Marco A. Fontelos

The Beads-on-String Structure of Viscoelastic Threads

By adding minute concentrations of a high-molecular-weight polymer, liquid jets or bridges collapsing under the action of surface tension develop a characteristic shape of uniform threads connecting spherical fluid drops. In this paper, high-precision measurements of this beads-on-string structure are combined with a theoretical analysis of the limiting case of large polymer relaxation times and high polymer extensibilities, for which the evolution can be divided into two distinct regimes. For times smaller than the polymer relaxation time over which the beads-on-string structure develops, we give a simplified local description, which still retains the essential physics of the problem. At times much larger than the relaxation time, we show that the solution consists of exponentially thinning threads connecting almost spherical drops. Both experiment and theoretical analysis of a one-dimensional model equation reveal a self-similar structure of the corner where a thread is attached to the neighbouring drops.

Drop dynamics after impact on a solid wall: Theory and simulations

We study the impact of a fluid drop onto a planar solid surface at high speed so that at impact, kinetic energy dominates over surface energy and inertia dominates over viscous effects. As the drop spreads, it deforms into a thin film, whose thickness is limited by the growth of a viscous boundary layer near the solid wall. Owing to surface tension, the edge of the film retracts relative to the flow in the film and fluid collects into a toroidal rim bounding the film. Using mass and momentum conservation, we construct a model for the radius of the deposit as a function of time. At each stage, we perform detailed comparisons between theory and numerical simulations of the Navier–Stokes equation.

Drop dynamics on the beads-on-string structure for viscoelastic jets: A numerical study

It is well known that a viscoelastic jet breaks up much more slowly than a Newtonian jet. Typically, it evolves into the so-called beads-on-string structure, where large drops are connected by thin threads. The slow breakup process provides the viscoelastic jet sufficient time to exhibit some new phenomena. The aim of this paper is to investigate the drop dynamics of the beads-on-string structure. This includes drop migration, drop oscillation, drop merging and drop draining. We will use a 1D Oldroyd-B model for the viscoelastic jet, and solve this model numerically by an explicit finite difference method. Close to exponential draining of the filament, we found that the variation of the axial elastic force in the filament is roughly four times larger than the variation of the capillary force with opposite sign. This fact implies that the elastic force is responsible for the drop migration and oscillation. Our study of the drop draining process shows that the elastic force also plays an important role here, allowing the liquid to flow from smaller drops into larger drops through the filament.

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 < α ≤ 1. The limiting case α → 0 corresponds to 2D Euler equations, and α = 1 corresponds to the surface quasi-geostrophic equation. The singularity is point-like, and it is approached in a self-similar manner.

MATHEMATICAL ANALYSIS OF A MODEL FOR THE INITIATION OF ANGIOGENESIS

In this paper we consider a nonlinear system of partial differential equations consisting of one parabolic equation and two ordinary differential equations in t. The system arises in a mathematical model of angiogenesis, a process of sprouting of new blood vessels from an existing vascular network. We prove that the system has a unique global solution and study its asymptotic behavior as

Capillarity driven spreading of power-law fluids

t\rightarrow \infty

Capillarity driven spreading of circular drops of shear-thinning fluid

. In particular, we show that stationary solutions are local attractors.

The Effect of Surface Tension on the Moore Singularity of Vortex Sheet Dynamics

Abstract We investigate the spreading of thin liquid films of power-law rheology. We construct an explicit travelling wave solution and source-type similarity solutions. We show that when the nonlinearity exponent λ for the rheology is larger than one, the governing dimensionless equation h t + ( h λ +2 | h xxx | λ −1 h xxx ) x =0 admits solutions with compact support and moving fronts. We also show that the solutions have bounded energy dissipation rate.

Photoionization effects in ionization fronts

We investigate the spreading of thin, circular liquid drops of power-law rheology. We derive the equation of motion using the thin film approximation, construct source-type similarity solutions, and compute the spreading rate, aparent contact angles and height profiles. In contrast with the spreading of Newtonian liquids, the contact line paradox does not arise for shear thinning fluids.

Break-up and no break-up in a family of models for the evolution of viscoelastic jets

We investigate the regularization of Moore’s singularities by surface tension in the evolution of vortex sheets and its dependence on the Weber number (which is inversely proportional to surface tension coefficient). The curvature of the vortex sheet, instead of blowing up at finite time t0, grows exponentially fast up to a O(We) limiting value close to t0. We describe the analytic structure of the solutions and their self-similar features and characteristic scales in terms of the Weber number in a O(We−1) neighborhood of the time at which, in absence of surface tension effects, Moore’s singularity would appear. Our arguments rely on asymptotic techniques and are supported by full numerical simulations of the PDEs describing the evolution of vortex sheets.