Ghost effect by curvature Providence, November 2011 M&MOCS - PowerPoint PPT Presentation

Ghost effect by curvature Providence, November 2011 M&MOCS – MATHEMATICS AND MECHANICS OF COMPLEX SYSTEMS Universit` a dell’Aquila - Italy Ghost effect by curvature – p. 1/39

Joint project with : L. Arkeryd R. Marra A. Nouri Ghost effect by curvature – p. 2/39

Ghost effect The macroscopic equations for the hydrodynamic fields are derived from the Boltzmann equation in the limit when the Knudsen number K goes to 0 . von Karman relation between K , R (Reynolds number) and M (Mach number): M ∼ KR . To keep the Reynolds number R finite, one has so take M = O ( K ) . This and the assumption that temperature T and density ρ are constant at the lowest order in K , as is well known, give the incompressible Navier-Stokes & Fourier equations as limit of the Boltzmann equation. What about ρ and T non constant at the lowest order in K ? Ghost effect by curvature – p. 3/39

Ghost effect The formal answer can be obtained by expanding the solution to the Boltzmann equation in K . The result is somewhat unexpected: the macroscopic equations differ from the incompressible Navier-Stokes equations by the presence of a thermal stress tensor and div u � = 0 . Moreover, the equation for the temperature differs from the standard heat equation. The responsible for this modification is the convective motion described by the velocity field u , which is O ( K ) , hence invisible in the limit K → 0 , but affects the temperature which O (1) in K . Ghost effect by curvature – p. 4/39

Ghost effect For this reason Y. Sone called this phenomenon Ghost Effect. REMARK: If one starts from the standard compressible Navier-Stokes-Fourier equations, instead of the Boltzmann equation, and takes the limit M → 0 without assuming ρ and T constant, there is no ghost effect. Ghost effect is a purely kinetic feature. The term ghost effect is used by Sone to refer more generally to situations where the macroscopic limiting equations contain finite modifications due to the presence of quantities which are infinitesimal in the limit. The ghost effect by curvature is an example of such situations. Ghost effect by curvature – p. 5/39

Ghost effect by curvature ε ∼ K . Boltzmann equation: : 5 7 8 9 = ; < 3 > ? @ 4 6 2 0 / . 1 A " # $ F E D C B & % ! ( ) * + , - ~ N O G M ' L H I J K j o p v k l q r s t u z w x y { | } n U T S R Q P m i h ^ V g Y X \ ] W _ ` a b c f d e [ Z Z ( v r , v θ , v z ) components of v in the basis ( e r , e θ , e z ) . L+D � � ∂F ∂r + v θ ∂F ∂F ∂z + v θ ∂F ∂F L − v r ∂θ + v z v r v θ r r ∂v r ∂v θ = 1 εQ ( F, F ) , z P Q ( f, g ) = 1 � � dn | n · ( v − v ∗ ) |× R 3 dv ∗ 2 S 2 f ′ ∗ g ′ + f ′ g ′ � � ∗ − f ∗ g − g ∗ f , θ r v ′ = v − nn · ( v − v ∗ ) , v ′ ∗ = v ∗ + nn · ( v − v ∗ ) . Q Y X Ghost effect by curvature – p. 6/39

Ghost effect by curvature 1 @ ? > = < ; : 9 8 7 6 4 3 2 5 0 / . ! F E D C B A " # % $ & ( ) * + , - ~ L O M N K I H ' G J q k l m n o p s r t u v w x y z { j | } P Q R S T U c Y V W X \ e ] ^ _ ` a b i h g f d Z [ Z Fix D = 1 . Change of variables L+D L y = r − L, x = − Lθ, v y = v r , v x = − v θ . z P Scaling: L = ε 2 1 θ r c > 0 specified later. c 2 , Q Y X Ghost effect by curvature – p. 7/39

Ghost effect by curvature ∂z + ε 2 � � = 1 ∂F ∂F ∂F ∂F ∂F − v y ζ ( y ) v x ∂x + v y ∂y + v z c 2 ζ ( y ) v x εQ ( F, F ) , v x ∂v y ∂v x 1 ζ ( y ) = . 1 + ε 2 c 2 y For simplicity, assume: ∂F ∂x ≡ 0 . Ghost effect by curvature – p. 8/39

Boundary conditions Rotating cylinders at fixed temperatures: Diffuse reflection boundary conditions. Notation: − | v − u | 2 ρ 2 T M ( ρ, T, u ; v ) = (2 πT ) 3 / 2 e On the boundaries: u · n ≡ u y = 0 . � 2 π � M ( i ) ( v ) = ˜ T ( i ) M (1 , T ( i ) , u ( i ) ; v ) , v y ˜ Mdv = 1 , i = 0 , 1 . v y > 0 Ghost effect by curvature – p. 9/39

Boundary conditions F (0 , z, v ) = ˜ M (0) α (0) ( F ) v y > 0 F (1 , z, v ) = ˜ M (1) α (1) ( F ) v y < 0 , � � α (0) ( F ) = − α (1) ( F ) = v y F (0 , z, v ) dv, v y F (1 , z, v ) dv, v y < 0 v y > 0 Zero mass flux at the boundary: � dvF ( y, z, v ) v y = 0 for y = 0 , 1 . Ghost effect by curvature – p. 10/39

Boundary conditions The inner cylinder is at temperature T = 1 and rotates with speed U (0) ; the outer cylinder is also at temperature T = 1 and rotates with speed U (1) . We assume √ M (0) = ˜ 2 πM (1 , 1 , ( U (0) , 0 , 0); v ) , √ M (1) = ˜ 2 πM (1 , 1 , ( U (1) 0 , 0); v ) . Ghost effect by curvature – p. 11/39

Low Mach number Due to the special geometry, we do not need all the components of the velocity field of the order of the Knudsen number but only the y and z components. With the notation ˆ v = ( v y , v z ) , we assume � dv ˆ vF ( y, z, v ) = O ( ε ) , low Mach number . The radial flux � dvv x F ( y, z, v ) is O (1) in ε but we will need it to be small, say of order δ : δ ≫ ε . Therefore we set U ( i ) = δU ( i ) , i = 0 , 1 , with U ( i ) = O (1) both in ε and in δ . Ghost effect by curvature – p. 12/39

Local equilbrium A current of order δ may produce temperature changes in the fluid of the same order and, consequently, density changes. Therefore one expects the solution at 0 -th order in ε to be the local equilibrium with ρ = 1 + δr , T = 1 + δτ : M δ = M (1 + δr, 1 + δτ, ( δU, 0 , 0); v ) , where δU ( y, z ) is the tangential component of the velocity field. We seek for the solution in the form F ( y, z, v ) = M δ + ε [Φ + R ] with Φ a truncated expansion in ε and R a suitable remainder. Ghost effect by curvature – p. 13/39

Expansion N � ε n − 1 F n . Φ = n =1 Plugging in the expansion one gets conditions on the F n which, up to the first order in ε require � | v − Ue x | 2 − 3(1 + δτ ) ρ 1 + u · v � �� + B· ˆ ∇ τ, F 1 = M δ 1 + δτ + τ 1 2(1 + δτ ) 2 2 Q ( M δ , B ) ≡ L δ B = ˜ B , v | v − Ue x | 2 − 5(1 + δτ ) ˜ B = M δ ˆ , 2(1 + δτ ) 2 Ghost effect by curvature – p. 14/39

Expansion ˆ ˆ ˆ ∇P = 0 , ∇ · [ˆ uρ ] = 0 , ∇ ≡ ( ∂ y , ∂ z ) ∇ U = ρ − 1 ˆ u · ˆ ∇ · ( η ˆ ˆ ∇ U ) , u − δ 2 c 2 U 2 e y = ρ − 1 � � u · ˆ − ˆ ∇P 2 + ˆ ∇ · ( η ˆ u ) + ˆ ∇ ˆ ∇ ˆ ∇ Σ ˆ � u = ρ − 1 P = (1 + δr )(1 + δτ ) ˆ ˆ vF 1 dv, � δ 2 �� Σ = ˆ σ 1 ˆ ∇ τ ⊗ ˆ ∇ τ + σ 2 ˆ ∇ U ⊗ ˆ � ∇ · ∇ U . P with η , σ 1 , σ 2 smooth functions of T . Ghost effect by curvature – p. 15/39

Expansion The equation for the temperature τ is 5 ∇ τ = ρ − 1 � ∇ U | 2 � u · ˆ ∇ ( κ ˆ ˆ ∇ τ ) + δη | ˆ 2 ˆ , with κ a smooth function of T . Ghost effect by curvature – p. 16/39

Limit for δ → 0 The macroscopic equations become much simpler in the limit δ → 0 . In order to keep the correction term in the equation u − δ 2 c 2 U 2 e y = ρ − 1 � � u · ˆ − ˆ ∇P 2 + ˆ ∇ · ( η ˆ u ) + ˆ ∇ ˆ ∇ ˆ ∇ Σ ˆ , L = ε 2 we set (with 1 c 2 ) c = δC with C and O (1) constant related to the curvature. Hence the scaling becomes ε 2 1 L = C 2 δ 2 . Ghost effect by curvature – p. 17/39

Limiting equations ˆ ∇ · ˆ u = 0 , u · ˆ ∇ U = η 0 ˆ ˆ ∆ U, ∇P 2 − 1 u · ˆ u + ˆ C 2 U 2 e y = η 0 ˆ ∇ ˆ ˆ ∆ˆ u, 5 u · ˆ ∇ τ = κ 0 ˆ 2 ˆ ∆ τ, with η 0 = η (1) , κ 0 = κ (1) . P 2 second order pressure. Ghost effect by curvature – p. 18/39

Boundary conditions We consider now the following boundary conditions: U (0 , z ) = U (0) , U (1 , z ) = U (1) , u (0 , z ) = ˆ ˆ u (1 , z ) = 0 , τ (0 , z ) = τ (1 , z ) = 0 . Ghost effect by curvature – p. 19/39

Bifurcation δ = 0 Set β = U (0) − U (1) . Laminar solution: U ℓ ( y ) = U (1) + β (1 − y ) , u ℓ = 0 , ˆ τ ℓ = 0 . There is β c such that, for β > β c (but close to β c ) the solution bifurcate into two non laminar solutions U ± ( y, z ) , ˆ u ± ( y, z ) , with τ = 0 . � U ± − U ℓ � = O ( β − β c ) , � ˆ u ± � = O ( β − β c ) . No such a bifurcation in planar Couette flow. The term C − 2 U 2 e y is responsible for the bifurcation. It is due the curvature which on the other hand goes to 0 in this scaling limit. Ghost effect. Ghost effect by curvature – p. 20/39

Bifurcation δ = 0 Linear analysis: Sone-Doi [SD] Theorem both for linear and non linear bifurcation in Arkeryd-R.E.-Marra-Nouri [AEMN] in preparation. Ghost effect by curvature – p. 21/39

Bifurcation δ � = 0 When δ � = 0 the system is slightly compressible. Bifurcation persists: there are solutions ( U δ u δ ± , τ δ ± , ˆ ± ) such that � ( U δ u δ ± , τ δ ± , ˆ ± ) − ( U ± , ˆ u ± , 0) � = O ( δ ) . Perturbation argument. Extra complications due to the (small) compressibility. Theorem proved in [AEMN] in H s -norms. Ghost effect by curvature – p. 22/39