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

Thermally induced non-equilibrium fluctuations: gravity and finite-size effects 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 IWNET, Røros, Norway, August 19-24, 2012

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

2 T  THERMAL FLUCTUATIONS IN FLUIDS 1 T T 1 T 2 L

at constant pressure: c p  ds dT T

Fluctuating Hydrodynamics Example: temperature evolution equation   ( at constant pressure ) ( v 0)    T       T       c T Q Q v Q   p   t  Linear phenomenological laws   Q 0 are valid only “on average”: “Fluctuating” heat equation    T           c T T 2 v Q   p  t         T T T t t ( , ), r v 0 v r ( , ), 0

Thermal fluctuations in equilibrium   T          c T 2 Q p  t Fluctuation-dissipation theorem:                Q t Q t k T t t 2 ( , ) r ( , ) r 2 ( r r ) ( ) i j ij B 0 k T 2            T q t T q aq t 2 B 0 , ,0 exp  c p

Thermal fluctuations in a temperature gradient T 1  T T L 1 2 T 2  g   L T 4  R Rayleigh number:  a α is thermal expansion coefficient ν is kinematic viscosity a = λ / ρ c p is thermal diffusivity

Fluid in temperature gradient    T           c T T 2 v Q   p   t        T T T t t ( , ), r v 0 v r ( , ), 0 Fluctuating heat equation:     T             c T T 2 v Q   p  0  t  Fluctuating Navier-Stokes equation at constant pressure: 1          v 2 v + S  t Coupling between heat mode and viscous mode through  T 0

Assumption: local equilibrium for noise correlations                Q t Q t k T t t 2 ( , ) r ( , ) r 2 ( r r ) ( ) i j ij B 0           S t S t r , r , ij kl                    k T t t 2 r r ij kl il jk B 0

Fluids in a temperature gradient               C t C A aq t A q t 2 2 ( ) 1 exp exp    T 0 c   c  T T 2 2 ( ) ( )  p  p A A 0 0  T     T a a q T a q 2 2 4 2 2 4 ( ) ( ) 0 0 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

              C t C A D q t A q t 2 2 ( ) 1 exp exp    T T 0 Toluene q= 2255 cm –1 ,  T= 220 K/cm Law, Segrè, Gammon, Sengers, Phys. Rev. A 41 , 816 (1990)

c   c  T 2 T 2 ( ) ( )  p A  p A T    T D D q 2 2 4   T D q ( ) 2 2 4 ( ) T T T Segrè, Gammon, Sengers, Law, Phys. Rev. A 45 , 714 (1992)

Thermal fluctuations in a binary fluid Decay rate of viscous fluctuations  q 2 Decay rate of thermal fluctuations aq 2 Decay rate of concentration fluctuations Dq 2 In liquids:  > a ฀ D Lewis number Le= a / D

Fluid mixtures in a concentration gradient  1 t           c c D c 2 v J   0   v 1        2 v S   t δ J is fluctuating mass-diffusion flux    c               J t J t k T D t t *   r r r r ( , ) ( , ) 2 ( ) ( ) i j ij   B 0   T P , Coupling between concentration mode and viscous mode through  c 0

A T c   c  T T 2 2 ( ) ( )  p p A  A T    T a a q 2 2 4   T a q ( ) 2 2 4 ( ) Segrè, Gammon, Sengers, Law, Phys. Rev. A 45 , 714 (1992)

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

NONEQUILIBRIUM CONCENTRATION FLUCTUATIONS EFFECT OF GRAVITY   1  S S 0      NE NE 4 q q   1 / RO     1   q T 4   g One-component:   RO  T  0 P     1   q c 4   g Mixture:    RO 0 D  c  T

Thermal fluctuations in a temperature gradient T 1  T T L 1 2 T 2  g   L T 4  R Rayleigh number:  a α is thermal expansion coefficient ν is kinematic viscosity a = λ / ρ c p is thermal diffusivity

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 T 1 L T 1 < T 2 T 2  g   L T 4  R (positive) Rayleigh number:  a α is thermal expansion coefficient ν is kinematic viscosity a = λ / ρ c p is thermal diffusivity

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 R c 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 T 1 L T 1 > T 2 T 2  g   L T 4  R (negative) Rayleigh number:  a α is thermal expansion coefficient ν is kinematic viscosity a = λ / ρ c p is thermal diffusivity

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

J.V. Sengers, J.M. Ortiz de Zárate Lecture Notes in Physics 584 ( Springer,2002), pp. 121-145 c = 0.50 % 1 c = 2.00 % c = 4.00 % 0.1 NE 0 ~ S NE / S 0.01 1E-3 ~ 1E-3 1E-4 1E-4 4 5 6 7 8 9 10 1E-5 0.1 1 10 q / q RO polystyrene-toluene solutions

polystyrene-toluene solution Δ T =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)

CS 2 C.J. Takacs, A. Vailati, R. Cerbino, S. Mazzoni, M. Giglio, D.S. Cannell PRL 106 , 244502 (2011)

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