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Marco A. Fontelos

Publications and source records attributed to Marco A. Fontelos.

At least 19 recordsLinked to original sources
arXiv2026

Evolution of viscous vortex filaments and soliton-type propagation

We study the evolution of a viscous incompressible fluid whose initial vorticity is supported on a smooth open curve. We show that, for $νt\ll 1$ and sufficiently small Reynolds number $Γ/ν$, the vorticity is described at leading order by a Lamb--Oseen type vortex concentrated around a curve evolving according to the binormal flow predicted by the localized induction approximation. The solution is written as an explicit leading-order profile plus a lower-order perturbation, which is controlled in a Morrey $\mathcal{M}^{\infty}$ norm. Then, we apply this construction to the Hasimoto soliton. In this case, the estimates are uniform with respect to the torsion parameter, allowing us to consider a large-torsion regime in which the soliton undergoes a macroscopic displacement. We show that the corresponding Navier--Stokes solution contains a localized portion of the kinetic-energy distribution, associated with the binormal velocity, which remains concentrated inside a moving physical region and undergoes an order-one displacement during an admissible time interval.

math.AP↗
arXiv2025

Phase field modelling of the growth and detachment of bubbles in a hydrogen electrolyzer

We develop and implement numerically a phase field model for the growth and detachment of a gas bubble resting on an electrode and being filled with hydrogen produced by water electrolysis. The bubble is surrounded by a viscous liquid, has a prescribed static contact angle and is also subject to gravitational forces. We compute, as a function of the static contact angle, the time at which the bubble detaches from the substrate and what volume it has at that time. We also investigate de dependence of the detachment time on other parameters such as the applied voltage and the hydrogen ion concentration at the fluid bulk.

cond-mat.soft↗
arXiv2025

Phase field modelling of the detachment of bubbles from a solid substrate

We develop and implement numerically a phase field model for the evolution and detachment of a gas bubble resting on a solid substrate and surrounded by a viscous liquid. The bubble has a static contact angle $θ$ and will be subject to gravitational forces. We compute, as a function of the static contact angle, the cricital Bond number over which bubbles detach from the substrate. Then, we perform similar studies for bubble resting on inclined substrates and bubbles under the action of an external flow. We provide approximate formulas for the critical Bond number under all these circumstances. Our method is also able to resolve the pinchoff of the bubble and the possible appearence of satellites.

physics.flu-dyn↗
arXiv2024

Spectral properties of Lévy Fokker--Planck equations

Hermite polynomials, which are associated to a Gaussian weight and solve the Laplace equation with a drift term of linear growth, are classical in analysis and well-understood via ODE techniques. Our main contribution is to give explicit Euclidean formulae of the fractional analogue of Hermite polynomials, which appear as eigenfunctions of a Lévy Fokker-Planck equation. We will restrict, without loss of generality, to radially symmetric functions. A crucial tool in our analysis is the Mellin transform, which is essentially the Fourier transform in logarithmic variable and which turns weighted derivatives into multipliers. This allows to write the weighted space in the fractional case that replaces the usual $L_r^2(\mathbb R^n, e^{|x|^2/4})$. After proving compactness, we obtain a exhaustive description of the spectrum of the Lévy Fokker--Planck equation and its dual, the fractional Ornstein--Uhlenbeck problem, which forms a basis thanks to the spectral theorem for self-adjoint operators. As a corollary, we obtain a full asymptotic expansion for solutions of the fractional heat equation.

math.AP↗
arXiv2023

Evolution of viscous vortex filaments and desingularization of the Biot-Savart integral

We consider a viscous fluid with kinematic viscosity $ν$ and initial data consisting of a smooth closed vortex filament with circulation $Γ$. We show that, for short enough time, the solution consists of a deformed Lamb-Oseen vortex whose center (a filament) follows the binormal flow dynamics plus leading order corrections that depend locally on the filament curvature and the nonlocal interactions with distant parts of the filament. In order to achieve this scale separation we require $Γ/ν$ to be sufficiently small.

math.AP↗
arXiv2020

Discrete self-similarity in the formation of satellites for viscous cavity break-up

The breakup of a jet of a viscous fluid with viscosity $μ_{1}$ immersed into another viscous fluid with viscosity $μ_{2}$ is considered in the limit when the viscosity ratio $λ=μ_{1}/μ_{2}$ is close to zero. We show that, in this limit, a transition from ordinary continuous selfsimilarity to discrete selfsimilarity takes place as $λ$decreases. The result being that instead of a single point breakup, the rupture of the inner jet occurs through the appearance of an infinite sequence of filaments of decreasing size that will eventually produce infinite sequences of bubbles of the inner fluid inside the outer fluid. The transition can be understood as the result of a Hopf bifurcation in the system of equations modelling the physical problem.

physics.flu-dyn↗
arXiv2020

ODE-methods in non-local equations

Non-local equations cannot be treated using classical ODE theorems. Nevertheless, several new methods have been introduced in the non-local gluing scheme of our previous article "On higher dimensional singularities for the fractional Yamabe problem: a non-local Mazzeo-Pacard program"; we survey and improve those, and present new applications as well. First, from the explicit symbol of the conformal fractional Laplacian, a variation of constants formula is obtained for fractional Hardy operators. We thus develop, in addition to a suitable extension in the spirit of Caffarelli--Silvestre, an equivalent formulation as an infinite system of second order constant coefficient ODEs. Classical ODE quantities like the Hamiltonian and Wrońskian may then be utilized. As applications, we obtain a Frobenius theorem and establish new Poho\vzaev identities. We also give a detailed proof for the non-degeneracy of the fast-decay singular solution of the fractional Lane-Emden equation.

math.AP↗
arXiv2019

On higher dimensional singularities for the fractional Yamabe problem: a non-local Mazzeo-Pacard program

We consider the problem of constructing solutions to the fractional Yamabe problem that are singular at a given smooth sub-manifold, and we establish the classical gluing method of Mazzeo and Pacard for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of an ODE, and thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. No traditional phase-plane analysis is available here. Instead, first, we provide a rigorous construction of radial fast-decaying solutions by a blow-up argument and a bifurcation method. Second, we use conformal geometry to rewrite this non-local ODE, giving a hint of what a non-local phase-plane analysis should be. Third, for the linear theory, we examine a fractional Schrödinger equation with a Hardy type critical potential. We construct its Green's function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a two-dimensional kernel as in the second order case.

math.AP↗
arXiv2017

Asymptotic decay and non-rupture of viscous sheets

For a nonlinear system of coupled PDEs, that describes evolution of a viscous thin liquid sheet and takes account of surface tension at the free surface, we show exponential $(H^1,\,L^2)$ asymptotic decay to the flat profile of its solutions considered with general initial data. Additionally, by transforming the system to Lagrangian coordinates we show that the minimal thickness of the sheet stays positive for all times. This result proves the conjecture formally accepted in the physical literature [1], that a viscous sheet can not rupture in finite time in absence of external forcing. Moreover, in absence of surface tension we find a special class of initial data for which the Lagrangian solution exhibits $L^2$-exponential decay to the flat profile.

math.AP↗
arXiv2017

Discrete self-similarity in interfacial hydrodynamics and the formation of iterated structures

The formation of iterated structures, such as satellite and sub-satellite 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. The result is an infinite sequence of ridges and filaments with similarity properties. The character of these discretely self-similar solutions as the result of a Hopf bifurcation from ordinarily self-similar solutions is also described.

physics.flu-dyn↗
arXiv2017

Thermal rupture of a free liquid sheet

We consider a free liquid sheet, taking into account the dependence of surface tension on temperature, or concentration of some pollutant. The sheet dynamics are described within a long-wavelength description. In the presence of viscosity, local thinning of the sheet is driven by a strong temperature gradient across the pinch region, resembling a shock. As a result, for long times the sheet thins exponentially, leading to breakup. We describe the quasi one-dimensional thickness, velocity, and temperature profiles in the pinch region in terms of similarity solutions, which posses a universal structure. Our analytical description agrees quantitatively with numerical simulations.

physics.flu-dyn↗
arXiv2015

A nonlocal diffusion problem on manifolds

In this paper we study a nonlocal diffusion problem on a manifold. These kind of equations can model diffusions when there are long range effects and have been widely studied in Euclidean space. We first prove existence and uniqueness of solutions and a comparison principle. Then, for a convenient rescaling we prove that the operator under consideration converges to a multiple of the usual Heat-Beltrami operator on the manifold. Next, we look at the long time behavior on compact manifolds by studying the spectral properties of the operator. Finally, for the model case of hyperbolic space we study the long time asymptotics and find a different and interesting behavior.

math.AP↗
arXiv2014

A stable self-similar singularity of evaporating drops: ellipsoidal collapse to a point

We study the problem of evaporating drops contracting to a point. Going back to Maxwell and Langmuir, the existence of a spherical solution for which evaporating drops collapse to a point in a self-similar manner is well established in the physical literature. The diameter of the drop follows the so-called $D^{2}$ law: the second power of the drop-diameter decays linearly in time. In this study we provide a complete mathematical proof of this classical law. We prove that evaporating drops which are initially small perturbations of a sphere collapse to a point and the shape of the drop converges to a self-similar ellipsoid whose center, orientation, and semi-axes are determined by the initial shape.

math.AP↗
arXiv2013

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 surrounding created by any metallic object. In his experiment, two streams of droplets are allowed to drip from separated nozzles into separated buckets, which are at the same time interconnected through the dripping needles. In this paper we build a microfluidic water dropper and demonstrate that the droplets get charged and break-up due to electrohydrodynamic instabilities. A comparison with recent simulations shows the dependence of the acquired charge in the droplets on different parameters of the system. The phenomenon opens a door to cheap and accessible transformation of pneumatic pressure into electrical energy and to an enhanced control in microfluidic and biophysical manipulation of capsules, cells and droplets via self-induced charging of the elements.

physics.flu-dyn↗
arXiv2012

Time decay of scaling critical electromagnetic Schrödinger flows

We obtain a representation formula for solutions to Schrödinger equations with a class of homogeneous, scaling-critical electromagnetic potentials. As a consequence, we prove the sharp $L^{1}\to L^{\infty}$ time decay estimate for the 3D-inverse square and the 2D-Aharonov-Bohm potentials.

math.AP↗
arXiv2008

The role of self-similarity in singularities of PDE's

We survey rigorous, formal, and numerical results on the formation of point-like singularities (or blow-up) for a wide range of evolution equations. We use a similarity transformation of the original equation with respect to the blow-up point, such that self-similar behaviour is mapped to the fixed point of a \textit{dynamical system}. We point out that analysing the dynamics close to the fixed point is a useful way of characterising the singularity, in that the dynamics frequently reduces to very few dimensions. As far as we are aware, examples from the literature either correspond to stable fixed points, low-dimensional centre-manifold dynamics, limit cycles, or travelling waves. For each "class" of singularity, we give detailed examples.

math-ph↗
arXiv2007

A catalogue of singularities

This paper is an attempt to classify finite-time singularities of PDEs. Most of the problems considered describe free-surface flows, which are easily observed experimentally. We consider problems where the singularity occurs at a point, and where typical scales of the solution shrink to zero as the singularity is approached. Upon a similarity transformation, exact self-similar behaviour is mapped to the fixed point of a {\it infinite dimensional dynamical system} representing the original dynamics. We show that the dynamics close to the fixed point is a useful way classifying the structure of the singularity. Specifically, we consider various types of stable and unstable fixed points, centre-manifold dynamics, limit cycles, and chaotic dynamics.

math.AP↗