This research activity results from the fact of being one of the systems where I study/ied pattern formation, creation of topological defects, etc...

The results presented here, have been mostly reported in some of the publications cited ⇒ here ⇐

Introduction

When temperature gradient is applied to a fluid system, there appear heat transport mechanisms in the system which try to return it to the equilibrium state. The mechanisms which are always involved are conduction and radiation. However, it could happen, that for values of the control parameter (which could be defined as the reduced temperature difference) higher than a threshold value, this mechanisms are not enough to keep the system without a flow stable. Consequently, the purely conductive state becomes unstable and the fluid begins to move (convection).

Systems heated from below in a homogeneous way were first considered by Bénard [H. Bénard. Rev. Gén. Sci. Pure Appl. 11 (1900) 1261]. We consider the temperature difference between the top and the bottom of the liquid layer as the experimental control parameter ΔT. For low values of ΔT the heat transport is done efficiently by conduction. Therefore it will be attained a linear temperature profile upon the depth z:

T(z)=T₀+ΔT z/h

Here h is the depth of the fluid layer and T₀ the temperature of the bottom (z=0).

This profile induces a density stratification of the fluid, which can be modelled by:

ρ(T)=ρ₀(1-αΔT)

(α is the volumetric thermal expansion coefficient)

In the Boussinesq approximation it is assumed that the variation of other parameters due to the temperature profile is small and does not influences on the dynamical regimes of the systems. Also, the stratification is considered only in the gradient of hydrostatic pressure term of the Navier Stokes equations.

The stratification of the density tries to destabilize the conductive state when fluctuations exist, due to the Archimedes buoyancy force. The destabilizing effect of buoyancy can be offset owing to the stabilizing effects of thermal diffusion and viscous dissipation. The buoyancy time is defined as:

τ_b=(h/gαΔT)^1/2

(g is the acceleration of gravity)

The stabilizing effects are governed by the following characteristic times:

τ_θ=h²/κ , τ_ν=h²/ν

(κ and ν are the thermal diffusivity and the kinematic viscosity, respectively. τ_θ is the time of stabilization of the temperature by conduction in a liquid layer of depth h)

Classically, the conduction convection bifurcation in a Rayleigh-Bénard system (where the surface tension effects are neglected) is characterized by the Rayleigh number Ra:

Ra=τ_θτ_ν/τ_b²

Then, if the Rayleigh number is bigger than a critical value (Ra_c), the bifurcation is crossed over. Therefore it is possible to define the control parameter ε as:

ε≡(Ra-Ra_c)/Ra_c=(ΔT-ΔT_c)/ΔT_c

When, the system is bounded from above by a fluid state, as in the case of a system open to the atmosphere, it is needed also to take into account the surface tension effects. This kind of convection is called Bénard-Marangoni convection. This introduces a new non-dimensional number called the Marangoni number Ma:

Ma=τ_θτ_ν/τ_σ²

where:

τ_σ=(h³ρ₀/|∂_Tσ|ΔT)^1/2

(ρ₀ is the density at the mean temperature and σ is the surface tension)

When both effects (bouyancy and surface tension) are important the system at threshold verifies the Nield equation [D. Nield J. Fluid Mech. 19 (1964) 341]. The Bénard-Marangoni convection bifurcation is slightly subcritical [M.F. Schatz et al. Phys. Rev. Lett. 75 (1995) 1938].

When heating is not homogeneous or not in the gravity direction, the threshold for having convection disappears and the focus is given to further bifurcations. This fact gives an added value to the corresponding experiments.

On the other hand, thermal convection in Liquid crystal systems is much more complicated to understand. The main point is that a liquid crystal is anisotropic in the viscosity, the thermal conductivity and so on. Then, there appear new interesting phenomena.

Collaborators

(in alphabetical order, all levels of collaboration: in the lab, permanent collaboration, temporary collaboration)

• Javier Burguete

• Sergio Casado

• Héctor L. Mancini

• Montserrat A. Miranda

• J. Salán

Results

The following movies come from conversion of old (long and sometimes defective) VHS videos, corresponding in all the cases to Bénard-Marangoni convection of silicone oil at different viscosities. In the same movie is possible that the images are at double speed (this is the case when in the images you can see several horizontal lines indicating fast-forward) with regard to the normal speed indicated. The original movies where taken by S. Casado and W. González-Viñas.

• Speed: 16x, intermediate aspect ratio. It is possible to see, among other interesting things, the division of a convective cell.

• Speed: 1x, small-intermediate aspect ratio. It is possible to see the effect of spatial filtering and focusing onto different planes. In the image below dark points represent hot spots, where the fluid goes upward, and the sides of the convective cells (bright) represent cold zones where the fluid goes downward.