f 1 M > ui t + u u i j < m [ ] c f 1 M > < m x j 2 c i 2 u i < m [ ] c j c f 1 M > < m x j 2 c i 2 = 0, [ c 2] 1 + f 1 M > c i [ c j c 2] 1 + f 1 M > c i p t p + u i + 2 < m 3 2 c ic 2 f 1 M > + 2 ui 3 t + u u i j < mc i 1 + f 1 M > x j δ u i ij < m x j 2 c2 1 + f 1 M > + 2 u i < mc i c j 1 + f 1 M > 3 x j = 0, p t + u p i + 2 < m 3 2 c ic 2 f 1 M > +p u i δ ijp u i + 2 x j 3 < mc ic j f 1 M > u i = 0, x j p t + u p i p u i + 2 < m 3 2 c ic 2 f 1 M > < mc ic j f 1 M > u i = x j Defining the fluid stresses, τ ij, and heat flux, h i, to be m < c i c j f 1 M >= τ ij, 45 m 2 < c ic 2 f 1 M >= h i, 46 the continuity, momentum, and energy equations for the first-order solution can be summarized as follows: t + u i = 0, 47 u i t + u u i j + 1 p 1 τ ij = 0, 48 x j x j 15

16 p t + u p i p u i + 2 h i τ u i ij = x j These are the Navier-Stokes equations, which describe the time evolution of the velocity moments, u i, and p. In order to complete the description of the first-order solution, all that remains is to determine f 1 and calculate expressions for the fluid stresses and heat flux. From Eq. 28 and using the fact that M = M,u i,θ, can write f 1 = τ [ M M t + v i + M uα u α t + v u α i + M ] θ θ t + v θ i, = τ [ M M t + [u i + c i ] + M uα u α t + [u i + c i ] u α + M θ θ t + [u i + c i ] θ ], = τ [ M M t + [u i + c i ] + M uα u α t + [u i + c i ] u α + 1 M p θ t + [u i + c i ] p p M 2 θ t + [u i + c i ] ], = τ [ M M p M 2 θ t + [u i + c i ] + M uα u α t + [u i + c i ] u α + 1 M p θ t + [u i + c i ] p ], 50 The next step is to use the zeroth-order moment equations i.e., the Euler equations to evaluate the convective derivatives of, u i, and p. This is an important approximation in the Chapman-Enskog technique. One can then rewrite the expression for f 1 given above as f 1 = τ [ M M p M 2 θ + 1 M θ u i 5 3 p u i + c i p + c i + M 1 p u α + c i u α x α ]. 51 Now, the derivatives of the Maxwellian must be evaluated. From Eq. 12 it follows that and hence ln M = ln 3 2 ln θ 1 c 2 [ 2 θ ln m 2π 3/2], 52 1 M M = 1, 53 1 M = c i c i = c i M u i θ u i θ = c i p, 54 1 M M θ = 3 2θ + c2 2θ 2 = 3 2p + 2 c 2 2p Substituting these expression into the preceding equation for f 1 can write f 1 = τ [ c2 u i + c i + c α 2p p 16 1 p u α + c i x α

17 = τ + 1 [ 2 3 2p + 2 c 2 2p p u i p + c i 2p 2c ic 2 5 2p c i p ] + p c u j ic j 3p c2 u i ]. 56 Noting that p c u j ic j = 2p c uj ic j + u i, x j the final expression for f 1 can be obtained and written as [ f 1 2 = τ 2p 2c αc 2 5 p 2p c α + x α 2p c uβ αc β + u α x α x β ] 3p c2 u α. 57 x α Note the change of indices. Finally, using the definitions of the fluid stresses and heat flux given by Eqs. 45 and 46, can write τ ij = m < c i c j f 1 M > [ = mτ < c i c j 2p c uβ αc β + u α ] x α x β 3p c2 u α M > x α = τ 2p m < c uβ ic j c α c β M > + u α τ x α 3p m < c ic j c 2 M > u α p 2 = τ 2p [δ ijδ αβ + δ iα δ jβ + δ iβ δ αj ] ui = τp + u j 2 x j 3 τp δ u α ij x α [ ui = µ + u j 2 ] x j 3 δ u α ij x α x β uβ + u α x α x β x α τ p 2 3p [δ ijδ ββ + 2δ iβ δ jβ ] u α x α 58 h i = m 2 < c ic 2 f 1 M > = m 2 τ < c ic 2 2 2p 2c αc 2 5 2p c α p M > x α = 2 τ 4p 2 m < c ic α c 4 M > p + 5τ x α 4p m < c ic α c 2 M > p x α = 2 τ p 3 4p 2 2 [35δ p iα] + 5τ p 2 x α 4p [δ p iαδ ββ + 2δ iβ δ αβ ] x α = 35τp p + 5τp 4 4 [3δ p iα + 2δ iα ] x α = 5τp p 2 = κ T, where µ = τp is the dynamic viscosity and κ = 5kτp/2m is the thermal conductivity. These expressions for the fluid stresses and heat fluxes are identical to those given in the textbook of 17 59

18 y u y = d/2 x y= d/2 u Figure 2: Couette flow between two oppositely moving flat plates. Gombosi [1], showing that a Chapman-Enskog-like expansion can be applied to the 13-moment equations to recover the Navier-Stokes equations i.e., the 13- and 20-moment equations contain the Navier-Stokes model. This does not mean that the moment closure method of Grad and the Chapman-Enskog method are equivalent! An important limitation of the BGK or relaxation time collision operator is revealed by determining the Prandtl number based on the transport coefficients, µ and κ, given above. By definition the Prandtl number is Pr = µc p κ. 60 For a monatomic ideal gas, the specific heat at constant pressure is C p = γr/γ 1 = 5R/2 = 5k/2m. Using this value for C p and the expressions for µ and κ given above can write Pr = 52mkτp = mkτp From this it can be seen that the use of a single relaxation time in the BGK collision operator is equivalent to assuming that the Prandtl number, which is essentially the ratio of relaxation time for the diffusion of momentum to the relaxation time for the diffusion of internal energy, is unity. As only one relaxation time is introduced in the model this should be expected. In actuality, most gases have a Prandtl number somewhat less than one. A value near 0.70 is typical. The use of a hard sphere inverse potential for the collision operator yields a Prandtl number of two thirds 2/ Higher-Order Solutions: Burnett and Super-Burnett Equations The Chapman-Enskog technique can be continued to include more and more terms in the perturbative expansion. To second-order in the small parameter, ǫ, the approximate solutions of the Boltzmann equation satisfy the Burnett and super-burnett equation, respectively, with a BGK collision operator [5 13]. 5 Couette Flow Consider planar subsonic incompressible flow between twp oppositely moving infinite plates as shown in Fig. 2. Three solutions to this problem will be considered: 18

19 i the Navier-Stokes continuum flow solution, ii a kinetic free-molecular flow solution, iii and a slip flow solution based on the mean free path method. The solution due to Lees [14] is also considered. A summary of the couette flow solution in each case now follows: 5.1 Navier-Stokes Continuum Flow Solution uy = u y d, 62 and τ xy = τ wall = µ u d, 63 uy = d/2 2 = 1, 64 u τ wall 1 2 u 2kT πm where Kn = λ/d and µ = 1 2 λ v = 1 2 λ 8kT /πm. 5.2 Kinetic Free-Molecular Flow Solution = 2Kn, 65 Assuming full a accommodation: and uy = 0, 66 τ xy = τ wall = 1 2 u 2kT πm, 67 uy = d/2 2 = 0, 68 u 1 2 u τ wall 2kT πm = 1, Slip Flow Solution Assuming full a accommodation σ = 1: and uy = u y d τ xy = τ wall = µ u d uy = d/2 2 = u τ wall 1 2 u 2kT πm λ 3 d λ 3 d, 70, , 72 3Kn = 2Kn 1 + 4, 73 3Kn 19

20 1 2 u y=d/2 / u0 0.1 Navier-Stokes Free-molecular flow Slip flow Mean Free Path Lees solution Kn = l/d Figure 3: Normalized flow velocity at the upper surface as a function of Knudsen number, Kn = λ/d, for planar subsonic Couette flow between parallel diffusively reflecting walls. Results are shown for Navier-Stokes, free-molecular flow, and slip flow models, as well as an approximation due to Lees txy 0.01 Navier-Stokes Free-molecular flow Slip flow Mean Free Path Lees solution Kn = l/d Figure 4: Normalized shear stress as a function of Knudsen number, Kn = λ/d, for planar subsonic Couette flow between parallel diffusively reflecting walls. Results are shown for Navier-Stokes, free-molecular flow, and slip flow models, as well as an approximation due to Lees. 5.4 Comparison of Solutions These three solutions are compared to the Lees solution in Figs. 3 and 4. 20

21 6 Shock-Wave Structure 6.1 Background The application of several moment closures to the prediction of planar shock structure for a monatomic gas is considered. Shock profile prediction is a challenging problem that features significant departures from LTE, yet it is unencumbered with difficulties associated with complex geometries and/or boundary condition prescription. For these reasons it is useful for evaluating the capabilities of moment methods. Included in the investigation are results for the 13- and 20- moment closures of Grad [15], which are based on expansions about a local Maxwellian, the 20- and 35-moment Gaussian-based closures recently proposed by Groth et al. [16, 17], and the 10- moment Gaussian closure considered in other recent studies by Brown et al. [18] and Levermore and Morokoff [19], theoretical descriptions for all of which are provided for the case of planar onedimensional flow. A higher-order point-implicit upwind finite-volume scheme, based on a MUSCLtype solution reconstruction procedure is used to solve the hyperbolic equation sets resulting from the closures and the predicted shock structures of the moment models are compared to those of the Navier-Stokes and Burnett equation, as well as DSMC calculations. 6.2 Moment Closures for One-Dimensional Planar Flows In classical kinetic theory, the state of a dilute gas is prescribed by a distribution function for the particle velocities, F, whose time evolution in phase space is governed by the Boltzmann equation. Neglecting external forces, the Boltzmann equation for a simple monatomic gas can be formulated in terms of the components of the particle velocity v i, position coordinates x i, and time t and written as F t + v F i = δf δt, 74 where δf/δt is the Boltzmann collision operator representing the time rate of change of the distribution function produced by inter-particle collisions. Refer to the texts by Chapman and Cowling [20] and Gombosi [1] for further details. A recent mathematical treatment of kinetic theory and the Boltzmann equation is given by Cercignani et al. [21]. In standard perturbative moment closure techniques the non-perturbative closures recently proposed by Levermore [22] will not be considered here, the complex problem of directly solving the Boltzmann equation in six-dimensional phase space is avoided and replaced with one of solving a system of generalized transport equations for various fluid quantities. Such a simplification is achieved by assuming that the distribution function can be approximated by a series expansion of the form F = W [ 1 + A + B α c α + C αβ c α c β + D αβγ c α c β c γ + E αβγδ c α c β c γ c δ + ], 75 where W is the weight function for the expansion, c i = v i u i is the random particle velocity, and u i is the average or bulk velocity of the particles. The expansion coefficients A, B α, C αβ, D αβγ, and E αβγδ, etc..., are in general taken to functions of the velocity moments of F having the form σ = mmfc α,x α,td 3 c = m < MF >, 76 where M = Mc and m is the particle mass. For M = 1, σ =, where is the mass density. Other higher-order velocity moments up to 5th-order can be defined as follows: P ij = m < c i c j F >, 77 21

22 Q ijk = m < c i c j c k F >, 78 K ijkl = m < c i c j c k c l F > 1 [P ijp kl + P ik P jl + P il P jk ], 79 S ijklm = m < c i c j c k c l c m F >, 80 where P ij is the symmetric generalized pressure tensor, Q ijk is the generalized heat flow tensor, and the fourth- and fifth-order tensors R ijkl and S ijklm represent the deviatoric components of the fourth- and fifth-order velocity moments, respectively. The hydrodynamic pressure, p, is related to the pressure tensor by the expression p = P ii /3 and the usual fluid-dynamic or deviatoric stress tensor, τ ij, is given by τ ij = pδ ij P ij. The heat flux vector, h i, is related to a simple contraction of the heat flow tensor and given by h i = Q ijj /2. The infinte series of Eq. 75 is truncated by adopting an appropriate closing approximation and the time evolution of the resulting finite set of velocity moments, and hence expansion coefficients, are determined by solving a set of moment equations. Non-equilibrium solutions to the Boltzmann equation can then be constructed from the solutions of the moment equations using the truncated series expansion relating the approximate distribution function to the predicted velocity moments. As noted in the introduction, the system of partial-differential equations PDEs describing the transport of various velocity moments that results from the application of a moment closure are quasi-linear and hyperbolic. They also contain source terms representing inter-particle collisional relaxation processes. These hyperbolic systems may be expressed in the following weak conservation form: U N t + x F N = S N, 81 where U N is the solution vector of conserved velocity moments, F N is a tensor representing the moment fluxes, and S N is the source vector describing the time rate of change of the velocity moments produced by collisional processes. The integer N has been introduced to indicate the number of dependent variables associated with the full three-dimensional form of the closure model. The specification of the source terms, S N, necessitates the mathematical modeling of the collision operator δf/δt. In the present numerical study, a two-scale relaxation-time approximation is adopted, as suggested by Levermore [22] and considered by Groth and Levermore [23]. In this simplified model, complicated integral expressions for the collision operator are replaced by δf δt = F M Pr 1 F G, 82 τ τ where τ is a characteristic relaxation time for the collision processes, Pr is the Prandtl number, and M and G are local Maxwellian and Gaussian distribution functions, respectively, given by M = exp 1 c 2, 83 3/2 m 2πp/ 2 p G = m2π 3/2 exp 1 1/2 2 Θ 1 αβ c αc β. 84 The characteristic relaxation time, which is the inverse of the collision frequency, ν = 1/τ, can be related to the dynamic viscosity, µ, by means of the relationship τ = µ/p. The symmetric tensor, Θ ij, is related to the pressure tensor by the expression Θ ij = P ij / and = detθ. The 22

23 Maxwellian, M, corresponds to the form for solutions of the Boltzmann equation under conditions of LTE i.e., δf/δt = 0. The Gaussian distribution, G, is a form for non-equilibrium solutions of the Boltzmann equation first deduced by Maxwell [24] and then re-discovered, or at least reconsidered, many years later by Schlüter [25] and then again by Holway [26]. It is a generalization of the Maxwellian that accounts for the influences of non-zero fluid stresses in a non-perturbative manner. Mathematical properties of the Gaussian are discussed by Levermore [22] and Groth et al. [16]. The collision operator of Eq. 82 is a two-time-scale extension of the BGK collision operator [2] that preserves the usual collisional invariants and, under equilibrium conditions, δf/δt = 0 and F = M as required. The model also permits the formal recovery of a physically correct Navier- Stokes limit for Prandtl numbers different from unity, provided the functional dependence of the viscosity coefficient is adequately modeled. For a monatomic gas, Pr = 2/3 and, in many cases, the viscosity can be taken to have a power-law dependence on the temperture, T, of the form µ = µ o T/T o ω, 85 where µ o and T o are reference values and the exponent ω depends on the form for the forces governing inter-particle collisional processes ω = 1 for Maxwell molecules. Although there are inaccuracies associated with relaxation-time approximations of this type, the model is thought to be sufficient for the applications considered herein. The remainder of this section will provide a brief summary of the moment closures considered in the present study, for the particular case of planar one-dimensional flows in the x-direction of a Cartesian coordinate system x,y,z. Included in the summary are the Maxwellian and Gaussian closures, Maxwellian-based 13- and 20-moment closures of Grad [15], and Gaussian-based 20- and 35-moment models described by Groth et al. [16,17]. In each case the approximate form for the distribution function, moment equations, and closing relationship are discussed. Note that for the one-dimensional case, the moment equations of Eq. 81 reduce to U N t where F N is the conservative flux vector Moment Maxwellian Closure + FN x = SN, 86 The most elementary moment closure is the 5-moment Maxwellian model for which N = 5. In this approximation, it is assumed that the gas is everywhere in LTE and that the phase-space velocity distribution function is given by the Maxwellian, M, which for one-dimensional planar flows has the form [ M = exp 1 c 2 x + 2c 2 ] y. 87 3/2 m 2πp/ 2 p The closing relationships implied by these assumptions are τ xx = 0, h x = 0, 88 and the resulting hyperbolic set of moment equations are the well-known Euler equations of compressible gas dynamics that describe the evolution of, u, and p. The solution vector, U 5, and source vector, S 5, of the Euler equations for a monatomic non-reacting gas can be expressed as U 5 = u 1 2 u p 23, S5 =

24 According to this inviscid model, the propagation speed of acoustical disturbances is a = 5p/3 and strict hyperbolicity of the Euler equations is ensured for > 0 and p > 0. Note that the discontinuous solutions of shock wave structure provided by the Maxwellian closure are fully understood and will not be considered herein. However, the Euler equations are used in the shock profile computations for the the higher-order moment closures to prescribe initial and boundary data and have therefore been briefly reviewed Moment Maxwellian-Based Closure The 13- and 20-moment closures of Grad [15] offer estimates of non-equilibrium solutions to the Boltzmann equation that are implicitly based on perturbations to the equilibrium Maxwellian solution. Both closures are based on a series expansion for F of the form given by Eq. 75 where the weight function is taken to be the Maxwellian i.e., W = M. In the one-dimensional case, the approximated form for the velocity distribution of the 13-moment model can be written as [ F 13 = M 1 h x c x a 3 x a x + P xx P yy c 2 x 3a 2 x a 2 x + h x c x c 2 x 5a 3 x a x a c2 y x a 2 x c2 y a 2 x ], 90 where M is given by Eq. 87, a x = p/, p = P xx + 2P yy /3, and closure has been achieved by assuming that Q xxx = 6 5 h x, Q xyy = 2 5 h x, 91 K xxxx = 3 τ2 xx, K xxyy = 1 τxx 2 2, 92 K yyyy = 3 τxx 2 4, 93 with τ xx = 2P xx P yy /3. For planar one-dimensional flows, there are five dependent variables associated with the 13-moment closure. The solution vector of the moment equations describing the transport of, u, P xx, P yy, and h x is U 13 = u u 2 + P xx P yy 1 2 u u3p xx + 2P yy + h x

25 The related source column vector, derived by evaluating the appropriate velocity moments of the two-scale relaxation-time model for the collision operator described above, is given by S 13 = τ P xx P yy 1 3τ P xx P yy 2 3τ up xx P yy Pr τ h x. 95 Note that, as with all moment closures, the 13-moment approximation has introduced additional transport equations beyond those appearing at the level of the Maxwellian for describing the higherorder velocity moments such as heat flux, h x. This is contrast to the Navier-Stokes and Burnett models which express these higher-order moment in terms of gradients of the lower-order quantities Moment Maxwellian-Based Closure The primary difference between the 13- and 20-moment Grad closures is that, in the 20-moment model, the closing relationship for the heat flow tensor expressed in terms of the heat flux vector, which for the one-dimensional case is represented by the relations given in Eq. 91, are not utilized and the full heat flow tensor is considered. The approximate form for the distribution function of the 20-moment closure is given by [ F 20 Qxxx = M 1 2a 3 + Q xyy cx x a 3 x a x + P xx P yy c 2 x 3a 2 x a 2 c2 y x a 2 x + Q xxx c 3 x 6a 3 x a 3 + Q yyy c x c 2 y x a 3 x a x a 2 x ], 96 and the solution and source vectors of the system of moment equations for the six dependent variables, u, P xx, P yy, Q xxx, and Q yyy may be written as U 20 = u u 2 + P xx P yy u 3 + 3uP xx + Q xxx up yy + Q xyy, 97 25

26 S 20 = τ P xx P yy 1 3τ P xx P yy 2 τ up xx P yy Pr τ Q xxx 1 3τ up xx P yy Pr τ Q xyy, 98 where, as for the 13-moment model, the source terms of Eq. 98 have been evaluated using the collision operator of Eq Moment Gaussian Closure The 10-moment Gaussian closure model may be constructed by assuming the the heat flow tensor, Q ijk, is identically zero and then further assuming that the distribution function, F, can be adequately represented by the Gaussian, G. For one-dimensional planar flows in the x-direction, the first assumption leads to the following closing relation: Q xxx = Q xyy = 0, 99 and the Gaussian simplifies to the well-known bi-maxwellian distribution which may be expressed as G = m 2π/ 3/2 1/2 P xx Pyy 2 [ exp 1 ] c 2 2 x + 2 c2 y. 100 P xx P yy Like the previous Maxwellian-based models, the moment equations of the Gaussian closure can be expressed in the weak conservation form of Eq. 86 where the column vectors of conserved velocity moments and sources are given by U 10 = S 10 = u u 2 + P xx P yy τ P xx P yy 1 3τ P xx P yy, Although the Gaussian closure does not account for the effects of heat transfer, it does provide a representation of adiabatic non-equilibrium flows that accounts for the effects of non-zero fluid 26

27 stresses in a non-perturbative manner i.e., unlike the expansions for the velocity distribution of the 13- and 20-moment closure, differences in the components of the pressure P xx and P yy are accounted for directly in the argument of the exponential of Eq. 100 such that G remains positive for all physically realistic values of P ij. It can also be shown that the one-dimensional form of the moment equations of the Gaussian model remain strictly hyperbolic for > 0 and P xx > 0. These and other more detailed mathematical and solution properties of the Gaussian closure have been explored in several recent studies [16,18,19,22]. Refer to these references for further details Moment Gaussian-Based Closure An alternate hierarchy of higher-order closure models to the Maxwellian-based heirarchy considered by Grad can be constructed by using the Gaussian as a weight function in place of the usual Maxwellian. Gassian-based closures of this type have been recently considered by Groth et al. [16, 27]. At the 20-moment level, the one-dimensional form for the approximate particle velocity distribution function is F 20 = G [ 1 Qxxx 2a 3 x + Q xyy a 3 x P xx cx P yy a x + Q xxx c 3 x 6a 3 x a 3 + Q xyy Pxx 2 c x c 2 y x a 3 x Pyy 2 a x a 2 x ], 103 where a x = P xx / and closure is provided by assuming that the deviatoric components of the fourth-order velocity moments vanish. The latter implies that K xxxx = K xxyy = K yyyy = 0, 104 in the one-dimensional case. The solution and source column vectors for the system of moment equations that results from this 20-moment Gaussian-based approximation are identical to those of the 20-moment Maxwellian-based closure given in Eqs. 97 and 98; the differences in the two quasi-linear equations sets appear in the moment flux vectors not shown here. In comparison to the Grad Maxwellian-based model, an important advantage of the Gaussian-based approximation is that the influences of the fluid stresses are incorporated directly in the weight function of the expansion, W = G, and, as such, the approximate form for the distribution function should remain valid for a larger range of flow conditions. It can also be shown that the eigenstructure of the 20- moment Gaussian-based model has a much simpler mathematical form than that of the Maxwellianbased model, making the Gaussian-based model more amenable to solution by numerical methods. It should also be mentioned that the Gaussian-based 20-moment model has similarities with the bi-maxwellian-based 16-moment closure considered in the studies of Oraevskii et al. [25], Chodura and Pohl [28], Demars and Schunk [29], and Barakat and Schunk [30]. In fact, for planar onedimensional flows, the two closures are identical in every respect Moment Gaussian-Based Closure A Gaussian-based 35-moment closure that accounts for variations in the deviatoric components of the fourth-order velocity moments can be constructed by closing at the level of the fifth-order velocity moments. Such a model was recently proposed and investigated in considerable detail by Groth et al. [16] and has been shown to possess several desirable mathematical properties. For one-dimensional planar flows, the closing relations relating the fifth-order velocity moments to the 27

28 lower order quantities can be written as S xxxxx = 10 P xxq xxx S xyyyy = 3 P xxq xyy, S xyyyy = 6 P yyq xyy + P yyq xxx and the truncated series expansion for the distribution function is given by [ F 35 = G 1 + K xxxx 8a 4 + K xxyy P xx x 2a 4 + K yyyy Pxx 2 x P yy 3a 4 x Pyy 2 Qxxx 2a 3 x Kxxxx 4a 4 x Kxxyy 2a 4 x + Q xyy a 3 x + K xxyy 2a 4 x P 2 xx P 2 yy P xx P yy P xx P yy cx + 2K yyyy 3a 4 x, 105, 106 a x c 2 x a 2 x c 2 y P 3 xx P 3 yy + Q xxx c 3 x 6a 3 x a 3 + Q xyy Pxx 2 c x c 2 y x a 3 x Pyy 2 a x a 2 x + K xxxx c 4 x 24a 4 x a 4 + K xxyy Pxx 2 c 2 x c 2 y x 2a 4 x Pyy 2 a 2 x a 2 x ] + K yyyy 6a 4 x Pxx 4 c 4 y Pyy 4 a 4 x a 2 x. 107 The solution and source vectors of the hyperbolic set of moment equations describing the transport of the nine dependent variables, u, P xx, P yy, Q xxx, Q yyy, K xxxx, K xxyy, and K yyyy for 35-moment model may be expressed as [ U 35 =,u,u 2 + P xx,p yy, u 3 + 3uP xx + Q xxx,up yy + Q xyy, u 4 + 6u 2 P xx + 3 P 2 xx + 4uQ xxx + K xxxx, u 2 P yy + P xxp yy + 2uQ xyy + K xxyy, 3 P yy 2 ] T + K yyyy, 108 S 35 = [ 0,0, 2 3τ P 1 xx P yy, 3τ P xx P yy, 2 τ up xx P yy Pr τ Q xxx, 1 3τ up xx P yy Pr τ Q xyy, 4 3τ 2P xx + P yy P xx P yy 28