scholarly journals Integrability aspects of the vortex filament equation for pseudo-null curves

2017 ◽  
Vol 14 (06) ◽  
pp. 1750090 ◽  
Author(s):  
José del Amor ◽  
Ángel Giménez ◽  
Pascual Lucas
Keyword(s):  
Space Form ◽  
Burgers Equation ◽  
Three Dimensional ◽  
Recursion Operator ◽  
Vortex Filament ◽  
Lorentzian Space ◽  
Null Curve ◽  
Vortex Filament Equation ◽  
Null Curves ◽  
The Relationship

An algebraic background in order to study the integrability properties of pseudo-null curve motions in a three-dimensional Lorentzian space form is developed. As an application, we delve into the relationship between the Burgers’ equation and the pseudo-null vortex filament equation. A recursion operator for the pseudo-null vortex filament equation is also provided.

On Bäcklund transformation and vortex filament equation for pseudo null curves in Minkowski 3-space

2016 ◽  
Vol 13 (06) ◽  
pp. 1650077 ◽  
Author(s):  
Milica Grbović ◽  
Emilija Nešović
Keyword(s):  
Evolution Equation ◽  
Bäcklund Transformation ◽  
Sufficient Conditions ◽  
Vortex Filament ◽  
Backlund Transformation ◽  
Burger's Equation ◽  
Null Curve ◽  
Vortex Filament Equation ◽  
Null Curves ◽  
The Given

In this paper, we introduce Bäcklund transformation of a pseudo null curve in Minkowski 3-space as a transformation mapping a pseudo null helix to another pseudo null helix congruent to the given one. We also give the sufficient conditions for a transformation between two pseudo null curves in the Minkowski 3-space such that these curves have equal constant torsions. By using the Da Rios vortex filament equation, based on localized induction approximation (LIA), we derive the vortex filament equation for a pseudo null curve and prove that the evolution equation for the torsion is the viscous Burger’s equation. As an application, we show that pseudo null curves and their Frenet frames generate solutions of the Da Rios vortex filament equation.

scholarly journals On involutes of order k of a null Cartan curve in Minkowski spaces

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2295-2305
Author(s):  
Muhammad Hanif ◽  
Hou Hua ◽  
Emilija Nesovic
Keyword(s):  
Minkowski Space ◽  
Vortex Filament ◽  
Null Curve ◽  
Minkowski Spaces ◽  
Vortex Filament Equation ◽  
The Relationship

In this paper, we define an involute and an evolving involute of order k of a null Cartan curve in Minkowski space En1 for n ? 3 and 1 ? k ? n-1. In relation to that, we prove that if a null Cartan helix has a null Cartan involute of order 1 or 2, then it is Bertrand null Cartan curve and its involute is its Bertrand mate curve. In particular, we show that Bertrand mate curve of Bertrand null Cartan curve can also be a non-null curve and find the relationship between the Cartan frame of a null Cartan curve and the Frenet or the Cartan frame of its non-null or null Cartan involute of order 1 ? k ? 2. We show that among all null Cartan curves in E31 , only the null Cartan cubic has two families of involutes of order 1, one of which lies on B-scroll. We also give some relations between involutes of orders 1 and 2 of a null Cartan curve in Minkowski 3-space. As an application, we show that involutes of order 1 of a null Cartan curve in E31 , evolving according to null Betchov-Da Rios vortex filament equation, generate timelike Hasimoto surfaces.

scholarly journals Null Curve Evolution in Four-Dimensional Pseudo-Euclidean Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
José del Amor ◽  
Ángel Giménez ◽  
Pascual Lucas
Keyword(s):  
Vector Fields ◽  
Evolution Equations ◽  
Space Form ◽  
Curve Evolution ◽  
Lie Bracket ◽  
Null Curve ◽  
Induction Equation ◽  
Null Curves ◽  
Local Vector ◽  
Local Symmetries

We define a Lie bracket on a certain set of local vector fields along a null curve in a 4-dimensional semi-Riemannian space form. This Lie bracket will be employed to study integrability properties of evolution equations for null curves in a pseudo-Euclidean space. In particular, a geometric recursion operator generating infinitely many local symmetries for the null localized induction equation is provided.

Pseudo null curve variations for Bishop frame in 3D semi-Riemannian manifold

2019 ◽  
Vol 16 (03) ◽  
pp. 1950043 ◽  
Author(s):  
Zehra Özdemi̇r
Keyword(s):  
Magnetic Field ◽  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Space Form ◽  
Charged Particles ◽  
Null Curve ◽  
Null Curves ◽  
The Magnetic Field ◽  
Killing Equations

In the present paper, the relation between invariants of the pseudo null curves and the variational vector fields of semi-Riemannian manifolds is introduced. After that, the Killing equations are written in terms of the Bishop curvatures along the pseudo null curve. By means of this approach, Killing equations make allow to interpret the movement of charged particles within the magnetic field. Afterwards, as an application, pseudo null magnetic curves are defined using the Killing variational vector field. The parametric representations of all pseudo null magnetic curves are determined in semi-Riemannian space form. Moreover, various examples of pseudo null magnetic curves are illustrated.

scholarly journals Vortex Filament Equation for a Regular Polygon in the Hyperbolic Plane

2021 ◽  
Vol 32 (1) ◽  
Author(s):  
Francisco de la Hoz ◽  
Sandeep Kumar ◽  
Luis Vega
Keyword(s):  
Center Of Mass ◽  
Point Of View ◽  
Vortex Filament ◽  
Complete Agreement ◽  
Euclidean Case ◽  
Runge Kutta Method ◽  
Vortex Filament Equation ◽  
The One ◽  
The Relationship ◽  
Planar Polygon

AbstractThe aim of this paper is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and both of its ends grow up exponentially, which makes the problem more challenging from a numerical point of view. However, using a finite difference scheme in space combined with a fourth-order Runge–Kutta method in time and fixed boundary conditions, we show that the numerical solution is in complete agreement with the one obtained by means of algebraic techniques. Second, as in the Euclidean case, we claim that, at infinitesimal times, the evolution of VFE for a planar polygon as the initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only can we compute the speed of the center of mass of the planar polygon, but the relationship also allows us to compare the time evolution of any of its corners with the evolution in the Euclidean case.

scholarly journals Dynamics of Single Droplet Splashing on Liquid Film by Coupling FVM with VOF

Processes ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 841
Author(s):  
Yuzhen Jin ◽  
Huang Zhou ◽  
Linhang Zhu ◽  
Zeqing Li
Keyword(s):  
Liquid Film ◽  
Weber Number ◽  
Stokes Equations ◽  
Numerical Study ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Single Droplet ◽  
Navier Stokes Equations ◽  
Volume Method ◽  
The Relationship

A three-dimensional numerical study of a single droplet splashing vertically on a liquid film is presented. The numerical method is based on the finite volume method (FVM) of Navier–Stokes equations coupled with the volume of fluid (VOF) method, and the adaptive local mesh refinement technology is adopted. It enables the liquid–gas interface to be tracked more accurately, and to be less computationally expensive. The relationship between the diameter of the free rim, the height of the crown with different numbers of collision Weber, and the thickness of the liquid film is explored. The results indicate that the crown height increases as the Weber number increases, and the diameter of the crown rim is inversely proportional to the collision Weber number. It can also be concluded that the dimensionless height of the crown decreases with the increase in the thickness of the dimensionless liquid film, which has little effect on the diameter of the crown rim during its growth.

scholarly journals A Field Measurement Based Wind Characteristics Analysis of a Typhoon in Near-Ground Boundary Layer

Atmosphere ◽  
2021 ◽  
Vol 12 (7) ◽  
pp. 873
Author(s):  
Dandan Xia ◽  
Liming Dai ◽  
Li Lin ◽  
Huaifeng Wang ◽  
Haitao Hu
Keyword(s):  
Turbulence Intensity ◽  
Field Measurement ◽  
Three Dimensional ◽  
Wind Data ◽  
Statistical Characteristics ◽  
Gust Factor ◽  
Characteristics Analysis ◽  
Wind Characteristics ◽  
The Mean ◽  
The Relationship

The field measurement was conducted to observe the wind field data of West Pacific typhoon “Maria” in this research. With the application of ultrasonic anemometers installed in different heights (10 m, 80 m, 100 m) of the tower, the three dimensional wind speed data of typhoon “Maria” was acquired. In addition, vane-type anemometers were installed to validate the accuracy of the wind data from ultrasonic anemometers. Wind characteristics such as the mean wind profile, turbulence intensity, integral length scale, and wind spectrum are studied in detail using the collected wind data. The relationship between the gust factor and turbulence intensity was also studied and compared with the existing literature to demonstrate the characteristics of Maria. The statistical characteristics of the turbulence intensity and gust factor are presented. The corresponding conclusion remarks are expected to provide a useful reference for designing wind-resistant buildings and structures.

A new approach to the bienergy and biangle of a moving particle lying in a surface of lorentzian space

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Talat Körpınar ◽  
Yasin Ünlütürk
Keyword(s):  
Vector Fields ◽  
Elastic Surface ◽  
Geometrical Characterization ◽  
Lorentzian Space ◽  
New Approach ◽  
Darboux Vector ◽  
Moving Particle ◽  
The Relationship ◽  
Surface Curve

AbstractIn this research, we study bienergy and biangles of moving particles lying on the surface of Lorentzian 3-space by using their energy and angle values. We present the geometrical characterization of bienergy of the particle in Darboux vector fields depending on surface. We also give the relationship between bienergy of the surface curve and bienergy of the elastic surface curve. We conclude the paper by providing bienergy-curve graphics for different cases.

Sign in / Sign up

Export Citation Format

Share Document