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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
– Thermodynamics = macrodynamics – Weakly nonlocal = there are more gradients
– Examples:
Guyer-Krumhans
Ginzburg-LandauCahn-Hilliard (- Frank)other phase field...
Classical Irreversible
Thermodynamics
Local equilibrium (~ there is no microstructure)
Beyond local equilibrium (nonlocality):
•in time (memory effects)•in space (structure effects)
dynamic variables?
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.
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)
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
Schrödinger-Madelung fluid
222
1),,(s
22
SchM
vv
2
8
1 2rSchM IP
0:s2
ss2
1 22
s
vIP
vP
(Fisher entropy)
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
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
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
Solution:
2
)(),(
2),(m),(s),,,,(s
22
e
vv
).,,()(),()( 1 vjPvj CmCs
Simplification:
0:)s(:)m( vIPv
.p
s,),,(,1m2e1
0vj
0:)2
)(p(
2
vIP
Pr
Coulomb-Mohr
vLPPP vr
isotropy: Navier-Stokes like + ...
Entropy inequality:
Properties
1 Other models: a) Goodman-Cowin
2)2)(p( 2r IP
h configurational force balance
b) Navier-Stokes type: somewhere
2)( s
2)(2
pt
spt
)(ln
2
11
N
S
t
s
unstable
stable
2 Coulomb-Mohr
nPnN r: NPS r:
222 )( stNS
3 solid-fluid(gas) transition
v)( relaxation (1D)
IP pr
4 internal spin: no corrections
Conclusions-- Phenomenological background
- for any statistical-kinetic theory- Kaniadakis (kinetic), Plastino (maxent)
-- Nontrivial material (in)stability- not a Ginzburg-Landau- phase ‘loss’