Biomedical subjects

Christophe Josserand

Publications and source records attributed to Christophe Josserand.

6 recordsLinked to original sources

PubMed2006

Weak turbulence for a vibrating plate: can one hear a Kolmogorov spectrum?

We study the long-time evolution of waves of a thin elastic plate in the limit of small deformation so that modes of oscillations interact weakly. According to the theory of weak turbulence (successfully applied in the past to plasma, optics, and hydrodynamic waves), this nonlinear wave system evolves at long times with a slow transfer of energy from one mode to another. We derive a kinetic equation for the spectral transfer in terms of the second order moment. We show that such a theory describes the approach to an equilibrium wave spectrum and represents also an energy cascade, often called the Kolmogorov-Zakharov spectrum. We perform numerical simulations that confirm this scenario.

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PubMed2006

Singular jets and bubbles in drop impact.

We show that when water droplets gently impact on a hydrophobic surface, the droplet shoots out a violent jet, the velocity of which can be up to 40 times the drop impact speed. As a function of the impact velocity, two different hydrodynamic singularities are found that correspond to the collapse of the air cavity formed by the deformation of the drop at impact. It is the collapse that subsequently leads to the jet formation. We show that the divergence of the jet velocity can be understood using simple scaling arguments. In addition, we find that very large air bubbles can remain trapped in the drops. The surprising occurrence of the bubbles for low-speed impact is connected with the nature of the singularities, and can have important consequences for drop deposition, e.g., in ink-jet printing.

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PubMed2005

Condensation of classical nonlinear waves.

We study the formation of a large-scale coherent structure (a condensate) in classical wave equations by considering the defocusing nonlinear Schrödinger equation as a representative model. We formulate a thermodynamic description of the classical condensation process by using a wave turbulence theory with ultraviolet cutoff. In three dimensions the equilibrium state undergoes a phase transition for sufficiently low energy density, while no transition occurs in two dimensions, in complete analogy with standard Bose-Einstein condensation in quantum systems. On the basis of a modified wave turbulence theory, we show that the nonlinear interaction makes the transition to condensation subcritical. The theory is in quantitative agreement with the numerical integration of the nonlinear Schrödinger equation.

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PubMed2005

Asymptotic behavior of the Rayleigh-Taylor instability.

We investigate long time numerical simulations of the inviscid Rayleigh-Taylor instability at Atwood number one using a boundary integral method. We are able to attain the asymptotic behavior for the spikes predicted by Clavin and Williams for which we give a simplified demonstration. In particular, we observe that the spike's curvature evolves as t(3), while the overshoot in acceleration shows good agreement with the suggested 1/t(5) law. Moreover, we obtain consistent results for the prefactor coefficients of the asymptotic laws. Eventually we exhibit the self-similar behavior of the interface profile near the spike.

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PubMed2004

Stability of giant vortices in quantum liquids.

We show how giant vortices can be stabilized for strong external potentials in Bose-Einstein condensates. We illustrate the formation of these vortices thanks to the Ginzburg-Landau dissipative dynamics for two typical potentials in two spatial dimensions. The giant vortex stability is studied for the particular case of a rotating cylindrical hard wall. Due to axial symmetry the minimization of the perturbed energy is simplified into a one dimensional relaxation dynamics. Solving this 1D minimization problem, we observe that giant vortices are either never stable, or only stable in a finite frequency range. Finally we obtain the marginal curve for the minimum frequency needed to observe a giant vortex.

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PubMed2002

Nonlinear growth of periodic patterns.

We study the growth of a periodic pattern in one dimension for a model of spinodal decomposition, the Cahn-Hilliard equation. We particularly focus on the intermediate region, where the nonlinearity cannot be neglected anymore, and before the coalescence dominates. The dynamics is captured through the standard technique of a solubility condition performed over a particular family of quasistatic solutions. The main result is that the dynamics along this particular class of solutions can be expressed in terms of a simple ordinary differential equation. The density profile of the stationary regime found at the end of the nonlinear growth is also well characterized. Numerical simulations correspond satisfactorily to the analytical results through three different methods and asymptotic dynamics are well recovered, even far from the region where the approximations hold.

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