scholarly journals A Derivation of Proca Equations on Cantor Sets: A Local Fractional Approach

2014 ◽  
Vol 10 ◽  
pp. 48-56 ◽  
Author(s):  
Victor Christianto ◽  
Biruduganti Rahul
Keyword(s):  
Maxwell Equations ◽  
High Energy Physics ◽  
Stokes Equations ◽  
High Energy ◽  
Cantor Sets ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Fractal Media ◽  
Vector Calculus ◽  
Proca Equations

In a recent paper published at Advances in High Energy Physics (AHEP) journal, Yang Zhao et al. derived Maxwell equations on Cantor sets from the local fractional vector calculus. It can be shown that Maxwell equations on Cantor sets in a fractal bounded domain give efficiency and accuracy for describing the fractal electric and magnetic fields. Using the same approach, elsewhere Yang, Baleanu & Tenreiro Machado derived systems of Navier-Stokes equations on Cantor sets. However, so far there is no derivation of Proca equations on Cantor sets. Therefore, in this paper we present for the first time a derivation of Proca equations and GravitoElectroMagnetic (GEM) Proca-type equations on Cantor sets. Considering that Proca equations may be used to explain electromagnetic effects in superconductor, We suggest that Proca equations on Cantor sets can describe electromagnetic of fractal superconductors; besides GEM Proca-type equations on Cantor sets may be used to explain some gravitoelectromagnetic effects of superconductor for fractal media. It is hoped that this paper may stimulate further investigations and experiments in particular for fractal superconductor. It may be expected to have some impact to fractal cosmology modeling too.

scholarly journals London-Proca-Hirsch Equations for Electrodynamics of Superconductors on Cantor Sets

2015 ◽  
Vol 4 ◽  
pp. 1-7 ◽  
Author(s):  
Victor Christianto
Keyword(s):  
Magnetic Fields ◽  
Maxwell Equations ◽  
High Energy Physics ◽  
High Energy ◽  
Cantor Sets ◽  
Vector Calculus ◽  
Electric And Magnetic Fields ◽  
Proca Equations ◽  
First Time ◽  
Energy Physics

In a recent paper published at Advances in High Energy Physics (AHEP) journal, Yang Zhao et al. derived Maxwell equations on Cantor sets from the local fractional vector calculus. It can be shown that Maxwell equations on Cantor sets in a fractal bounded domain give efficiency and accuracy for describing the fractal electric and magnetic fields. However, so far there is no derivation of equations for electrodynamics of superconductor on Cantor sets. Therefore, in this paper I present for the first time a derivation of London-Proca-Hirsch equations on Cantor sets. The name of London-Proca-Hirsch is proposed because the equations were based on modifying Proca and London-Hirsch’s theory of electrodynamics of superconductor. Considering that Proca equations may be used to explain electromagnetic effects in superconductor, I suggest that the proposed London-Proca-Hirsch equations on Cantor sets can describe electromagnetic of fractal superconductors. It is hoped that this paper may stimulate further investigations and experiments in particular for fractal superconductor. It may be expected to have some impact to fractal cosmology modeling too.

scholarly journals Systems of Navier-Stokes Equations on Cantor Sets

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Xiao-Jun Yang ◽  
Dumitru Baleanu ◽  
J. A. Tenreiro Machado
Keyword(s):  
Fluid Flow ◽  
Bounded Domain ◽  
Stokes Equations ◽  
Cantor Sets ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Fractal Media ◽  
Vector Calculus

We present systems of Navier-Stokes equations on Cantor sets, which are described by the local fractional vector calculus. It is shown that the results for Navier-Stokes equations in a fractal bounded domain are efficient and accurate for describing fluid flow in fractal media.

scholarly journals Finding Exact Forms on a Thermodynamic Manifold

2018 ◽  
Author(s):  
Chao Ju ◽  
Mark Stalzer
Keyword(s):  
Maxwell Equations ◽  
Stokes Equations ◽  
Differential Forms ◽  
Thermodynamic System ◽  
Navier Stokes ◽  
State Variables ◽  
Navier Stokes Equations ◽  
Vector Calculus ◽  
Cotangent Space ◽  
Classical Thermodynamics

Because only two variables are needed to characterize a simple thermodynamic system in equilibrium, any such system is constrained on a 2D manifold. Of particular interest are the exact 1-forms on the cotangent space of that manifold, since the integral of exact 1-forms is path-independent, a crucial property satisfied by state variables such as internal energy dE and entropy dS. Our prior work [1] shows that given an appropriate language of vector calculus, a machine can re-discover the Maxwell equations and the incompressible Navier-Stokes equations from data. In this paper, We enhance this language by including differential forms and show that machines can re-discover the equation for entropy dS given data. Since entropy appears in various fields of science in different guises, a potential extension of this work is to use the machinery developed in this paper to let machines discover the expressions for entropy from data in fields other than classical thermodynamics.

Poiseuille equation for steady flow of fractal fluid

2016 ◽  
Vol 30 (22) ◽  
pp. 1650128 ◽  
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Steady Flow ◽  
Cross Sections ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Fractal Media ◽  
Vector Calculus ◽  
Continuous Models ◽  
Incompressible Viscous Fluids ◽  
Poiseuille Equation

Fractal fluid is considered in the framework of continuous models with noninteger dimensional spaces (NIDS). A recently proposed vector calculus in NIDS is used to get a description of fractal fluid flow in pipes with circular cross-sections. The Navier–Stokes equations of fractal incompressible viscous fluids are used to derive a generalization of the Poiseuille equation of steady flow of fractal media in pipe.

Numerical Simulation of Electromagnetic Stirring Effect on Flow Pattern of Metal Melt

2011 ◽  
Vol 264-265 ◽  
pp. 1574-1579
Author(s):  
H. Namaki ◽  
S. Hossein Seyedein ◽  
M.R. Afshar Moghadam ◽  
R. Ghasemzadeh
Keyword(s):  
Flow Pattern ◽  
Maxwell Equations ◽  
Flow Model ◽  
Stokes Equations ◽  
Electromagnetic Stirring ◽  
Navier Stokes ◽  
Simultaneous Solution ◽  
Navier Stokes Equations ◽  
Stirring Effect ◽  
Good Agreement

In this study, a mathematical model was developed to simulate 2-D axisymmetric melt flow under magnetic field in a cylindrical container. The modeling of this process required the simultaneous solution of the turbulent Navier-Stokes equations together with Maxwell equations. The flow pattern in liquid bath was obtained using a two-equation κ-є turbulent flow model, which was further used to obtain the solute distribution. The governing differential equations were solved numerically using finite volume based finite difference method. The computed results, were found to be in good agreement with the measurements reported in the literature. The effect of stirring parameters on temperature homogeneity of the melt have been discussed and presented.

scholarly journals On three-dimensional incompressible Navier-Stokes fluid on cantor sets in spherical Cantor type co-ordinate system

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 853-858
Author(s):  
Zhi-Jun Meng ◽  
Yao-Ming Zhou ◽  
Dong-Mu Mei
Keyword(s):  
Heat Conduction ◽  
Stokes Equations ◽  
Heat Conduction Problem ◽  
Three Dimensional ◽  
Cantor Sets ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Stokes Fluid ◽  
Incompressible Navier Stokes Equation

This paper addresses the systems of the incompressible Navier-Stokes equations on Cantor sets without the external force involving the fractal heat-conduction problem vial local fractional derivative. The spherical Cantor type co-ordinate method is used to transfer the incompressible Navier-Stokes equation from the Cantorian co-ordinate system into the spherical Cantor type co-ordinate system.

Control Of Processes Of Ignition And Combustion Stabilization In The Supersonic Combustion Chamber

2016 ◽  
Vol 11 (2) ◽  
pp. 46-55
Author(s):  
Olga Vankova ◽  
Marat Goldfeld ◽  
Natalya Fedorova
Keyword(s):  
Combustion Chamber ◽  
Stokes Equations ◽  
High Energy ◽  
Supersonic Combustion ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Model Calculations ◽  
Detailed Chemistry ◽  
Sst Turbulence Model ◽  
The Stability

In the paper, results of mathematical modeling of a flow in the supersonic combustion chamber are presented, which have been performed under the conditions of burning initiation by means of an electronic bunch of high energy on the basis of the offered ignition model. Calculations are carried out on the basis of the Reynolds averaged Navier – Stokes equations supplemented by the k–ω SST turbulence model and detailed chemistry kinetics. As a result of numerical modeling, it has been shown that in a frame of the offered model it is possible to predict the ignition of mixture at low stagnation temperatures. The numerical results confirm the experimental data. It is shown that the choice of the optimum scheme of stabilization and the stabilizer geometry allows one to get the flame propagation over all the channel and to provide the stability of combustion even at high flow Mach numbers. The offered mathematical model has allowed defining the conditions of ignition

Application of Lorentz Force Velocimetry to Open Channel Flow

2011 ◽  
Vol 690 ◽  
pp. 99-102 ◽  
Author(s):  
Xiao Dong Wang ◽  
Rico Klein ◽  
Yuri Kolesnikov ◽  
Andre Thess
Keyword(s):  
Lorentz Force ◽  
Maxwell Equations ◽  
Open Channel ◽  
Stokes Equations ◽  
Magnetic System ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Electrically Conducting ◽  
Lorentz Force Velocimetry ◽  
The Relationship

Lorentz Force Velocimetry (LFV) is a noncontact method for flow measurement in electrically conducting fluid, especially in high temperature, opaque and aggressive molten metal. The principle is based on exposing the flow to a magnetic system and measuring the drag force acting upon it [1]. The aim of present paper is to study the application of LFV for open channel liquid metal flows, to numerically obtain the relationship between the measured Lorentz force and the flow rate. This provides the calibrating criterion of LFV. To this end, we firstly investigate a metal bar with different cross-section shapes passing through the magnetic system; Secondly, we study the relationship by a multiphysics numerical model fully coupling Navier-Stokes equations and Maxwell equations.

scholarly journals Well-posedness of the Navier—Stokes—Maxwell equations

2014 ◽  
Vol 144 (1) ◽  
pp. 71-86 ◽  
Author(s):  
Pierre Germain ◽  
Slim Ibrahim ◽  
Nader Masmoudi
Keyword(s):  
Maxwell Equations ◽  
Stokes Equations ◽  
Mild Solutions ◽  
A Priori ◽  
Local Existence ◽  
Large Data ◽  
Navier Stokes ◽  
Well Posedness ◽  
Scale Invariant ◽  
Navier Stokes Equations

We study the local and global well-posedness of a full system of magnetohydrodynamic equations. The system is a coupling of the incompressible Navier—Stokes equations with the Maxwell equations through the Lorentz force and Ohm's law for the current. We show the local existence of mild solutions for arbitrarily large data in a space similar to the scale-invariant spaces classically used for Navier—Stokes. These solutions are global if the initial data are small enough. Our results not only simplify and unify the proofs for the space dimensions 2 and 3, but also refine those in [8]. The main simplification comes from an a prioriLt2 (Lx∞) estimate for solutions of the forced Navier—Stokes equations.

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