arXiv subjects

M. A. Fontelos

Publications and source records attributed to M. A. Fontelos.

18 recordsLinked to original sources

arXiv2024

Evolution of vortex filaments and reconnections in the Gross-Pitaevski equation and its approximation by the binormal flow equation

The evolution of a vortex line following the binormal flow equation (i.e. with a velocity proportional to the local curvature in the direction of the binormal vector) has been postulated as an approximation for the evolution of vortex filaments in both the Euler system for inviscid incompressible fluids and the Gross-Pitaevski equation in superfluids. We address the issue of whether this is a suitable approximation or not and its degree of validity by using rigorous mathematical methods and direct numerical simulations. More specifically, we show that, as the vortex core thickness goes to zero, the vortex core moves (at leading order and for long periods of time) with a velocity proportional to its local curvature and the binormal vector to the curve. The main idea of our analysis lies in a reformulation of the Gross-Pitaevski equation in terms of associated velocity and vorticity fields that resemble the Euler system written in terms of vorticity in its weak form. We also present full numerical simulations aimed to compare Gross-Pitaevski and binormal flow in various physical situations of interest such as the periodic evolution of deformed vortex rings and the reconnection of vortex filaments.

physics.flu-dyn↗

arXiv2024

Controllability for a non-local formulation of surface gravity waves

In this paper, we study the approximate controllability of a system governed by an evolution problem known as the sloshing problem. This problem involves a spatial, nonlocal differential operator inherent in the dynamics of a two-dimensional, incompressible, non-viscous fluid within a confined domain. Our work establishes unique continuation results that enable the application of source control localized in an interior domain, allowing the aforementioned controllability.

math.AP↗

arXiv2022

Stability of selfsimilar solutions to the fragmentation equation with polynomial daughter fragments distribution

We study fragmentation equations with power-law fragmentation rates and polynomial daughter fragments distribution function $p(s)$. The corresponding selfsimillar solutions are analysed and their exponentially decaying asymptotic behaviour and $C^{\infty }$ regularity deduced. Stability of selfsimilar solutions (under smooth exponentially decaying perturbations), with sharp exponential decay rates in time are proved, as well as $C^{\infty }$ regularity of solutions for $t>0$. The results are based on explicit expansion in terms of generalized Laguerre polynomials and the analysis of such expansions. For perturbations with power-law decay at infinity stability is also proved. Finally, we consider real analytic $p(s)$.

math-ph↗

arXiv2022

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. For sufficiently effective clumping $1 \ge γ> 1/2$, the coagulation equation produces an infinitely large cluster in finite time (a process known as the gel transition). Using a combination of analytical methods and numerics, we calculate the anomalous scaling dimensions of the main cluster growth, calling into question results much used in the literature. Apart from the solution branch which originates from the exactly solvable case $γ= 1$, we find a new branch of solutions near $γ= 1/2$, which violates scaling relations widely believed to hold universal.

cond-mat.soft↗

arXiv2022

On global in time self-similar solutions of Smoluchowski equation with multiplicative kernel

We study the similarity solutions (SS) of Smoluchowski coagulation equation with multiplicative kernel $K(x,y)=(xy)^{s}$ for $s<\frac{1}{2}$. When $s<0$% , the SS consists of three regions with distinct asymptotic behaviours. The appropriate matching yields a global description of the solution consisting of a Gamma distribution tail, an intermediate region described by a lognormal distribution and a region of very fast decay of the solutions to zero near the origin. When $s\in \left( 0,\frac{1}{2}\right) $, the SS is unbounded at the origin. It also presents three regions: a Gamma distribution tail, an intermediate region of power-like (or Pareto distribution) decay and the region close to the origin where a singularity occurs. Finally, full numerical simulations of Smoluchowski equation serve to verify our theoretical results and show the convergence of solutions to the selfsimilar regime.

math-ph↗

arXiv2021

Lateral instability in streamers discharges

In this paper we study the lateral instability of streamer discharges using a continuum discharge model. We observe similarities to Kelvin-Helmholz instability in fluids. In strong electric fields, lateral long wave length perturbations can grow while small wave length perturbations diminish during the discharge evolution. We perform numerical simulations and carry out asymptotic analysis of the instability. Our results could explain the growth observed in experimental discharges.

physics.plasm-ph↗

arXiv2012

The onset of tree-like patterns in negative streamers

We present the first analytical and numerical studies of the initial stage of the branching process based on an interface dynamics streamer model in the fully 3-D case. This model follows from fundamental considerations on charge production by impact ionization and balance laws, and leads to an equation for the evolution of the interface between ionized and non-ionized regions. We compare some experimental patterns with the numerically simulated ones, and give an explicit expression for the growth rate of harmonic modes associated with the perturbation of a symmetrically expanding discharge. By means of full numerical simulation, the splitting and formation of characteristic tree-like patterns of electric discharges is observed and described.

physics.plasm-ph↗

arXiv2011

Electric discharge contour dynamics model: the effects of curvature and finite conductivity

In this paper we present the complete derivation of the effective contour model for electrical discharges which appears as the asymptotic limit of the minimal streamer model for the propagation of electric discharges, when the electron diffusion is small. It consists of two integro-differential equations defined at the boundary of the plasma region: one for the motion and a second equation for the net charge density at the interface. We have computed explicit solutions with cylindrical symmetry and found the dispersion relation for small symmetry-breaking perturbations in the case of finite resistivity. We implement a numerical procedure to solve our model in general situations. As a result we compute the dispersion relation for the cylindrical case and compare it with the analytical predictions. Comparisons with experimental data for a 2-D positive streamers discharge are provided and predictions confirmed.

physics.plasm-ph↗

arXiv2011

Ant foraging and minimal paths in simple graphs

Ants are known to be able to find paths of minimal length between the nest and food sources. The deposit of pheromones while they search for food and their chemotactical response to them has been proposed as a crucial element in the mechanism for finding minimal paths. We investigate both individual and collective behavior of ants in some simple networks representing basic mazes. The character of the graphs considered is such that it allows a fully rigorous mathematical treatment via analysis of some markovian processes in terms of which the evolution can be represented. Our analytical and computational results show that in order for the ants to follow shortest paths between nest and food, it is necessary to superimpose to the ants' random walk the chemotactic reinforcement. It is also needed a certain degree of persistence so that ants tend to move preferably without changing their direction much. It is also important the number of ants, since we will show that the speed for finding minimal paths increases very fast with it.

q-bio.PE↗

arXiv2010

Contour dynamics model for electric discharges

A contour dynamics model for electrical discharges is obtained and analyzed. The model is deduced as the asymptotic limit of the minimal streamer model for the propagation of electric discharges, in the limit of small electron diffusion. The dispersion relation for a non planar 2-D discharge is calculated. The development and propagation of finger-like patterns are studied and their main features quantified.

physics.plasm-ph↗

arXiv2006

Theory of the collapsing axisymmetric cavity

We investigate the collapse of an axisymmetric cavity or bubble inside a fluid of small viscosity, like water. Any effects of the gas inside the cavity as well as of the fluid viscosity are neglected. Using a slender-body description, we show that the minimum radius of the cavity scales like $h_0 \propto t'^α$, where $t'$ is the time from collapse. The exponent $α$ very slowly approaches a universal value according to $α=1/2 + 1/(4\sqrt{-\ln(t')})$. Thus, as observed in a number of recent experiments, the scaling can easily be interpreted as evidence of a single non-trivial scaling exponent. Our predictions are confirmed by numerical simulations.

physics.flu-dyn↗

arXiv2006

Photoionization effects in streamer discharges

In this paper we study the effects of photoionization processes on the propagation of both negative and positive streamer discharges. We show that negative fronts accelerate in the presence of photoionization events. The appearance and propagation of positive streamers travelling with constant velocity is explained as the result of the combined effects of photoionization and electron diffusion. The photoionization range plays an important role for the selection of the velocity of the streamer as we show in this work.

physics.plasm-ph↗

arXiv2005

Singularities on charged viscous droplets

We study the evolution of charged droplets of a conducting viscous liquid. The flow is driven by electrostatic repulsion and capillarity. These droplets are known to be linearly unstable when the electric charge is above the Rayleigh critical value. Here we investigate the nonlinear evolution that develops after the linear regime. Using a boundary elements method, we find that a perturbed sphere with critical charge evolves into a fusiform shape with conical tips at time $t_0$, and that the velocity at the tips blows up as $(t_0-t)^α$, with $α$ close to -1/2. In the neighborhood of the singularity, the shape of the surface is self-similar, and the asymptotic angle of the tips is smaller than the opening angle in Taylor cones.

math-ph↗

arXiv2005

On the mechanism of branching in negative ionization fronts

We explain a mechanism for branching of a planar negative front. Branching occurs as the result of a balance between the destabilizing effect of impact ionization and the stabilizing effect of electron diffusion on ionization fronts. The dispersion relation for transversal perturbation is obtained analytically and reads: $s = |k|/[2 (1 + |k|)] - D |k|^2$, where $D$, which is assumed to be small, is the ratio between the electron diffusion coefficient and the intensity of the externally imposed electric field. We estimate the spacing $λ$ between streamers in a planar discharge and deduce a scaling law $λ\sim D^{1/3}$.

physics.flu-dyn↗

arXiv2004

Very viscous drops cannot break up

We consider an axisymmetric, freely suspended fluid drop with surface tension, whose viscosity is so large that both inertia and forcing by an external fluid can be ignored. We show that whatever be the initial condition, pinchoff can never occur.

physics.flu-dyn↗

arXiv2004

Capillarity driven spreading of circular drops of shear-thinning fluid

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.

math-ph↗

arXiv2003

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, for which the evolution can be divided into two distinct regimes. This excludes the very late stages of the evolution, for which the polymers have become fully stretched. For times smaller than the polymer relaxation time, over which the beads-on-string structure develops, we give a simplified local description, which still contains the full complexity 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.

cond-mat.mtrl-sci↗

arXiv2003

Analysis of the self-similar spreading of power law fluids

We consider the equation that models the spreading of thin liquid films of power-law rheology. In particular, we analyze the existence and uniqueness of source-type self-similar solutions in planar and circular symmetries. We find that for shear-thinning fluids there exist a family of such solutions representing both finite and zero contact angle drops and that the solutions with zero contact angle are unique. We also prove the existence of traveling waves in one space dimension and classify them.

math-ph↗