On the existence of leapfrogging pair of circular vortex filaments

We propose and analyze a system of nonlinear partial differential equations describing the motion of a pair of vortex filaments. As a preliminary analysis, we first consider the case when the filaments are arranged as straight and parallel lines, and explicitly solve the system to show that the motion of the lines resemble that of point vortices moving in a plane. Then, we consider the motion of a pair of coaxial circular vortex filaments. We show that in this case, the system can be reduced to a two‐dimensional Hamiltonian system. Based on this formulation, we give a condition for the initial configuration and parameters of the filaments for leapfrogging to occur, and prove that the condition is in fact necessary and sufficient.     

Invariant tori in perturbed three vortex motion

The motion of three point vortices in an ideal fluid in a plane comprises a Hamiltonian dynamical system – one that is completely integrable, so it exhibits numerous periodic orbits, and quasiperiodic orbits on invariant tori. Certain perturbations of three vortex dynamics, such as three vortex motion in a half-plane, are also Hamiltonian, but not completely integrable. Yet these perturbed systems may still have periodic trajectories and invariant tori close to those for the unperturbed dynamics. Extending recent work by the authors [4], invariant 2-tori approximating those for the unperturbed system are located and analyzed using a combination of classical analysis, asymptotics, and Hamiltonian methods. It is shown that the results and approximation methods used are applicable to several perturbations of three vortex dynamics such as three vortices in a half-plane, the restricted four vortex problem in the plane, and three coaxial vortex rings in 3-space. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Parabolic Type Equations and Markov Stochastic Processes on Adeles

In this paper we study the Cauchy problem for new classes of parabolic type pseudodifferential equations over the rings of finite adeles and adeles. We show that the adelic topology is metrizable and give an explicit metric. We find explicit representations of the fundamental solutions (the heat kernels). These fundamental solutions are transition functions of Markov processes which are adelic analogues of the Archimedean Brownian motion. We show that the Cauchy problems for these equations are well-posed and find explicit representations of the evolution semigroup and formulas for the solutions of homogeneous and non-homogeneous equations.     

Finite-time Collapse of Three Point Vortices in the Plane

We investigate the finite-time collapse of three point vortices in the plane utilizing the geometric formulation of three-vortexmotion from Krishnamurthy, Aref and Stremler (2018) Phys. Rev. Fluids 3, 024702. In this approach, the vortex system is described in terms of the interior angles of the triangle joining the vortices, the circle that circumscribes that triangle, and the orientation of the triangle. Symmetries in the governing geometric equations of motion for the general three-vortex problem allow us to consider a reduced parameter space in the relative vortex strengths. The well-known conditions for three-vortex collapse are reproduced in this formulation, and we show that these conditions are necessary and sufficient for the vortex motion to consist of collapsing or expanding self-similar motion. The geometric formulation enables a new perspective on the details of this motion. Relationships are determined between the interior angles of the triangle, the vortex strength ratios, the (finite) system energy, the time of collapse, and the distance traveled by the configuration prior to collapse. Several illustrative examples of both collapsing and expanding motion are given.     

Motion of a vortex filament in the half-space

A model equation for the motion of a vortex filament immersed in three-dimensional, incompressible and inviscid fluid is investigated as a preliminary attempt to model the motion of a tornado. We solve an initial–boundary value problem in the half-space, where we impose a boundary condition in which the vortex filament is allowed to move on the boundary.     

Dynamics and self-propulsion of a spherical body shedding coaxial vortex rings in an ideal fluid

We describe a model for the dynamic interaction of a sphere with uniform density and a system of coaxial circular vortex rings in an ideal fluid of equal density. At regular intervals in time, a constraint is imposed that requires the velocity of the fluid relative to the sphere to have no component transverse to a particular circular contour on the sphere. In order to enforce this constraint, new vortex rings are introduced in a manner that conserves the total momentum in the system. This models the shedding of rings from a sharp physical ridge on the sphere coincident with the circular contour. If the position of the contour is fixed on the sphere, vortex shedding is a source of drag. If the position of the contour varies periodically, propulsive rings may be shed in a manner that mimics the locomotion of certain jellyfish. We present simulations representing both cases.     

On a dynamical system related to fluid mechanics

We introduce a dynamical system strictly related to fluid mechanics and similar to the classical N point vortex system. In the first part we analyze the qualitative behavior of the time evolution and in particular we show the properties of collapse and chaoticity. In the second part of the paper we investigate the relation of the dynamical system with a system of N concentrated large enough smoke rings in an incompressible and inviscid fluid, with axial symmetry and without swirl. We prove the rigorous connection between the two models at time zero for any N. The extension of the same result to any time is obtained only for a smoke ring alone, while for the general case it is just a matter of conjecture. Received June 1998; Revised November 1998     

An Improved Vortex Lattice Method for Nonstationary Problems

The vortex lattice method is improved for modeling nonlinear highly nonstationary processes appearing in an interaction of bodies that undergo an irregular motion in a proximity of solid boundaries with large-scale vortex structures. We show that keeping the condition of freezing the vortex buildups in the medium leads, in the vortex lattice method, to eliminating the arbitrariness in the calculated time step, singularity radius, and the buffer-zone radius.For the ensemble of discrete vortices that model the surface of tangential discontinuity of the velocity, we propose an economical method for solving the Cauchy problem. The method decreases the discretization error related to the replacement of this surface with a system of discrete vortices.A test for the improved vortex lattice method has been conducted for a nonstationary nonlinear problem on nonharmonic angular oscillations of a wing in a stationary medium near a solid surface in the case where there is no gap between the wing and the surface.By using the improved vortex lattice method, one succeeded, for the first time, in obtaining a solution of a problem of this type that converges from the numerical point of view. A comparison of the obtained results with known experimental data shows a good agreement.     

Parametric instability of a many point-vortex system in a multi-layer flow under linear deformation

The paper deals with a dynamical system governing the motion of many point vortices located in different layers of a multi-layer flow under external deformation. The deformation consists of generally independent shear and rotational components. First, we examine the dynamics of the system’s vorticity center. We demonstrate that the vorticity center of such a multi-vortex multi-layer system behaves just like the one of two point vortices interacting in a homogeneous deformation flow. Given nonstationary shear and rotational components oscillating with different magnitudes, the vorticity center may experience parametric instability leading to its unbounded growth. However, we then show that one can shift to a moving reference frame with the origin coinciding with the position of the vorticity center. In this new reference frame, the new vorticity center always stays at the origin of coordinates, and the equations governing the vortex trajectories look exactly the same as if the vorticity center had never moved in the original reference frame. Second, we studied the relative motion of two point vortices located in different layers of a two-layer flow under linear deformation. We analyze their regular and chaotic dynamics identifying parameters resulting in effective and extensive destabilization of the vortex trajectories.     

Vortex pairs and dipoles

Point vortices have been extensively studied in vortex dynamics. The generalization to higher singularities, starting with vortex dipoles, is not so well understood.We obtain a family of equations of motion for inviscid vortex dipoles and discuss limitations of the concept. We then investigate viscous vortex dipoles, using two different formulations to obtain their propagation velocity. We also derive an integro-differential for the motion of a viscous vortex dipole parallel to a straight boundary.     

Effective Dynamics of a Magnetic Vortex in a Local Potential

We study a simplified mean field model of superconductor dynamics in the presence of impurities or for variable superconductor depth. This model is given by the gradient-flow version of the Ginzburg-Landau equations (Gorkov-Eliashberg equations) with an addition of a potential term. We find a dynamical law of motion of the vortex center, involving the potential, such that for datum close to a (static) magnetic vortex the solution is close, for all times, to a magnetic vortex whose center obeys this law.     

A Third-Order Dispersive Flow for Closed Curves into Kähler Manifolds

This article is devoted to studying the initial value problem for a third-order dispersive equation for closed curves into Kähler manifolds. This equation is a geometric generalization of a two-sphere valued system modeling the motion of vortex filament. We prove the local existence theorem by using geometric analysis and classical energy method.     

Vortex Interaction in a Ducted Flow

The objective of this study is to describe the structure of pipe flow by considering it as a superposition of many axisymmetric vortex rings. In knowing the unsteady gross feature of pipe flow, the investigation on vortex interactions is important. As a first step to the goal, we investigate the nonlinear interaction among vortex rings whose number is three at most. The interaction among vortex rings of equal circulation is here investigated. Momentum and energy conservation of the present vortex ring system are also discussed to know a better understanding of the perturbed pipe flow. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical simulation of the interaction between vortex ring and bubble plume

In present paper a three-dimensional Vortex-In-Cell method with two-way coupling effect was developed to study the bubble plume entrainment by a vortex ring. In this method the continuous flow was calculated by the three-dimensional Vortex-In-Cell method and the bubbles are tracked through bubble motion equation. Two-way coupling effect between continuous flow and dispersed bubbles is considered by introducing a vorticity source term, which is induced by the change of void fraction gradient in each computational cell. After validated by the comparison between experimental measurements and simulation results for the motion of vortex rings and the rising velocity of bubble plume, present method is implemented to simulate the interaction between an evolving vortex ring and a rising bubble plume. It was found that there is little effect of the bubble entrainment to the total circulation of vortex ring while the effect of bubble entrainment to the vortex ring structure is quite obvious. The bubble entrainment by the vortex ring not only changed the vorticity distribution in the vortex structure, but also displaced the positions of the vortex cores. The vorticity in the lower vortex core of the vortex ring decreases more than that in the upper vortex core of the vortex ring while the vortex core in the upper part of the vortex ring is displaced to the center of vortex ring by the entrained bubbles. Smaller bubbles are easier to be entrained by the large scale vortex structure and the transportation distance is in inverse proportion to bubble diameter.     

Integrability of the Problem of the Motion of a Cylinder and a Vortex in an Ideal Fluid

In this paper, we obtain a nonlinear Poisson structure and two first integrals in the problem of the plane motion of a circular cylinder and n point vortices in an ideal fluid. This problem is a priori not Hamiltonian; specifically, in the case n= 1 (i.e., in the problem of the interaction of a cylinder with a vortex) it is integrable.     

Dipole approximation in three-vortex dynamics

We solve the problem of the relative motion of two nearby vortices (a dipole pair) and a third vortex for different current functions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 3, pp. 409–416, March, 2007.     

Interaction of point and dipole vortices in an incompressible liquid

We discuss the interaction between point vortices and point dipole vortices in two-dimensional ideal hydrodynamics and show that the equations of motion of the interacting point and point dipole vortices are exactly integrable. We find exact solutions for all possible parameter values characterizing the vortices and for arbitrary initial conditions and establish the regimes of vortex motion.     

Transfer of Gorenstein dimensions along ring homomorphisms

A central problem in the theory of Gorenstein dimensions over commutative noetherian rings is to find resolution-free characterizations of the modules for which these invariants are finite. Over local rings, this problem was recently solved for the Gorenstein flat and the Gorenstein projective dimensions; here we give a solution for the Gorenstein injective dimension. Moreover, we establish two formulas for the Gorenstein injective dimension of modules in terms of the depth invariant; they extend formulas for the injective dimension due to Bass and Chouinard.     

Properties of the solution of the adjoint problem describing the motion of viscous fluids in a tube

For the Navier-Stokes equations, we study a solution invariant with respect to a oneparameter group and modeling a nonstationary motion of two viscous fluids in a cylindrical tube; the fluid layer near the tube wall can be viewed as a lubricant. The motion is due to a nonstationary pressure drop. We obtain a priori estimates for the velocities in the layers. We find a stationary state of the system and show that it is the limit state as t → ∞ provided that the pressure gradient in one of the fluids stabilizes with time. We solve the inverse problem of finding the pressure gradients and the velocity field from a known flow rate.