Invariant Forms of Conservation Equations and Some Examples of Their Exact Solutions

2014 ◽  
Vol 136 (3) ◽  
Author(s):  
Siavash H. Sohrab
Keyword(s):  
Exact Solutions ◽  
Momentum Conservation ◽  
Immiscible Fluids ◽  
Enskog Equation ◽  
Cauchy Stress ◽  
Liquid Droplets ◽  
Scale Invariant ◽  
Conservation Equations ◽  
Cylindrically Symmetric ◽  
Laminar And Turbulent Flow

A scale-invariant model of statistical mechanics is described leading to invariant Boltzmann equation and the corresponding invariant Enskog equation of change. A modified form of Cauchy stress tensor for fluid is presented such that in the limit of vanishing intermolecular spacing, all tangential forces vanish in accordance with perceptions of Cauchy and Poisson. The invariant forms of mass, thermal energy, linear momentum, and angular momentum conservation equations derived from invariant Enskog equation of change are described. Also, some exact solutions of the conservation equations for the problems of normal shock, laminar, and turbulent flow over a flat plate, and flow within a single or multiple concentric spherical liquid droplets made of immiscible fluids located at the stagnation point of opposed cylindrically symmetric gaseous finite jets are presented.

Invariant Forms of Conservation Equations and Some Examples of Their Exact Solutions

2014 ◽  
Author(s):  
Siavash H. Sohrab
Keyword(s):  
Momentum Conservation ◽  
Enskog Equation ◽  
Cauchy Stress ◽  
Cauchy Stress Tensor ◽  
Scale Invariant ◽  
Spherical Droplet ◽  
Conservation Equations ◽  
Invariant Model ◽  
Cylindrically Symmetric ◽  
Tangential Forces

A scale-invariant model of statistical mechanics is described leading to invariant Boltzmann equation and the corresponding invariant Enskog equation of change. A modified form of Cauchy stress tensor for fluid is presented such that in the limit of vanishing intermolecular spacing all tangential forces vanish in accordance with perceptions of Cauchy and Poisson. The invariant forms of mass, thermal energy, linear momentum, and angular momentum conservation equations derived from invariant Enskog equation of change are described. Also, some exact solution of the conservation equations for the problems of normal shock, flow over a flat plate, and flow within a spherical droplet located at the stagnation point of opposed cylindrically-symmetric gaseous jets are presented.

Invariant Forms of Conservation Equations for Reactive Fields and Hydro-Thermo-Diffusive Theory of Laminar Flames

2014 ◽  
Vol 137 (1) ◽  
Author(s):  
Siavash H. Sohrab
Keyword(s):  
Angular Momentum ◽  
Propagation Speed ◽  
Premixed Flame ◽  
Laminar Flames ◽  
Enskog Equation ◽  
Scale Invariant ◽  
Propagation Models ◽  
Conservation Equations ◽  
Invariant Model ◽  
Good Agreement

A scale-invariant model of statistical mechanics is described leading to invariant Enskog equation of change that is applied to derive invariant forms of conservation equations for mass, thermal energy, linear momentum, and angular momentum in chemically reactive fields. Modified hydro-thermo-diffusive theories of laminar premixed flames for (1) rigid-body and (2) Brownian-motion flame propagation models are presented and are shown to be mathematically equivalent. The predicted temperature profile, thermal thickness, and propagation speed of laminar methane–air premixed flame are found to be in good agreement with existing experimental observations.

scholarly journals Tight focusing of light beams: a set of exact solutions

2016 ◽  
Vol 472 (2195) ◽  
pp. 20160538 ◽  
Author(s):  
John Lekner
Keyword(s):  
Exact Solutions ◽  
Focal Plane ◽  
Bessel Functions ◽  
Momentum Conservation ◽  
Focal Region ◽  
Beam Direction ◽  
Lommel Functions ◽  
Tight Focusing ◽  
Causal Propagation ◽  
Light Beams

Exact solutions of Maxwell's equations representing light beams are explored. The solutions satisfy all of the physical requirements of causal propagation and of energy, momentum and angular momentum conservation. A set of solutions can be found from a proto-beam by an imaginary translation along the beam direction. The proto-beam is given analytically in terms of the Bessel functions J 0 , J 1 and the Lommel functions U 0 , U 1 , or equivalently in terms of products of the spherical Bessel functions and Legendre polynomials. The complex wavefunction has rings of zeros in the focal plane. Localization of the focal region is to within about one half of the vacuum wavelength.

scholarly journals On certain new and exact solutions of the Emden-Fowler equation and Emden equation via invariant variational principles and group invariance

1991 ◽  
Vol 32 (4) ◽  
pp. 457-468 ◽  
Author(s):  
O. P. Bhutani ◽  
K. Vijayakumar
Keyword(s):  
Differential Equation ◽  
Exact Solutions ◽  
Lie Group ◽  
Variational Principles ◽  
First Integrals ◽  
Noether Theorem ◽  
Repeated Application ◽  
Scale Invariant ◽  
Emden Equation ◽  
Fowler Equation

AbstractAfter formulating the alternate potential principle for the nonlinear differential equation corresponding to the generalised Emden-Fowler equation, the invariance identities of Rund [14] involving the Lagrangian and the generators of the infinitesimal Lie group are used for writing down the first integrals of the said equation via the Noether theorem. Further, for physical realisable forms of the parameters involved and through repeated application of invariance under the transformation obtained, a number of exact solutions are arrived at both for the Emden-Fowler equation and classical Emden equations. A comparative study with Bluman-Cole and scale-invariant techniques reveals quite a number of remarkable features of the techniques used here.

scholarly journals STATIC CYLINDRICALLY SYMMETRIC INTERIOR SOLUTIONS IN f(R) GRAVITY

2012 ◽  
Vol 27 (25) ◽  
pp. 1250138 ◽  
Author(s):  
M. SHARIF ◽  
SADIA ARIF
Keyword(s):  
Energy Density ◽  
Exact Solutions ◽  
Perfect Fluid ◽  
Functional Form ◽  
Ricci Scalar ◽  
The Other ◽  
New Exact Solutions ◽  
Cylindrically Symmetric ◽  
Theory Of Gravity ◽  
Interior Solutions

We investigate some exact static cylindrically symmetric solutions for a perfect fluid in the metric f(R) theory of gravity. For this purpose, three different families of solutions are explored. We evaluate energy density, pressure, Ricci scalar and functional form of f(R). It is interesting to mention here that two new exact solutions are found from the last approach, one is in particular form and the other is in the general form. The general form gives a complete description of a cylindrical star in f(R) gravity.

MAPPING OF A DETERMINISTIC DYNAMICS WITH QUENCHED VARIABLES INTO A STOCHASTIC PROBLEM WITH COGNITIVE MEMORY

Fractals ◽  
1995 ◽  
Vol 03 (03) ◽  
pp. 471-481
Author(s):  
R. CAFIERO ◽  
A. GABRIELLI ◽  
M. MARSILI ◽  
L. PIETRONERO
Keyword(s):  
Stochastic Dynamics ◽  
Immiscible Fluids ◽  
Quenched Disorder ◽  
Invasion Percolation ◽  
Scale Invariant ◽  
New Approach ◽  
Deterministic Dynamics ◽  
Irreversible Dynamics ◽  
Disordered Porous Media ◽  
With Memory

Irreversible dynamics in media with quenched disorder seems to be the essential mechanism for a variety of phenomena like fracture propagation or displacement of immiscible fluids in disordered porous media. This problem does not seem to be treatable along the standard theoretical schemes. Recently a new approach has been introduced that allows mapping of a deterministic dynamics with quenched disorder like that of Invasion Percolation into a stochastic dynamics with memory. This memory consists essentially in a cognitive process that modifies the probability distribution of quenched variables conditionally to all previous events. This approach, together with the FST and the corresponding analysis of the scale invariant dynamics, provides a new framework to understand the self organization and to compute the critical exponents of problems like Invasion Percolation and Bak and Sneppen.

On rotating charged dust in general relativity

1978 ◽  
Vol 362 (1710) ◽  
pp. 329-340 ◽  
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Differential Rotation ◽  
Single Equation ◽  
Charged Dust ◽  
Newtonian Physics ◽  
Coupled Equations ◽  
Cylindrically Symmetric ◽  
Axis Of Symmetry

The problem of charged dust rotating about an axis of symmetry is considered both in Newtonian physics and in general relativity. The Newtonian problem is reduced to a single equation in the case of constant rotation, and to two coupled equations in the case of differential rotation, and some explicit cylindrically symmetric solutions are obtained. In general relativity some new cylindrically symmetric exact solutions for constant rotation are derived, and the problem of differential rotation is reduced to four coupled equations for four unknowns.

scholarly journals Multiphase Flow in the Gap Between Two Rotating Cylinders

2020 ◽  
Vol 328 ◽  
pp. 02017
Author(s):  
Milada Kozubková ◽  
Jana Jablonská ◽  
Marian Bojko ◽  
František Pochylý ◽  
Simona Fialová
Keyword(s):  
Multiphase Flow ◽  
Dynamic Properties ◽  
Immiscible Fluids ◽  
Immiscible Liquids ◽  
Hydrodynamic Bearing ◽  
Concentric Cylinders ◽  
Laminar And Turbulent Flow ◽  
Application Potential ◽  
The Stability ◽  
Future Technologies

The research of liquids composed of two (or more) mutually immiscible components is a new emerging area. These liquids represent new materials, which can be utilized as lubricants, liquid seals or as fluid media in biomechanical devices. The investigation of the problem of immiscible liquids started some years ago and soon it was evident that it will have a great application potential. Recently, there has been an effort to use ferromagnetic or magnetorheological fluids in the construction of dumpers or journal bearings. Their advantage is a significant change in dynamic viscosity depending on magnetic induction. In combination with immiscible liquids, qualitatively new liquids can be developed for future technologies. In our case, immiscible fluids increase the dynamic properties of the journal hydrodynamic bearing. The article focuses on the stability of single-phase and subsequently multiphase flow of liquids in the gap between two concentric cylinders, one of which rotates. The aim of the analysis was to study the effect of viscosity and density on the stability/instability of the flow, which is manifested by Taylor vortices. Methods of experimental and mathematical analysis were used for the analysis in order to verify mathematical models of laminar and turbulent flow of immiscible liquids.

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