From the Boltzmann equation to incompressible viscous hydrodynamics - PowerPoint PPT Presentation

From the Boltzmann equation to incompressible viscous hydrodynamics Diogo Ars´ enio CNRS & Universit´ e Paris Diderot (Paris 7) 10 th International Conference on Operations Research Partial Differential Equations Session La Habana 6-9 March 2012

Fluid dynamics D. Ars´ enio Hydrodynamic limits

Fluid dynamics ( t , x , v ) ∈ [0 , ∞ ) × Ω × R D Particle number density: F ( t , x , v ) ≥ 0 Ω ⊂ R D , D ≥ 2 ( D =1) D. Ars´ enio Hydrodynamic limits

Fluid dynamics ( t , x , v ) ∈ [0 , ∞ ) × Ω × R D Particle number density: F ( t , x , v ) ≥ 0 Ω ⊂ R D , D ≥ 2 ( D =1) 2 e − | v − u | 2 � � statistical ρ Maxwellian distribution: F ( t , x , v ) = 2 θ D equilibrium (2 πθ ) D. Ars´ enio Hydrodynamic limits

Fluid dynamics ( t , x , v ) ∈ [0 , ∞ ) × Ω × R D Particle number density: F ( t , x , v ) ≥ 0 Ω ⊂ R D , D ≥ 2 ( D =1) 2 e − | v − u | 2 � � statistical ρ Maxwellian distribution: F ( t , x , v ) = 2 θ D equilibrium (2 πθ ) The incompressible Navier-Stokes-Fourier system: ∂ t u + u · ∇ x u − ν ∆ x u = −∇ x p ∇ x · u = 0 D +2 ( ∂ t θ + u · ∇ x θ ) − κ ∆ x θ = 0 2 D. Ars´ enio Hydrodynamic limits

Fluid dynamics ( t , x , v ) ∈ [0 , ∞ ) × Ω × R D Particle number density: F ( t , x , v ) ≥ 0 Ω ⊂ R D , D ≥ 2 ( D =1) 2 e − | v − u | 2 � � statistical ρ Maxwellian distribution: F ( t , x , v ) = 2 θ D equilibrium (2 πθ ) The incompressible Navier-Stokes-Fourier system: ∂ t u + u · ∇ x u − ν ∆ x u = −∇ x p ∇ x · u = 0 D +2 ( ∂ t θ + u · ∇ x θ ) − κ ∆ x θ = 0 2 The Boltzmann equation: ( ∂ t + v · ∇ x ) F ( t , x , v ) = B ( F , F ) ( t , x , v ) D. Ars´ enio Hydrodynamic limits

The Boltzmann collision operator D. Ars´ enio Hydrodynamic limits

The Boltzmann collision operator S D − 1 ( F ′ F ′ � � B ( F , F ) ( t , x , v ) = ∗ − FF ∗ ) b ( v − v ∗ , σ ) d σ dv ∗ R D F ′ = F ( t , x , v ′ ) , F ′ ∗ = F ( t , x , v ′ ∗ ) , F ∗ = F ( t , x , v ∗ ) v ′ = v + v ∗ + | v − v ∗ | − | v − v ∗ | ∗ = v + v ∗ v ′ σ, σ 2 2 2 2 D. Ars´ enio Hydrodynamic limits

The Boltzmann collision operator S D − 1 ( F ′ F ′ � � B ( F , F ) ( t , x , v ) = ∗ − FF ∗ ) b ( v − v ∗ , σ ) d σ dv ∗ R D F ′ = F ( t , x , v ′ ) , F ′ ∗ = F ( t , x , v ′ ∗ ) , F ∗ = F ( t , x , v ∗ ) v ′ = v + v ∗ + | v − v ∗ | − | v − v ∗ | ∗ = v + v ∗ v ′ σ, σ 2 2 2 2 v + v ∗ = v ′ + v ′ � (conservation of momentum) ∗ | v | 2 + | v ∗ | 2 = | v ′ | 2 + | v ′ ∗ | 2 (conservation of energy) D. Ars´ enio Hydrodynamic limits

The Boltzmann collision operator S D − 1 ( F ′ F ′ � � B ( F , F ) ( t , x , v ) = ∗ − FF ∗ ) b ( v − v ∗ , σ ) d σ dv ∗ R D F ′ = F ( t , x , v ′ ) , F ′ ∗ = F ( t , x , v ′ ∗ ) , F ∗ = F ( t , x , v ∗ ) v ′ = v + v ∗ + | v − v ∗ | − | v − v ∗ | ∗ = v + v ∗ v ′ σ, σ 2 2 2 2 D. Ars´ enio Hydrodynamic limits

The Boltzmann collision operator S D − 1 ( F ′ F ′ � � B ( F , F ) ( t , x , v ) = ∗ − FF ∗ ) b ( v − v ∗ , σ ) d σ dv ∗ R D F ′ = F ( t , x , v ′ ) , F ′ ∗ = F ( t , x , v ′ ∗ ) , F ∗ = F ( t , x , v ∗ ) v ′ = v + v ∗ + | v − v ∗ | − | v − v ∗ | ∗ = v + v ∗ v ′ σ, σ 2 2 2 2 5 hypotheses: binary collisions (rarefied gas) localization in time and space of collisions elastic collisions micro-reversibility of collisions molecular chaos D. Ars´ enio Hydrodynamic limits

The Boltzmann collision operator S D − 1 ( F ′ F ′ � � B ( F , F ) ( t , x , v ) = ∗ − FF ∗ ) b ( v − v ∗ , σ ) d σ dv ∗ R D F ′ = F ( t , x , v ′ ) , F ′ ∗ = F ( t , x , v ′ ∗ ) , F ∗ = F ( t , x , v ∗ ) v ′ = v + v ∗ + | v − v ∗ | − | v − v ∗ | ∗ = v + v ∗ v ′ σ, σ 2 2 2 2 D. Ars´ enio Hydrodynamic limits

The Boltzmann collision operator S D − 1 ( F ′ F ′ � � B ( F , F ) ( t , x , v ) = ∗ − FF ∗ ) b ( v − v ∗ , σ ) d σ dv ∗ R D F ′ = F ( t , x , v ′ ) , F ′ ∗ = F ( t , x , v ′ ∗ ) , F ∗ = F ( t , x , v ∗ ) v ′ = v + v ∗ + | v − v ∗ | − | v − v ∗ | ∗ = v + v ∗ v ′ σ, σ 2 2 2 2 The collision kernel: b ( v − v ∗ , σ ) = b ( | v − v ∗ | , cos θ ) ≥ 0 D. Ars´ enio Hydrodynamic limits

The Boltzmann collision operator S D − 1 ( F ′ F ′ � � B ( F , F ) ( t , x , v ) = ∗ − FF ∗ ) b ( v − v ∗ , σ ) d σ dv ∗ R D F ′ = F ( t , x , v ′ ) , F ′ ∗ = F ( t , x , v ′ ∗ ) , F ∗ = F ( t , x , v ∗ ) v ′ = v + v ∗ + | v − v ∗ | − | v − v ∗ | ∗ = v + v ∗ v ′ σ, σ 2 2 2 2 The collision kernel: b ( v − v ∗ , σ ) = b ( | v − v ∗ | , cos θ ) ≥ 0 Hard spheres: b ( v − v ∗ , σ ) = | v − v ∗ | ∈ L 1 loc D. Ars´ enio Hydrodynamic limits

The Boltzmann collision operator S D − 1 ( F ′ F ′ � � B ( F , F ) ( t , x , v ) = ∗ − FF ∗ ) b ( v − v ∗ , σ ) d σ dv ∗ R D F ′ = F ( t , x , v ′ ) , F ′ ∗ = F ( t , x , v ′ ∗ ) , F ∗ = F ( t , x , v ∗ ) v ′ = v + v ∗ + | v − v ∗ | − | v − v ∗ | ∗ = v + v ∗ v ′ σ, σ 2 2 2 2 The collision kernel: b ( v − v ∗ , σ ) = b ( | v − v ∗ | , cos θ ) ≥ 0 Hard spheres: b ( v − v ∗ , σ ) = | v − v ∗ | ∈ L 1 loc Intermolecular forces deriving from an inverse power potential: 1 φ ( r ) = r s − 1 , s > 2 b ( v − v ∗ , σ ) = | v − v ∗ | γ b 0 (cos θ ), γ = s − 5 s − 1 sin D − 2 θ b 0 (cos θ ) ≈ 1 2 ∈ L 1 θ 1+ ν / loc , ν = s − 1 D. Ars´ enio Hydrodynamic limits

The Boltzmann collision operator S D − 1 ( F ′ F ′ � � B ( F , F ) ( t , x , v ) = ∗ − FF ∗ ) b ( v − v ∗ , σ ) d σ dv ∗ R D F ′ = F ( t , x , v ′ ) , F ′ ∗ = F ( t , x , v ′ ∗ ) , F ∗ = F ( t , x , v ∗ ) v ′ = v + v ∗ + | v − v ∗ | − | v − v ∗ | ∗ = v + v ∗ v ′ σ, σ 2 2 2 2 The collision kernel: b ( v − v ∗ , σ ) = b ( | v − v ∗ | , cos θ ) ≥ 0 Hard spheres: b ( v − v ∗ , σ ) = | v − v ∗ | ∈ L 1 loc Intermolecular forces deriving from an inverse power potential: 1 φ ( r ) = r s − 1 , s > 2 b ( v − v ∗ , σ ) = | v − v ∗ | γ b 0 (cos θ ), γ = s − 5 s − 1 sin D − 2 θ b 0 (cos θ ) ≈ 1 2 ∈ L 1 θ 1+ ν / loc , ν = s − 1 long-range interactions ⇒ grazing collisions ⇒ non-integrable kernel D. Ars´ enio Hydrodynamic limits

Microscopic-macroscopic link Conservation laws D. Ars´ enio Hydrodynamic limits

Microscopic-macroscopic link Conservation laws ( ∂ t + v · ∇ x ) F ( t , x , v ) = B ( F , F ) ( t , x , v ) D. Ars´ enio Hydrodynamic limits

Microscopic-macroscopic link Conservation laws ( ∂ t + v · ∇ x ) F ( t , x , v ) = B ( F , F ) ( t , x , v ) Macroscopic variables: � density: ρ ( t , x ) = R D F ( t , x , v ) dv � bulk velocity: ρ u ( t , x ) = R D F ( t , x , v ) v dv R D F ( t , x , v ) | v − u ( t , x ) | 2 � temperature: ρθ ( t , x ) = dv D D. Ars´ enio Hydrodynamic limits

Microscopic-macroscopic link Conservation laws ( ∂ t + v · ∇ x ) F ( t , x , v ) = B ( F , F ) ( t , x , v ) Macroscopic variables: � density: ρ ( t , x ) = R D F ( t , x , v ) dv � bulk velocity: ρ u ( t , x ) = R D F ( t , x , v ) v dv R D F ( t , x , v ) | v − u ( t , x ) | 2 � temperature: ρθ ( t , x ) = dv D   1   � v Microscopic conservation laws: R D B ( F , F ) ( t , x , v )  dv = 0     | v | 2  2 D. Ars´ enio Hydrodynamic limits

Microscopic-macroscopic link Conservation laws ( ∂ t + v · ∇ x ) F ( t , x , v ) = B ( F , F ) ( t , x , v ) Macroscopic variables: � density: ρ ( t , x ) = R D F ( t , x , v ) dv � bulk velocity: ρ u ( t , x ) = R D F ( t , x , v ) v dv R D F ( t , x , v ) | v − u ( t , x ) | 2 � temperature: ρθ ( t , x ) = dv D D. Ars´ enio Hydrodynamic limits

Microscopic-macroscopic link Conservation laws ( ∂ t + v · ∇ x ) F ( t , x , v ) = B ( F , F ) ( t , x , v ) Macroscopic variables: � density: ρ ( t , x ) = R D F ( t , x , v ) dv � bulk velocity: ρ u ( t , x ) = R D F ( t , x , v ) v dv R D F ( t , x , v ) | v − u ( t , x ) | 2 � temperature: ρθ ( t , x ) = dv D Macroscopic conservation laws:  ∂ t ρ + ∇ x · ( ρ u ) = 0    ∂ t ( ρ u ) + ∇ x · ( ρ u ⊗ u + P ) = 0 � ρ | u | 2 � �� ρ | u | 2 � � 2 + D 2 + D  ∂ t 2 ρθ + ∇ x · 2 ρθ u + Pu + q = 0   � stress tensor: P ( t , x ) = R D F ( t , x , v )( v − u ) ⊗ ( v − u ) dv R D F ( t , x , v )( v − u ) | v − u | 2 dv � thermal flux: q ( t , x ) = D. Ars´ enio Hydrodynamic limits

Hydrodynamic regimes Compressible Euler D. Ars´ enio Hydrodynamic limits

Hydrodynamic regimes Compressible Euler ( ∂ t + v · ∇ x ) F ǫ ( t , x , v ) = 1 B ( F ǫ , F ǫ ) ( t , x , v ) ǫ ↑ Knudsen number ≈ mean free path D. Ars´ enio Hydrodynamic limits

Hydrodynamic regimes Compressible Euler ( ∂ t + v · ∇ x ) F ǫ ( t , x , v ) = 1 B ( F ǫ , F ǫ ) ( t , x , v ) ǫ ↑ Knudsen number ≈ mean free path � t ǫ , x � Hyperbolic scaling: F ǫ ( t , x , v ) = F ǫ , v D. Ars´ enio Hydrodynamic limits