Weakly nonlocal heat conduction

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Weakly nonlocal heat conduction modeling
memory and structure with nonequilibrium
thermodynamics Peter Ván HAS, RIPNP,
Department of Theoretical Physics

• Introduction
• memory and structure?
• different heat equations
• Memory Cattaneo-Vernote
• Structure Guyer-Krumhansl
• Hierarchy of heat equations - balances
• Discussion

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Why nonequilibrium thermodynamics?

• Thermodynamics

science of temperature
Thermodynamics science of macroscopic
energy changes
general framework of any Thermodynamics
(?) macroscopic (?) continuum (?)
theories

• General framework
• Second Law
• fundamental balances
• objectivity - frame indifference

reversibility special limit
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Nonlocalities
Restrictions from the Second Law. change of the
entropy current change of the entropy
Change of the constitutive space
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Weakly nonlocal memory inertia
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Non-homogeneous equilibrium structure
s
a
Single well
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s
a
Double well two phases
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Stable mixed sructure twinning in shape memory
alloys Initial and boundary conditions!
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Second Law
basic balances

• basic state
• constitutive state
• constitutive functions

Second law
(universality)
Constitutive theory
Method Liu procedure - solving the Liu
equations
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Ginzburg-Landau (thermodynamic, relocalized)
constitutive state space
constitutive functions
local state
Liu procedure (Farkass lemma)
?
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Basic state, constitutive state and constitutive
functions
Heat conduction Extended Thermodynamics

• basic state

• constitutive state

• constitutive functions

Heat conduction
Fourier
Cattaneo-Vernote
Guyer-Krumhansl
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Weakly nonlocal extended thermodynamics
state space
constitutive space
constitutive functions
local state
Liu procedure (Farkass lemma)
It is not solvable! Currents and forces?
solution?
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extended (Gyarmati) entropy
concave entropy (stability)
entropy current (Nyíri) (B current multiplier)
plausible general (dETdS qTJ)
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Solution, conditions (e.g. L11 p.d.)
gradients
Guyer-Krumhansl equation new
terms applications? Liu?
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Weakly nonlocal extended thermodynamics (again)
state space
constitutive space
constitutive functions
Specific questions
Why?
balance form
Why?
locality
Closure?
hierarchy
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Weakly nonlocal extended thermodynamics (again)
state space
constitutive space
constitutive functions
First order nonlocality
Liu procedure (Farkass lemma)
Liu equations
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Lius theorem
Conditions
Usage
Consequences
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Balance form evolution local s, H
Liu equations
Dissipation inequality
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New independent variables! potential
structure no dissipation
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Weakly nonlocal extended thermodynamics (again)
state space
constitutive space
constitutive functions
Second order nonlocality local state s(e,q)
Liu procedure (Farkass lemma)
Liu equations
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Once more
extended entropy
current multiplier
Closed (trivial)
Almost balance
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• Discussion
• Kinetic phenomenological
• Universality independent on the micro-modell
• Constructivity Liu force-current systems
• Origin of balances?
• Closure
• C(weakly nonlocal in time)?
• Second Law

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References General Gyarmati, I., The wave
approach of thermodynamics and some problems of
non-linear theories, Journal of Non-Equilibrium
Thermodynamics, 1977, 2, p233-260. Müller, I.
and Ruggeri, T. Rational Extended
Thermodynamics, Springer Verlag, 1998, Springer
Tracts in Natural Philosophy V 37, New
York-etc. Jou, D. , Casas-Vázquez, J. and Lebon,
G., Extended Irreversible Thermodynamics,
Springer Verlag, 2001, Berlin-etc., 3rd, revised
edition. Heat conduction Cimmelli, V. A. and
Ván, P., The effects of nonlocality on the
evolution of higher order fluxes in
non-equilibrium thermodynamics, Journal of
Mathematical Physics, 2005, 46, p112901,
(cond-mat/0409254). Ciancio, V. , Cimmelli, V.
A. and Ván, P., On the evolution of higher order
fluxes in non-equilibrium thermodynamics, 2006,
(cond-mat/0407530).
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Thank you for your attention!