Quasi-linear versus potential-based formulations of force-flux - - PowerPoint PPT Presentation

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Quasi-linear versus potential-based formulations of force-flux - - PowerPoint PPT Presentation

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Quasi-linear versus potential-based formulations of force-flux relations and the GENERIC for irreversible processes: comparisons and examples Markus Htter (1,*) Bob Svendsen (2) (1) Eindhoven University of Technology Materials Technology

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Quasi-linear versus potential-based formulations of force-flux relations and the GENERIC for irreversible processes: comparisons and examples

Markus Hütter (1,*) Bob Svendsen (2)

(1) Eindhoven University of Technology Materials Technology http://www.mate.tue.nl (2) RWTH Aachen Material Mechanics http://www.aices.rwth-aachen.de

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  • relations between forces and fluxes (e.g. …)
  • (quasi-)linear relations [1,2]
  • perturbation theory
  • fluctuation-dissipation theorem
  • dissipation potential [3]

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[1] de Groot, Mazur, 1962. Non-equilibrium Thermodynamics. [2] Lifshitz, Pitaevskii, 1981. Physical Kinetics. Vol. 10, Landau and Lifshitz Series on Theoretical Physics. [3] Šilhavý, 1997. The Mechanics and Thermodynamics of Continuous Media.

irreversible dynamics

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motivation

  • Fluid mechanics (hydrodyn.; fluctuating hydrodyn.)
  • transport processes (thermal, diffusion, electric, thermophoretic, …)
  • relativistic hydrodynamics (special and general relativity)
  • Kinetic theory of gases
  • Suspensions
  • two-phase flow, LCPs
  • crystallization (flow-induced)
  • Solid mechanics
  • elasticity, viscoplasticity, anisotropic yielding
  • damage mechanics
  • dislocation reactions
  • Complex fluids with structural variables (tensor, distr. fct.)
  • dumbbell
  • reptation
  • pompom, XPP
  • Modeling
  • coarse graining, multi-scale simulations
  • mean-field approximations

Han Meijer Marc Geers Bob Svendsen

Page 2 10-10-2012

Theo Tervoort Hans Christian Öttinger

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SLIDE 4
  • entropy production (2nd law)
  • quasi-linear relations
  • is irrelevant
  • Page 3

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comparison

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  • entropy production (2nd law)
  • dissipation potential
  • positivity and convexity

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comparison

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SLIDE 6
  • entropy production (2nd law)
  • linear relation
  • dissipation potential

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close to equilibrium: linear response

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SLIDE 7
  • →
  • construction of
  • close to equilibrium :
  • general:

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general

?

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SLIDE 8
  • →
  • counter example (1): conflict with convexity

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general

?

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SLIDE 9
  • →
  • counter example (2): non-dissipative irreversible dynamics
  • slip (Schowalter derivative) [1,2,3]

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general

?

[1] Wapperom, Hulsen, J. Rheol. 42, 999, 1998. [2] Dressler et al., Rheol. Acta 38, 117, 1999. [3] Öttinger, Beyond Equilibrium Thermodynamics, 2005.

→ →

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SLIDE 10
  • →
  • counter example (3) [condition]

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general

?

[1] Wapperom, Hulsen, J. Rheol. 42, 999, 1998. [2] Dressler et al., Rheol. Acta 38, 117, 1999. [3] Öttinger, Beyond Equilibrium Thermodynamics, 2005.

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  • generalization of Helmholtz theorem

with

  • one can show

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more elegant procedure

[1] Edelen, 1973 [2] Edelen, 1986 [3] Šilhavý, 1997, Ch. 12.

if j,f is symmetric; especially: j = φ,f

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SLIDE 12
  • generalization of Helmholtz theorem

with

  • quasi-linear relation

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more elegant procedure

[1] Edelen, 1973 [2] Edelen, 1986 [3] Šilhavý, 1997, Ch. 12.

(1) (2) (2)

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Grmela, Öttinger, PRE 56 (1997), 6620. Öttinger, Grmela, PRE 56 (1997), 6633. Öttinger, Beyond Equilibrium Thermodynamics, 2005.

reversible irreversible

GENERIC

G eneral E quation for the N on- E quilibrium R eversible- I rreversible C oupling Poisson

  • perator

friction matrix

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  • quasi-linear form
  • dissipation-potential representation

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GENERIC

Grmela, Öttinger, PRE 56 (1997), 6620. Öttinger, Grmela, PRE 56 (1997), 6633. Öttinger, Beyond Equilibrium Thermodynamics, 2005.

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  • mapping between GENERIC and force-flux relations

relation between force and entropy derivative with independent of entropy production: quasi-linear form:

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GENERIC

Edwards, 1998. Öttinger, Beyond Equilibrium Thermodynamics, 2005.

→ →

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  • mapping between GENERIC and force-flux relations

relation between force and entropy derivative with independent of entropy production: dissipation-potential form:

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GENERIC

Edwards, 1998. Öttinger, Beyond Equilibrium Thermodynamics, 2005.

→ →

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SLIDE 17

fluctuations irreversible reversible

(N. B. Wecando)

slow fast

separation of time scales

coarse graining: projection operators

Öttinger, PRE, 1998. Öttinger, Beyond Equilibrium Thermodynamics, 2005.

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  • forces and fluxes
  • possible potential-representation

1. 2. 3.

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example 1: heat conduction

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SLIDE 19
  • forces and fluxes
  • possible potential-representation

1. here: 2. Grmela (2010)

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example 2: chemical reaction(s)

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  • relations between forces and fluxes (e.g. …)
  • (quasi-)linear relations [1,2]
  • perturbation theory
  • fluctuation-dissipation theorem
  • dissipation potential [3]

Page 19 10-10-2012

Hütter, Svendsen, manuscript submitted to Contin. Mech. Thermodyn., 2012.

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