Softening of edges of solids by surface tension

Surface tension tends to minimize the area of interfaces between pieces of matter in different thermodynamic phases, be they in the solid or the liquid state. This can be relevant for the macroscopic shape of very soft solids, and lead to a roughening of initially sharp edges. We calculate this effect for a neo-Hookean elastic solid, with assumptions corresponding to actual experiments, namely the case where an initially sharp edge is rounded by the effect of surface tension felt when the fluid surrounding the soft solid (and so surface tension) is changed at the solid/liquid boundary. We consider two opposite limits where the analysis can be carried to the end, the one of a shallow angle and the one of a very sharp angle. Both cases yield a discontinuity of curvature in the state with surface tension although the initial state had a discontinuous slope

A non-equilibrium system in a steady state: wind waves in the open ocean

We derive scaling laws for the steady spectrum of wind excited waves, assuming two inviscid fluids (air and water) and no surface tension, an approximation valid at large speeds. In this limit there exists an unique (small) dimensionless parameter $\epsilon$, the ratio of the mass densities of the two fluids, air and water, independently of the wind speed. The smallness of $\epsilon$ allows to derive some important average properties of the wave system. The average square slope of the waves is of order $|ln (\epsilon^2)|^{-1}$, a small but not very small quantity. This supports the often used assumption of small nonlinearity in the wave-wave interaction. We introduce an equation to be satisfied by the two-point correlation of the height fluctuations.Comment: 6 pages, 0 figur

Fortelling catastrophes?

A generic saddle-node bifurcation is proposed to modelize fast transitions of finite amplitude arising in geophysical (and perhaps other) contexts, when they result from the intrinsic dynamics of the system. The fast transition is generically preceded by a precursor phase which is less rapid, that we characterize. In this model, if an external source of noise exist, the correlation length of the fluctuations increases before the transition, and its spectrum tends to drift towards lower frequencies. This change in the fluctuations could be a way of detecting catastrophic events before they happen.Comment: This paper contains 7p. and 4 figures. It is also submitted at the Paris Conference intitled "Science du Nonlineaire", 16-18 Mars 2011, and will be published in "Comptes Rendus du Nonmlineaire, Non-lin\'eaire publications Paris France, 03 201

Take-off of small Leidenfrost droplets

We put in evidence the unexpected behaviour of Leidenfrost droplets at the later stage of their evaporation. We predict and observe that, below a critical size $R_l$, the droplets spontaneously take-off due to the breakdown of the lubrication regime. We establish the theoretical relation between the droplet radius and its elevation. We predict that the vapour layer thickness increases when the droplets become smaller. A satisfactory agreement is found between the model and the experimental results performed on droplets of water and of ethanol.Comment: Accepted for publication in Phys. Rev. Lett. (2012