Anomalous dimensions of the Smoluchowski coagulation equation.

The coagulation (or aggregation) equation was introduced by Smoluchowski in 1916 to describe the clumping together of colloidal particles through diffusion, but has been used in many different contexts as diverse as physical chemistry, chemical engineering, atmospheric physics, planetary science, and economics. The effectiveness of clumping is described by a kernel K(x,y), which depends on the sizes of the colliding particles x,y. We consider kernels K=(xy)^{γ}, but any homogeneous function can be treated using our methods.

Discrete Self-Similarity in Interfacial Hydrodynamics and the Formation of Iterated Structures.

The formation of iterated structures, such as satellite and subsatellite drops, filaments, and bubbles, is a common feature in interfacial hydrodynamics. Here we undertake a computational and theoretical study of their origin in the case of thin films of viscous fluids that are destabilized by long-range molecular or other forces. We demonstrate that iterated structures appear as a consequence of discrete self-similarity, where certain patterns repeat themselves, subject to rescaling, periodically in a logarithmic time scale.

A note on the self-similar solutions to the spontaneous fragmentation equation.

We provide a method to compute self-similar solutions for various fragmentation equations and use it to compute their asymptotic behaviours. Our procedure is applied to specific cases: (i) the case of mitosis, where fragmentation results into two identical fragments, (ii) fragmentation limited to the formation of sufficiently large fragments, and (iii) processes with fragmentation kernel presenting a power-like behaviour.

The microfluidic Kelvin water dropper.

The so-called "Kelvin water dropper" is a simple experiment demonstrating the spontaneous appearance of induced free charge in droplets emitted through a tube. As Lord Kelvin explained, water droplets spontaneously acquire a net charge during detachment from a faucet due to the presence of electrical fields in their surroundings created by any metallic object. In his experiment, two streams of droplets are allowed to drip from separate nozzles into separate buckets, which are, at the same time, interconnected through the dripping needles.

Flower patterns in drop impact on thin liquid films.

We describe experimentally the formation of a pattern for drop impacts on thin liquid films for a large range of impact parameters. Using the shallow-water approximation, we are able to explain the main mechanisms leading to these patterns: it consists in the linear instability of the self-similar axisymmetric radial solution of the equations. Agreement between the experiments and the theory is remarkably good, leading, in particular, to the prediction that the most unstable fold number scales like (We/h∞)2/7.

Mechanism of branching in negative ionization fronts.

When a strong electric field is applied to nonconducting matter, narrow channels of plasma called streamers may form. Branchlike patterns of streamers have been observed in anode directed discharges. We explain a mechanism for branching as the result of a balance between the destabilizing effect of impact ionization and the stabilizing effect of electron diffusion on ionization fronts.

Ionization fronts in negative corona discharges.

We use a hydrodynamic minimal streamer model to study negative corona discharge. By reformulating the model in terms of a quantity called a shielding factor, we deduce laws for the evolution in time of both the radius and intensity of the ionization fronts. We also compute the evolution of the front thickness under the conditions for which it diffuses due to the geometry of the problem and show its self-similar character.

Evidence of singularities for a family of contour dynamics equations.

In this work, we show evidence of the existence of singularities developing in finite time for a class of contour dynamics equations depending on a parameter 0 < alpha 0 corresponds to 2D Euler equations, and alpha = 1 corresponds to the surface quasi-geostrophic equation. The singularity is point-like, and it is approached in a self-similar manner.

Drops: the collapse of capillary jets.

The appearance of fluid filaments during the evolution of a viscous fluid jet is a commonly observed phenomenon. It is shown here that the break-up of such a jet subject to capillary forces is impossible through the collapse of a uniform filament.