Browsing by Author "Rica, Sergio"
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- ItemAxisymmetric self-similar finite-time singularity solution of the Euler equations(2023) Cádiz Carvajal, Rodrigo Esteban; Martinez Arguello Diego; Rica, SergioAbstract Self-similar finite-time singularity solutions of the axisymmetric Euler equations in an infinite system with a swirl are provided. Using the Elgindi approximation of the Biot–Savart kernel for the velocity in terms of vorticity, we show that an axisymmetric incompressible and inviscid flow presents a self-similar finite-time singularity of second specie, with a critical exponent ν. Contrary to the recent findings by Hou and collaborators, the current singularity solution occurs at the origin of the coordinate system, not at the system’s boundaries or on an annular rim at a finite distance. Finally, assisted by a numerical calculation, we sketch an approximate solution and find the respective values of ν. These solutions may be a starting point for rigorous mathematical proofs.
- ItemBúsqueda de soluciones singulares en tiempo finito de las ecuaciones de Euler para fluidos incompresibles y no viscosos(2023) Martínez Argüello, Diego Fernando; Rica, Sergio; Pontificia Universidad Católica de Chile. Facultad de FísicaEn el contexto del problema de la globalidad de las ecuaciones Euler (EE) para fluidos incompresibles e invíscidos, se presenta un procedimiento para obtener soluciones singulares en un tiempo finito tc y asintóticas en t → tc para las EE en 3D con simetría axial. La singularidad se induce con un ansatz autosimilar para la dependencia radio-temporal, caracterizado por un exponente autosimilar, ν, por encontrar. Junto con el uso de una expansión en el ángulo polar, las EE axisimétricas se transforman en un conjunto autónomo e infinito de ecuaciones diferenciales ordinarias para las amplitudes de la expansión. Este sistema dinámico se resuelve numéricamente para diferentes truncaturas, N, de la expansión ajustando el exponente ν según las condiciones de borde. Como resultado, las soluciones para diferentes N forman una secuencia convergente tanto en el exponente ν ≈ 2.0 como en las amplitudes autosimilares. Además, se muestra que las soluciones son estables, calculando aproximadamente su conjunto infinito de autovalores de inestabilidad y concluyendo que ν ≈ 2 es el único exponente estable para N → ∞. Esta secuencia convergente indica que existe una solución que resuelve el sistema dinámico exacto, para N → ∞, y el flujo asociado sería una solución singular en tiempo finito de las EE que sugiere su no-globalidad.
- ItemEvidence of a finite-time pointlike singularity solution for the Euler equations for perfect fluids(2024) Martinez-Arguello, Diego; Rica, SergioThis paper investigates the evolution of the Euler equations near a potential blow-up solution. We employ an approach where this solution exhibits second-type self-similarity, characterized by an undetermined exponent v. This exponent can be seen as a nonlinear eigenvalue, determined by the solution of a self-similar partial differential equation with appropriate boundary conditions. Specifically, we demonstrate the existence of an axisymmetric solution of the Euler equations by expanding the axial vorticity using associated Legendre polynomials as a basis. This expansion results in an infinite hierarchy of ordinary differential equations, which, when truncated up to a certain order N-& lowast;, allows for the numerical resolution of a finite set of ordinary differential equations. Through this numerical analysis, we obtain a solution that satisfies the appropriate boundary conditions for a specific value of the exponent v. By exploring various truncations, we establish a sequence in N & lowast; for the parameter v(N & lowast;), providing evidence of the convergence of the exponent v. Our findings suggest a self-similar exponent v approximate to 2, presenting a promising path for a numerical or analytical approach indicating that v may indeed be exactly 2.
- ItemPotential anisotropic finite-time singularity in the three-dimensional axisymmetric Euler equations(2022) Rica, SergioThe search of finite-time singularity solutions of Euler equations is considered for the case of an incompressible and inviscid fluid. Under the assumption that a finite-time blowup solution may be spatially anisotropic as time goes by such that the flow contracts more rapidly into one direction than into the other, it can be shown that the dynamics of an axially symmetric flow with swirl may be approximated to a simpler hyperbolic system. By using the method of characteristics, it can be shown that generically the velocity flow exhibits multivalued solutions appearing on a rim at a finite distance from the axis of rotation, which displays a singular behavior in the radial derivatives of velocities. Moreover, the general solution shows a genuine blow-up, which is also discussed. This singularity is generic for a vast number of smooth finite-energy initial conditions and is characterized by a local singular behavior of velocity gradients and accelerations.
- ItemStrong turbulence for vibrating plates: Emergence of a Kolmogorov spectrum(2019) During, Gustavo; Josserand, Christophe; Krstulovic, Giorgio; Rica, SergioIn fluid turbulence, energy is transferred from one scale to another by an energy cascade that depends only on the energy-dissipation rate. It leads by dimensional arguments to the Kolmogorov 1941 (K41) spectrum. Here we show that the normal modes of vibrations in elastic plates also manifest an energy cascade with the same K41 spectrum in the fully nonlinear regime. In particular, we observe different patterns in the elastic deformations such as folds, developable cones, and even more complex stretching structures, in analogy with spots, swirls, vortices, and other structures in hydrodynamic turbulence. We show that the energy cascade is dominated by the kinetic contribution and that the stretching energy is at thermodynamical equilibrium. We characterize this energy cascade, the validity of the constant energy-dissipation rate over the scales. Finally, we discuss the role of intermittency using the correlation functions that exhibit anomalous exponents.
- ItemSuperflow passing over a rough surface: Vortex nucleation(2024) Frisch, Thomas; Nazarenko, Sergey; Rica, SergioThe transition from a free vortex superflow to a dissipative vortex flow in a superfluid remains an open problem, in particular because in real experiments, microscopic asperities on the walls play a major role since the relevant scales in a superfluid motion are basically of atomic scale. Here we model a superflow using the Gross-Pitaevskii mean-field equation in a domain with a rigid sinusoidal boundary. We demonstrate, analytically and numerically, the existence of a well-defined critical velocity for vortex nucleation. Additionally, we discuss the intrinsic mechanism of vortex nucleation by the appearance of a solitary wave characterized by a depletion of the wave function near the boundary.
- ItemVariational estimates for the speed propagation of fronts in a nonlinear diffusive Fisher equation(2022) Benguria, Rafael D.; Depassier, M. Cristina; Rica, SergioWe examine non-linear diffusive front propagation in the frame of the Fisher-type equation: dtu = dx (D(u)dxu)+ u(1 - u). We study the problem of a sudden jump in diffusivity motivated by models of glassy polymers. It is shown that this problem differs substantially from the problem of front propagation in the usual Fisher equation which was solved by Kolmogorov, Petrovsky, and Piskunov (KPP) in 1937. As in the Fisher, Kolmogorov, Petrovsky, Piskunov (FKPP) problem, the asymptotic dynamics of the non linear diffusive front propagation is reduced to the study of a nonlinear ordinary differential equation with adequate boundary conditions. Since this problem does not allow an exact result for the propagation speed, we use a variational approach to estimate the front speed and compare it with direct time-dependent numerical simulations showing an excellent agreement.