NonEquilibrium Thermodynamics of Flowing Systems: 2

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1 *Following the development in Beris and Edwards, 1994, Section 9.2 NonEquilibrium Thermodynamics of Flowing Systems: 2 Antony N. Beris Schedule: Multiscale Modeling and Simulation of Complex Fluids Center for Scientific Computing and Mathematical Modeling University of Maryland, College Park 1. 4/13/07, 9:30 am Introduction. One mode viscoelasticity. 2. 4/13/07, 10:15 am Coupled transport: Two-fluid model*. 3. 4/14/07, 2:00 pm Modeling under constraints: Liquid crystals. 4. 4/14/07, 3:00 am Non-homogeneous systems: Surface effects.

2 Coupled Transport in a Viscoelastic Fluid Two approaches: Single and two-fluid system Formalism can tell you what it CAN be, but not what it ACTUALLY is! (Comparison with underlying microscopic theory is necessary) The cruder the structure, the easier to work out the predictions but also the more cloudy those predictions are Single fluid model: coarser; 2-fluid model: finer Next: Develop general equations

3 Single Fluid Model: Variables For an incompressible, inhomogeneous (variable polymer concentration, n=chain number density is variable) system we have ρ 1, the polymer density (n=n A ρ 1 /MW 1 ) v, the velocity s, the entropy density (alternatively, T, temperature) c, the conformation tensor where C = <RR> (second moment of the end-to-end distribution function) = nc At equilibrium, c=k B T/K where K is the equilibrium equivalent entropic elastic energy constant of the polymer chain

4 Single Fluid Hamiltonian The Hamiltonian (extended Helmholtz free energy of the system) is assumed to have the form: V ( 2 ) 1 2 ρ e m A= v + a + a where a e is the elastic free energy density corresponding to a dilute solution: dv ( KC) ( nk T ) a 1 e = 2 ( KtrC nkbtlog det ) B and a m represents the mixing energy density (approximated by a Flory-Huggins) term: a = k T( nlogϕ + n log(1 ϕ)) m B s where n s is the solvent number density and φ is the polymer volume fraction: ϕ = ( nn ) ( nn + n ) s

5 Single Fluid Poisson Bracket: Reversible Equations For an isothermal system, we get the standard reversible dynamics for an elastic medium together with a convection equation for the polymer density: D Dt D ρ Dt v D 1 Dt ρ = 0 = p + T T C v C C v = 0 T a T = 2C C T

6 Single Fluid: Dissipation Structure Defined for two arbitrary functionals F, G by the bilinear functional [F,G] (to within an entropy correction term): η s δf δf δg δg [ FG, ] + β + β dω 2 δv β δv δv β δv δf δg Λβγε d Ω δc δc δf δg Dβ β dω δρ 1 δρ 1 δf δg δg δf Eβ γ Cγλ β + γ Cγλ β dω δc λ δρ 1 δc λ δρ 1 B β γε δ F δg C ε Cεκ dω δc λ δc κβ β γ γλ

7 Single Fluid: Final Equations D ρ Dt D ρ1 Dt D Dt a δ H = D + E T ( 1 ) 2 β β β ε βε δρ1 ( η ( )) s v = p+ T + v + v β β β β β C v C C v = Λ β γ γβ γ γ β βγε δ H δc δ H + C E + C B T T a T = 2C C ( 1 ) 2 γ γ βε ε γ γ βε κ εκ δρ1 γε

8 Single Fluid Formalism The general formalism leads to new terms to the polymer mass balance and conformation evolution equations: In the polymer mass balance: A new driving force appears proportional to the gradient to the polymer stress In the polymer conformation evolution equation: Two new terms appear, involving second derivatives of the chemical potential and the stress In addition, there are other dependencies (n hidden with C) Moreover, many uncertainties still remain (too many adjustable parameters) and the nonnegative entropy production is hard to ascertain in the general case

9 Particular Case: D=-1/2E=1/4B=Dδ D ρ Dt a D ρ1 Dt δ H = D T ε ε δρ1 ( η ( )) s v = p+ T + v + v β β β β β D Dt C v C C v = Λ β γ γβ γ γ β βγε δ H 2C D T T a T = 2C C γ γ β κ βκ δρ1 δ H δc γε

10 Two- Fluid Model: Variables For an incompressible, inhomogeneous (variable polymer concentration, n=chain number density is variable) system we have (keeping ρ=ρ 1 +ρ 2 =constant) ρ 1, the polymer density (n=n A ρ 1 /MW 1 ); ρ 2, the solvent density g 1 =ρ 1 v 1, the polymer momentum density; g 2 =ρ 2 v 2 s, the entropy density (alternatively, T, temperature) c, the conformation tensor where C = <RR> (second moment of the end-to-end distribution function) = nc At equilibrium, c=k B T/K where K is the equilibrium equivalent entropic elastic energy constant of the polymer chain

11 Two-Fluid Hamiltonian The Hamiltonian (extended Helmholtz free energy of the system) is assumed to have the form: V ( ) 2ρ1 1 2ρ2 2 e m A= v + v + a + a dv where a e is the elastic free energy density corresponding to a dilute solution: ( KC) ( nk T ) a 1 e = 2 ( KtrC nkbtlog det ) B and a m represents the mixing energy density (approximated by a Flory-Huggins) term: a = k T( nlogϕ + n log(1 ϕ)) m B s where n s is the solvent number density and φ is the polymer volume fraction: ϕ = ( nn ) ( nn + n ) s

12 ρ Two- Fluid Poisson Bracket: Reversible Equations For an isothermal system, we get the standard reversible dynamics for 2 interpenetrating continua of which one is an elastic medium v ρ1 + v = C β t 1ρ1 0 ( ) + v v = p + T 1 1 1β β 1 1 β β ρ2 ρ ( ) ( ) + v = 2ρ2 0 ( ) + v C v C C v = γ 1γ β γ 1 γβ γ γ 1β 0 T ae T = 2C C v + v v = p 2 2 2β β 2 2

13 Transformation of Variables To introduce the dissipation terms it is first necessary to make a transformation of variables ρ ρ ρ ρ ρ = + ; = ρ ρ ρ ρ g ρ + = g1+ g2 + v; g = g1+ g2 = v ρ ρ ρ where ρ ρ g ρ v ; g ρ v ; v = v ; ρ + v v ρ = v v therefore δ H δg δ H = v; = v δg +

14 Two- Fluid: Reversible Equations in g v =0 g+ Transformed Variables ρ ( ) (( ) ) + v ρ + 1 ϕ v a ρ = 0 + v g + v g + g v + g v = p+ T β 1β 1 2β 2 1β 1β 2β 2β β β ( v1 g1 ) ( v2 g2 ) g1 v1 g2 v2 ( T ) + (1 ϕ) ϕ + (1 ϕ) ϕ = (1 ϕ) Π β β β β β β β β β β β C β t ( ) + v C v C C v = γ 1γ β γ 1 γβ γ γ 1β 0 T ae T = 2C C

15 Two-Fluid: Dissipation Structure Defined for two arbitrary functionals F, G by the bilinear functional [F,G] (to within an entropy correction term): [ F, G] η s δf δf δg δg + β + β dω 2 δg β δg + + δg + β δg + δf δg Zβ dω δg δg β δf δg Λβγε d Ω δc δc β γε

16 Two- Fluid: Final Momentum and Conformation Equations g+ ( ) + v g + v g + g v + g v = β 1β 1 2β 2 1β 1β 2β 2β ( η ( )) s p+ T + v + v β β β β β g ( ) ( ) + (1 ϕ) v g ϕ v g + (1 ϕ) g v ϕg v = β 1β 1 β 2β 2 1β 1β 2β 2β ( ) Z v (1 ϕ) Π T β β β β β C β ( ) + v C v C C v = Λ γ 1γ β γ 1 γβ γ γ 1β βγε δ H δc γε

17 Two- Fluid: Small Differential Inertia ϕ β β γ γ 1 (1 ) ( ) v = Z Π T and therefore, substituting this relationship into the polymer density equation, we have: ρ (( ) ( )) + v ρ = ϕ ρ Z Π T β β γ βγ

18 Two- Fluid Formalism: Conclusions The general formalism leads to specific new terms to the polymer mass balance and conformation evolution equations: In the polymer mass balance: A new driving force appears proportional to the gradient to the polymer stress In the polymer conformation evolution equation: The reference velocity with respect to which it is calculated is the polymer phase velocity In addition, there are other dependencies (n hidden with C) The 2-fluid equation leaves no uncertainties! It has been confirmed from microscopic theory (Curtiss and Bird, 1996).

19 Applications Coupled mass/momentum transport in a dilute polymer system: Two-fluid model. Apostolakis MV, Mavrantzas VG, Beris AN Stress gradient-induced migration effects in the Taylor-Couette flow of a dilute polymer solution J. NON-NEWTONIAN FLUID MECH. 102: (2002) Non-homogeneous systems: Surface Effects on the Rheology and Chain Conformation in Dilute Polymer Solutions. Mavrantzas VG, Beris AN A hierarchical model for surface effects on chain conformation and rheology of polymer solutions. I. General formulation JOURNAL OF CHEMICAL PHYSICS 110: (1999) II. Application to a neutral surface JOURNAL OF CHEMICAL PHYSICS 110: (1999)