research article fundamentals of the thermohydrogravidynamic...

Hindawi Publishing CorporationInternational Journal of GeophysicsVolume 2013, Article ID 519829, 39 pageshttp://dx.doi.org/10.1155/2013/519829

Research ArticleFundamentals of the Thermohydrogravidynamic Theory ofthe Global Seismotectonic Activity of the Earth

Sergey V. Simonenko

V.I. Il’ichev Pacific Oceanological Institute, Far Eastern Branch of Russian Academy of Sciences,43 Baltiyskaya Street, Vladivostok 690041, Russia

Correspondence should be addressed to Sergey V. Simonenko; [email protected]

Received 7 November 2012; Accepted 10 April 2013

Academic Editor: Umberta Tinivella

Copyright © 2013 Sergey V. Simonenko.This is an open access article distributed under theCreativeCommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The article presents the fundamentals of the cosmic geophysics (representing the deterministic thermohydrogravidynamic theoryintended for earthquakes prediction) based on the author’s generalized differential formulation of the first law of thermodynamicsextending the classical Gibbs’ formulation by taking into account (along with the classical infinitesimal change of heat 𝛿𝑄 and theclassical infinitesimal change of the internal energy 𝑑𝑈

𝜏) the infinitesimal increment of the macroscopic kinetic energy 𝑑𝐾

𝜏, the

infinitesimal increment of the gravitational potential energy 𝑑𝜋𝜏, the generalized expression for the infinitesimal work 𝛿𝐴np,𝜕𝜏 done

by the nonpotential terrestrial stress forces (determined by the symmetric stress tensorT) acting on the boundary of the continuumregion 𝜏, and the infinitesimal increment𝑑𝐺 of energy due to the cosmic and terrestrial nonstationary energy gravitational influenceon the continuum region 𝜏 during the infinitesimal time 𝑑𝑡. Based on the established generalized differential formulation of thefirst law of thermodynamics, the author explains the founded cosmic energy gravitational genesis of the strong Chinese 2008 andthe strong Japanese 2011 earthquakes.

1. Introduction

Theproblem of the long-term predictions of the strong earth-quakes [1, 2] is the significant problem of the modern geo-physics. The analysis of the period 1977–1985 revealed [3] thestrongly nonrandom tendencies in the earthquake-inducedgeodetic changes (owing to the mass redistribution of mate-rial inside the Earth) related to the change of the Earth’srotation and the Earth’s gravitational field.The analysis of theperiod 1977–1993 (characterized by 11015 major earthquakes)revealed [4] the strong earthquakes’ tendency to increasethe Earth’s spin (rotational) energy. The analysis of the sameperiod 1977–1993 revealed [5] “an extremely strong tendencyfor the earthquakes to decrease the global gravitationalenergy” confirming the inherent relation of the earthquakeswith the transformation of the Earth’s gravitational energyinto the seismic wave energy and frictional heat.The previousanalysis of the principal geological features of the past 400 ×10

6 years revealed [6] the geological evidence for a pulsatinggravitation related to periodic variation of the Earth’s radiusduring the geological evolution of the Earth.The combination

of satellite and gravimetric data revealed [7] the free-airanomalies of the Earth’s gravitational field.

It is well known that “the deterministic prediction of thetime of origin, hypocentral (or epicentral) location, andmag-nitude of an impending earthquake is an open scientific prob-lem” [8]. It was conjectured [8] that the possible earthquakeprediction and warning must be carried out on a determinis-tic basis.However, it was pointed out [8]with some regret thatthe modern “study of the physical conditions that give rise toan earthquake and the processes that precede a seismic rup-ture of an ordinary event are at a very preliminary stage and,consequently, the techniques of prediction of time of origin,epicentre, and magnitude of an impending earthquake nowavailable are below standard”. The authors [8] argued that “anew strong theoretical scientific effort is necessary to try tounderstand the physics of the earthquake”. It was conjectured[8] that the present level of knowledge of the geophysicalprocesses “is unable to achieve the objective of a deterministicprediction of an ordinary seismic event, but it certainly will ina more or less distant future tackle the problem with seri-ousness and avoiding scientifically incorrect, wasteful, and

2 International Journal of Geophysics

inconclusive shortcuts, as sometimes has been done”. Sgrignaand Conti conjectured [8] that “it will take long time (maybe years, tens of years, or centuries) because this approachrequires a great cultural, financial, and organizational efforton an international basis”. It was conjectured [8] that a pos-sible contribution to a deterministic earthquake predictionapproach is related to observations and physical modellingof earthquake precursors to formulate, in perspective, “aunified theory able to explain the causes of its genesis, andthe dynamics, rheology, and microphysics of its preparation,occurrence, postseismic relaxation, and interseismic phases”.It was conjectured [8] that “the study of the physical condi-tions that give rise to an earthquake and of the processes thatprecede a seismic rupture is at a very preliminary stage and,consequently, the techniques of prediction available at themoment are below standard”. However, Sgrigna and Contibelieve [8] that “it should be better to pursue the deterministicprediction approach even if a reliable deterministic methodof earthquake prediction will presumably be available only inthe more distant future”.

It was pointed out [9] that the gravity changes (derivedfrom regional gravity monitoring data in China from 1998 to2005) exhibited noticeable variations before the occurrenceof two large earthquakes in 2008 in the areas surround-ing Yutian (Xinjiang) and Wenchuan (Sichuan). A recentresearch [10] by Zhan and his colleagues demonstrated thatsignificant gravity changes were observed before all nine largeearthquakes that ruptured within or near mainland Chinafrom 2001 to 2008. It was pointed out [9] that the past experi-ence and empirical data showed that “earthquakes typicallyoccur within one to two years after a period of significantgravity changes in the region in question”. It was concluded[9] that the “additional research is needed to remove thesubjective nature in the determination of the timeframe of aforecasted earthquake”.

The need of the thermohydrogravidynamic approach [11,12] is confirmed by previous studies [3–5, 13] and by notice-able variations of gravitational field identified [9, 10] beforestrong earthquakes in China from 2001 to 2008. The neces-sity to consider the gravitational field (during the strongearthquakes) is also related to the observations of the slowgravitational [14, 15] ground waves resulting from strongearthquakes and spreading out from the focal regions [16, 17]of earthquakes. Lomnitz pointed out [16] that the gravita-tional groundwaves (related to great earthquakes) “have beenregularly reported for many years and remain a controversialsubject in earthquake seismology”. Richter presented [18] thedetailed analysis of these observations and made the con-clusion that “there is almost certainly a real phenomenon ofprogressing or standing waves seen on soft ground in themeizoseismal areas of great earthquakes”. Lomnitz presented[17] the real evidence of the existence of the slow gravitationalwaves in sedimentary layers during strong earthquakes. Thefundamental connections of the geodynamics, seismicity,and volcanism with gravitation (and the slow gravitationalground waves resulting from strong earthquakes) are pre-sented in the works [19–22].

It was conjectured [23] that the recent destructive earth-quakes occurred in Sichuan (China, 2008), Italy (2009), Haiti

(2010), Chile (2010), New Zealand (2010), and Japan (2011)“have shown that, in present state, scientific researchers haveachieved little or almost nothing in the implementationof short- and medium-term earthquake prediction, whichwould be useful for disaster mitigation measures”. It wasconjectured [23] that “this regrettable situation could beascribed to the present poor level of achievements in earth-quake forecast”. It was pointed out [23] that “although manymethods have been claimed to be capable of predicting earth-quakes (as numerous presentations on earthquake precursorsregularly show at every international meeting), the problemof formulating such predictions in a quantitative, rigorous,and repeatable way is still open”. It was formulated [23] that“another problem of practical implementation of earthquakeforecasting could be due to the lack of common understand-ing and exchange of information between the scientific com-munity and the governmental authorities that are responsiblefor earthquake damagemitigation in each country: they oper-ate in two different environments, they aim at different tasks,and they generally speak two different languages”. It waspointed out [23] that “the way how seismologists should for-mulate their forecasts and how they should transfer them todecision-makers and to the public is still a tricky issue”. It wasclearly formulated [23] that “the formulation of probabilisticearthquake forecasts with large uncertainties in space andtime and very low probability levels is still difficult to be usedby decision-making people”. It was conjectured [23] that “inreal circumstances the authorities deal with critical problemsrelated to the high cost of evacuating the population from anarea where the scientificmethods estimate an expected rate ofdestructive earthquake as one in many thousand days, whilethey require much more deterministic statements”. In thespecial issue [23] of the International Journal of Geophysics,Console et al. assessed the status of the art of earthquake fore-casts and their applicability. They invited authors “to reportmethods and case studies that could concretely contribute or,at least seemed promising, to improve the present frustratingsituation, regarding the practical use of earthquake forecasts”[23].

In this article, by accepting with gratitude the personalinvitation from Dr. Reem Ali and Dr. Radwa Ibrahim (rep-resenting the Editorial Office of the International Journalof Geophysics) to submit an article to the special issue on“GeophysicalMethods for Environmental Studies”, the authorpresents the fundamentals of the thermohydrogravidynamictheory intended for deterministic prediction of earthquakes.The thermohydrogravidynamic theory is based on the estab-lished generalized differential formulation [11, 12, 24–26] ofthe first law of thermodynamics (formoving rotating deform-ing compressible heat-conducting stratified macroscopiccontinuum region 𝜏 subjected to the nonstationary Newto-nian gravity):

𝑑𝑈𝜏+ 𝑑𝐾

𝜏+ 𝑑𝜋

𝜏= 𝛿𝑄 + 𝛿𝐴np,𝜕𝜏 + 𝑑𝐺 (1)

extending the classical formulation [27] by taking intoaccount (along with the classical infinitesimal change of heat𝛿𝑄 and the classical infinitesimal change of the internalenergy 𝑑𝑈

𝜏) the infinitesimal increment of the macroscopic

International Journal of Geophysics 3

kinetic energy𝑑𝐾𝜏, the infinitesimal increment of the gravita-

tional potential energy𝑑𝜋𝜏, the generalized expression for the

infinitesimal work 𝛿𝐴np,𝜕𝜏 done by the nonpotential terres-trial stress forces (determined by the symmetric stress tensorT) acting on the boundary of the continuum region 𝜏, andthe infinitesimal increment 𝑑𝐺 of energy due to the cosmicand terrestrial nonstationary energy gravitational influenceon the continuum region 𝜏 during the infinitesimal time 𝑑𝑡.

In Section 2 we begin by considering the inherent phys-ical incompleteness of the classical expression [28, 29] forthe macroscopic kinetic energy per unit mass 𝜀

𝑘defined (in

classical nonequilibrium thermodynamics) as the sum of themacroscopic translational kinetic energy per unit mass 𝜀

𝑡=

(1/2)k2 of the mass center of a continuum region and themacroscopic internal rotational kinetic energy per unit mass𝜀𝑟= (1/2)𝜃𝜔

2, where k is the speed of the mass center of asmall continuum region, 𝜔 is an angular velocity of internalrotation [29, 30], and 𝜃 is an inertiamoment per unitmass of asmall continuum region [28]. The classical de Groot andMazur expression has inherent physical incompleteness [24,31] related to the questionable assumption about the rigid-like rotation of a small continuum region. The classical deGroot and Mazur expression [28] does not consider thenonequilibrium component of the macroscopic velocity fieldrelated to the velocity shear defined by the rate of straintensor 𝑒

𝑖𝑗. In Section 2 the macroscopic kinetic energy per

unit mass 𝜀𝑘is presented [31] as a sum of the macroscopic

translational kinetic energy per unit mass 𝜀𝑡= (1/2)k2 of the

mass center of a continuum region, the classical macroscopicinternal rotational kinetic energy per unit mass 𝜀

𝑟[28], the

new macroscopic internal shear kinetic energy per unit mass𝜀𝑠[31], and the new macroscopic internal kinetic energy of

shear-rotational coupling per unitmass 𝜀coup𝑠,𝑟

[31] with a smallcorrection. The presented expression for 𝜀

𝑘and its particular

form for homogeneous continuum regions of spherical andcubical shapes generalized [31] the classical de Groot andMazur expression in classical nonequilibrium thermodynam-ics [28, 29] by taking into account the newmacroscopic inter-nal shear kinetic energy per unit mass 𝜀

𝑠, which expresses the

kinetic energy of irreversible dissipative shear motion, andalso the new macroscopic internal kinetic energy of shear-rotational coupling per unit mass 𝜀coup

𝑠,𝑟, which expresses the

kinetic energy of local coupling between irreversible dissi-pative shear and reversible rigid-like rotational macroscopiccontinuum motions.

Following the “Statistical thermohydrodynamics of irre-versible strike-slip-rotational processes” [11] and the “Ther-mohydrogravidynamics of the Solar System” [12], inSection 2.2 we present the generalized differential formula-tion of the first law of thermodynamics (in the Galilean frameof reference) for nonequilibrium shear-rotational states of thedeformed finite one-component individual continuum (char-acterized by the symmetric stress tensor T) region 𝜏 movingin the nonstationary gravitational field. In Section 2.3 wepresent the generalized differential formulation [11, 12] ofthe first law of thermodynamics (in the Galilean frame ofreference) for nonequilibrium shear-rotational states of thedeformed finite individual region 𝜏 of the compressible

viscous Newtonian one-component continuum moving inthe nonstationary gravitational field.We present the generali-zation [11, 12] of the classical [27] expression 𝛿𝐴np,𝜕𝜏 =−𝛿𝑊 = −𝑝𝑑𝑉 by taking into account (for Newtonian con-tinuum) the infinitesimal works 𝛿𝐴

𝑐and 𝛿𝐴

𝑠, respectively,

of acoustic and viscous Newtonian forces acting during theinfinitesimal time interval 𝑑𝑡 on the boundary surface 𝜕𝜏 ofthe individual continuum region 𝜏 bounded by the contin-uum boundary surface 𝜕𝜏. Based on the generalized dif-ferential formulation of the first law of thermodynamics, inSection 2.4 we present the analysis [11, 12] of the gravitationalenergy mechanism of the gravitational energy supply intothe continuum region 𝜏 owing to the local time increase ofthe potential𝜓 of the gravitational field inside the continuumregion 𝜏 subjected to the nonstationary Newtonian gravita-tional field.

Following the “Statistical thermohydrodynamics of irre-versible strike-slip-rotational processes” [11] and the “Ther-mohydrogravidynamics of the Solar System” [12], in Section 3we present the fundamentals of the cosmic energy gravita-tional genesis of earthquakes. Using the evolution equationof the total mechanical energy of themacroscopic continuumregion 𝜏 (of the compressible viscousNewtonian continuum),we demonstrate the physical adequacy [11, 12] of the rota-tionalmodel [2] of the earthquake focal region for the seismiczone of the Pacific Ring. We present the thermodynamicfoundation [11, 12] of the classical deformational (shear)model [1] of the earthquake focal region for the quasi-uni-formmedium of the Earth’s crust characterized by practicallyconstant viscosity. We present the generalized thermohydro-gravidynamic shear-rotational model [11, 12] of the earth-quake focal region by taking into account the classical macro-scopic rotational kinetic energy [28, 29], the macroscopicnonequilibrium kinetic energies [24, 31], and the externalcosmic energy gravitational influences [12, 25, 26] on the focalregion of earthquakes.

In Section 4 we present the fundamentals of the cosmicgeophysics [12] applicable for the planets of the Solar System.In Section 4.1 we consider the energy gravitational influenceson the Earth of the inner planets and the outer planets of theSolar System. In Section 4.1.1 we present the relation for theenergy gravitational influences (on the Earth) of the innerand the outer planets in the second approximation of theelliptical orbits of the planets of the Solar System. InSection 4.1.2we present the evaluation of the relativemaximalplanetary instantaneous energy gravitational influences onthe Earth in the approximation of the circular orbits of theplanets of the Solar System. In Section 4.1.3 we present theevaluation of the relative maximal planetary integral energygravitational influences on the Earth in the approximationof the circular orbits of the planets of the Solar System. InSection 4.2 we consider the energy gravitational influenceson the Earth of the Moon. In Section 4.2.1 we present theevaluation of the relativemaximal instantaneous energy grav-itational influence of the Moon on the Earth in the secondapproximation of the elliptical orbits of the Earth and theMoon around the combined mass center 𝐶

3,MOON of theEarth and the Moon. In Section 4.2.2. we present the evalua-tion of the maximal integral energy gravitational influence of


the Moon on the Earth in the approximation of the ellipticalorbits of the Earth and the Moon around the combined masscenter 𝐶

3,MOON of the Earth and the Moon. In Section 4.3we demonstrate the reality of the cosmic energy gravitationalgenesis of preparation and triggering of earthquakes owingto the energy gravitational influence on the Earth of theMoon and the planets of the Solar System. In Section 4.3.1we demonstrate the real cosmic energy gravitational genesisof preparation of earthquakes by considering the energygravitational influence on the Earth of Venus. In Section 4.3.2we demonstrate the real cosmic energy gravitational genesisof triggering of the preparing earthquakes. In Section 4.4 wedemonstrate the cosmic energy gravitational genesis of theseismotectonic activity induced by the nonstationary cosmicenergy gravitational influences on the Earth of the Sun, theMoon, Venus, Jupiter, and Mars. In Section 4.4.1 we presentthe evaluations of the time periodicities of the maximal(instantaneous and integral) energy gravitational influenceson the Earth of the Sun-Moon system, Venus, Jupiter, andMars. In Section 4.4.2 we present the empirical time period-icities of the seismotectonic activity for various regions of theEarth. In Section 4.4.3 we present the set of the time peri-odicities of the periodic global seismotectonic (and vol-canic) activity and the global climate variability of the Earthinduced by the different combinations of the cosmic energygravitational influences on the Earth of the Sun and theMoon, Venus, Jupiter, and Mars. In Section 4.5 we presentthe evidence of the cosmic energy gravitational genesis of thestrong Chinese 2008 earthquakes. In Section 4.6 we presentthe evidence of the cosmic energy gravitational genesis of thestrongest Japanese earthquakes near the Tokyo region. InSection 5wepresent the summary ofmain results and conclu-sion.

2. The Generalized Formulation ofthe First Law of Thermodynamics forMoving Rotating DeformingCompressible Heat-Conducting MacroscopicIndividual Continuum Region 𝜏 Subjectedto the Nonstationary NewtonianGravitational Field

2.1. The Generalized Expression for the Macroscopic KineticEnergy of a Small Continuum Region in NonequilibriumTher-modynamics. De Groot and Mazur defined the macroscopickinetic energy per unit mass 𝜀

𝑘as [28] the sum of the

macroscopic translational kinetic energy per unit mass 𝜀𝑡=

(1/2)k2 of a continuum region (particle) mass center and themacroscopic internal rotational kinetic energy per unit mass𝜀𝑟= (1/2)𝜃𝜔

2:

𝜀𝑘= 𝜀

𝑡+ 𝜀

𝑟=1

2k2

+1

2𝜃𝜔

2

, (2)

where k is the speed of the mass center of a small continuumregion, 𝜔 is an angular velocity of internal rotation [29], and𝜃 is an inertia moment per unit mass of a small continuumregion [28]. Gyarmati’s definition [29] of the macroscopic

𝑋3

𝑋2𝑋1

𝐾

𝐾

𝑥1 𝑥2

𝑥3

𝐶

𝜏

𝜇3

𝜇2

𝜇1

r

𝜈1𝜈2

𝜈3

g

rcVc

P𝛿r

Figure 1: Cartesian coordinate system 𝐾 of a Galilean frame ofreference and the continuum region mass center-affixed Lagrangiancoordinate system𝐾.

kinetic energy per unit mass is analogous to de Grootand Mazur’s one. The classical de Groot and Mazur’s andGyarmati’s definition (2) of the macroscopic kinetic energyper unit mass for a shear flows has some inherent physicalincompleteness associated with the assumption about therigid-like rotation of the continuum region with the angularvelocity vector 𝜔. This definition is based on the assump-tion of local thermodynamic equilibrium since it does notconsider the nonequilibrium shear component of the macro-scopic continuum motion related to the rate of strain tensor𝑒𝑖𝑗. However, the assumption of local thermodynamic equilib-

rium, as noted by de Groot and Mazur [28], may be justifiedonly by reasonable agreement of the experimental resultswiththe theoretical deductions based on this assumption.

Landau and Lifshitz defined [32] the macroscopic inter-nal energy of a small macroscopic continuum region as thedifference between the total kinetic energy of the continuumregion and kinetic energy of the translational macroscopicmotion of the continuum region. According to Landau andLifshitz’s definition [32] of the macroscopic internal energy,the term (1/2)𝜃𝜔2 in the expression (2) is the internal energyof the macroscopic (hydrodynamic) continuum motion. Theclassical definition [28, 29] of the macroscopic internalrotational kinetic energy per unit mass (1/2)𝜃𝜔2 is consistentwith the Landau and Lifshitz’s definition of the macroscopicinternal energy.We shall use further the Landau and Lifshitz’sdefinition [32] of the macroscopic internal energy.

Following the works [11, 12, 24–26], we shall presentthe foundation of the generalized expression for the macro-scopic kinetic energy in nonequilibrium thermodynamics.We shall assume that 𝜏 is a small individual continuum region(domain) bounded by the closed continual boundary surface𝜕𝜏 considered in the three-dimensional Euclidean space withrespect to a Cartesian coordinate system𝐾. We shall considerthe small continuum region 𝜏 in a Galilean frame of referencewith respect to a Cartesian coordinate system 𝐾 centred atthe origin 𝑂 and determined by the axes𝑋

1,𝑋

2, and𝑋

3(see

Figure 1).The unit normal 𝐾-basis coordinate vectors triad 𝜇1, 𝜇2,

and 𝜇3 is taken in the directions of the axes 𝑋1, 𝑋2, and 𝑋3,


respectively. The 𝐾-basis vector triad is taken to be right-handed in the order 𝜇1, 𝜇2, and 𝜇3; see Figure 1. g is the localgravity acceleration.

An arbitrary point 𝑃 in three-dimensional physical spacewill be uniquely defined by the position vector r = 𝑋

𝑖𝜇𝑖≡

(𝑋1, 𝑋

2, and 𝑋

3) originating at the point 𝑂 and terminating

at the point 𝑃. The continuum region-affixed Lagrangiancoordinate system𝐾 (with the axes𝑥

1,𝑥

2, and𝑥

3) is centered

to the mass center 𝐶 of the continuum region 𝜏. The axes𝑥1, 𝑥

2, and 𝑥

3are taken parallel to the axes 𝑋

1, 𝑋

2, and

𝑋3, respectively: the axis 𝑥

𝑖parallel to the axis 𝑋

𝑖, where

𝑖 = 1, 2, 3. The unit normal 𝐾-basis coordinate vector triad^1, ^

2, and ^

3is taken in the directions of the axes 𝑥

1, 𝑥

2,

and 𝑥3, respectively. The 𝐾-basis vector triad is taken to be

right-handed in the order ^1, ^

2, and ^3.Themathematical dif-

ferential of the position vector r, 𝛿r ≡ 𝑥𝑖^𝑖≡ (𝑥

1, 𝑥

2, and 𝑥

3),

expressed in terms of the coordinates 𝑥𝑖(𝑖 = 1, 2, 3) in the

𝐾-coordinate system, originates at the mass centre 𝐶 of the

continuum region 𝜏 and terminates at the arbitrary point 𝑃of the continuum region.

The position vector rc of the mass center 𝐶 of the con-tinuum region 𝜏 in the 𝐾-coordinate system is given by thefollowing expression:

rc =1

𝑚𝜏

∭𝜏

r𝜌 𝑑𝑉, (3)

where

𝑚𝜏= ∭

𝜏

𝜌 𝑑𝑉 (4)

is the mass of the continuum region 𝜏, 𝑑𝑉 is the mathemat-ical differential of physical volume of the continuum region,𝜌 ≡ 𝜌(r, 𝑡) is the local macroscopic density of mass distribu-tion, r is the position vector of the continuumvolume𝑑𝑉, and𝑡 is the time.The speed of themass centre𝐶 of the continuumregion 𝜏 is defined by the following expression:

V𝑐=𝑑rc𝑑𝑡

=∭

𝜏

k𝜌 𝑑𝑉

𝑚𝜏

, (5)

where k = 𝑑r/𝑑𝑡 is the hydrodynamic velocity vector and theoperator 𝑑/𝑑𝑡 = (𝜕/𝜕𝑡) + k ⋅ ∇ denotes the total derivativefollowing the continuum substance [33]. The relevant three-dimensional fields such as the velocity and the localmass den-sity (and also the first and the second derivatives of the rele-vant fields) are assumed to vary continuously throughout theentire continuumbulk of the continuum region 𝜏.The instan-taneous macroscopic kinetic energy of the continuum region𝜏 (bounded by the continuum boundary surface 𝜕𝜏) is thesum of the kinetic energies of small parts constituting thecontinuum region 𝜏when the number of the parts, 𝑛, tends toinfinity and the maximum from their volumes tends to zero[33]:

𝐾𝜏≡ ∭

𝜏

𝜌k2

2𝑑𝑉, (6)

where k is the local hydrodynamic velocity vector, 𝜌 is thelocal mass density, and 𝑑𝑉 is the mathematical differential of

physical volume of the continuum region. We use the com-mon Riemann’s integral here and everywhere.

For the analysis of the relative continuum motion in thephysical space in the vicinity of the position vector rc of themass centre𝐶we have the Taylor series expansion (consistentwith the Helmholtz’s theorem [30, 34]) of the hydrodynamicvelocity vector k(r) for each time moment 𝑡:

k (rc + 𝛿r) = k (rc) + 𝜔 (rc) × 𝛿r

+

3

∑

𝑖,𝑗=1

𝑒𝑖𝑗(rc) 𝛿𝑟𝑗𝜇𝑖

+1

2

3

∑

𝑖,𝑗,𝑘=1

𝜕2V

𝑖

𝜕𝑋𝑗𝜕𝑋

𝑘

𝛿𝑟𝑗𝛿𝑟

𝑘𝜇𝑖+ kres,

(7)

where k(r) ≡ (V1(r), V

2(r), V

3(r)) is the hydrodynamic velocity

vector at the position vector r, 𝛿r ≡ r − rc ≡ (𝛿𝑟1, 𝛿𝑟2, 𝛿𝑟3) ≡(𝑥

1, 𝑥

2, 𝑥

3) is the differential of the position vector r,

𝜔 (r) ≡ 12(∇ × k (r)) = (𝜔

1, 𝜔

2, 𝜔

3) (8)

is the angular velocity of internal rotation (a half of the vortic-ity vector) in the 𝐾-coordinate system at the position vectorr,

𝜔V (r) ≡ (∇ × k (r)) (9)

is the local vorticity in the 𝐾-coordinate system at the posi-tion vector r,

𝑒𝑖𝑗(r) = 1

2(𝜕V

𝑖(r)

𝜕𝑋𝑗

+𝜕V

𝑗(r)

𝜕𝑋𝑖

) (10)

is the rate of strain tensor in the 𝐾-coordinate system at theposition vector r, (𝑖, 𝑗 = 1, 2, 3),

∇ ≡ 𝜇1𝜕

𝜕𝑋1

+ 𝜇2

𝜕

𝜕𝑋2

+ 𝜇3

𝜕

𝜕𝑋3

(11)

is the gradient operator, and

kres =3

∑

𝑖=1

𝑤𝑖𝜇𝑖

(12)

is the small residual part of the Taylor series expansion (7),where 𝑤

𝑖= 𝑂(𝑑

3

𝜏), (𝑖 = 1, 2, 3), 𝑑

𝜏= sup

𝐴,𝐵∈𝜕𝜏

√(r(𝐴, 𝐵))2 isthe diameter of the continuum region 𝜏 and the vector r(𝐴, 𝐵)originates at point 𝐴 and terminates at point 𝐵 of the surface𝜕𝜏. The linear of 𝛿r terms of the Taylor series expansion (7) ispresented in the classical form [33].


Substituting formula (7) into the formula (6) and integrat-ing by parts, then we obtain the following expression [24, 31]:

𝐾𝜏= 𝐾

𝑡+ 𝐾

𝑟+ 𝐾

𝑠+ 𝐾

coup𝑠,𝑟

+ 𝐾res

=1

2𝑚

𝜏V2

𝑐+1

2

3

∑

𝑖,𝑘=1

𝐼𝑖𝑘𝜔𝑖(rc) 𝜔𝑘 (rc)

+1

2

3

∑

𝑖,𝑗,𝑘=1

𝐽𝑗𝑘𝑒𝑖𝑗(rc) 𝑒𝑖𝑘 (rc)

+

3

∑

𝑖,𝑗,𝑘,𝑚=1

𝜀𝑖𝑗𝑘𝐽𝑗𝑚𝜔𝑖(rc) 𝑒𝑘𝑚 (rc) + 𝐾res,

(13)

where𝑚𝜏is the mass of the continuum region 𝜏 and 𝐼

𝑖𝑘is the

ik-component of the classical inertia tensor depending on themass distribution in the continuum region 𝜏 under consider-ation:

𝐼𝑖𝑘= ∭

𝜏,𝐾

(𝛿𝑖𝑘(

3

∑

𝑗=1

𝑥2

𝑗) − 𝑥

𝑖𝑥𝑘)𝜌𝑑𝑉, (14)

where 𝑥𝑖, 𝑥

𝑘are the 𝑖, 𝑘-components of the vector 𝛿r, respec-

tively, in the𝐾-coordinate system, 𝛿𝑖𝑘is the Kronecker delta

tensor, 𝜀𝑖𝑗𝑘

is the third-order permutation symbol, and 𝐽𝑗𝑘is

the 𝑗, 𝑘-component classical centrifugal tensor dependingon the mass distribution in the continuum region 𝜏 underconsideration:

𝐽𝑗𝑘= ∭

𝜏,𝐾

𝑥𝑗𝑥𝑘𝜌 𝑑𝑉, (15)

𝐾res = 𝑂(𝑑7

𝜏) is a small residual part of the macroscopic ki-

netic energy after substituting the Taylor series expansion (7)into formula (6).

Formula (13) states that the macroscopic kinetic energy𝐾

𝜏of the small continuum region 𝜏 is the sum of the macro-

scopic translational kinetic energy 𝐾𝑡of the continuum re-

gion 𝜏

𝐾𝑡=1

2𝑚

𝜏V2c , (16)

the macroscopic internal rotational kinetic energy 𝐾𝑟of the

continuum region 𝜏

𝐾𝑟=1

2

3

∑

𝑖,𝑘=1

𝐼𝑖𝑘𝜔𝑖(rc) 𝜔𝑘 (rc)

≡1

2𝐼𝑖𝑘𝜔𝑖(rc) 𝜔𝑘 (rc) ,

(17)

the macroscopic internal shear kinetic energy 𝐾𝑠of the con-

tinuum region 𝜏

𝐾𝑠=1

2

3

∑

𝑖,𝑗,𝑘=1

𝐽𝑗𝑘𝑒𝑖𝑗(rc) 𝑒𝑖𝑘 (rc)

≡1

2𝐽𝑗𝑘𝑒𝑖𝑗(rc) 𝑒𝑖𝑘 (rc) ,

(18)

and themacroscopic kinetic energy of shear-rotational coupl-ing 𝐾coup

𝑠,𝑟of the continuum region 𝜏

𝐾coup𝑠,𝑟

=

3

∑


𝜀𝑖𝑗𝑘

𝐽𝑗𝑚

𝜔𝑖(rc) 𝑒𝑘𝑚 (rc)

≡ 𝜀𝑖𝑗𝑘𝐽𝑗𝑚𝜔𝑖(rc) 𝑒𝑘𝑚 (rc) .

(19)

The macroscopic internal rotational kinetic energy 𝐾𝑟is

the classical [28, 29] kinetic energy of reversible (equilibrium)rigid-like macroscopic rotational continuum motion. Themacroscopic internal shear kinetic energy 𝐾

𝑠expresses the

kinetic energy of irreversible (nonequilibrium) shear contin-uummotion related to the rate of strain tensor 𝑒

𝑖𝑗.Themacro-

scopic internal kinetic energy of the shear-rotational coupling𝐾

coup𝑠,𝑟

expresses the kinetic energy of the local couplingbetween irreversible deformation and reversible rigid-likerotation.

Thededuced expression (13) for𝐾𝜏confirms the postulate

[35] that the velocity shear (𝑒𝑖𝑗

̸= 0) represents an additionalenergy source taking into account the Evans, Hanley, andHess’s extended formulation [35] of the first law of thermo-dynamics for nonequilibrium deformed states of continuummotion.The energies𝐾

𝑟,𝐾

𝑠,𝐾coup

𝑠,𝑟, and𝐾res are the Galilean

invariants with respect to different inertial 𝐾-coordinatesystems.

We obtained [31] from (13) the following expression forthe macroscopic kinetic energy per unit mass 𝜀

𝑘= 𝐾

𝜏/𝑚

𝜏:

𝜀𝑘= 𝜀

𝑡+ 𝜀

𝑟+ 𝜀

𝑠+ 𝜀

coup𝑠,𝑟

+ 𝜀res

=1

2V2c +

1

2

3

∑

𝑖,𝑘=1

𝜃𝑖𝑘𝜔𝑖𝜔𝑘+1

2

3

∑

𝑖,𝑗,𝑘=1

𝛽𝑗𝑘𝑒𝑖𝑗𝑒𝑖𝑘

+

3

∑


𝜀𝑖𝑗𝑘𝛽𝑗𝑚𝜔𝑖𝑒𝑘𝑚

+ 𝜀res,

(20)

where

𝜃𝑖𝑘=

𝐼𝑖𝑘

𝑚𝜏

=𝐼𝑖𝑘

∭𝜏

𝜌 𝑑𝑉(𝑖, 𝑘 = 1, 2, 3) (21)

is the ik-component of the classical inertia tensor per unitmass of the continuum region 𝜏,

𝛽𝑖𝑘=

𝐽𝑖𝑘

𝑚𝜏

=𝐽𝑖𝑘

∭𝜏

𝜌 𝑑𝑉(𝑖, 𝑘 = 1, 2, 3) (22)

is the ik-component of the classical centrifugal tensor per unitmass of the continuum region 𝜏,

𝜀𝑡=

𝐾𝑡

𝑚𝜏

=1

2V2c (23)

is the macroscopic translational kinetic energy per unit massof the continuum region 𝜏 (moving as a whole at speed Vc ofthe mass center of the continuum region 𝜏),

𝜀𝑟=

𝐾𝑟

𝑚𝜏

=1

2𝜃𝑖𝑘𝜔𝑖𝜔𝑘

(24)


is the macroscopic internal rotational kinetic energy per unitmass of the continuum region 𝜏,

𝜀𝑠=

𝐾𝑠

𝑚𝜏

=1

2𝛽𝑗𝑘𝑒𝑖𝑗𝑒𝑖𝑘

(25)

is themacroscopic internal shear kinetic energy per unitmassof the continuum region 𝜏,

𝜀coup𝑠,𝑟

=𝐾

coup𝑠,𝑟

𝑚𝜏

= 𝜀𝑖𝑗𝑘𝛽𝑗𝑚𝜔𝑖𝑒𝑘𝑚

(26)

is the macroscopic internal kinetic energy of the shear-rota-tional coupling per unit mass (of the continuum region 𝜏),and 𝜀res = 𝑂(𝑑

4

𝜏) is the residual correction.The energies 𝜀

𝑟, 𝜀

𝑠,

𝜀coup𝑠,𝑟

, and 𝜀res are the Galilean invariants with respect to dif-ferent inertial 𝐾-coordinate systems. We have 𝜀

𝑟= 𝑂(𝑑

2

𝜏),

𝜀𝑠= 𝑂(𝑑

2

𝜏), 𝜀coup

𝑠,𝑟= 𝑂(𝑑

2

𝜏), and 𝜀res = 𝑂(𝑑

4

𝜏), when 𝑑

𝜏→ 0,

where 𝑑𝜏is the defined diameter of the continuum region 𝜏.

For a homogeneous continuum region of simple form(sphere or cube) we have

𝐼𝑖𝑘= 𝐼𝛿

𝑖𝑘, 𝐽

𝑗𝑘= 𝐽𝛿

𝑗𝑘. (27)

Formula (17) for the macroscopic internal rotational kineticenergy𝐾

𝑟is reduced to the classical expression [29]

𝐾𝑟=1

2𝐼𝜔

2

, (28)

where 𝜔2 = 𝜔21+ 𝜔

2

2+ 𝜔

2

3. Formula (18) for the macroscopic

internal shear kinetic energy 𝐾𝑠is reduced to the expression

[24, 31]

𝐾𝑠=1

2𝐽𝑒

𝑖𝑗𝑒𝑖𝑗≡1

2𝐽(𝑒

𝑖𝑗)2

, (29)

which is proportional to the local kinetic energy dissipationrate per unit mass 𝜀dis = 2](𝑒𝑖𝑗)

2 in an incompressible viscousNewtonian continuum, where ] is the molecular viscosity.The macroscopic internal kinetic energy of shear-rotationalcoupling 𝐾coup

𝑠,𝑟vanishes for the homogeneous continuum

region 𝜏 of the form of the sphere or cube. The macroscopickinetic energy 𝐾

𝜏for the homogeneous continuum region 𝜏

of the shape of sphere or cube is given by following expression[24, 31]

𝐾𝜏=1

2𝑚

𝜏V2c +

1

2𝐼𝜔

2

+1

2𝐽(𝑒

𝑖𝑗)2

+ 𝐾res. (30)

Hence, the macroscopic kinetic energy per unit mass 𝜀𝑘for

the homogeneous continuum sphere or cube 𝜏 is expressed asthe sum of explicit terms [24, 31]

𝜀𝑘=1

2V2c +

1

2𝜃𝜔

2+1

2𝛽(𝑒

𝑖𝑗)2

+ 𝜀res, (31)

where 𝜀𝑡= (1/2)V2c is the macroscopic translational kinetic

energy per unit mass of the continuum region 𝜏; 𝜃 = 𝐼/𝑚𝜏;

𝛽 = 𝐽/𝑚𝜏; 𝜀

𝑟= (1/2)𝜃𝜔

2 is the classical [28, 29] macroscopicinternal rotational kinetic energy per unit mass of the contin-uum region 𝜏; 𝜀

𝑠= (1/2)𝛽(𝑒

𝑖𝑗)2 is the macroscopic internal

shear kinetic energy per unit mass of the homogeneous con-tinuum sphere or cube 𝜏 [24, 31].

We have the following expression for the macroscopicinternal kinetic energy 𝐾int of the homogeneous continuumregion 𝜏 of the shape of sphere or cube [24, 31]

𝐾int =1

2𝐼𝜔

2

+1

2𝐽(𝑒

𝑖𝑗)2

+ 𝐾res. (32)

Themacroscopic internal kinetic energy per unit mass 𝜀int forthe homogeneous continuum region 𝜏 of the shape of sphereor cube is given by the sum of explicit terms [24, 31]:

𝜀int =1

2𝜃𝜔

2+1

2𝛽(𝑒

𝑖𝑗)2

+ 𝜀res. (33)

Compare formula (31) with the de Groot andMazur’s def-inition (2). Expression (31) is reduced to deGroot andMazur’sdefinition (2) under condition

𝑒𝑖𝑗= 0 (𝑖, 𝑗 = 1, 2, 3) (34)

of local thermodynamic equilibrium. Therefore, we can con-clude that the definition (2) of themacroscopic kinetic energyper unitmass 𝜀

𝑘in classical nonequilibrium thermodynamics

[28, 29] is based on the assumption 𝑒𝑖𝑗= 0 of local thermody-

namic equilibrium [24, 31, 35].The obtained formula (20) for 𝜀

𝑘and its particular form

(31) (obtained for homogeneous continuum regions of spher-ical and cubical shapes) generalized [24, 31] the classical deGroot and Mazur expression (2) in classical nonequilibriumthermodynamics [28, 29] by taking into account the irre-versible dissipative shear component of the macroscopiccontinuummotion related to the rate of strain tensor 𝑒

𝑖𝑗. The

expression (20) for 𝜀𝑘contains the new macroscopic internal

shear kinetic energy per unit mass 𝜀𝑠, which expresses the

kinetic energy of irreversible dissipative shear motion, andalso the newmacroscopic internal kinetic energy of the shear-rotational coupling per unit mass 𝜀coup

𝑠,𝑟, which expresses the

kinetic energy of local coupling between irreversible dissi-pative shear and reversible rigid-like rotational macroscopiccontinuum motions.

The macroscopic internal shear kinetic energy per unitmass (for homogeneous continuum regions of spherical andcubical shapes)

𝜀𝑠=1

2𝛽(𝑒

𝑖𝑗)2 (35)

is proportional to the kinetic energy viscous dissipation rateper unit mass

𝜀dis,𝑠 = 2](𝑒𝑖𝑗)2 (36)

in an incompressible viscous Newtonian continuum charac-terized by the kinematic viscosity ]. We have shown [24] thatthe proportionality

𝜀𝑠∼ 𝜀dis,𝑠 = 2](𝑒𝑖𝑗)

2 (37)

is the basis of the established association [36, 37] betweena structure and an order (and, hence, the associated macro-scopic kinetic energy), on the one hand, and irreversible dissi-pation, on the other hand, for the dissipative structures of tur-bulence in viscous Newtonian fluids.


2.2. The Generalized Differential Formulation of the FirstLaw of Thermodynamics (in the Galilean Frame of Reference)for Nonequilibrium Shear-Rotational States of the DeformedOne-Component Individual Finite Continuum Region (Char-acterized by the Symmetric Stress Tensor T) Moving in theNonstationary Newtonian Gravitational Field. Following theworks [11, 12, 25, 26], we shall present the foundation ofthe generalized differential formulation of the first law ofthermodynamics (in the Galilean frame of reference) fornonequilibrium shear-rotational states of the deformed finiteone-component individual continuum region (characterizedby the symmetric stress tensor T) moving in the nonsta-tionary Newtonian gravitational field. We shall considerthe deformed finite one-component individual continuumregion in nonequilibrium shear-rotational states character-ized by the following condition:

𝑒𝑖𝑗

̸= 0 (𝑖, 𝑗 = 1, 2, 3) . (38)

Considering the graphical methods in the thermodynamicsof fluids, Gibbs [27] formulated the first law of thermody-namics for the fluid body (fluid region) as follows (in Gibbs’designations):

𝑑𝜀 = 𝑑𝐻 − 𝑑𝑊, (39)

where 𝑑𝜀 is the differential of the internal thermal energy ofthe fluid body, 𝑑𝐻 is the differential change of heat across theboundary of the fluid body related to the thermal molecularconductivity (associated with the corresponding external orinternal heat fluxes), and 𝑑𝑊 = 𝑝𝑑𝑉 is the differential workproduced by the considered fluid body on its surroundings(surrounding fluid) under the differential change 𝑑𝑉 of thefluid region (of volume 𝑉) characterized by the thermody-namic pressure 𝑝.

The formulation [32] of the first law of thermodynamicsfor the general thermodynamic system (material region) isgiven by the equivalent form (in Landau’s and Lifshitz’s desig-nations [32])

𝑑𝐸 = 𝑑𝑄 − 𝑝𝑑𝑉, (40)

where 𝑑𝐴 = −𝑝𝑑𝑉 is the differential work produced by thesurroundings (surroundings of the thermodynamic system)on the thermodynamic system under the differential change𝑑𝑉 of volume 𝑉 of the thermodynamic system characterizedby the thermodynamic pressure 𝑝; 𝑑𝑄 is the differential heattransfer (across the boundary of the thermodynamic system)related to the thermal interaction of the thermodynamic sys-tem and the surroundings (surrounding environment); 𝐸 isthe energy of the thermodynamic system, which should con-tain (as supposed [32]) the kinetic energy of the macroscopiccontinuum motion.

We shall use the differential formulation of the first law ofthermodynamics [28] for the specific volume 𝜗 = 1/𝜌 of thecompressible viscous one-component deformed continuumwith no chemical reactions:

𝑑𝑢

𝑑𝑡=𝑑𝑞

𝑑𝑡− 𝑝

𝑑𝜗

𝑑𝑡− 𝜗Π : Grad k, (41)

where 𝑢 is the specific (per unit mass) internal thermalenergy, 𝑝 is the thermodynamic pressure, Π is the viscousstress tensor, k is the hydrodynamic velocity of the continuummacrodifferential element [28], and 𝑑𝑞 is the differentialchange of heat across the boundary of the continuum region(of unit mass) related to the thermal molecular conductivitydescribed by the heat equation [28]:

𝜌𝑑𝑞

𝑑𝑡= − div J

𝑞, (42)

where J𝑞is the heat flux [28]. The viscous stress tensor Π is

taken from the decomposition of the pressure tensor P [28]:

Ρ = 𝑝𝛿 +Π, (43)

where 𝛿 is the Kronecker delta tensor.Considering the Newtonian viscous stress tensor PV ≡ Π

of the compressible viscous Newtonian continuum with thecomponents [29]

Π𝑖𝑗= {(

2

3]𝜌 − 𝜂V) div k} 𝛿𝑖𝑗 − 2]𝜌𝑒𝑖𝑗, (44)

the differential formulation (41) of the first law of thermody-namics (for the continuum region (of unit mass) of the com-pressible viscous Newtonian one-component deformed con-tinuum with no chemical reactions) can be rewritten as fol-lows:

𝑑𝑢

𝑑𝑡=𝑑𝑞

𝑑𝑡− 𝑝

𝑑𝜗

𝑑𝑡+ (]

2−2

3]) (div k)2 + 2](𝑒

𝑖𝑗)2

, (45)

where ] = 𝜂/𝜌 is the coefficient of the molecular kinematic(first) viscosity and ]

2= 𝜂V/𝜌 is the coefficient of the molec-

ular volume (second) viscosity [38]. The first and the secondterms in the right-hand side of relation (45) are analogous tothe corresponding respective first and second terms in theright-hand side of the classical formulations (39) and (40).The third term in the right-hand side of relation (45)

𝑑𝑞𝑖,𝑐= (

𝜂]

𝜌−2

3]) (div k)2𝑑𝑡 (46)

is related to the “internal” heat induced during the time inter-val 𝑑𝑡 by viscous-compressible irreversibility [24].The fourthterm in the right-hand side of relation (45)

𝑑𝑞𝑖,𝑠= 2](𝑒

𝑖𝑗)2

𝑑𝑡 (47)

is related to the “internal” heat induced during the time inter-val 𝑑𝑡 by viscous-shear irreversibility [24]. The differentialformulation (45) of the first law of thermodynamics (for thecontinuum element of the compressible viscous Newtonianone-component deformed continuumwith no chemical reac-tions) takes into account (in addition to the classical terms)the viscous-compressible irreversibility and viscous-shearirreversibility inside the continuum element of the compress-ible viscous Newtonian one-component deformed contin-uum with no chemical reactions.


Using the differential formulation (41) of the first lawof thermodynamics [28] for the total derivative 𝑑𝑢/𝑑𝑡 (fol-lowing the liquid substance) of the specific (per unit mass)internal thermal energy 𝑢 of a compressible viscous one-component deformed continuumwith no chemical reactions,the heat equation (42) [28], the general equation of contin-uum movement [29]

𝑑k

𝑑𝑡=

1

𝜌divT + g (48)

for the deformed continuum characterized by the symmetricstress tensorT of general form (in particular, with the compo-nents [29]

𝑇𝑖𝑗= −{𝑝 + (

2

3]𝜌 − 𝜂V) div k} 𝛿𝑖𝑗 + 2]𝜌𝑒𝑖𝑗 (49)

for the compressible viscousNewtonian one-component con-tinuum) and taking into account the time variations of thepotential𝜓of the nonstationary gravitational field (character-ized by the local gravity acceleration vector g = −∇𝜓) insidean arbitrary finite macroscopic individual continuum region𝜏, we derived [11, 12] the generalized differential formulation(for theGalilean frame of reference) of the first law of thermo-dynamics (for moving rotating deforming compressible heat-conducting stratified macroscopic continuum region 𝜏 sub-jected to the nonstationary Newtonian gravitational field):

𝑑 (𝐾𝜏+ 𝑈

𝜏+ 𝜋

𝜏)

= 𝑑𝑡∬𝜕𝜏

(k ⋅ (n ⋅ T)) 𝑑Ωn

− 𝑑𝑡∬𝜕𝜏

(J𝑞⋅ n) 𝑑Ωn + 𝑑𝑡∭

𝜏

𝜕𝜓

𝜕𝑡𝜌 𝑑𝑉,

(50)

where

𝛿𝐴np,𝜕𝜏 = 𝑑𝑡∬𝜕𝜏

(k ⋅ (n ⋅ T)) 𝑑Ωn (51)

is the differential work done during the infinitesimal timeinterval 𝑑𝑡 by nonpotential stress forces (pressure, compress-ible, and viscous forces for Newtonian continuum) acting onthe boundary surface 𝜕𝜏of the continuumregion 𝜏;𝑑Ωn is thedifferential element (of the boundary surface 𝜕𝜏 of the contin-uum region 𝜏) characterized by the external normal unit vec-tor n; t = n ⋅T is the stress vector [29],T = −P [29], whereP isthe pressure tensor characterized (in particular, for themodelof the compressible viscous Newtonian continuum charac-terized by the coefficients of kinematic viscosity ] and thevolume viscosity 𝜂V) by components (obtained from(49))

𝑃𝑖𝑗= {𝑝 + (

2

3]𝜌 − 𝜂V) div k} 𝛿𝑖𝑗 − 2]𝜌𝑒𝑖𝑗;

𝛿𝑄 = −𝑑𝑡∬𝜕𝜏

(J𝑞⋅ n) 𝑑Ωn

(52)

is the differential change of heat of themacroscopic individualcontinuum region 𝜏 related to the thermalmolecular conduc-tivity of heat across the boundary 𝜕𝜏 of the continuum region

𝜏, where J𝑞is the heat flux [28] (across the element 𝑑Ωn of the

continuum boundary surface 𝜕𝜏);

𝜋𝜏≡ ∭

𝜏

𝜓𝜌𝑑𝑉 (53)

is the macroscopic potential energy (of the macroscopicindividual continuum region 𝜏) related to the nonstationarypotential 𝜓 of the gravitational field;

𝑈𝜏≡ ∭

𝜏

𝑢𝜌 𝑑𝑉 (54)

is the classical microscopic internal thermal energy of themacroscopic individual continuum region 𝜏;

𝐾𝜏= ∭

𝜏

𝜌k2

2𝑑𝑉 (55)

is the instantaneous macroscopic kinetic energy of the mac-roscopic individual continuum region 𝜏. The instantaneousmacroscopic kinetic energy 𝐾

𝜏is given by the relation (13)

[24, 31] for the small macroscopic individual continuumregion 𝜏.

The generalized differential formulation (50) of the firstlaw of thermodynamics can be rewritten as follows [11, 12]:

𝑑𝑈𝜏+ 𝑑𝐾

𝜏+ 𝑑𝜋

𝜏= 𝛿𝑄 + 𝛿𝐴np,𝜕𝜏 + 𝑑𝐺 (56)

extending the classical [12] formulations (39) and (40)

𝑑𝑈 = 𝛿𝑄 − 𝑝𝑑𝑉, (𝑑𝜀 ≡ 𝑑𝑈, −𝛿𝑊 = −𝑝𝑑𝑉) (57)

by taking into account (along with the classical infinitesimalchange of heat 𝛿𝑄 and the classical infinitesimal change of theinternal energy 𝑑𝑈

𝜏≡ 𝑑𝑈) the infinitesimal increment of the

macroscopic kinetic energy 𝑑𝐾𝜏, the infinitesimal increment

of the gravitational potential energy 𝑑𝜋𝜏, the generalized

infinitesimalwork𝛿𝐴np,𝜕𝜏 done on the continuumregion 𝜏bythe surroundings of 𝜏, and the infinitesimal amount 𝑑𝐺 ofenergy [11, 12]

𝑑𝐺 = 𝑑𝑡∭𝜏

𝜕𝜓

𝜕𝑡𝜌 𝑑𝑉 (58)

added (or lost) as the result of the Newtonian nonstationarygravitational energy influence on the continuum region 𝜏during the infinitesimal time interval 𝑑𝑡.

The generalized differential formulation (50) of the firstlaw of thermodynamics can be rewritten as follows [11, 12]:

𝑑𝐸𝜏

𝑑𝑡=

𝑑

𝑑𝑡(𝐾

𝜏+ 𝑈

𝜏+ 𝜋

𝜏)

= ∭𝜏

(1

2k2

+ 𝑢 + 𝜓)𝜌𝑑𝑉

= ∬𝜕𝜏

(k ⋅ (n ⋅ T)) 𝑑Ωn

−∬𝜕𝜏

(J𝑞⋅ n) 𝑑Ωn +∭

𝜏

𝜕𝜓

𝜕𝑡𝜌 𝑑𝑉.

(59)


The equivalent generalized differential formulations (50),(56), and (59) of the first law of thermodynamics take intoaccount the following factors:

(1) the classical heat thermal molecular conductivity(across the boundary 𝜕𝜏 of the macroscopic contin-uum region 𝜏) related to the classical infinitesimalchange of heat 𝛿𝑄:

𝛿𝑄 = −𝑑𝑡∬𝜕𝜏

(J𝑞⋅ n) 𝑑Ωn, (60)

(2) the classical infinitesimal change of the internalenergy 𝑑𝑈

𝜏of the macroscopic continuum region 𝜏:

𝑑𝑈𝜏≡ 𝑑∭

𝜏

𝑢𝜌 𝑑𝑉, (61)

(3) the established [11, 12] infinitesimal increment of themacroscopic kinetic energy 𝑑𝐾

𝜏of the macroscopic

continuum region 𝜏:

𝑑𝐾𝜏= 𝑑∭

𝜏

𝜌k2

2𝑑𝑉, (62)

(4) the established [11, 12] infinitesimal increment of thegravitational potential energy 𝑑𝜋

𝜏of themacroscopic

continuum region 𝜏:

𝑑𝜋𝜏= 𝑑∭

𝜏

𝜓𝜌𝑑𝑉, (63)

(5) the established [11, 12] generalized infinitesimal work𝛿𝐴np,𝜕𝜏 done on themacroscopic continuum region 𝜏by the surroundings of 𝜏:

𝛿𝐴np,𝜕𝜏 = 𝑑𝑡∬𝜕𝜏

(k ⋅ (n ⋅ T)) 𝑑Ωn, (64)

(6) the established [11, 12] infinitesimal amount 𝑑𝐺 ofenergy added (or lost) as the result of the Newtoniannonstationary gravitational energy influence on themacroscopic continuum region 𝜏 during the infinites-imal time interval 𝑑𝑡:

𝑑𝐺 = 𝑑𝑡∭𝜏

𝜕𝜓

𝜕𝑡𝜌 𝑑𝑉. (65)

The generalized differential formulations (50), (56), and(59) of the first law of thermodynamics (given for theGalileanframe of reference) are valid for nonequilibrium shear-rotational states of the deformed finite individual continuumregion (characterized by the symmetric stress tensor T in thegeneral equation (48) of continuummovement [29]) movingin the nonstationary gravitational field. The generalized dif-ferential formulations (50) and (56) of the first law of ther-modynamics [11, 12] are the generalizations of the classicalformulations (39) and (40) of the first law of thermodynamicstaking into account: (1) the generalized expression (51) for thedifferential work 𝛿𝐴np,𝜕𝜏 done during the infinitesimal timeinterval 𝑑𝑡 by nonpotential stress forces acting on the bound-ary surface 𝜕𝜏 of the individual continuum region 𝜏 and (2)

the time variations of the potential 𝜓 of the nonstationarygravitational field inside the individual continuum region 𝜏due to the deformation of the individual continuum region 𝜏and due to the external (terrestrial and cosmic) gravitationalinfluence on the individual continuum region 𝜏moving in thetotal (internal + external) nonstationary gravitational field.

2.3. The Generalized Differential Formulation of the First Lawof Thermodynamics (in the Galilean Frame of Reference) forNonequilibriumShear-Rotational States of theDeformed FiniteIndividual Region of the Compressible Viscous NewtonianOne-Component Continuum Moving in the NonstationaryGravitational Field. There is evidence [39] that the rocks ofthe Earth’s crust at protracted loadings may be considered asfluids characterized by the very high viscosity. According tothe classical viewpoint [39], the local mechanism of creationof the earthquakes is related to the release of the accumulatedpotential energy of the elastic deformation during the suddenlocal break (i.e., the discontinuous shear) of the Earth’s crust(or the sudden increase of fluidity in the local region ofthe Earth’s crust) accompanied by viscous relaxation andgeneration of seismic waves. It was conjectured [40] that“more punctual and refined methods of the mathematicalanalysis are obligatory” for “the practical assessment of theseismic hazard”.

Taking into account the established [31] conception of themacroscopic internal shear kinetic energy (per unit mass) 𝜀

𝑠

related to the rate of medium deformation (i.e., with the rateof strain tensor 𝑒

𝑖𝑗= 𝑑𝜀

𝑖𝑗/𝑑𝑡, where 𝜀

𝑖𝑗is the deformation

tensor [30]), we have elucidated [41] from the viewpoint ofnonequilibrium thermodynamics the mechanism of genera-tion of seismic waves from the separate deformed finite zoneof the Earth’s crust. The proportionality (37) takes place alsofor deformed compressible finite region of the Earth’s crust forsudden rise of fluidity (in a local region of the Earth’s crust)related to the local sudden medium deformation in the sepa-rate seismic zones of the seismic activity. Taking into accountthe established [31] proportionality (37), we have assumed[31] that the accumulated potential energy of the elastic defor-mation (related to the deformation tensor 𝜀

𝑖𝑗) converts to the

macroscopic internal shear kinetic energy 𝐾𝑠(related to the

rate of strain tensor 𝑒𝑖𝑗) in the seismic zone simultaneously

with the damping of 𝐾𝑠by viscous dissipation and radiation

of seismic waves during several oscillations. In Section 3 weshall evaluate this mechanism on the basis of the generalizeddifferential formulation (50) of the first law of thermodynam-ics for nonequilibrium shear-rotational states of the deformedfinite individual continuum region (characterized by thesymmetric stress tensorT)moving in the nonstationary grav-itational field.

Following the works [11, 12, 25, 26], we shall presentthe foundation of the generalized differential formulation ofthe first law of thermodynamics for nonequilibrium shear-rotational states of the deformed finite individual region ofthe compressible viscous Newtonian one-component con-tinuum moving in the nonstationary gravitational field. Thegeneralized differential formulation (50) of the first lawof thermodynamics (formulated for the Galilean frame of


reference) is valid for arbitrary symmetric stress tensor T, inparticular for nonequilibrium shear-rotational states of thedeformed finite individual region of the compressible viscousNewtonian one-component continuum moving in the non-stationary gravitational field. The coefficient of molecularkinematic (first, shear) viscosity ] = 𝜂/𝜌 and the coefficient ofmolecular volume (second) viscosity ]

2= 𝜂V/𝜌 are assumed

to vary for each time moment 𝑡 as an arbitrary continuousfunctions of the Cartesian space (three-dimensional) coordi-nates.

The differential work 𝛿𝐴np,𝜕𝜏 for the Newtonian symmet-ric stress tensor T (characterized by the components (49)) ispresented by three explicit terms [11, 12]:

𝛿𝐴np,𝜕𝜏 = 𝛿𝐴𝑝 + 𝛿𝐴𝑐 + 𝛿𝐴 𝑠

= −𝑑𝑡∬𝜕𝜏

𝑝 (k ⋅ n) 𝑑Ωn

− 𝑑𝑡∬𝜕𝜏

(2

3𝜂 − 𝜂V) div k (k ⋅ n) 𝑑Ωn

+ 𝑑𝑡∬𝜕𝜏

2𝜂 V𝛽𝑛𝛼𝑒𝛼𝛽

𝑑Ωn,

(66)

where

𝛿𝐴𝑝= −𝑑𝑡∬

𝜕𝜏

𝑝 (k ⋅ n) 𝑑Ωn (67)

is the differential work of the hydrodynamic pressure forcesacting on the boundary surface 𝜕𝜏 of the individual contin-uum region 𝜏 (bounded by the continuum boundary surface𝜕𝜏) during the infinitesimal time interval 𝑑𝑡;

𝛿𝐴𝑐= −𝑑𝑡∬

𝜕𝜏

(2

3𝜂 − 𝜂V) div k (k ⋅ n) 𝑑Ωn (68)

is the differential work (related to the combined effects of theacoustic compressibility, molecular kinematic viscosity, andmolecular volume viscosity) of the acoustic (compressible)pressure forces acting on the boundary surface 𝜕𝜏 of theindividual continuum region 𝜏 during the infinitesimal timeinterval 𝑑𝑡;

𝛿𝐴𝑠= 𝑑𝑡∬

𝜕𝜏

2𝜂 V𝛽𝑛𝛼𝑒𝛼𝛽

𝑑Ωn (69)

is the differential work of the viscous Newtonian forces(related to the combined effect of the velocity shear, that is, thedeformation of the continuum region 𝜏, and the molecularkinematic viscosity) acting on the boundary surface 𝜕𝜏 of theindividual continuum region 𝜏 during the infinitesimal timeinterval 𝑑𝑡.

Along with (45) the differential formulation of the firstlaw of thermodynamics [28] for the total derivative 𝑑𝑢/𝑑𝑡(following the continuum substance) of the internal thermalenergy per unit mass 𝑢 of the one-component deformed con-tinuum with no chemical reactions, the thermohydrody-namic theory [28] contains additionally the equations of themass and momentum balances:

𝜕𝜌

𝜕𝑡= − div 𝜌k, (70)

𝜌𝑑k

𝑑𝑡= −Grad𝑝 + 𝜂Δk + (1

3𝜂 + 𝜂])Grad div k. (71)

The generalized differential formulation (50) of the firstlaw of thermodynamics (together with the generalized differ-ential work 𝛿𝐴np,𝜕𝜏 given by the expression (66)) is valid fornonequilibrium shear-rotational states of the deformed finiteindividual region of the compressible viscous Newtonianone-component continuum moving in the nonstationarygravity field. The coefficient of molecular kinematic (first,shear) viscosity ] = 𝜂/𝜌 and the coefficient of molecular vol-ume (second) viscosity ]

2= 𝜂V/𝜌 are assumed to vary for each

timemoment 𝑡 as an arbitrary continuous functions of Carte-sian space (three-dimensional) coordinates.

The generalized differential formulation (50) takes intoaccount the dependences of the hydrodynamic pressure onthe hydrodynamic vorticity𝜔V and on the rate of strain tensor𝑒𝑖𝑗by means of the component 𝛿𝐴

𝑝(in the expression (66)

for 𝛿𝐴np,𝜕𝜏) given by the expression (67). The presence of thethird term 𝛿𝐴

𝑠(given by the expression (69) and related to

the combined effect of the molecular kinematic viscosity andthe deformation of the continuum region 𝜏 defined by therate of strain tensor 𝑒

𝛼𝛽) in the expression (66) for 𝛿𝐴np,𝜕𝜏

generalizes essentially the classical formulations (39) and (40)of the first law of thermodynamics by taking into account thedifferential work of the viscous Newtonian forces acting onthe boundary continuum surface 𝜕𝜏 of the individual contin-uum region 𝜏.

The general equation (48) of continuum movement [29]for the compressible viscousNewtonian one-component con-tinuum is reduced to the following equation:

𝜌𝑑k

𝑑𝑡= −Grad𝑝 + 𝜂Δk

+ (1

3𝜂 + 𝜂])Grad div k

+ (Grad 𝜂) ⋅ e

− div k Grad(23𝜂 − 𝜂]) + g,

(72)

where (Grad 𝜂) ⋅ e is the internal multiplication of the vector(Grad 𝜂) and the rate of strain tensor e in accordance withthe corresponding definition [29]. Equation (72) generalizesthe Navier-Stokes equation (71) (given for g = 0) by takinginto account the dependences of the coefficient of molecularkinematic viscosity ] = 𝜂/𝜌 and the coefficient of molecularvolume viscosity ]

2= 𝜂V/𝜌 on the space (three-dimensional)

Cartesian coordinates.The relevant example for illustration of the significance

of the term 𝛿𝐴𝑠(in the expression (66) for the differential

work 𝛿𝐴np,𝜕𝜏) is related to the thermodynamic consideration[12] of the processes of the energy exchange [42] betweenthe oceans and the lithosphere of the Earth. According tothe expression (69) for the term 𝛿𝐴

𝑠, the energy exchange

between the oceans (and the atmosphere) and the lithosphereof the Earth is possible only under the presence of the me-dium acoustic compressibility (i.e., div k ̸= 0) and themediumdeformations (i.e., 𝑒

𝛼𝛽̸= 0) in the boundary regions of fluid

(in the oceans), air (in the atmosphere), and the compress-ible deformed lithosphere of the Earth. According to thegeneralized expression (66) for the differential work 𝛿𝐴np,𝜕𝜏,


the energy exchange between the oceans (and the atmo-sphere) and the lithosphere of the Earth is impossible fornondeformed (𝑒

𝛼𝛽= 0) and noncompressible (div k = 0) lith-

osphere.We have the evolution equation for the total mechanical

energy (𝐾𝜏+ 𝜋

𝜏) of the deformed finite individual macro-

scopic continuum region 𝜏 [11, 12]:

𝑑

𝑑𝑡(𝐾

𝜏+ 𝜋

𝜏) =

𝑑

𝑑𝑡∭

𝜏

(1

2k2

+ 𝜓)𝜌𝑑𝑉

= ∭𝜏

𝑝 div k 𝑑𝑉

+∭𝜏

(2

3𝜂 − 𝜂V) (div k)

2

𝑑𝑉

−∭𝜏

2](𝑒𝑖𝑗)2

𝜌 𝑑𝑉

+∬𝜕𝜏

(k ⋅ (n ⋅ T)) 𝑑Ωn

+∭𝜏

𝜕𝜓

𝜕𝑡𝜌 𝑑𝑉

(73)

obtained from the generalized differential formulation (50)for the compressible viscous Newtonian one-componentcontinuummoving in the nonstationary gravitational field. InSection 3 we shall use the evolution equation (73) of the totalmechanical energy to found the rotational, shear, and theshear-rotational models of the earthquake focal region.

2.4. Cosmic and Terrestrial Energy Gravitational Genesis of theSeismotectonic (and Volcanic) Activity of the Earth Induced bythe Combined Cosmic and Terrestrial Nonstationary EnergyGravitational Influences on an Arbitrary Individual Contin-uumRegion 𝜏 (of the Earth) and by theNonpotential TerrestrialStress Forces Acting on the Boundary Surface 𝜕𝜏 of the Indi-vidual Continuum Region 𝜏. Following the works [11, 12], weshall present the physical mechanisms of the energy fluxes tothe continuum region 𝜏 related to preparation of earthquakes.The equivalent generalized differential formulations (50)and (59) of the first law of thermodynamics show that thenonstationary gravitational field (related to the nonstationarygravitational potential 𝜓) gives the following gravitationalenergy power:

𝑊gr (𝜏) = ∭𝜏

𝜕𝜓

𝜕𝑡𝜌 𝑑𝑉 =

𝑑𝐺

𝑑𝑡(74)

associated with the gravitational energy power of the totalcombined (external cosmic and terrestrial and internalrelated to the macroscopic continuum region 𝜏) gravitationalfield. If themacroscopic continuum region 𝜏 is not very large,consequently, it cannot induce the significant time variationsto the potential 𝜓 of the gravity field inside the continuumregion 𝜏. According to the equivalent generalized differentialformulations (50) and (59) of the first law of thermodynamicsand to the evolution equation (73) for the total mechanical

energy (𝐾𝜏+ 𝜋

𝜏) of the deformed finite individual macro-

scopic continuum region 𝜏, the energy power of the grav-itational field may produce the fractures in the continuumregion 𝜏. We shall consider this aspect in Section 3.

The generalized differential formulation (59) of the firstlaw of thermodynamics and the expression (74) for thegravitational energy power 𝑊gr(𝜏) show that the local timeincrease of the potential 𝜓 of the gravitational field is thegravitational energy mechanism of the gravitational energysupply into the continuum region 𝜏. Really, the local timeincrease of the potential 𝜓 of the gravitational field inside thecontinuum region 𝜏 (𝜕𝜓/𝜕𝑡 > 0) supplies the gravitationalenergy into the continuum region 𝜏. Consequently, accordingto the generalized differential formulation (59) and to theevolution equation (73), the total energy (𝐾

𝜏+𝑈

𝜏+𝜋

𝜏) of the

continuum region 𝜏 and the totalmechanical energy (𝐾𝜏+𝜋

𝜏)

of the continuum region 𝜏 are increased if 𝜕𝜓/𝜕𝑡 > 0.According to the generalized differential formulation (59)

of the first law of thermodynamics and to the evolutionequation (73), the gravitational energy supply into the con-tinuum region 𝜏may induce the formation of fractures in thecontinuum region 𝜏 related to the production of earthquake.This conclusion corresponds to the observations [1, 3–5, 9, 10,13] of the identified anomalous variations of the gravitationalfield before strong earthquakes.

According to the generalized differential formulation (59)of the first law of thermodynamics and to the evolution equa-tion (73), the supply of energy into the continuum region 𝜏may occur also by means of the work

𝐴np,𝜕𝜏 = ∫𝑡

𝑡0

𝑑𝑡∬𝜕𝜏

(k ⋅ (n ⋅ T)) 𝑑Ωn (75)

done by nonpotential stress forces (pressure, compressible,and viscous forces for Newtonian continuum) acting on theboundary surface 𝜕𝜏 of the continuum region 𝜏 during thetime interval (𝑡 − 𝑡

0).

The considered mechanisms of the energy supply to theEarth’s macroscopic continuum region 𝜏 should result in theirreversible process of the splits formation in the rocks relatedto the generation of the high-frequency acoustic waves fromthe focal continuum region 𝜏 before the earthquake. Takingthis into account, the sum 𝛿𝐴

𝑐+𝛿𝐴

𝑠in the expression (45) is

interpreted [11, 12] as the energy flux (related to the compress-ible and viscous forces acting on the boundary surface 𝜕𝜏 ofthe continuum region 𝜏) [38]

𝛿𝐹vis,𝑐 = 𝛿𝐴𝑐 + 𝛿𝐴 𝑠 (76)

directed across the boundary 𝜕𝜏 (see Figures 2 and 3) of thecontinuum region 𝜏.

The considered mechanisms of the energy supply to theEarth’s macroscopic continuum region 𝜏 should result in thesignificant increase of the energy flux 𝛿𝐹vis,𝑐 of the geo-acous-tic energy from the focal region 𝜏 before the earthquake. Thededuced conclusion is in a good agreement with the results ofthe detailed experimental studies [43].


𝜏1

𝜏2

𝐹1(𝜏1)

(𝜕𝜏)1

𝑅1(𝜏)

(𝜕𝜏)2𝑅1(𝜏)

𝜕𝜏 = (𝜕𝜏)1 ∪ (𝜕𝜏)2

Figure 2: The macroscopic continuum region 𝜏 containing twosubsystems 𝜏

1and 𝜏

2interacting on the surface 𝐹

1(𝜏) of the

tangential jump of the continuum velocity.

𝜏int

𝜏ext

𝜕𝜏

𝜕𝜏𝑖

Figure 3: The macroscopic continuum region 𝜏 consisting ofthe subsystems 𝜏int and 𝜏ext interacting on the surface 𝜕𝜏𝑖 of thecontinuum velocity jump.

3. Generalized ThermohydrogravidynamicShear-Rotational and Classical Shearand Rotational Models of the EarthquakeFocal Region

3.1. The Generalized Thermohydrogravidynamic Shear-Rota-tional and the Classical Shear (Deformational) Models ofthe Earthquake Focal Region Based on the Generalized Dif-ferential Formulation of the First Law of Thermodynamics.Following the works [11, 12], we shall present the foundationof the generalized thermohydrogravidynamic shear-rota-tional model of the earthquake focal region based on thegeneralized differential formulation (59) of the first law ofthermodynamics. Using the evolution equation (73) of thetotal mechanical energy of the subsystem 𝜏 (the macroscopiccontinuum region 𝜏) of the Earth, we shall show now that theformation of fractures (modeling by the jumps of the contin-uum velocity on some surfaces) is related to irreversible dissi-pation of themacroscopic kinetic energy and the correspond-ing increase of entropy. We consider at the beginning theanalysis of formation of the main line flat fracture (associated

with the surface 𝐹1(𝜏) of the continuum velocity jump) inside

the macroscopic continuum region 𝜏 (bounded by the closedsurface 𝜕𝜏). The macroscopic continuum region 𝜏 may bedivided into two subsystems 𝜏

1and 𝜏

2by continuingmentally

the surface 𝐹1(𝜏) by means of surface 𝑅

1(𝜏) crossing the

surface 𝜕𝜏 of themacroscopic region 𝜏.The surface of the sub-system 𝜏

1consists of the surface (𝜕𝜏)

1(which is the part of the

surface 𝜕𝜏) and the surfaces 𝐹1(𝜏) and 𝑅

1(𝜏). The surface of

the subsystem 𝜏2consists of the surface (𝜕𝜏)

2(which is the

part of the surface 𝜕𝜏) and the surfaces 𝐹1(𝜏) and 𝑅

1(𝜏).

Using the formulation (73), we have the evolution equa-tions for the total mechanical energy of the macroscopicsubsystems 𝜏

1and 𝜏

2:

𝑑

𝑑𝑡(𝐾

𝜏1

+ 𝜋𝜏1

)

=𝑑

𝑑𝑡∭

𝜏1

(1

2k2

+ 𝜓)𝜌𝑑𝑉

= ∭𝜏1

𝑝 div k 𝑑𝑉

+∭𝜏1

(2


2

𝑑𝑉

−∭𝜏1

2](𝑒𝑖𝑗)2

𝜌 𝑑𝑉

+∬(𝜕𝜏)1

(k ⋅ (n ⋅ T)) 𝑑Ωn

+∬𝐹1(𝜏)

(k1(𝜏

1) ⋅ (𝜁

1⋅ T)) 𝑑Σ𝜁

1

+∬𝑅1(𝜏)

(k1(𝜏

1) ⋅ (𝜁

1⋅ T)) 𝑑Σ𝜁

1

+∭𝜏1

𝜕𝜓


𝑑

𝑑𝑡(𝐾

𝜏2

+ 𝜋𝜏2

)

=𝑑

𝑑𝑡∭

𝜏2

(1

2k2

+ 𝜓)𝜌𝑑𝑉

= ∭𝜏2

𝑝 div k 𝑑𝑉

+∭𝜏2

(2


2

𝑑𝑉

−∭𝜏2

2](𝑒𝑖𝑗)2

𝜌 𝑑𝑉

+∬(𝜕𝜏)2

(k ⋅ (n ⋅ T)) 𝑑Ωn

−∬𝐹1(𝜏)

(k1(𝜏

2) ⋅ (𝜁

1⋅ T)) 𝑑Σ

−𝜁1

−∬𝑅1(𝜏)

(k1(𝜏

2) ⋅ (𝜁

1⋅ T)) 𝑑Σ

−𝜁1

+∭𝜏2

𝜕𝜓

𝜕𝑡𝜌 𝑑𝑉, (77)


where 𝜁1is the external unit normal vector of the surface

(of the subsystem 𝜏1) presented by surfaces 𝐹

1(𝜏) and 𝑅

1(𝜏)

and −𝜁1is the external unit normal vector of the surface (of

the subsystem 𝜏2) presented also by surfaces 𝐹

1(𝜏) and 𝑅

1(𝜏).

Adding (77) (by using the equality 𝑑Σ𝜁1

= 𝑑Σ−𝜁1

of theelements of area of surfaces 𝐹

1(𝜏) and 𝑅

1(𝜏)) we get the evo-

lution equation for the total mechanical energy (𝐾𝜏+ 𝜋

𝜏) =

(𝐾𝜏1

+𝐾𝜏2

+𝜋𝜏1

+𝜋𝜏2

) of the macroscopic region 𝜏 consistingof subsystems 𝜏

1and 𝜏

2interacting on the surface 𝐹

1(𝜏) of the

tangential jump of the continuum velocity:

𝑑

𝑑𝑡(𝐾

𝜏+ 𝜋

𝜏)

=𝑑

𝑑𝑡∭

𝜏

(1

2k2

+ 𝜓)𝜌𝑑𝑉

= ∭𝜏

𝑝 div k 𝑑𝑉

+∭𝜏

(2


2

𝑑𝑉

−∭𝜏

2](𝑒𝑖𝑗)2

𝜌 𝑑𝑉

+∬𝜕𝜏

(k ⋅ (n ⋅ T)) 𝑑Ωn

+∬𝐹1(𝜏)

((k1(𝜏

1) − k

1(𝜏

2)) ⋅ (𝜁

1⋅ T)) 𝑑Σ𝜁

1

+∭𝜏

𝜕𝜓


(78)

where k1(𝜏

1) is the vector of the continuum velocity on the

surface 𝐹1(𝜏) in the subsystem 𝜏

1, and k

1(𝜏

2) is the vector of

the continuum velocity on the surface 𝐹1(𝜏) in the subsystem

𝜏2.The evolution equation (78) takes into account the total

mechanical energy (𝐾𝜏+𝜋

𝜏) of themacroscopic region 𝜏 con-

sisting of subsystems 𝜏1and 𝜏

2interacting on the surface𝐹

1(𝜏)

of the tangential jump of the continuum velocity. The firstterm in the right-hand side (of (78)) describes the evolutionof the total mechanical energy of themacroscopic continuumregion 𝜏 due to the continuum reversible compressibility;the second and third terms express the dissipation of themacroscopic kinetic energy by means of the irreversiblecontinuum compressibility and the velocity shear. The formsof three primary terms in the right-hand side (of (78)) arerelated to the considered model of the compressible viscousNewtonian continuum. The fourth, fifth, and sixth termsin the right-hand side (of (78)) are the universal terms forarbitrary model of continuum characterized by symmetricalstress tensor T. The fourth term expresses the power

𝑊np,𝜕𝜏 =𝛿𝐴np,𝜕𝜏

𝑑𝑡= ∬

𝜕𝜏

(k ⋅ (n ⋅ T)) 𝑑Ωn (79)

of external (for the continuum region 𝜏) nonpotential stressforces acting on the boundary surface 𝜕𝜏 of the macroscopiccontinuum region 𝜏. The fifth term expresses the power ofexternal (for the continuum region 𝜏) forces on different sides

of the surface 𝐹𝑖(𝜏) characterized by the velocity jump during

the fracture formation. The sixth term in (78) presents thepower of the total mechanical energy added (or lost) as theresult of the Newtonian nonstationary gravitational energyinfluence on the macroscopic continuum region 𝜏 related tovariations of the potential 𝜓 of the combined gravitationalfield in the continuum region 𝜏.

Consider (78) for one continuum velocity jump on thenonstationary surface𝐹

1(𝜏)during the time interval (𝑡, 𝑡+Δ𝑡).

Taking into account the form of the fifth term on the right-hand side of the evolution equation (78), we obtained [11, 12]the expression for the work 𝛿𝐴np,𝐹

1(𝜏)

(done during the timeinterval (𝑡, 𝑡+Δ𝑡) by the external (for the continuum region 𝜏)nonpotential stress forces acting on different sides of thevelocity jump on the surface 𝐹

1(𝜏)):

𝛿𝐴np,𝐹1(𝜏)

= ∫

𝑡+Δ𝑡

𝑡

(∬𝐹1(𝜏)

((k1(𝜏

1) − k

1(𝜏

2)) ⋅ (𝜁

1⋅ T)) 𝑑Σ

𝜁1

)𝑑𝑡,

(80)

which is reduced to the following expression:

𝛿𝐴np,𝐹1(𝜏)

= ∬𝐹1(𝜏)

(∫

𝑡+Δ𝑡

𝑡

(k1(𝜏

1) − k

1(𝜏

2)) ⋅ (𝜁

1⋅ T) 𝑑𝑡) 𝑑Σ𝜁

1

.

(81)

To test the formula (81), let us calculate the energy𝛿𝐴np,ΔΣ, which dissipates during formation of the surfacedislocation on the small surface ΔΣ during the time interval(𝑡, 𝑡 + Δ𝑡). Using the theorem about the average value andintegrating the internal integral on time, we obtained fromrelation (81) for 𝐹

1(𝜏) = ΔΣ the following relation [11, 12]:

𝛿𝐴np,ΔΣ

= ∬ΔΣ

(w (𝜁1, 𝑡 + Δ𝑡) − w (−𝜁

1, 𝑡 + Δ𝑡)) ⋅ ⟨(𝜁

1⋅ T)⟩ 𝑑Σ𝜁

1

,

(82)

where ⟨(𝜁1⋅T)⟩ is the average value of the stress vector for the

element of area 𝑑Σ𝜁1

of the two-side surface ΔΣ, andw(𝜁1, 𝑡 +

Δ𝑡) and w(−𝜁1, 𝑡 + Δ𝑡) are the vectors of the continuum dis-

placement on different sides of the element of area 𝑑Σ𝜁1

of thetwo-side surface ΔΣ in the points characterized by normalunit vectors 𝜁

1and−𝜁

1. Using the obvious expression for “lin-

ear” time average ⟨(𝜁1⋅ T)⟩

⟨(𝜁1⋅ T)⟩ = 1

2(p (𝜁

1, 𝑡) − p (−𝜁

1, 𝑡 + Δ𝑡)) (83)

as the arithmetical average of the values of the stress vectorsp on the different sides from the surface of the jump of thecontinuum velocity, we obtained [11, 12] the expression for


the elementary work of the external nonpotential stress forceson the two-side surface ΔΣ of dislocation:

𝛿𝐴np,ΔΣ

=1

2∬

ΔΣ

(w (𝜁1, 𝑡 + Δ𝑡) − w (−𝜁

1, 𝑡 + Δ𝑡))

⋅ (p (𝜁1, 𝑡) + p (−𝜁

1, 𝑡 + Δ𝑡)) 𝑑Σ𝜁

1

.

(84)

This expressionwas obtained [44] in the frame of the classicallinear approach to formation of surface dislocations in rigidcompressible continuum on the small area of surface ΔΣ. It isclear that the assumption (83) is valid only forweak tangentialjumps of the continuum displacement. Consequently, wecan consider the expression (80) as the natural nonlineargeneralization of expression (84) for arbitrary surface𝐹

1(𝜏) of

dislocation and for strong tangential jumps of the continuumdisplacement on the surface 𝐹

1(𝜏) of dislocation. The work

(80) of the external (for the continuum region 𝜏) nonpotentialstress forces should be negative.The sufficient energy 𝛿𝐸

𝑑,𝐹1(𝜏)

needed for formation of the surface 𝐹1(𝜏) of dislocation is

equal to the work of the internal forces in the macroscopiccontinuum region 𝜏. The energy 𝛿𝐸

𝑑,𝐹1(𝜏)

should be positiveand equal to the expression (80) with the sign “−”:

𝛿𝐸𝑑,𝐹1(𝜏)

= − 𝛿𝐴np,𝐹1(𝜏)

= −∫

𝑡+Δ𝑡

𝑡

(∬𝐹1(𝜏)

((k1 (𝜏1) −k1 (𝜏2)) ⋅ (𝜁1 ⋅ T)) 𝑑Σ𝜁1

)𝑑𝑡

> 0.

(85)

The formulae (80), (84), and (85) are obtained (taking intoaccount the generalized differential formulation (50) of thefirst law of thermodynamics) for the model of continuumcharacterized by an arbitrary symmetrical stress tensor T.

The macroscopic internal shear kinetic energy (𝐾𝑠)𝜏1

(ofthe subsystem 𝜏

1), themacroscopic internal rotational kinetic

energy (𝐾𝑟)𝜏1

(of the subsystem 𝜏1), and the macroscopic

kinetic energy of shear-rotational coupling (𝐾coup𝑠,𝑟

)𝜏1

(of thesubsystem 𝜏

1) are the significant components of the macro-

scopic internal shear-rotational kinetic energy (𝐾𝑠-𝑟)𝜏1

[11, 12,24, 31]:

(𝐾𝑠-𝑟)𝜏1

= (𝐾𝑟)𝜏1

+ (𝐾𝑠)𝜏1

+ (𝐾coup𝑠,𝑟

)𝜏1

, (86)

taken into account (along with the classical internal thermalenergy 𝑈

𝜏1

of the macroscopic continuum region 𝜏1, the

macroscopic potential energy 𝜋𝜏1

of the macroscopic con-tinuum region 𝜏

1, and the macroscopic translational kinetic

energy (𝐾𝑡)𝜏1

= (1/2)𝑚𝜏1

(V𝑐)2

𝜏1

of the continuum region 𝜏1

(of a mass𝑚𝜏1

) moving as a whole at speed equal to the speed(V

𝑐)𝜏1

of the center ofmass of the continuum region 𝜏1) in the

generalized differential formulation (50) of the first law ofthermodynamics for the macroscopic continuum region 𝜏

1.





, and the macroscopic kinetic energy of shear-rotational coupling (𝐾coup

𝑠,𝑟)𝜏2

are the significant componentsof the macroscopic internal shear-rotational kinetic energy(𝐾

𝑠-𝑟)𝜏2

[11, 12, 24, 31]:

(𝐾𝑠-𝑟)𝜏2

= (𝐾𝑟)𝜏2

+ (𝐾𝑠)𝜏2

+ (𝐾coup𝑠,𝑟

)𝜏2

, (87)

taken into account (along with the classical internal ther-mal energy 𝑈

𝜏2

, the macroscopic potential energy 𝜋𝜏2

,and the macroscopic translational kinetic energy (𝐾

𝑡)𝜏2

=

(1/2)𝑚𝜏2

(V𝑐)2

𝜏2

of the continuum region 𝜏2(of a mass 𝑚

𝜏2

)moving as a whole at the speed (V

𝑐)𝜏2

of the center of mass ofthe continuum region 𝜏

2) in the generalized differential for-

mulation (50) of the first law of thermodynamics for themac-roscopic continuum region 𝜏

2.





, the macroscopic kinetic energy of shear-rota-tional coupling (𝐾coup

𝑠,𝑟)𝜏1

, the macroscopic translational ki-netic energy (𝐾

𝑡)𝜏1

= (1/2)𝑚𝜏1

(V𝑐)2

𝜏1

, themacroscopic poten-tial energy 𝜋

𝜏1

, the macroscopic internal shear kinetic energy(𝐾

𝑠)𝜏2

(of the subsystem 𝜏2), the macroscopic internal rota-

tional kinetic energy (𝐾𝑟)𝜏2

, the macroscopic kinetic energyof shear-rotational coupling (𝐾coup

𝑠,𝑟)𝜏2

, themacroscopic trans-lational kinetic energy (𝐾

𝑡)𝜏2

= (1/2)𝑚𝜏2

(V𝑐)2

𝜏2

, and themacroscopic potential energy 𝜋

𝜏2

are the significant energycomponents taken into account in the presented thermo-hydrogravidynamic shear-rotational model described by theevolution equation (78) for the total mechanical energy (𝐾

𝜏+

𝜋𝜏) of the macroscopic region 𝜏 consisting of interacting

subsystems 𝜏1and 𝜏

2.

3.2. The Rotational Model of the Earthquake Focal RegionBased on the Generalized Differential Formulation of the FirstLaw of Thermodynamics. Following the works [11, 12], weshall present the foundation of the rotational model [2] ofthe earthquake focal region for the seismic zone of the PacificRing. It was noted [2] that the studies of the dislocationmodels of the focal regions of strong earthquakes showed thebad correspondencewith themodel of flat endless dislocationin the uniform continuum [45–47]. The analysis [2] showedthat the conditions exist to realize the rotational mechanismrelated to the rotation of the geoblocks by means of thestress forces related to the Earth rotation in the vicinity ofthe seismic zone of the Pacific Ring. It was noted [2] thatthe rotational mechanism can be more real compared to theconventionalmechanism related to the formation of themainline flat fracture inside the focal region.

Let us consider the energy thermodynamic analysis ofthe rotational mechanism [2] of the earthquake focal region,related to formation of the circular continuum velocity jumprevealed in the form of circular dislocation after relaxation ofthe seismic process in the earthquake focal region.The devel-oped and tested (in this section) mathematical formalism ofdescription of the main line flat fracture may be generalizedon the closed surfaces of the continuum velocity jumps.Following the rotational model [2] of the earthquake focal


region, we consider the separate geoblock 𝜏int of the seismiczone. If the external influences of the nonstationary gravita-tional forces (on the geoblock 𝜏int) and the nonpotential stressforces (on the boundary 𝜕𝜏

𝑖of the geoblock 𝜏int) exceed the

certain critical value then the geoblock may rotate and sliprelative to the surrounding fine plastic layer (subsystem) 𝜏extwith the tangential continuumvelocity jumpon the boundarysurface 𝜕𝜏

𝑖of the geoblock 𝜏int. We assume that fine plastic

layer (subsystem) 𝜏ext is limited by external surface 𝜕𝜏 of theconsidered thermodynamic system 𝜏 consisting of themacro-scopic subsystems 𝜏int and 𝜏ext.

Using the evolution equation (73) of the total mechanicalenergy of the subsystem 𝜏, we obtained [11, 12] the evolutionequations for the total mechanical energy of the macroscopicsubsystems 𝜏int and 𝜏ext:

𝑑

𝑑𝑡(𝐾

𝜏int+ 𝜋

𝜏int)

=𝑑

𝑑𝑡∭

𝜏int

(1

2k2

+ 𝜓)𝜌𝑑𝑉

= ∭𝜏int

𝑝 div k 𝑑𝑉

+∭𝜏int

(2


2

𝑑𝑉

−∭𝜏 int

2](𝑒𝑖𝑗)2

𝜌 𝑑𝑉

+∬𝜕𝜏𝑖

(kint (𝜕𝜏𝑖) ⋅ (m ⋅ T)) 𝑑Σm

+∭𝜏int

𝜕𝜓


𝑑

𝑑𝑡(𝐾

𝜏ext+ 𝜋

𝜏ext)

=𝑑

𝑑𝑡∭

𝜏ext

(1

2k2

+ 𝜓)𝜌𝑑𝑉

research article fundamentals of the thermohydrogravidynamic...

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