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Philippe Beltrame

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Pattern formation in spherical Rayleigh-Bénard convection


The spherical Rayleigh-Bénard convection is commonly encountered in sciences like astrophysics and geophysics. It is especially a paradigm for the application of the equivariant bifurcation theory (here the symmetry group is O(3)). The theory using the symmetry group of the problem can not only predict the possible (relative) equilibrium solutions but also to point out intermittent like behavior:  solution spends a very long time near equilibrium and suddenly switch to the next equilibrium in a short time. For the geophysical time scale, such behavior is reminiscent of the quasi-periodic inversion of the Earth’s magnetic pole which depends on the convection motion in the outer core.


Here, we focus on convective motion of silicone oil in a spherical gap under a central force field due to the dielectrophoretic effect. This set-up is used for the ESA GEOFLOW-experiment which is currently running on the ISS (International Space Station). The microgravity environment is an unique opportunity to corroborate the O(3)-equivariant bifurcation theory.

Although the central force differs from the gravity field we obtain the same typical stationary solution or rotating wave as in literature. Furthermore, we are able to show the same kind of intermittent-like dynamics can occur.

Eddy-Current Testing (ECT)


The detection of cracks using ECT is a common technique in the industry: inspection of wings of airplanes, or tubes in nuclear power plants. The challenge is to detect, as early as possible, very thin crack (thickness e ≃ 10μm). There is two emphases:

  1. scale problem: thickness of crack ≪other crack dimensions ,

  2. erratic structure of the crack.

To tackle this problem, the ideal crack is modeled by a current dipole surface. The mathematical formulation leads to an integral equation with a so-called ‘hypersingular’ kernel. We point out that this singularity is only due to the electrostatic part of the Green’s kernel, i.e. the quadrupole nature of the crack. This simple physical picture is helpful to adapt regularization techniques required for the numerical computation.
This modeling proves its efficiency and can be easily improved by introducing “small” parameters as: the local conductivity (due to current leakage between both crack faces) and the small thickness e.
The drawback of this singular integral formulation is the requirement of surface smoothness. The study of irregular crack surface is an open problem.

Pattern formation in microfluidics


Microfluidic components and devices are useful as well for engineering as for biological analyzers. The dynamics of such a fluid are mainly governed by surface tension, viscosity (energy dissipation) and interaction with the substrate. The goal of this research is to answer to the following questions :

  1. *Pattern by dewetting: for which field is the drop/holes structure regular?

  2. *Dynamics of droplets, liquid ridges and rivulets on a heterogeneous surface: do they remain stuck, do they slide (mean velocity), do they breakup?...

  3. *How to transport micro particle in pores lattice with zero mean flow?

My approach is to use/develop bifurcation tools to explore the rich and complex saptio-temporal patterns.

Non-equilibrium and preferential flow in porous media


Preferential flow refers to the often rapid movement of water and solutes through porous media. In unsaturated soil, it may due to local heterogeneities such as wormholes, rootholes or fractures. The flow dynamics through a macropore is complex (intermittent flow) even chaotic. Then, constitutive equation characterizing the hydraulic properties of the macropore, as it is usually employed, fails to represent this flow scenarios.

My current research focuses on the description of the liquid film dynamics in macropore taking into account the interaction between the soil and the liquid (wettability) and also a free surface. The resulting equation is non-linear and may display complex scenarios. My approach is to use/develop bifurcation tools to explore the rich and complex flow scenarios.

A «spontaneous» preferential flow may also appear due to front instability in the soil matrix. Recently, Cueto-Felgueroso and Juanes propose a phase field model in order to take into account a macroscopic surface tension effect at the front. I am applying this model to stratified media in which the fingering instability is enhanced. I am using the bifurcation tools to explore the solutions of this phase field model.

Cueto-Felgueroso and Juanes, WRR 45, W10409,2009