Post on 12-Jan-2016
What is thermodynamics and what is it for? II. Continuum physics – constitutive theory
Peter Ván HAS, RIPNP, Department of Theoretical Physics
– Introduction – Constitutive space and constitutive functions– Classical irreversible thermodynamics– Weakly non-local extensions
• Internal variables, heat conduction and fluids
– Discussion
Centre of Nonlinear Studies, Tallinn, Estonia, 19/6/2006.
Thermo-Dynamic theory
)a(fa Dynamic law:
,...),c,v(a
1 Statics (equilibrium properties)
S
aa
S,,
T
1
e
S
2 Dynamics
0)a(f)a(DSa)a(DS)a(S
1 + 2 + closed system
S is a Ljapunov function of the equilibrium of the dynamic law
Constructive application:
)()),(()(0)()( aDSaaDSLafafaDS force current
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: – fundamental balances– objectivity - frame indifference– Second Law
Basic state, constitutive state and constitutive functions:
ee q
– basic state:(wanted field: T(e))
e
)(Cq),( eeC
Heat conduction – Irreversible Thermodynamics
),( ee ))(),(( eTeT T q )())(),((),( eTeTeTee q
Fourier heat conduction:
But: qq LT qqq 21LLT Cattaneo-VernoteGuyer-Krumhansl
– constitutive state:– constitutive functions:
,...),,,,( 2eeeee ???
1)
)(C ),( v C
Local state – Euler equation
0
0
Pv
v
2)
– basic state:– constitutive state:– constitutive function:
Fluid mechanics
Nonlocal extension - Navier-Stokes equation:v
se
p1
),,()()( 2
IP
vIvvP 2))((),( p
But: 22)( IP prKor
),,,( 2 vC),( v
)(CP
Korteweg fluid
fa
a
s
a
sLa
Internal variable
– basic state: aa– constitutive state:
– constitutive function:
A) Local state - relaxation
da
dsLff
da
ds 0
3)
B) Nonlocal extension - Ginzburg-Landau
aaa 2,,
),( aaa
sL
alaslaaasaas )('ˆ,
2)(ˆ),( 2 e.g.
)(Cf
)0)('ˆ( as
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
Second Law:
aa ja basic balances ,...),( va
– basic state:– constitutive state:– constitutive functions:
a
)C(aj,...),,(C aaa
weakly nonlocalSecond law:
0)()( sCCs J
Constitutive theory
Method: Liu procedure
(universality)
(and more)
Irreversible thermodynamics:
0
J
0ja
sa
– basic state:
– constitutive state:– constitutive functions:
a
Jj ,, sa
),( aa C
primary!!Liu procedure (Farkas lemma):
A) Liu equations:
0a
j
a
J0
aa
ass ,,
)(),()('ˆ),(
),(ˆ),(
0 ajaajaaJ
aaa
aas
ss
Te
s qqJ
Heat conduction: a=e
B) Dissipation inequality:
0'ˆ
a
jjs
s aa0
12
TTT
What is explained:
The origin of Clausius-Duhem inequality: - form of the entropy current - what depends on what
Conditions of applicability!!
- the key is the constitutive space
Logical reduction:
the number of independent physical assumptions!
Mathematician: ok but…Physicist:
no need of such thinking, I am satisfied well and used to my analogiesno need of thermodynamics in general
Engineer:consequences??
Philosopher: …Popper, Lakatos:
excellent, in this way we can refute
Ginzburg-Landau (variational):
dVaasas ))(2
)(ˆ()( 2
))('ˆ( aasla – Variational (!) – Second Law?– ak
aassa )('ˆ
sla a
Weakly nonlocal internal variables
dVaasas ))(2
)(ˆ()( 2
sla a
Ginzburg-Landau (thermodynamic, relocalized)
),,( 2aaa
J),,( sf
Liu procedure (Farkas’s lemma)
)(as
0' fss J
constitutive state space
constitutive functions
fa 0 Js
),( aa J
?
local state
a
saaaa
),(),( BJ
0')(' sfss BB
a
sL
a
sLa 2211
'' 2221 sLsLf B
'' 1211 sLsL B
isotropy
))('( aasla
current multiplier
Ginzburg-Landau (thermodynamic, non relocalizable)
fa
0 Js
),,( 2aaa
J),,( sf
Liu procedure (Farkas’s lemma)
),( aas ),()()( 0 aaCfa
sC
jJ
0
fa
s
a
ss
a
s
a
sLa
state space
constitutive functions 0 fa
Weakly nonlocal extended thermodynamics
),,,,( 2qqq ee
J),,( sG
Liu procedure (Farkas’s lemma):
),( qes
),,( qqJ e
0
Gs
e
ss q
qJ
constitutive space
constitutive functions
0 qe
0 Js0Gq
solution?
local state:
),( qe state space
qqmqq ),(2
1)(),( 0 eeses
qqqBqqJ ),,(),,( ee
extended (Gyarmati) entropy
entropy current (Nyíri)(B – current multiplier)
0)(:
qmBqIB
Ge
ss
qqmB 2221 LLG qqIB 1211 LL
e
s
qqIqqqm 22211211 LLe
sLL
gradientGuyer-Krumhansl equation
Korteweg fluids (weakly nonlocal in density, second grade)
),,( v C ),,,( v wnlC
)(),(),( CCCs PJ
Liu procedure (Farkas’s lemma):
constitutive state
constitutive functions
0 v
0)()( CCs J0Pv )C(
...J)(ess ),(),( ess
),( v basic state
0:s2
ss2
1 22
s
vIP
rv PPP
reversible pressurerP
Potential form: nlr U P
)()( eenl ssU Euler-Lagrange form
Variational origin
Schrödinger-Madelung fluid2
22),(
SchM
SchMs
2
8
1 2IP rSchM
(Fisher entropy)
Bernoulli equation
Schrödinger equation
v ie
Thermodynamics = theory of material stability
Ideas:– Phase transitions in gradient systems?In quantum fluids:– There is a family of equilibrium (stationary) solutions.
0v .constEUU SchM – There is a thermodynamic Ljapunov function:
dVEUL
22
22
1
2),(
v
v
semidefinite in a gradient (Soboljev ?) space
2
xD)(xU
2
Mov1.exe
Conclusions- Dynamic stability, Ljapunov function???- Universality – independent on the micro-modell- Constructivity – Liu + force-current systems- Variational principles: an explanation
Second Law
Problems, perspectives: objectivity (material frame indifference):
mechanics (hyperstress and strain)!electrodynamics (special relativity)
),,( aaaC
But: heat conduction, two component fluids (sand), Cahn-Hilliard, complex Ginzburg-Landau, Korteweg-de Vries, …. , weakly non-local statistical physics, …
Thank you for your attention!