Vortex Solutions Research Articles

The integrable Ermakov structure and elliptic vortex solution in the inviscid gas-liquid two-phase flow

The integrable Ermakov structure and elliptic vortex solution in the inviscid gas-liquid two-phase flow

Planar vortices in a Lorentz symmetry breaking environment

We revisit planar vortex solutions within a model derived from the dimensional reduction of scalar electromagnetism, with a quartic potential incorporating the Carroll–Field–Jackiw term. We explore analytically and numerically the influence of a Lorentz symmetry-breaking constant field on these configurations and our analysis shows how this constant field can produce a new asymptotic behaviour characterised by damped oscillations of the electric and magnetic fields on the edge of the vortices.

Weakly Compressible Approximation of the Taylor–Green Vortex Solution

ABSTRACTThe Taylor–Green vortex represents an exact solution of the Navier–Stokes equations in . In this work, an approximation of this solution in two spatial dimensions is proposed for weakly compressible flows. These flows are characterized by small compressibility (or, equivalently, by a small Mach number) and are often employed in computational fluid dynamics to approximate the behaviour of incompressible Newtonian fluids. In this framework, the proposed solution is expected to be a useful benchmark for numerical solvers that implement the weakly compressibility approximation. To this end, some numerical examples are reported in the final section of this work.

Superconducting multi-vortices and a novel BPS bound in chiral perturbation theory

We derive a novel BPS bound from chiral perturbation theory minimally coupled to electrodynamics at finite isospin chemical potential. At a critical value of the isospin chemical potential, a system of three first-order differential field equations (which implies the second-order field equations) for the gauge field and the hadronic profile can be derived from the requirement to saturate the bound. These BPS configurations represent magnetic multi-vortices with quantized flux supported by a superconducting current. The corresponding topological charge density is related to the magnetic flux density, but is screened by the hadronic profile. Such a screening effect allows to derive the maximal value of the magnetic field generated by these BPS magnetic vortices, being Bmax = 2, 04 × 1014 G. The solution for a single BPS vortex is discussed in detail, and some physical consequences, together with the comparison with the magnetic vortices in the Ginzburg-Landau theory at critical coupling, are described.

Spatial momentum separation for optical vortex via phase-assembled metasurfaces

Expanding the spatial degrees of freedom is regarded to be the most innovative and effective solution for optical vortex encoding/decoding technology. A phase-assembled metasurface is proposed here to separate the momentum of the optical vortex into different spatial locations, and the incident high-order optical vortex is encoded in 18 channels. By combining both the propagation phase and the Pancharatnam-Berry (PB) phase, the fundamental Gaussian mode (solid spot) positions and other higher-order orbital angular momentum modes (hollow mode spots) positions at different channels are used to achieve the encryption of all-optical information. The scheme of grouping the mode spots on different channels into solid and hollow points greatly reduces the difficulty of encoding/decoding a high-order optical vortex. In this way, the combination of 2 incident high-order optical vortexes can represent 256 hexadecimal numbers from 00 to FF. We envision this work will open avenues for information encryption, multidimensional data storage, wavefront manipulation, and advanced orbital angular momentum detection technologies.

High-throughput production of low-cost hydrophobic and oleophilic mullite fiber sponges for high-temperature PM filtration

Particulate matter (PM) from high-temperature emissions like chemical plants, coal stoves and vehicle exhausts poses a gravel challenge to human health. To address this issue, researchers have explored various fiber filters, yet the bulk struggle to withstand high temperatures. In this study, mullite fiber sponges were developed utilizing low-cost materials and Kármán vortex solution blow spinning, using surfactants to improve the spinnability of the sol. Optimized sponges demonstrate ultralight (19 mg cm−3), temperature-resistant reversible compressibility (50% strain) and a water contact angle of 135°. These sponges exhibited exceptional thermal insulation (thermal conductivity: 0.0256 W m−1 K−1) and performed well in high-temperature air filtration. At 800 °C, the mullite sponge with a base weight of 35 mg cm−2, achieved an average filtration efficiency of 98.18 % and 99.57 % for PM2.5 and PM2.5−10, respectively, with a quality value of 0.98 Pa-1 at a wind speed of 4 cm s−1. This low-cost mullite fiber sponge offers a promising avenue for designing high-performance filtration materials.

Analytic approach for computation of topological number of integrable vortex on torus

Detailed structures of vortices on a torus are discovered by performing an analytic method to calculate the vortex number. We focus on analytic vortex solutions to the Chern-Simons-Higgs theory, whose governing equation is the so-called Jackiw-Pi equation. The equation is one of the integrable vortex equations and is reduced to Liouville’s equation. The requirement of continuity of the Higgs field strongly restricts the characteristics and the fundamental domain of the vortices. Also considered are the decompactification limits of the vortices on a torus, in which “flux loss” phenomena occasionally occur.

A two-cell vortex model

A two-cell vortex solution to the Navier–Stokes equations is derived in this article. The theoretical analysis provides expressions for the velocity components, vorticity, and pressure distributions of the two-cell vortex flow. A key feature of this new modeling approach is the presence of a “vortex eye” region, which is absent in classical single-cell vortex models. The derived tangential velocity profile is kept general, allowing for the recovery of the single-cell profile as a special case. The model parameters are constrained by the boundary conditions inherent to two-cell vortices. The model explicitly demonstrates that the two-cell vortex flow exhibits two regions of recirculating flow structures in the meridional plane. The analysis shows that near the vortex core, fluid descends from above and diverges outward. This downward flow meets the incoming flow from the vortex's outer region at a distance known as the “eye-wall” radius. The merging of these two streams results in an upward deflection, generating the two-cell fluid motion. Unlike in single-cell vortices, where vorticity reaches its maximum at the vortex center, the present results indicate that in two-cell vortices, the vorticity peaks at the eye-wall radius.

The vacuum-core vortex in relativistic perfect fluids

The governing equations in non-relativistic (conventional) compressible fluid flows are derived as a low-energy limit in relativistic flows. This suggests that exact solutions obtained in non-relativistic fluid dynamics possess their relativistic counterparts. As an example of such solutions, we consider a stationary potential flow and examine the relativistic effect on a vacuum-core (hollow-core) vortex solution in compressible fluid flows. The vacuum-core vortex solution is an exact solution in stationary potential flows, which is also true in relativistic flows. We construct the vacuum-core vortex solution in relativistic fluid flows and discuss the differences and similarities between non-relativistic and relativistic flows. We show that the vacuum-core radius in relativistic flows is larger than the one in non-relativistic flows for a fixed polytropic exponent and depends on the transonic speed (local sound speed) in the flow field. We also calculate various physical quantities such as density, pressure, and sound speed as functions of radius r from the center of the core and compare them with those in non-relativistic flows.

Extremization to fine tune physics informed neural networks for solving boundary value problems

We propose a novel method for fast and accurate training of physics-informed neural networks (PINNs) to find solutions to boundary value problems (BVPs) and initial boundary value problems (IBVPs). By combining the methods of training deep neural networks (DNNs) and Extreme Learning Machines (ELMs), we develop a model which has the expressivity of DNNs with the fine-tuning ability of ELMs. We showcase the superiority of our proposed method by solving several BVPs and IBVPs which include linear and non-linear ordinary differential equations (ODEs), partial differential equations (PDEs) and coupled PDEs. The examples we consider include a stiff coupled ODE system where traditional numerical methods fail, a 3+1D non-linear PDE, Kovasznay flow and Taylor–Green vortex solutions to incompressible Navier–Stokes equations and pure advection solution of 1+1 D compressible Euler equation.The Theory of Functional Connections (TFC) is used to exactly impose initial and boundary conditions (IBCs) of (I)BVPs on PINNs. We propose a modification to the TFC framework named Reduced TFC and show a significant improvement in the training and inference time of PINNs compared to IBCs imposed using TFC. Furthermore, Reduced TFC is shown to be able to generalize to more complex boundary geometries which is not possible with TFC. We also introduce a method of applying boundary conditions at infinity for BVPs and numerically solve the pure advection in 1+1 D Euler equations using these boundary conditions.

The role of axial angular momentum in exact non-linear solutions of multipolar spherical and cylindrical vortices

The time-dependent acceleration of fluid particles in an arbitrary superposition of multipolar spherical and cylindrical vortices in presence of a background rotational rigid flow is the gradient of the difference between their kinetic energy and the product of their total axial angular momentum times the angular velocity of the background flow. If the potential energy is defined as the time-dependent field whose minus gradient equals the acceleration of the flow, then the total energy, defined as the sum of kinetic and potential energies, equals the total axial angular momentum times the angular velocity of the background flow. The total energy of a fluid particle has therefore a linear relationship with the azimuthal velocity field. The role of the axial angular momentum in these oscillating flows is explained also in the classical Lagrangian and Hamiltonian descriptions of motion. The linear relation between total energy and angular momentum makes possible that the free parameters of these flows may be obtained, in a way similar to that developed in the quantum mechanics description of motion, as the eigenvalues of linear operators acting on some components of the spherical modes. Beltrami flow solutions in prolate spheroidal geometry for axisymmetric flows are also discussed.

N-Symmetric Interaction of N Hetons, II: Analysis of the Case of Arbitrary N

This paper seeks and examines N-symmetric vortical solutions of the two-layer geostrophic model for the special case when the vortices (or eddies) have vanishing summed strength (circulation anomaly). This study is an extension [Sokolovskiy et al. Phys. Fluids 2020, 32, 09660], where the general formulation for arbitrary N was given, but the analysis was only carried out for N=2. Here, families of stationary solutions are obtained and their properties, including asymptotic ones, are investigated in detail. From the point of view of geophysical applications, the results may help interpret the propagation of thermal anomalies in the oceans.

Baryons as vortexes on the η′ domain wall

We show that the recent construction of Nf = 1 baryons on the η′ domain wall can be understood as vortexes of the principal effective theory — the Chern-Simons-Higgs theory — on a 2+1-dimensional sheet. This theory has a series of vortex solutions, and the vortex with unit topological charge naturally spins Nc/2, which coincides with the spin of the one-flavor baryon in QCD. Since the Nc scaling of the vortexes is the same as that of baryons, the latter can be regarded as the former. By virtue of the particle-vortex symmetry, the dual Zhang-Hansson-Kivelson theory indicates that the quark carries topological charge 1/Nc and obeys fractional statistics. The generalization to arbitrary Nf is also discussed.

Vortex solutions in a binary immiscible Bose-Einstein condensate

We consider the mean-field vortex solutions and their stability within a two-component Bose-Einstein condensate in the immiscible limit. A variational approach is employed to study a system consisting of a majority component which contains a single quantized vortex and a minority component which fills the vortex core. We show that a super-Gaussian function is a good approximation of the two-component vortex solution for a range of atom numbers of the infilling component by comparing the variational solutions to the full numerical solutions of the coupled Gross-Pitaevskii equations. We subsequently examine the stability of the vortex solutions by perturbing the infilling component away from the center of the vortex core, thereby demonstrating their stability to small perturbations. Finally, we consider the dynamics of infilled vortices. Published by the American Physical Society 2024

Apparently superluminal superfluids

We consider the superfluid phase of a specific renormalizable relativistic quantum field theory. We prove that, within the regime of validity of perturbation theory and of the superfluid effective theory, there are consistent and regular vortex solutions where the superfluid’s velocity field as traditionally defined smoothly interpolates between zero and arbitrarily large superluminal values. We show that this solution is free of instabilities and of superluminal excitations. We show that, in contrast, a generic vortex solution for an ordinary fluid does develop an instability if the velocity field becomes superluminal. All this questions the characterization of a superfluid velocity field as the actual velocity of “something”.

Three-dimensional quasi-geostrophic vortices with vanishing amount of potential vorticity anomaly

We provide exact solutions of axisymmetric neutral vortices, that is, vortices with zero amount of potential vorticity anomaly (PVA), in three-dimensional (3D) quasi-geostrophic (QG) flows with distributed vorticity and density stratification. These solutions are linear combinations of PVA spherical layer-modes, which consist of conveniently normalized spherical Bessel functions of order 0, truncated by a zero of the spherical Bessel function of order 1. It is shown that, depending on the superposition of the different spherical layer-modes, some vortices remain axisymmetrically robust to small amplitude PVA perturbations, while other vortices evolve to stable QG tripoles. The robust axisymmetric 3D QG vortices analyzed here do not generate exterior potential flow and may provide an explanation to the persistence of baroclinic eddies in the ocean.

Uniqueness of Positive Vorticity Solutions to the 2d Euler Equations on Singular Domains

Uniqueness of Positive Vorticity Solutions to the 2d Euler Equations on Singular Domains

Isentropic Swirling Flow Through Supersonic Nozzles - Part I (free Vortex Flow)

An analytical model has been developed and an approximate solution for the free vortex swirling supersonic flow through nozzles. Earlier work by Mager, is potential flow based. In the present model swirl number has been redefined to confine the flow properties within the valid range. Flow properties are always determined by the total conditions (P0 and a0), nozzle profile and swirl intensity as in the case of one-dimensional flow. Integral form of the continuity equation has been used in order to satisfy the boundary conditions, since radial velocity component is ignored. For free vortex case, it has been shown that there exists a minimum radius along the length of the nozzle at which the density is zero and flow seems to be void causing reasonable blockage to the flow at throat. In defining the non-dimensional swirl number (β), axial velocity at each section was employed for non-dimensionalization as compared to Mager’s definition of using a global constant to find swirl number (α). The variations of performance parameters of the nozzle such as co-efficient of discharge (Cd) thrust efficiency (ηs) and impulse efficiency (ηI) and flow field parameters such as density, pressure, velocity etc. are presented as a function of swirl numbers (α, β) and compared with the available experimental data. The swirl numbers (α, β) are also correlated.

Three-Dimensional Unsteady Axisymmetric Viscous Beltrami Vortex Solutions to the Navier–Stokes Equations

This paper is aimed at eliciting consistency conditions for the existence of unsteady incompressible axisymmetric swirling viscous Beltrami vortices and explicitly constructing solutions that obey the conditions as well as the Navier–Stokes equations. By Beltrami flow, it is meant that vorticity, i.e., the curl of velocity, is proportional to velocity at any local point in space and time. The consistency conditions are derived for the proportionality coefficient, the velocity field and external force. The coefficient, whose dimension is of [length−1], is either constant or nonconstant. In the former case, the well-known exact nondivergent three-dimensional unsteady vortex solutions are obtained by solving the evolution equations for the stream function directly. In the latter case, the consistency conditions are given by nonlinear equations of the stream function, one of which corresponds to the Bragg–Hawthorne equation for steady inviscid flow. Solutions of a novel type are found by numerically solving the nonlinear constraint equation at a fixed time. Time dependence is recovered by taking advantage of the linearity of the evolution equation of the stream function. The proportionality coefficient is found to depend on space and time. A phenomenon of partial restoration of the broken scaling invariance is observed at short distances.

Structure dominated two-dimensional turbulence: formation, dynamics and interactions of dipole vortices

Two dimensional turbulence in geophysical fluids and plasma physics tends to be spotty, intermittent and rich in large scale structures such as coherent vortices or zonal flows, due to various mechanisms of self organization. Nonlinear solutions that rely on the vanishing of nonlinearity, especially the dipole vortex solution, stand out as key aspects of this structure dominated turbulence state. Using numerical simulations, it is demonstrated that an initial condition with a small number of high intensity turbulent patches, evolves towards a state dominated by coherent structures, and in particular dipole vortices, as each patch is organized into a finite number of dipole vortices that are ejected from this initially active region. In order to study the details of this process, an initial condition of two Gaussian peaks of the stream function is considered, and it was shown to result in a Chaplygin–Lamb dipole if the peaks have the same amplitude, or a Flierl–Stern–Whitehead dipole that rotates in the direction implied by the excess of vorticity if they do not. Analytical estimates for the velocity, the radius and the radius of curvature of the resulting dipole vortex is given in terms of the peaks and widths of the initial conditions. These are then verified by a detailed comparison of the analytical form of the vorticity of the dipole vortex and its numerical realization. It is argued that since these coherent structures are spared from the strong shear forces normally exerted by the nonlinearities, and can coexist with other localized solutions, or large scale flow patterns, they provide the backbone of the structure dominated or ‘sporadic’ turbulent state in two dimensions, on top of which other structures, waves and instabilities can develop. In order to elucidate these, a number of collision scenarios are considered. It is also shown that a simple two point vortex approximation to a dipole vortex seems to be appropriate for describing their evolution far from each-other, or for computing head on collisions between two or more dipole vortices, but not in the case of close or grazing collisions or their interaction with a nontrivial large scale flow.