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.

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.

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 Physical Approach to Solving the Mathematical Navier-Stokes Problem

2019 ◽  
Author(s):  
Nikolai Kislov
Keyword(s):  
Fluid Flow ◽  
Stokes Equations ◽  
Stokes Problem ◽  
Differential Forms ◽  
Flow Regimes ◽  
Navier Stokes ◽  
Balance Equations ◽  
Fluid System ◽  
Navier Stokes Equations ◽  
The Continuum

Instead of modeling based on using an infinitesimal fluid element, which is treated as a continuous medium, we consider fluid flow in a fluid system as a model gas flow in a model gas system identical to the fluid system. We assigned to the model gas properties, which differ from the properties typically assigned to the ideal gas. In our approach, we mimic the movement of each particle/molecule composing the model gas and then gather that movement into macro quantities characterizing the fluid flow. We formulated integro-differential balance equations for mass, momentum, and energy applied to any non-moving point in three-dimensional space occupied by the model gas and operable from the continuum through the rarefied to the ballistic flow regimes. In parallel, we worked toward understanding how our derived integro-differential equations of the balance may relate to the existing Navier-Stokes equations. Since Navier-Stokes equations are limited to the continuum and collision-dominated flow regimes, we reduced the derived mass and momentum integro-differential balance equations by if during the period between sequential collisions, the relative change of any property value or any parameter characterizing the model gas is insignificant. Then by applying the method of vector differentiation, we converted the mass and momentum integro-differential balance equations into corresponding vector differential balance equations. We were surprised to observe that our converted differential forms look identically to the Navier-Stokes equations. This finding has led us to the conclusion that, in the collision-dominated flow regime, the formulated integro-differential forms of the balance are exact implicit solutions for corresponding Navier-Stokes equations. This finding also suggests that the proposed matching vector integro-differential forms of the mass, momentum, and energy balance, which can be solved by computer-implemented methods with no difficulty, may represent a physical problem described by the Navier-Stokes equations. We also provided five additional validation tests demonstrating the feasibility of the proposed method.

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.

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.

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.

Effects of stator splitter blades on aerodynamic performance of a single-stage transonic axial compressor

2020 ◽  
Vol 14 (4) ◽  
pp. 7369-7378
Author(s):  
Ky-Quang Pham ◽  
Xuan-Truong Le ◽  
Cong-Truong Dinh
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Axial Compressor ◽  
Aerodynamic Performance ◽  
Numerical Results ◽  
Single Stage ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Operating Stability ◽  
Length Coverage

Splitter blades located between stator blades in a single-stage axial compressor were proposed and investigated in this work to find their effects on aerodynamic performance and operating stability. Aerodynamic performance of the compressor was evaluated using three-dimensional Reynolds-averaged Navier-Stokes equations using the k-e turbulence model with a scalable wall function. The numerical results for the typical performance parameters without stator splitter blades were validated in comparison with experimental data. The numerical results of a parametric study using four geometric parameters (chord length, coverage angle, height and position) of the stator splitter blades showed that the operational stability of the single-stage axial compressor enhances remarkably using the stator splitter blades. The splitters were effective in suppressing flow separation in the stator domain of the compressor at near-stall condition which affects considerably the aerodynamic performance of the compressor.

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