Thermally induced non-equilibrium fluctuations: gravity and - - PowerPoint PPT Presentation

Thermally induced non-equilibrium fluctuations: gravity and finite-size effects

IWNET, Røros, Norway, August 19-24, 2012 Jan V. Sengers Institute for Physical Science and Technology University of Maryland, College Park, MD 20742 José M. Ortiz de Zárate

• Depto. Física Aplicada I

Universidad Complutense, Madrid

Outline

• 1. Introduction: statement of the problem
• 2. Non-equilibrium fluctuating hydrodynamics
• 3. Light-scattering experiments
• 4. Gravity effect on non-equilibrium fluctuations
• 5. Gravity and finite-size effects near R-B instability
• 6. Gravity and finite-size effects far away from R-B

instability

THERMAL FLUCTUATIONS IN FLUIDS

L T1 T2

1 2

T T 

p

c ds dT T 

at constant pressure:

Example: temperature evolution equation

(at constant pressure)

p

T c T t               v Q

T       Q Q

  Q

“Fluctuating” heat equation Linear phenomenological laws are valid only “on average”:

2 p

T c T T t                   v Q

Fluctuating Hydrodynamics

( 0)   v

( , ), ( , ), T T T t t       r v v r

Thermal fluctuations in equilibrium

2 p

T c T t             Q

Fluctuation-dissipation theorem:

2 B

( , ) ( , ) 2 ( ) ( )

i j ij

Q t Q t k T t t      



        r r r r

   

 

2 2 B

, ,0 exp

p

k T T q t T q aq t c   



 

Thermal fluctuations in a temperature gradient

T1 T2 L Rayleigh number:

4

L T R a     g 

α is thermal expansion coefficient ν is kinematic viscosity a = λ/ρcp is thermal diffusivity

1 2

T T 

2 p

T c T T t                      v Q

2 p

T c T T t                   v Q

Fluid in temperature gradient

( , ), ( , ), T T T t t       r v v r

2

1 +

t 

   

   



v

v S

Fluctuating heat equation: Fluctuating Navier-Stokes equation at constant pressure: Coupling between heat mode and viscous mode through T0

Assumption: local equilibrium for noise correlations

   

  

  

B

, , 2

ij kl ij kl il jk

S t S t k T t t         



         r r r r

2 B

( , ) ( , ) 2 ( ) ( )

i j ij

Q t Q t k T t t      



        r r r r

 

   

2 2

( ) 1 exp exp

T

C t C A aq t A q t



         

2 2 2 2 4 2 2 4

( ) ( ) ( ) ( )

p p T

c c T T A A T a a q T a q



        

Fluids in a temperature gradient

T.R. Kirkpatrick, J.R. Dorfman and E.G.D. Cohen, Phys. Rev. A 26, 995 (1982),

• D. Ronis and I. Procaccia, Phys. Rev. A 26, 1812 (1982),

B.M. Law and J.V. Sengers, J. Stat. Phys. 57, 531 (1989).

Bragg-Williams condition

 

   

2 2

( ) 1 exp exp

T T

C t C A D q t A q t



         

Toluene q=2255 cm–1, T=220 K/cm

Law, Segrè, Gammon, Sengers,

• Phys. Rev. A 41, 816 (1990)

2 2 2 4

( ) ( )

p T T T

c T A T D D q     

2 2 2 4

( ) ( )

p T

c T A T D q



   

Segrè, Gammon, Sengers, Law, Phys. Rev. A 45, 714 (1992)

Thermal fluctuations in a binary fluid

In liquids:>a฀D Lewis number Le=a/D Decay rate of viscous fluctuations q2 Decay rate of thermal fluctuations aq2 Decay rate of concentration fluctuations Dq2

Fluid mixtures in a concentration gradient

2

1 t            v v S

2

1 c c D c t              v J

δJ is fluctuating mass-diffusion flux

Coupling between concentration mode and viscous mode through c0

* B ,

( , ) ( , ) 2 ( ) ( )

i j ij T P

c J t J t k T D t t                        r r r r

2 2 2 4

( ) ( )

p T

c T A T a a q     

2 2 2 4

( ) ( )

p

c T A T a q



   

Segrè, Gammon, Sengers, Law, Phys. Rev. A 45, 714 (1992)

AT

P.N. Segrè, R. Schmitz, J.V. Sengers, Physica A 195, 31 (1993)

NONEQUILIBRIUM CONCENTRATION FLUCTUATIONS EFFECT OF GRAVITY

  

      

4 RO NE NE

/ 1 1 q q S S

4 RO

1

T

q c D c              g

4 RO

1

P

q T T             g

One-component: Mixture:

Thermal fluctuations in a temperature gradient

T1 T2 L Rayleigh number:

4

L T R a     g 

α is thermal expansion coefficient ν is kinematic viscosity a = λ/ρcp is thermal diffusivity

1 2

T T 

J.M. Ortiz de Zaráte, J.V. Sengers,

Sol i d cur ve:

R=1700

Dashed cur ve:

R=0

Dot t ed cur ve:

R=25, 000

Thermal fluctuations in a temperature gradient: Heated from below T1 T2

L Rayleigh number:

4

L T R a     g 

α is thermal expansion coefficient ν is kinematic viscosity a = λ/ρcp is thermal diffusivity

T1 <T2

(positive)

Shadowgraphy

J.R. de Bruyn. E. Bodenschatz, S.W. Morris, S.P. Trainoff, Y. Hu, D.S. Cannell, G. Ahlers,

• Rev. Sci. Instrum. 67, 2043 (1996)

J.Oh, J.M.Ortiz de Zárate, J.V.Sengers, G.Ahlers

• Phys. Rev. E 69, 021106 (2004)

c c

R R R   

Oh, Ortiz de Zárate, Sengers, Ahlers, Phys. Rev. E 69, 021106 (2004)

Oh, Ortiz de Zárate, Sengers, Ahlers, Phys. Rev. E 69, 021106 (2004)

Oh, Ortiz de Zárate, Sengers, Ahlers, Phys. Rev. E 69, 021106 (2004)

Thermal fluctuations in a temperature gradient: Heated from above T1 T2

L Rayleigh number:

4

L T R a     g 

α is thermal expansion coefficient ν is kinematic viscosity a = λ/ρcp is thermal diffusivity

T1 > T2

(negative)

Vailati, Cerbino, Mazzoni, Giglio, Nikolaenko, Takacs, Cannell, Meyer, Smart, Applied Optics 45, 2155 (2006)

0.1 1 10 1E-5 1E-4 1E-3 0.01 0.1 1

~ ~ SNE / S

NE

c = 0.50 % c = 2.00 % c = 4.00 %

q/qRO

4 5 6 7 8 9 10 1E-4 1E-3

J.V. Sengers, J.M. Ortiz de Zárate Lecture Notes in Physics 584 (Springer,2002), pp. 121-145 polystyrene-toluene solutions

polystyrene-toluene solution ΔT=17.40 K

• A. Vailati, R. Cerbino, S. Mazzoni, C.J. Takacs, D.S. Cannell, M. Giglio

Nature Communications 2, article #290 (19 April, 2011)

• A. Vailati, R. Cerbino, S. Mazzoni,

C.J. Takacs, D.S. Cannell, M. Giglio Nature Communications 2, article #290 (19 April, 2011)

C.J. Takacs, A. Vailati, R. Cerbino, S. Mazzoni, M. Giglio, D.S. Cannell PRL 106, 244502 (2011)

CS2

Conclusions

• Validity of non-equilibrium fluctuating

hydrodynamics has been confirmed experimentally by light scattering and shadowgraphy

• Thermal fluctuations exhibit always a strong non-

equilibrium enhancement

• Non-equilibrium fluctuations are always long range

encompassing the entire system

• Non-equilibrium fluctuations on earth are affected by

gravity

• Non-equilibrium fluctuations are affected by the

finite size of the system