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Boundary Conditions for the Polyatomic Gases B. Rahimi H. Struchtrup Workshop on Moment Methods in Kinetic Theory II , 2014 Workshop on Moment Methods in Kinetic Theo B. Rahimi, H. Struchtrup (UVic) Boundary Conditions for the Polyatomic Gases / 8

R19 set of equations The regularized 19 (R19) set of equations consists of 19 PDEs for optimized moments ρ, v i , θ, ∆ θ, σ ij , q i , ∆ q i , B + , B − � � , ij and u 0 , 0 And 3 constitute equations for B + ij , B − ijk . R19 is the polyatomic counterpart of the monatomic’s R13 equations. Workshop on Moment Methods in Kinetic Theo B. Rahimi, H. Struchtrup (UVic) Boundary Conditions for the Polyatomic Gases / 8

Grad’s 36 phase density The generalized Grad’s 36 phase density for polyatomic molecules is � λ 0 , 0 + λ 0 , 0 C i + λ 1 , 0 C 2 + λ 0 , 0 < ij > C < i C j > + λ 0 , 1 e int f | 36 = f int i C i C 2 + λ 0 , 1 + λ 1 , 0 C i e int + λ 1 , 0 < ij > C 2 C < i C j > + λ 2 , 0 C 4 i i � + λ 0 , 0 < ijk > C < i C j C k > + λ 0 , 1 < ij > C < i C j > e int + λ 1 , 1 C 2 e int , This should reproduce the set of 36 raw moments { ρ, ρθ tr , ρθ int , σ ij , q i , tr , q i , int , u 1 , 0 ij , u 2 , 0 , u 0 , 0 ijk , u 0 , 1 ij , u 1 , 1 } , Therefore, λ 0 , 0 = 4 u 1 , 1 + u 2 , 0 , λ 0 , 1 = − u 1 , 1 8 − 3(2+ δ ) θ tr 2 δθ − 3(5 − δ ) θ tr + 5 δρθ 3 + 15 and ... 8 ρθ 2 4 θ 2 δθ 2 Workshop on Moment Methods in Kinetic Theo B. Rahimi, H. Struchtrup (UVic) Boundary Conditions for the Polyatomic Gases / 8

Wall boundary condition The wall boundary condition ˜ f ( c ) is � χ [(1 − ζ ) f tr , w + ζ f int , w ] + (1 − χ ) f ∗ n . ( c − v w ) ≻ 0 , | 36 n . ( c − v w ) ≺ 0 . f | 36 Two equilibrium distribution functions � 3 2 exp ρ I , w � � 2 θ w C 2 � 1 − 1 f tr = , 2 πθ w m � � �� f int = ρ w C 2 1 1 − 1 2 ) exp 2 + e int . m 3 Γ ( 1+ δ θ w 2 θ ( δ +3) / 2 (2 π ) w Workshop on Moment Methods in Kinetic Theo B. Rahimi, H. Struchtrup (UVic) Boundary Conditions for the Polyatomic Gases / 8

Boundary conditions The macroscopic boundary conditions are obtained by multiplying the wall � � ˜ distribution function by f � � 2 + I 2 /δ � � 5 Pr qint δ Pr qtr I 2 /δ � C 2 C 2 C y , C x C y , C y , C y 2 − 2 C 2 + I 2 /δ − 14+ δ 1 − 14 � � � � � , C x C y 2 θ , C y C y C y , C x C x C y . δ Be odd in the normal component of the particle velocity. Fluxes should be prescribed not the variables. Workshop on Moment Methods in Kinetic Theo B. Rahimi, H. Struchtrup (UVic) Boundary Conditions for the Polyatomic Gases / 8

Boundary conditions stress tensor � � θ + u 0 , 0 � 5 Pr qint q x + δ Pr qtr ∆ q x χ 2 xyy σ xy = − Υ V s + , √ √ 2 − χ π 5 ( 5 Pr qint + δ Pr qtr ) 2 θ heat flux, � � B − + (3 + δ ) (140+ δ (32+ δ )+(14 − δ ) δζ ) χ (56 − δ (1 − ζ )) 2 B − q y = − yy (2 − χ ) πθ 312 4(14+ δ )(42+25 δ ) B + + δ (4 − δ (1 − ζ )) + [(1 − ζ ) δ − 4] yy − (2+ δ (1 − ζ )) θ ( ρ ∆ θ − σ yy ) + δ (1 − ζ ) 4(42+25 δ ) B + ρθ 2 312 4 2 √ �� + Υ (4 + δζ ) ( θ − θ w ) − (1 − ζ ) ( δθ + 3∆ θ ) − V 2 � θ s 2 √ θ w = where, Υ = ρ w δ B + (14 − δ )(3+ δ ) yy + B + − B − 2 + σ yy θ + ρ (2 θ − ∆ θ ) 2 B − yy − − √ √ . 3 3 3 2 2 θ 2(14+ δ )(42+25 δ ) θ 156 θ 2 2(42+25 δ ) θ Workshop on Moment Methods in Kinetic Theo B. Rahimi, H. Struchtrup (UVic) Boundary Conditions for the Polyatomic Gases / 8

Boundary conditions Heat flux difference � 15 Pr qint Υ(1 − ζ ) √ � 5(6+ δ ) Pr qint − 6 δ Pr qtr χ 2 B + ∆ q y = θ ∆ θ − yy (2 − χ ) πθ 2 δ Pr qtr (14+ δ ) 4 Pr qtr (42+25 δ ) 5(42+25 δ ) Pr qint − 6(14 − δ ) Pr qtr 5 Pr qint − 4 Pr qtr 2 Pr qtr (14+ δ ) ρθ 2 B − + [3 + δ ] yy − 4 Pr qtr (14+ δ )(42+25 δ ) 5(40 − δ ) Pr qint − 125 δ Pr qtr B − + 5(12+ δ ) Pr qint +125 δ Pr qtr B + + 312 δ Pr qtr 312 δ Pr qtr 5(6 − δ ) Pr qint +12 δ Pr qtr 5 Pr qint − 6 Pr qtr − ρθ ∆ θ − 4 Pr qtr (14+ δ ) θσ yy 4 δ Pr qtr (14+ δ ) √ Υ Pr q tr V 2 � − θ s − (5 Pr q int − 4 Pr q tr ) θ 2 Pr qtr (14+ δ ) (5 Pr q int ζ − 4 Pr q tr ) ( θ − θ W )]] xy , u 0 , 0 yyy and u 0 , 0 Also, BCs for B − xxy . Workshop on Moment Methods in Kinetic Theo B. Rahimi, H. Struchtrup (UVic) Boundary Conditions for the Polyatomic Gases / 8

Thanks for listening ! More info: B. Rahimi and H. Struchtrup: Capturing non-equilibrium phenomena in rarefied polyatomic gases: A high-order macroscopic model, Phys. Fluids 26, 052001 (2014). Workshop on Moment Methods in Kinetic Theo B. Rahimi, H. Struchtrup (UVic) Boundary Conditions for the Polyatomic Gases / 8