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

Abstract

A scale-invariant model o f statistical mechanics is described leading to invariant Boltz mann equation and the corresponding invariant Enskog equation o f change. A modified form o f Cauchy stress tensor fo r fluid is presented such that in the limit o f vanishing intermolecular spacing, all tangential forces vanish in accordance with perceptions o f Cauchy and Poisson. The invariant form s o f mass, thermal energy, linear momentum, and angu lar momentum conservation equations derived from invariant Enskog equation o f change are described. Also, some exact solutions o f the conservation equations fo r the problems o f normal shock, laminar, and turbulent flow over aflat plate, and flow within a single or multiple concentric spherical liquid droplets made o f immiscible fluids located at the stagnation point o f opposed cylindrically symmetric gaseous finite jets are presented. Jo u rn a l o f E n erg y R eso u rces T e c h n o lo g y C o p y rig h t © 2014 by A S M E SEPTEMBER 2014, Vol. 136 / 032002-1 GALACTIC CLUSTER EGD (J +5) GALAXY EPD (J+4) PLANET EHD (J +3) HYDRO-SYSTEM EFD (J +2) FLUID ELEMENT EED (J+l) EDDY ECD (J) CLUSTER EMD (J -1) MOLECULE EAD (J -2) ATOM ESD (J -3) SUB-PARTICLE EKD (J-4) PHOTON ETD (J -5) TACHYON UNIVERSE LGD (J +11/2) GALACTIC CLUSTER LPD (J + 9/2) GALAXY LHD (J + 7/2) PLANET LFD (J + 5/2) HYDRO-SYSTEM LED (J + 3/2) FLUID ELEMENT LCD (J +1/2) EDDY LMD (J -1/2) CLUSTER LAD (J -3/2) MOLECULE LSD (J -5/2) ATOM LKD (J -7/2) SUB-PARTICLE LTD (J -9/2) PHOTON F ig . 1 A s c a l e -i n v a r i a n t m o d e l o f s t a t i s t i c a l m e c h a n i c s . E q u i l i b r i u m -/ i -d y n a m i c s o n t h e l e f t -h a n d s i d e a n d n o n e q u i i i b r i u m l a m i n a r -/ i -d y n a m i c s o n t h e r i g h t -h a n d s i d e f o rNomenclature cp = specific heat at constant pressure (J/kg K) D = diffusion coefficient (m2/s)

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

“…Following the classical methods [24][25][26][27][28][53][54][55][56][57] the invariant definition of density pp, and velocity of atom u,y;, element v,/j, and system v/p at the scale fl are [49,52] P ip = n ip m ip = m ip l/^du,/), u;/; = vimp/(_i (1)…”

“…Similarities between such statistical fields shown in Fig. 1 resulted in recent introduction of a scale-invariant model of statistical mechanics [44] and its application to the fields of thermodynamics [45][46][47], fluid mechanics [48,49], and quantum mechanics [50][51][52].…”