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One and two component weakly nonlocal fluids
Peter VánBUTE, Department of Chemical Physics
– Nonequilibrium thermodynamics - weakly nonlocal theories
– One component fluid mechanics - quantum (?) fluids
– Two component fluid mechanics - granular material
– Conclusions
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– Thermodynamics = macrodynamics – Weakly nonlocal = there are more gradients
– Examples:
Guyer-Krumhans
Ginzburg-LandauCahn-Hilliard (- Frank)other phase field...
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Classical Irreversible
Thermodynamics
Local equilibrium (~ there is no microstructure)
Beyond local equilibrium (nonlocality):
•in time (memory effects)•in space (structure effects)
dynamic variables?
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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.
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Nonequilibrium thermodynamics
aa ja basic balances ,...),( va
– basic state:– constitutive state:– constitutive functions:
a
)C(aj,...),,(C aaa
weakly nonlocalSecond law:
0)C()C(s ss j
Constitutive theory
Method: Liu procedure, Lagrange-Farkas multipliersSpecial: irreversible thermodynamics
(universality)
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Example 1: One component weakly nonlocal fluid
),,,(C vv ),,,,(Cwnl vv
)C(),C(),C(s Pjs
Liu procedure (Farkas’s lemma):
constitutive state
constitutive functions
0 v
0)C()C(s s j0Pv )C(
... Pvjs2
)(s),(s2
e
vv 2
),(s),,(s2
e
vv
),( v basic state
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Schrödinger-Madelung fluid
222
1),,(s
22
SchM
vv
2
8
1 2rSchM IP
0:s2
ss2
1 22
s
vIP
vP
(Fisher entropy)
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Potential form: Qr U P
Bernoulli equation
)()( eeQ ssU Euler-Lagrange form
Schrödinger equation
Remark: Not only quantum mechanics- more nonlocal fluids- structures (cosmic)- stability (strange)
Alkalmazás
Oscillator
v ie
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Example 2: Two component weakly nonlocal fluid
2211density of the solid componentvolume distribution function
),,( v
),,,,,( vv C
constitutive functions
)C(),C(),C(s s Pj
basic state
constitutive state
00 v
0Pv )C(0)C()C(s s j
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Constraints: )3(),2(),2(),1(),1(
.)(
,)(
,)(
,s
,s
,s
,s
,s
,s
s54s
s5s
s5s
5
4
3
2
1
0PIj
0Pj
0Pj
0
vv
v
v
.s
,s
,s
,s
0
0
0
0
isotropic, second order
Liu equations
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Solution:
2
)(),(
2),(m),(s),,,,(s
22
e
vv
).,,()(),()( 1 vjPvj CmCs
Simplification:
0:)s(:)m( vIPv
.p
s,),,(,1m2e1
0vj
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0:)2
)(p(
2
vIP
Pr
Coulomb-Mohr
vLPPP vr
isotropy: Navier-Stokes like + ...
Entropy inequality:
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Properties
1 Other models: a) Goodman-Cowin
2)2)(p( 2r IP
h configurational force balance
b) Navier-Stokes type: somewhere
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2)( s
2)(2
pt
spt
)(ln
2
11
N
S
t
s
unstable
stable
2 Coulomb-Mohr
nPnN r: NPS r:
222 )( stNS
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3 solid-fluid(gas) transition
v)( relaxation (1D)
IP pr
4 internal spin: no corrections
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Conclusions-- Phenomenological background
- for any statistical-kinetic theory- Kaniadakis (kinetic), Plastino (maxent)
-- Nontrivial material (in)stability- not a Ginzburg-Landau- phase ‘loss’