A Low Mach Number Limit of a Dispersive Navier-Stokes System - PowerPoint PPT Presentation

Transition Regime Models A Low Mach Number Limit of a Dispersive Navier-Stokes System Konstantina Trivisa Joint work with C.D. Levermore and Weiran Sun HYP-2012, Padova

Transition Regime Models Outline What is our objective? Transition regime models. Main result. Local well-posedness results for the DNS system and the ghost effect system . Strategy for the rigorous proof of the low Mach number limit of the DNS: Take into account the anti-symmetric structure of the high-order dispersive term. Establish uniform bounds for the solution by considering the interaction of its fast and slow motion parts. A priori estimates for the slow motion ψ ǫ = ( ǫ p ǫ , ǫ u ǫ , θ ǫ ) . A priori estimates for the fast motion ( p ǫ , u ǫ ) . Local decay of the energy of the fast motion.

Transition Regime Models Our objective Establish a low Mach number limit for classical solutions over the whole space of a compressible fluid dynamic system that includes dispersive corrections to the Navier-Stokes equations. ⇓ The limiting system is a so-called ghost effect system (Sone 2002) which is not derivable from the Navier-Stokes system of gas dynamics but is derivable from kinetic equations.

Transition Regime Models The long term objective This work is part of a research program that aims to identify fluid dynamic regimes and to construct a unified model that captures them all. Such a model can also be useful in transition regimes when classical fluid equations are inadequate to describe the dynamics of fluids while computations using kinetic models are expensive.

Kinetic equations Evolution of systems consisting of a large number of particles. Mathematical formulation: partial differential equations for the particle distribution function f ( t , x , v ) : t: time; x: position; v: velocity; f ≥ 0.

Classical examples of kinetic equations: Boltzmann equation (gas dynamics): ∂ t f + v · ∇ x f = C [ f ]; f = f ( t , x , v ) : particle distribution function; C [ f ] : collision among particles; Example: � � � � f ( v ′ ) f ( v ′ C [ f ] = 1 ) − f ( v ) f ( v 1 ) b ( v , v 1 , ω ) d ω d v 1 . R 3 S 2

Classical macroscopic regime: Many collisions, systems close to local equilibrium. Measurement: mean free path Kn = macroscopic length . Kn: Knudsen number; Macroscopic regime: small Knudsen number regime.

Classical example of fluid equations: Navier-Stokes equation ∂ t ρ + ∇ x · ( ρ u ) = 0 , conservation of mass ∂ t ( ρ u ) + ∇ x · ( ρ u ⊗ u ) + ∇ x p = ∇ x · Σ , conservation of momentum ∂ t ( ρ e ) + ∇ x · ( ρ eu ) + ∇ x · ( pu ) = ∇ x · (Σ u ) − ∇ x · q , conservation of energy ρ : mass density p : pressure Σ : viscous stress tensor ρ u : momentum density 2 ρ | u | 2 + D ρ e = 1 2 ρθ : energy density q : heat flux ( p , Σ , q ) = ( p , Σ , q )( ρ, u , θ, ∇ x θ ) : constitutive relation x u , ∇

Kinetic equation = ⇒ fluid equations? Boltzmann equation x f = 1 ∂ t f + v · ∇ Kn C [ f ] ρ = � f � , ← Kn → 0 → ⇓ ρ u = � vf � , ρ e = � 1 2 | v | 2 f � . Navier-Stokes equation ∂ t ρ + ∇ x · ( ρ u ) = 0 , ∂ t ( ρ u ) + ∇ x · ( ρ u ⊗ u ) + ∇ x p = ∇ x · Σ , ∂ t ( ρ e ) + ∇ x · ( ρ eu ) + ∇ x · ( pu ) = ∇ x · (Σ u ) − ∇ x · q ,

Transition Regime Models Transition Regime Plenty collisions but not enough to drive the system very close to local equilibrium; Computations using full kinetic equations are expensive. Classical macroscopic systems are inaccurate. Purpose of developing transition regime models Bridge the gap between kinetic equations and classical macroscopic equations.

Transition Regime Models Transition regime models Kinetic equations ⇓ Transition regime models ⇓ Classical macroscopic equations

Transition Regime Models Construction and analysis of interior equations: Higher order equations: adding higher order correction terms; Larger moment systems: including more moments of the density function. Construction of appropriate boundary conditions: Boundary conditions of macroscopic equations should match the given boundary conditions of the underlying kinetic equations. Boundary layer analysis.

Transition Regime Models Thermal induced flow Thermal induced flow: Maxwell: 1879 ρ∂ t u + ∇ x p − ∇ x · Σ + τ ∇ x ∆ x θ = 0 . Kogan, Galkin and Fridlender: nonlinear thermal stress, 1976. Aoki, Sone, Sugimoto, Takata ...

Transition Regime Models Ghost effect regime Ghost effect regime: small bulk velocity: U = ǫ u p = p 0 + ǫ p 1 small fluctuations in pressure field: large variation in temperature/density field: ∇ x ρ, ∇ x θ ∼ O (1) Classical fluid equations like NS are not accurate in the ghost effect regime.

Transition Regime Models Ghost-effect system Ghost effect system ∇ x ( ρθ ) = 0 , ∂ t ρ + ∇ x · ( ρ u ) = 0 , (1) x · ˜ ∂ t ( ρ u ) + ∇ x · ( ρ u ⊗ u ) + ∇ x P = ∇ x · Σ + ∇ Σ , ∂ t ( ρθ ) + ∇ x · ( ρθ u ) = −∇ x · q , Constitutive relation � � x u ) ⊤ − 2 Σ = µ ( θ ) ∇ x u + ( ∇ 3( ∇ x · u ) I , � � � � x θ − 1 x θ − 1 ˜ ∇ 2 x θ | 2 I Σ = τ 1 ( θ ) 3(∆ x θ ) I + τ 2 ( θ ) ∇ x θ ⊗ ∇ 3 |∇ , q = − k ( θ ) ∇ x θ.

Transition Regime Models Main Question Question: Can one in some sense unify the classical fluid equations and the ghost-effect system? ⇓ Higher order equations: dispersive Navier-Stokes. Reference: Levermore: Gas Dynamics Beyond Navier-Stokes .

Transition Regime Models Main Question Main questions: Well-posedness of the DNS system? Recovery of the ghost effect system in the ghost effect regime?

Transition Regime Models Main Question Main results: Well-posedness of the DNS system? Result: local well-posedness in Sobolev spaces; Reference: Ph.D. Thesis of Weiran Sun (2009). Recovery of a ghost effect system in the ghost effect regime? Result: DNS converges to a ghost effect system in the low Mach number limit. Reference: Levermore, Sun, Trivisa SIMA (2012).

Transition Regime Models The model Dispersive Navier-Stoke system (DNS): ∂ t ρ + ∇ x · ( ρ u ) = 0 , x · ˜ ρ∂ t u + ρ u · ∇ x u + ∇ x p ( ρ, θ ) = ∇ x · Σ + ∇ Σ (2) 3 2 ρ∂ t θ + 3 x · u = (Σ + ˜ 2 ρ u · + ∇ x θ + p ∇ Σ) : ∇ x u + ∇ x · ˜ q − ∇ x · q , ( ρ, u , θ )( x , 0) = ( ρ in , u in , θ in )( x ) , space variable x ∈ R 3 . time variable t ∈ [0 , ∞ ) , bulk velocity: u ∈ R 3 , Density: ρ, temperature: θ.

Transition Regime Models Constitutive relations Constitutive relations Ideal gas law: p ( ρ, θ ) = ρθ. Viscous stress tensor: x u ) ⊤ − 2 Σ = µ ( θ )( ∇ x u + ( ∇ 3( ∇ x · u ) I ) , µ ( θ ) > 0 . Heat flux: q ( θ ) = − κ ( θ ) ∇ x θ, κ ( θ ) > 0 . The total energy density: ρ e = 1 2 ρ | u | 2 + 3 2 ρθ.

Transition Regime Models Constitutive relations The quantities ˜ Σ and ˜ q denote dispersive corrections to the stress tensor and heat flux respectively and are given by

Transition Regime Models Constitutive relations x · ˜ Dispersive term in the velocity equation ∇ Σ where x θ − 1 ˜ Σ = τ 1 ( ρ, θ )( ∇ 2 3(∆ x θ ) I ) x θ − 1 x θ | 2 I ) + τ 2 ( ρ, θ )( ∇ x θ ⊗ ∇ 3 |∇ x u ) ⊤ − ( ∇ x u ) ⊤ ∇ + τ 3 ( ρ, θ )( ∇ x u ( ∇ x u ) (3) Dispersive term in the temperature equation ∇ x · ˜ q where q = τ 4 ( ρ, θ )(∆ x u + 1 ˜ 3 ∇ x ∇ x · u ) x u ) ⊤ − 2 + τ 5 ( ρ, θ ) ∇ x θ · ( ∇ x u + ( ∇ 3( ∇ x · u ) I ) � x u ) ⊤ � + τ 6 ( ρ, θ ) ∇ x u − ( ∇ · ∇ x θ. (4) τ 1 , . . . , τ 6 are C ∞ functions of their variables.

Transition Regime Models Constitutive relations One feature of the DNS system is that it possesses an entropy structure provided the transport coefficients in ˜ Σ and ˜ q satisfy τ 4 = θ τ 2 θ + 2 τ 5 � τ 4 � θ 2 = ∂ θ (5) 2 τ 1 , , θ 2 such that � τ 1 Σ : ∇ x u q ·∇ x θ � x u ) ⊤ − 2 �� ˜ θ +˜ = ∇ x · 2 θ ∇ x θ · ∇ x u + ( ∇ 3( ∇ x · u ) I . θ 2

Transition Regime Models Entropy equation Entropy density � ρ � η = ρ log . θ 3 / 2 Entropy equation � η u + q + ˜ q � � Σ x u − q � ∂η + ∇ x · = − θ : ∇ θ 2 · ∇ x θ θ � ˜ � Σ x u − ˜ q − θ : ∇ θ 2 · ∇ x θ . Total entropy is formally dissipated by the dispersive NS system in the same way as in the NS system over domains without boundaries.

Transition Regime Models Local well-posedness The proof of the local well-posedness of the DNS system follows using the classical energy method for hyperbolic-parabolic systems . Although we have third-order dispersive terms, the leading orders of these terms form an anti-symmetric structure. ⇓ Therefore they do not hamper the usual L 2 - H s estimates. The rest of the dispersive terms are of orders up to two. Although this is the same order as the dissipation, they do not introduce extra difficulties because they are of order O ( ǫ 2 ) while the dissipative terms are of order O ( ǫ ). Here we do need the viscosity coefficient µ ( θ ) and κ ( θ ) to be bounded away from zero when θ is bounded from below.