1. Rotating D’Arcy Convection Manel Naspreda Martí Departament de Física Fonamental Universitat de Barcelona Group of Statistical Physics Revisor: Michael C. Cross, Caltech University (California) USA H.Philibert Gaspard Darcy (1803-1858)

2. Outline: Motivation Historical Introduction Basic equations Rotating Rayleigh-Bénard convection D’Arcy convection and rotating D’Arcy convection Conclusions

3. Motivation: In rotating Rayleigh- Bénard convection there are predictions of chaos (wall state) in terms of the angular velocity and with small Prandtl number that it doesn’t work because of the mean flow. On the other hand, in D’Arcy convection there is not mean flow! Petrol!! Analytical motivation!!! RcRc  KL instability stripes wall state 1202 no pattern

4. Historical Introduction: Kuppers and Lorz (1969) predicted that there is some a critical value of the dimensionless angular velocity,  c, when the stripes become unstable (KL instability), rotating an angle  c  60º. Busse and Heikes (1980) proposed a dynamical model that was introducing tree amplitudes. Yuhai Tu and M. Cross (1992) introduced in the last model the spatial dependences and predicted spatiotemporal chaos and within the amplitude equation one can see that the scaling in time and in space are:

5. Actually, the experiments are not in a very well agreement with those predictions!!

6. Basic equations Navier-Stokes equation: Continuum equation, incompressible equation:

7. Density profile: where  is the fluid’s coefficient of thermal expansion at a constant pressure: and  T is the temperature measured from the temperature from the mid- depth: (T 1 –T 2 ) / 2. From the Navier-Stokes equation ( ), it is found that the pressure field is:

8. Temperature profile ( ):

9. Balance momentum equation: Temperature evolution equation(Fourier Law): Rotating Rayleigh-Bénard Convection: d T1T1 T2T2

10. where it has been considered the thermal diffusion time d 2 / , the pressure in units of d/ , the temperature in units of  /  gd 3 ( is the kinematic viscosity, g is the gravitational acceleration and  is the effective thermal diffusivity). The adimentional constants are:

13. D’Arcy Convection Hypotesis: Isotropic porous medium Average porous size is sufficiently large The boundaries conditions are the same as Rayleigh- Bénard convection, unrealisitc free slip boundary conditions. It’s Rayleigh-Bénard convection, but the fluid is flowing in a porous media.

14. Momentum balance equation (D’Arcy approximation): K is called the permeability of the porous medium, it is a positive constant and depends on the viscosity of the fluid and structure of the pores. Acceleration term is neglected compared to the damping term. No mean flow!! The other equations do not change.

15. Then if the time is measured in units of d 2 / , the temperature in units of  K/  gd, and the pressure in terms of K , one can find:

16. Linear Analysis Antsatz: Free slip b.c.: Critical Rayleigh number:

17. Amplitude equation Define an expansion parameter: Expand every quantity in terms of this parameter:

18. In a multi-scale analysis: wherer the slow variables scales as: X=  1/2 x; Y=  1/4 y; T=  t From the matricial equation: + some algebra...

19. where the differents parameters are:  0 =(2  ) -1 ;  0 =  -1 ; g 0 =1.

20. Rotating D’Arcy convection Coriolis force contribution in balance momentum: Z-component of the vorticity: The dynamics are described by:

21. Linear Analysis Antsatz + f.s.b.c: Critical Rayleigh number:

22. Amplitude equation Define the perturbative parameter: The way is the same as the D’Arcy convection, in sense of the perturbative theory and the seperation the magnitudes in seperate scales. The dynamics is described by: Some algebra, and...

23. where the differents parameters are:

24. Conclusions: Not enough time to work in this field. In this framework, it has shown that there is an invariant in the sign of the angular velocity!!

25. Future works: Introduce the time dependence and the non-linear term in the momentum balance equation, to find the possible efect of the sign of the angular velocity. Some computer simulations, to check the analytical framework. Introduction of more than one amplitude in the seperate scale spaciotemporal dependence, tending to BH.

26. References Li Ning and Robert E. Ecke, PRE, 47, 5 (1993). Yuhai Tu and M. C. Cross, PRL, 69, 17 (1992). Yuchou Hu, Robert E. Ecke and Guenter Ahlers, PRL, 74, 25 (1995). E. Y. Kuo and M. C. Cross, PRL, 47, 4 (1993). M. C. Cross, M. Louie and D. Meiron (unpublished).