Weakly nonlocal heat conduction – modeling memory and structure with nonequilibrium thermodynamics...

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

general framework of anyThermodynamics (?) macroscopic (?)

continuum (?) theories

Thermodynamics science of macroscopic energy changes

Thermodynamics

science of temperature

Why nonequilibrium thermodynamics?

reversibility – special limit

General framework: – Second Law – fundamental balances– objectivity - frame indifference

Space Time

Strongly nonlocal

Space integrals Memory functionals

Weakly nonlocal

Gradient dependent

constitutive functions

Rate dependent constitutive functions

Relocalized

Current multipliers Internal variables

Nonlocalities:

Restrictions from the Second Law.change of the entropy currentchange of the entropy

Change of the constitutive space

Weakly nonlocal memory: inertia

0)( 00 TTTTTCE QQQQ

TTe qqqq 0

TT0

Q

2 4 6 8 10t

1

2

3

4

5

T

T05T03T01T0.5T00.1

0)( 0 TTTCTC

.1

,1

,1,00

CT

.1)0(';0)0('

1

1

1,00

TT

CT

2 4 6 8 10t

1

2

3

4

5

T

T05T03T01T0.5T00.1

2 4 6 8 10t

1

2

3

4

5

6

T

T05T03T01T0.5T00.1

0)( 0 TTTCTC

Non-homogeneous equilibrium: structure

))('( uuslu s

a

-1

-0.5

0

0.5

1

x

0

10

20

30

40

50

t

-1

-0.5

0

0.5

1

u

-1

-0.5

0

0.5

1

x

-1 -0.5 0.5 1x

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

u

Single well

-1 -0.5 0.5 1

0.2

0.4

0.6

0.8

1

-1 -0.5 0.5 1x

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

u

-1

-0.5

0

0.5

1

x

0

10

20

30

t

-1

-0.5

0

0.5

1

u

-1

-0.5

0

0.5

1

x

-1 -0.5 0.5 1x

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

u

Double well: two phases

-1 -0.5 0.5 1

-0.05

-0.025

0.025

0.05

0.075

s

a

-1

-0.5

0

0.5

1

x

0

10

20

30

40

50

t

-1

-0.5

0

0.5

1

u

-1

-0.5

0

0.5

1

x

-1

-0.5

0

0.5

1

x

0

1

2

3

4

5

t

-1

-0.5

0

0.5

1

u

-1

-0.5

0

0.5

1

x

-1 -0.5 0.5 1x

-1

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

u

Stable mixed sructure: twinning in shape memory alloys

Initial and boundary conditions!

Second Law:

aa ja basic balances ,...),( va

– basic state:– constitutive state:– constitutive functions:

a

)C(aj,...),,(C aaa

Second law:

0)()( sCCs J

Constitutive theory

Method: Liu procedure + … - solving the Liu equations

(universality)

Ginzburg-Landau (thermodynamic, relocalized)

),,( 2aaa

J),,( sf

Liu procedure (Farkas’s lemma)

)(as

0' fss J

constitutive state space


fa 0 Js

),( aa J

?

local state

Basic state, constitutive state and constitutive functions:

ee q

– basic state: e

)(Cq),( eeC

Heat conduction – Extended Thermodynamics

T q

Heat conduction:

T qq

qqq lT

Cattaneo-Vernote

Guyer-Krumhansl

– constitutive state:– constitutive functions:

Fourier

,...),,( qeeC

Weakly nonlocal extended thermodynamics

),,,,( 2qqq ee

J),,( sG

Liu procedure (Farkas’s lemma):

),( qes

),,( qqJ e

0

Gs

e

ss q

qJ

constitutive space


0 qe

0 Js0 Gq

solution?

local state:

),( qe state space

It is not solvable!Currents and forces?

qmqq 2

1)(),( 0 eses

qqqBqqJ ),,(),,( ee

extended (Gyarmati) entropy

entropy current (Nyíri)(B – current multiplier)

plausiblegeneral (dE=TdS ~ q=TJ)

concave entropy (stability)

0)(:

qmBqIB Ge

ss

qqmB 2221 LLG

qqIB 1211 LLe

s

qqIqqqm 22211211 LLe

sLL

gradients

Guyer-Krumhansl equation+ new termsapplications?Liu?

qqqq llT

Solution, conditions (e.g. L11 p.d.)

Weakly nonlocal extended thermodynamics (again)

),,,,( 2qqq ee

J),,( sG

constitutive space


0 qe

0 Js0 Gq

),( qe state space

0 Gq

Specific questions:

0 Hq balance form Why?

...

0 ΨHhierarchy Closure?

locality Why?


),,,( qq ee

J),,( sG

constitutive space


0 qe

0 Js0 Gq

),( qe state space

0

0

G

e

q

q

First order nonlocality


0

0

IJ

J

qqq

q

sGs

Gs

e

ee

0 GsGsseGs eee qqqqq qIJJ

Liu equations

Liu’s theorem:

n

iii

n

iii

n

iii

n

iii

n

iiii

ii

11

11

1

,

0

,

AB

ABp

ApBp

ApBp

Usage:

Conditions:

Consequences:

0

0

IJ

J

qqq

q

sGs

Gs

e

ee

qHHqH q :),( eeG e

Balance form evolution + local s, H:

),( qJ e

0 GsGsseGs eee qqqqq qIJJ

0 qHIJHJ qqqq sses eee

Liu equations:

Dissipation inequality:

0 qHIJHJ qqqq sses eee

HqJ qsdsdd e

s

se

qH

q

J

JHqJ

:),(

“New” independent variables!

potential structureno dissipation

0

0

HIJ

HJ

qqq

q

ss

s

e

ee


),,,,( 2qqq ee

J),,( sG

constitutive space


0 qe

0 Js0 Gq

),( qe state space

0

0

G

e

q

q

Second order nonlocality

+ local state: s(e,q)


0

),,,(

Gss

ee

e qqJ

qqJ Liu equations

qqqBqqJ ),,,(),,,( eeee

0q)(

mBqIB

qJ q

Gs

Gss

e

e

current multiplier

qmqq 2

1)(),( 0 eses extended entropy

qqmB 2221 LLG

qqIB 1211 LLe

s

qqIqqqm 22211211 LLe

sLL

Once more:

Almost balance:

Closed (trivial)

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

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 NaturalPhilosophy 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).

Thank you for your attention!

Weakly nonlocal heat conduction – modeling memory and structure with nonequilibrium thermodynamics...

Documents

Transcript of Weakly nonlocal heat conduction – modeling memory and structure with nonequilibrium thermodynamics...