Hydrodynamics, Thermodynamics, and Mathematics

Transcription

1 Hydrodynamics, Thermodynamics, and Mathematics Hans Christian Öttinger Department of Mat..., ETH Zürich, Switzerland

2 Thermodynamic admissibility and mathematical well-posedness 1. structure of equations 2. example: hydrodynamics 3. solving vs. coarse graining 4. numerical solutions 5. boundary conditions Menu

3 Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications for solutions 2. example: hydrodynamics 3. solving vs. coarse graining 4. numerical solutions 5. boundary conditions

4 Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications for solutions 2. example: hydrodynamics & diffusive mass flux 3. solving vs. coarse graining 4. numerical solutions 5. boundary conditions

5 Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications for solutions 2. example: hydrodynamics & diffusive mass flux 3. solving vs. coarse graining & structure preservation in eliminating degrees of freedom 4. numerical solutions 5. boundary conditions

6 Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications for solutions 2. example: hydrodynamics & diffusive mass flux 3. solving vs. coarse graining & structure preservation in eliminating degrees of freedom 4. numerical solutions & discrete version of structure 5. boundary conditions

7 Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications for solutions 2. example: hydrodynamics & diffusive mass flux 3. solving vs. coarse graining & structure preservation in eliminating degrees of freedom 4. numerical solutions & discrete version of structure 5. boundary conditions & boundary thermodynamics

8 Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications for solutions 2. example: hydrodynamics & diffusive mass flux 3. solving vs. coarse graining & structure preservation in eliminating degrees of freedom 4. numerical solutions & discrete version of structure 5. boundary conditions & boundary thermodynamics

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15 GENERIC Structure General equation for the nonequilibrium reversible-irreversible coupling metriplectic structure (P. J. Morrison, 1986) Lx ( ) dx Lx ( ) dt δe ( x) δx + Mx ( ) δs( x) Mx ( ) δx L antisymmetric, Jacobi identity δs ( x) δx δe( x) δx M Onsager/Casimir symm., positive-semidefinite

16 Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications for solutions 2. example: hydrodynamics & diffusive mass flux 3. solving vs. coarse graining & structure preservation in eliminating degrees of freedom 4. numerical solutions & discrete version of structure 5. boundary conditions & boundary thermodynamics

17 Nonequilibrium Thermodynamics ( ρ( r), Mr ( ), ε( r), structural variables) mass density momentum density internal energy density balance equations structured time-evolution equations (GENERIC)

18 Hydrodynamics E 1 -- Mr ( ) ρ( r) ε ( r ) d r S s( ρ( r), ε( r) ) d r L is associated with the Lie group of space transformations: unique construction by following arguments: ρ, s M are scalar densities is a (contravariant) vector density M C D C T separation of mechanical (C) and thermodynamic (D) effects C: well-known for relaxational and transport processes and given E D: viscosity, thermal conductivity, and diffusion coefficient

19 Diffusion in Hydrodynamics C T r ---- v r ---- T r2 -- v 2 v ---- v r T v α ---- r v v α M ---- ρ reference velocity (constant) constant

20 Diffusion in Hydrodynamics C T r ---- v r ---- T r2 -- v 2 v ---- v r T v α ---- r v v α M ---- ρ reference velocity (constant) constant δe δx 1 --v 2 2 v 1

21 Hydrodynamic Equations ρ t M t s ---- t ---- ( M jρ) r ---- r 1 --MM ρ jρv + p1 + τ Ms + j s + dissipation r ρ s ρ -- µ T s ε µ µ ( v v) T j D' µ -- α ρ rt rt -- j s 1 T -- j q + ( µ α) ρ T -- j

22 Properties of Hydrodynamic Equations Momentum density and mass flux are two different quantities. Both momentum density and mass flux are conserved quantities. The conservation law for the mass flux is equivalent to the conservation law for the booster (expressing center-of-mass motion). Angular momentum is a conserved quantity. A rigidly rotating fluid is a solution of the hydrodynamic equations. Galilean invariance is satisfied. A new type of inertial forces occurs in the driving force for diffusion.

23 Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications for solutions 2. example: hydrodynamics & diffusive mass flux 3. solving vs. coarse graining & structure preservation in eliminating degrees of freedom 4. numerical solutions & discrete version of structure 5. boundary conditions & boundary thermodynamics

24 GENERIC Structure General equation for the nonequilibrium reversible-irreversible coupling metriplectic structure (P. J. Morrison, 1986) Lx ( ) dx Lx ( ) dt δe ( x) δx + Mx ( ) δs( x) Mx ( ) δx L antisymmetric, Jacobi identity δs ( x) δx δe( x) δx M Onsager/Casimir symm., positive-semidefinite

25 Statistical Mechanics: Entropy relative entropy

26 Statistical Mechanics: Friction Matrix τ 1 f M jk Π k e il 0 Qt f Π j x dt Green-Kubo Π j x x j k B 0 1 M jk k B τ τ Π j τ Π k x Einstein cf. D ( x) 2 2 t τ 1 τ2

27 Newton s (Hamilton s) equations of motion or Liouville equation Boltzmann s equation birth of irreversibility! Hilbert Grad Chapman-Enskog more irreversibility? Hydrodynamics

28 From Grad s 10-Moment Equations to Navier-Stokes ρ ( vρ) t r v v v π t r ρ r T v m T π:κ t r 3ρk σ t ---- ( vσ) κ π π κ T r ( π:κ) 3 π p1 + σ κ 1 --σ τ ---- v r T Singular perturbation theory: σ η κ + κ T ( 1:κ)1 η pτ

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30 Functional Integrals For a systematic calculation of the coarse grained hydrodynamic entropy: ΩρT (, ) ρ 2m det 1 + σ -- 3 exp ln d r p σ Functional integrals occur most naturally in systematic coarse graining!

31 Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications for solutions 2. example: hydrodynamics & diffusive mass flux 3. solving vs. coarse graining & structure preservation in eliminating degrees of freedom 4. numerical solutions & discrete version of structure 5. boundary conditions & boundary thermodynamics

32 Symplectic Integrators J. Moser (1968): dx L δe t dt δx reproduces values of integration scheme at times n t Alternatives: canonical transformations in each time step Lagrangian formulation (variational principle instead of symplectic structure)

33 GENERIC Integrators dx L δe t δs + M dt δx t δx L δs δe t M δx t δx L antisymmetric, Jacobi identity M t Onsager/Casimir symm., positive-semidefinite

34 Thermodynamic admissibility and mathematical well-posedness Menu 1. structure of equations & geometry, implications for solutions 2. example: hydrodynamics & diffusive mass flux 3. solving vs. coarse graining & structure preservation in eliminating degrees of freedom 4. numerical solutions & discrete version of structure 5. boundary conditions & boundary thermodynamics

35 Boundary Thermodynamics concentration excess at interface distance from interface 3d variables time evolution sources time evolution 2d variables boundary conditions

36 GENERIC Structure General equation for the nonequilibrium reversible-irreversible coupling dx Lx ( ) dt δe ( x) δx + Mx ( ) δs ( x) δx da dt δa dx δx dt δa L δe δa M δs δx δx δx δx { A, E} + [ AS, ] Poisson & dissipative brackets

37 reservoir (P 1 ) wall V W Example: Diffusion Cell reservoir (P 2 ) System: Solute particle number densities P in the bulk p at the wall wall Brenner & Ganesan, Phys. Rev. E 61 (2000) 6879 hco, Phys. Rev. E 73 (2006) δs P k δp B ln-----, P 0 δs k p δp B ln---- p 0 k B [ A, B] D b ---- δa δb P 3 r rδp rδp d + D s ---- δa r δp V W ---- δb r δp pd 2 r + ν δa s Ω δa δb Ω δb pd 2 r Ω 1 W δp δp δp δp ν s : ad/desorption rate

38 Diffusion Cell: Results [ A, S] V δa δp dp dt d 3 r irr δa dp d 2 r + δp dt W irr V A J irr d 2 r open boundaries: A J irr δa P A n D wall: δp b J r irr 0 evolution equations: dp dt dp dt P ---- D r b r p P ---- D r s n D r b r boundary condition on wall: P n D b r HP ν s pln p H p 0 P 0 characteristic length scale

39 Example: Wall Slip n System Ω 2 Ω 1 Ω n Ω 3 bulk variables ρ, M, s Ψ( Q) R tethered subchains (semiloops and tails) tangential on wall segment cdf temporary network model boundary variables cs, ψ( R) entropy density subchain cdf

40 Wall Slip: Building Blocks Entropy: trivially available Energy: internal energies depend on Poisson bracket: from a natural action of space transformations that leave the wall invariant s s ρ ŝ net ( Ψ) d 3 Q s teth ( c, ψ) d 3 R Dissipative bracket: [ A, B] heat conduction (bulk, wall) + polymer relaxation (bulk, wall) TT 1 T --δa δa δb δb d 2 r Kapitza resistance δs T δs T δs T δs Ω 1 R K

41 Wall Slip: Results wall τ shear n ( τ + p1) T ψ s teth d 3 R + s T R ψ r c t c eq av,eq τ s c r c eq c τ s av τ s av strength of binding Yarin & Graham, JoR (1998) requires ψ (or ansatz!) av τ s ψ( R) d 3 R τ s ( R) v slip wall τ shear v slip ck B T τ lnψ s ψd 3 R R Wall-slip law: Giesekus-type wall τ shear ψ c eq k B T ψ eq ln d 3 R R ψ eq wall τ shear av c eq k B Tτ s v slip R eq ln ψ ψ eq R R R eq av R v slip τ s

42 Next Steps Free boundaries Moving interfaces Viscoelastic interfaces More general relations between bulk and boundary variables Variables characterizing the geometry of interfaces Functional calculus Statistical mechanics

43 Thermodynamic admissibility and mathematical well-posedness 1. structure of equations & geometry, implications for solutions 2. example: hydrodynamics & diffusive mass flux 3. solving vs. coarse graining & structure preservation in eliminating degrees of freedom 4. numerical solutions & discrete version of structure 5. boundary conditions & boundary thermodynamics Thanks to Miroslav Grmela, Howard Brenner, Henning Struchtrup, Mario Liu

44 GENERIC with Fluctuations dx Lx ( ) dt δe( x) δx δs( x) dw Mx ( ) t + + Bx ( ) δx dt δ k B Mx ( ) δx dw t dt Bx ( ) Bx ( ) T 2k B Mx ( ) fluctuation-dissipation theorem dw dw t t' δ( t t' )1 dt dt' Itô Wiener process

45 GENERIC with Fluctuations dx Lx ( ) dt δe( x) δx δs( x) dw Mx ( ) t + + Bx ( ) δx dt δ k B Mx ( ) δx dw t dt Bx ( ) Bx ( ) T 2k B Mx ( ) fluctuation-dissipation theorem dw dw t t' δ( t t' )1 dt dt' Itô Wiener process Some important issues: Partial stochastic differential equations Fluctuation renormalization (long-time tails, mode-coupling)