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’