quasi-linear versus potential -based formulations of … · quasi-linear versus potential -based...
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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 TechnologyMaterials Technologyhttp://www.mate.tue.nl
(2) RWTH AachenMaterial Mechanicshttp://www.aices.rwth-aachen.de
• relations between forces and fluxes (e.g. …)
• (quasi-)linear relations [1,2]
• perturbation theory• fluctuation-dissipation theorem
• dissipation potential [3]
Page 110-10-2012
[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
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
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Theo Tervoort
Hans Christian Öttinger
• entropy production (2nd law)
• quasi-linear relations
• is irrelevant
•
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comparison
• entropy production (2nd law)
• dissipation potential
• positivity and convexity
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comparison
• entropy production (2nd law)
• linear relation• dissipation potential
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close to equilibrium: linear response
• →
• construction of
• close to equilibrium :• general:
Page 610-10-2012
general
?
• →
• counter example (1): conflict with convexity
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general
?
• →
• 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.
→ →
• →
• 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.
• 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
• 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)
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 operator
frictionmatrix
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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.
• mapping between GENERIC and force-flux relations
relation between force and entropy derivativewith independent of
entropy production:
quasi-linear form:
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GENERIC
Edwards, 1998.Öttinger, Beyond Equilibrium Thermodynamics, 2005.
→
→
• mapping between GENERIC and force-flux relations
relation between force and entropy derivativewith independent of
entropy production:
dissipation-potential form:
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GENERIC
Edwards, 1998.Öttinger, Beyond Equilibrium Thermodynamics, 2005.
→
→
fluctuationsirreversible
reversible
(N. B. Wecando)
slow
fast
separation of time scales
coarse graining: projection operators
Öttinger, PRE, 1998.Öttinger, Beyond Equilibrium Thermodynamics, 2005.
• forces and fluxes
• possible potential-representation
1.2.3.
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example 1: heat conduction
• forces and fluxes
• possible potential-representation
1. here:2. Grmela (2010)
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example 2: chemical reaction(s)
• relations between forces and fluxes (e.g. …)
• (quasi-)linear relations [1,2]
• perturbation theory• fluctuation-dissipation theorem
• dissipation potential [3]
Page 1910-10-2012Hütter, Svendsen, manuscript submitted to Contin. Mech. Thermodyn., 2012.
summary