Dynamics of Dissipative Viscous Cylindrical Collapse with Full …downloads.hindawi.com/journals/ahep/2018/9250786.pdf · 2019-07-30 · the hydrostatic periods of astral models

Research ArticleDynamics of Dissipative Viscous Cylindrical Collapse withFull Causal Approach in 119891(119877) Gravity

G Abbas and H Nazar

Department of Mathematics The Islamia University of Bahawalpur Bahawalpur Pakistan

Correspondence should be addressed to G Abbas abbasg91yahoocom

Received 3 September 2018 Revised 28 September 2018 Accepted 2 October 2018 Published 24 October 2018

Guest Editor Subhajit Saha

Copyright copy 2018 G Abbas and H Nazar This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited The publication of this article was funded by SCOAP3

The idea of this article is to examine the effects on dynamics of dissipative gravitational collapse in nonstatic cylindrical symmetricgeometry by usingMisner-Sharp concept in framework of metric 119891(119877) gravity theory In this interest we extended our study to thedissipative dark source case in both forms of heat flow and the free radiation streaming Moreover the role of different quantitiessuch as heat flux bulk and shear viscosity in the dynamical equation is evaluated in thorough version The dynamical equation isthen coupled with full causal transportation equations in the context of Israel-Stewart formalism The present scheme explains thephysical consequences of the gravitational collapse and that is given in the decreasing form of inertial mass density which dependson thermodynamics viscousheat coupling factors in background of 119891(119877) theory of gravity It is very interesting to tell us that themotives of this theory are reproduced for 119891(119877) = 119877 into general theory of relativity that has been done earlier

1 Introduction

A multiple work has been done in the significance of rela-tivistic geometry of the collapsing star to the general relativitytheory and higher order curvature theories In this scenariotoo many open challenges and curious issues still exist inmodern physics gravitational physics cosmological physicsrelativistic dense objects group of galaxies neutron star andblack holes Currently our aim is to discuss the gravitationalcollapse of the relativistic dense object in the presence ofdissipative term

In this arrangement the formation of the dense star ishighly explosive and exhaustible due to the highly inmost pullof gravity and this process is terminated with the externalpressure of the interior fuel However once the object haswakened due to its thermodynamical fission reaction thereis no any extended thermodynamical burning and therewill be an interminable gravitating collapse The geometryof the object is made up by its explosive mass which isconstantly gravitational and it is stimulating by the interiorcore of the object due to the gravitating interaction betweentheir particles Oppenheimer and Snyder [1] made an activitystride in the field of gravitating collapse and examined the

nitty gritty work in regard to the gravitational collapse issuesIn this order after the midway of the 20th century a briefsensible evaluation was recognized by Misner and Sharp[2] with isotropic matter distribution for the interior ofthe imploding object as well as in vacuum of the outsideobject Vaidya [3] explained the gravitational collapse withthe radiating source

Numerous models still exist to explain the enigmatic roleof the dark energy in a system In this inference we usethe higher order modified gravity theories such as 119891(119877)119891(119866) and 119891(119877 119879) gravity theories it can provide the well-established framework in the significance of rushing evolu-tion of the universe Capozziello [4] studied the notable issuesfor the accelerated growth of the universe and quintessence inthe Friedmann Robertson Walker (FRW) model with highercurvature gravity theories Sharif and Yousaf [5] exploredthe stability of the anisotropic fluid distribution in nonstaticspherically symmetric geometry with119891(119877)metric formalismMak and Harko [6] investigated the notional indicationsthroughmany astrophysical estimations and found the signif-icant natural dissimilarities in the pressure Herrera and San-tos [7] determined the impacts on imperfect fluid formationwith astral spherical object for the behavior of restful rotation

HindawiAdvances in High Energy PhysicsVolume 2018 Article ID 9250786 13 pageshttpsdoiorg10115520189250786

2 Advances in High Energy Physics

Webber [8] examined the solutions of pressure anisotropyin dense models with electromagnetic field Chakraborty etal [9] analyzed the gears of tangential pressure and alongwith radial pressure in stellar structures by quasisphericalparadigms in the form of collapse Garattini [10] evaluatedthe formation of exposed singularity throughmass limitationin metric 119891(119877) theory of gravity Sharif and his coresearchers[11ndash18] studied the various consequences of energy densityinhomogeneity and Weyl tensor with gravitational collapsein dense objects with General relativity theory as well as inmodified theories of gravity Cognola et al [19] introducedthe results of spherical imploding object for the black hole in119891(119877) context by the parameter of positive Ricci invariant 119877Capozziello et al [20] examined the modified Lane-Emdenequation in context of metric 119891(119877) gravity theory and studiedthe hydrostatic periods of astral models Copeland et al[21] analyzed the several concepts for the rushing growthof universe Amendola et al [22] discussed the feasibleparadigms of metric 119891(119877) gravity in the context of bothEinstein and Jordan frames

Chandrasekhar [23] introduced the impacts on the con-stancy of dynamics with perfect fluid collapsing geometryHerrera et al [24] discussed the results of dynamical dise-quilibrium with spherically symmetric nonadiabatic collapseand found the heat flow out from the gravitating source in theform of radial heat flux Abbas et al [25] presented the resultsof compact star models in anisotropic fluid distributionwith cylindrical symmetric static spacetime geometry Makand Harko [26] studied the kind of exact solutions of fieldequations by considering the spherical symmetric structureand also conferred the energy density along with tangentialand radial pressure that are limited and increased in the coreof the imperfect fluid object Rahaman et al [27] prolongedthe Krori-Barua solution with spherical symmetric staticspacetime for the investigation of charge imperfect fluiddistribution Herrera and his collaborators [28ndash33] did workin different directions such as self-gravitating dens modelsdiscussed with local anisotropic matter distribution in thecracking spherical symmetric structure in context of equa-tion of state with perturbed approach They established thevarious results in gravitational collapse for the dissipative caseby using Misner and Sharp approach such gravitating sourceundergoes in form of free radiation streaming and diffusionapproximations They investigated the expansion free con-ditions for the spherically symmetric anisotropic dissipativecollapsing matter Further Herrera and his collaborators [34ndash38] have evaluated the Lake algorithm in static sphericallysymmetric anisotropic configuration for the determination ofnew formalism of two functions (instead of one) to generateall possible results They studied the results of structurescalars 119884119879119865 for the charged dissipative anisotropic collapsinggeometry in presence of cosmological constant They havealso done detailed work on the dynamical disequilibriumwith spherically symmetric locally anisotropic fluid whichcollapses adiabatically under the expansion-free conditionIn few investigations they have used the noncasual as wellas casual approaches to discuss the dynamics of gravitatingsource Nolan [39] examined the effects on gravitationalcollapse for counter rotating dust cloud with cylindrical

paradigms and found the naked singularity Hayward [40]produced the results of cylindrical geometry for black holesgravitational waves and cosmic strings

In this paper we analyze the self-gravitational collapsewith cylindrical symmetric geometry in context of metric119891(119877) gravity theory The main persistence of this study is toexamine the viscous dissipative collapse in the form of radialheat flow and free radiation streaming in cylindrical nonstaticspacetime with full causal approach

The proposal of this work is as follows In Section 2we express the cylindrical symmetric isotropic collapsingmatter distribution and its related variables in context of119891(119877)theory of gravity Section 3 consists of Einstein modified fieldequations matching conditions are given for the interior andexterior manifolds119872minus and119872+ and the dynamical equationthat was accompanied with physical parameters such as heatflux bulk and shear viscosity Section 4 is devoted to thetransportation equations attained in frame of119872119897119897119890119903-Israel-Stewart formalism [41ndash43] then transportation equations areassociated with dynamical equation Finally the conclusionsof the study have been discussed in the last section

2 Self-Gravitational Collapsing Geometry andRelated Conventions

In this section we consider a locally isotropic collapsingfluid with full causal description in context of Misner-Sharp approach formed by cylindrically symmetric nonstaticinterior and exterior geometry of the dissipative fluid whichundergoes dissipation in the form of heat flow and freestreaming radiation In general relativity the Einstein-Hilbert(EH) action can be expressed as follows

119878119864119867 = 12120581 int 1198894119909radicminus119892119877 (1)

The extended formulation of the Einstein-Hilbert (EH) in119891(119877) context is119878119898119900119889119894119891 = 12120581 int1198894119909radicminus119892 (119891 (119877) + 119871 (119898119886119905119905119890119903)) (2)

where 119891(119877) is arbitrary function of Ricci scalar 119877 120581 is cou-pling constant and 119871 (119898119886119905119905119890119903) the Lagrangian matter density ofthe action The following modified 119891(119877) field equations areobtained by the variation of (2) with respect to 119892120572120573119865 (119877)119877120572120573 minus 12119891 (119877) 119892120572120573 minus nabla120572nabla120573119865 (119877) + 119892120572120573nabla120572nabla120572119865 (119877)

= 120581119879120572120573(3)

where 119865(119877) = 119889119891(119877)119889119877 nabla120572nabla120572 is the Drsquo Alembert oper-ator nabla120572 denotes the covariant derivative and 119879120572120573 is theisotropic energy-momentum tensorThe above equations canbe expressed in Einstein tensor as given below

119866120572120573 = 120581119865 (119879119898120572120573 + 119879119863120572120573) (4)

Advances in High Energy Physics 3

where

119879119863120572120573 = 1120581 [119891 (119877) minus 119877119865 (119877)2 119892120572120573 + nabla120572nabla120573119865 (119877)minus 119892120572120573nabla120572nabla120572119865 (119877)]

(5)

represent the action energy-momentum tensor By consid-ering the comoving coordinates bounded by cylindricallysymmetric nonstatic spacetime geometry which contains thedissipative nature of the fluid defines by interior metric

119889119904minus2 = minus11986021198891199052 + 11986121198891199032 + 11986221198891206012 + 11986321198891199112 (6)

Here 119860 = 119860(119905 119903) 119861 = 119861(119905 119903) 119862 = 119862(119905 119903) 119863 = 119863(119905 119903) are themetric functions We labeled the coordinates 1199090 = 119905 1199091 = 1199031199092 = 120601 and 1199093 = 119911

The tensor 119879minus120572120573 of the dissipative collapsing matter distri-bution is119879minus120572120573 = (120583 + 119901 + Π)119881120572119881120573 + (119901 + Π) 119892120572120573 + 119902120572119881120573 + 119902120573119881120572

+ 120598119897120572119897120573 + 120587120572120573 (7)

where 120583 is the energy density 119901 is the isotropic pressure Πthe bulk viscosity 119902120572 the heat flux 120587120572120573 is the shear viscosity 120598the radiation density and 119881120572 the four velocity of the fluidFurthermore 119897120572 denotes radial null four vector The aboveextents gratify the results as below

119881120572119881120572 = minus1119881120572119902120572 = 0119897120572119881120572 = minus1119897120572119897120572 = 0

(8)

120587120583]119881] = 0120587[120583]] = 0120587120572120572 = 0

(9)

In the general nonreversible thermodynamics taken into [4445]

120587120572120573 = minus2120578120590120572120573Π = minus120577Θ (10)

wherever 120578 is the factor of shear viscosity and 120577 denotes thefactor of bulk viscosity 120590120572120573 and Θ are the shear tensor andheat flow expansion respectively

The shear tensor 120590120572120573 is defined by

120590120572120573 = 119881(120572120573) + 119886(120572119881120573) minus 13Θℎ120572120573 (11)

where the acceleration 119886120572 and the expansionΘ are recognizedby

119886120572 = 119881120572120573119881120573Θ = 119881120572

120572 (12)

and ℎ120572120573 = 119892120572120573 + 119881120572119881120573 is the projector onto the hypersurfaceorthogonal to the four velocity let us define the followingquantities for the given metric

119881120572 = 119860minus11205751205720 119902120572 = 119902119861minus11205751205721 119897120572 = 119860minus11205751205720 + 119861minus11205751205721

(13)

where 119902 depending on 119905 and 119903 Also it trails from (8) so that

1205870120572 = 012058711 = minus212058722 = minus212058733 (14)

In a more explicit formation we can write

120587120572120573 = Ω(120594120572120594120573 minus 13ℎ120572120573) (15)

where 120594120572 is a unit four vector along the radial directionfulfilling the conditions

120594120572120594120572 = 1120594120572119881120572 = 0120594120572 = 119861minus11205751205721

(16)

and Ω = (32)12058711From the information of (13) we obtain the following

nonnull components of shear tensor

12059011 = 11986123119860 [Σ1 minus Σ3] 12059022 = 11986223119860 [Σ2 minus Σ1] 12059033 = 1198632

3119860 [Σ3 minus Σ2] (17)

where

120590120572120573120590120572120573 = 1205902 = 131198602[Σ21 + Σ22 + Σ23] (18)

where

Σ1 = 119861 minus 119862 Σ2 = 119862 minus 119863Σ3 = 119863 minus 119861

(19)

Here dot denotes derivative wrt 119905By using (12) and (13) we get

1198861 = 119860 Θ = 1119860 (119861 + 119862 + 119863)

(20)

where prime represents derivative with respect to 119903


3 Modified Field Equations in 119891(119877) GravityThe modified field equation (4) for cylindrical symmetricspacetime in 119891(119877) metric formalism gives the followingsystem of equations

(119860119861)2 [minus11986210158401015840119862 minus 11986310158401015840

119863 + 1198611015840119861 (1198621015840119862 + 1198631015840

119863 ) minus 11986210158401198631015840

119862119863 ]

+ (119861119862 + 119861119863 + 119862119863) = 120581119865 [(120583 + 120598)1198602

+ 1198602

120581 minus(119891 (119877) minus 119877119865 (119877)2 )

minus 1198602(119861 + 119862 + 119863) + 119865101584010158401198612

+ 11986510158401198612 (1198621015840

119862 + 1198631015840

119863 minus 1198611015840119861 )]

(21)

minus ( 1198621015840119862 + 1015840

119863 minus 1198621015840119861119862 minus 1198631015840

119861119863 minus 1198601015840

119860119862 minus 1198601015840

119860119863 )

= 120581119865 [minus (119902 + 120598)119860119861 + 1120581 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )] (22)

minus (119861119860)2 (119862 + 119863 + 119862119863 minus 119860119862 minus 119860119863) + (119862

10158401198631015840

119862119863+ 11986010158401198621015840119860119862 + 11986010158401198631015840

119860119863 ) = 120581119865 [(119901 + Π + 120598 + 2Ω3 )1198612

+ 1198612120581 (119891 (119877) minus 119877119865 (119877)2 ) + 1198602

+ 1198602(119862 + 119863 minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 )]

(23)

minus (119862119860)2 [119861 + 119863 minus 119860 (119861 + 119863) + 119861119863] + (119862119861)

2

sdot [11986010158401015840

119860 + 11986310158401015840

119863 minus 1198601015840

119860 (1198611015840119861 minus 1198631015840

119863 ) minus 11986110158401198631015840

119861119863 ]

= 120581119865 [(119901 + Π minus Ω3 )1198622

minus 1198622120581 minus(119891 (119877) minus 119877119865 (119877)2 ) minus 1198602+ 119865101584010158401198612

+ 1198602(119860 minus 119861 minus 119862 minus 119863)

+ 11986510158401198612 (1198601015840

119860 minus 1198611015840119861 + 1198621015840119862 + 1198631015840

119863 )]

(24)

minus (119863119860)2 [119861 + 119862 minus 119860 (119861 + 119862) + 119861119862] + (119863119861 )

2

sdot [11986010158401015840

119860 + 11986210158401015840119862 minus 1198601015840

119860 (1198611015840119861 minus 1198621015840119862 ) minus 11986110158401198621015840119861119862 ]= 120581119865 [(119901 + Π minus Ω3 )1198632

minus 1198632

120581 minus(119891 (119877) minus 119877119865 (119877)2 ) minus 1198602+ 119865101584010158401198612


+ 11986510158401198612 (1198601015840

119860 minus 1198611015840119861 + 1198621015840119862 + 1198631015840

119863 )]

(25)

The energy of gravitating source per specific length definedby cylindrical symmetric space-time is given by [40 46ndash48]

119864 = (1 minus 119897minus2nabla119886119903nabla119886119903)8 (26)

For a cylindrical symmetric paradigms by killing vectors thecircumference radius 120588 and specific length 119897 and moreoverarial radius 119903 are described in [40 46ndash48]1205882 = 120585(1)119886120585119886(1) 1198972 = 120585(2)119886120585119886(2) so that 119903 = 120588119897

In the whole interior region C-energy has the followingform [16]

119898(119903 119905) = 119864119897= 1198978+ 18119863 [ 11198602

(119862 + 119863)2 minus 11198612 (1198621198631015840 + 1198621015840119863)2] (27)

31 Matching Conditions We consider an exterior spacetimeover 3119863 hypersurface Σ that describes the 4-dimensionalmanifold119872+ in cylindrically symmetric geometry [16 49] isgiven by

1198891199042+ = minus(minus2 (]) ) 119889]2 minus 2119889]119889

+ 2 (1198891206012 + 12057421198891198852) (28)

where (]) is the total mass and ] is the usual retarded timecoordinate while 120574 is a constant having the dimension of1length Moreover the interior spacetime of manifold 119872minus

over the boundary hypersurface Σ is

119896minus (119905 119903) = 119903 minus 119903Σ = 0 (29)

Here 119903Σ is constant The interior metric of119872minus over boundinghypersurface becomes

1198891199042minus Σ997904997904997904 minus1198891205912 + 11986221198891206012 + 11986321198891199112 (30)


where

119889120591 Σ997904997904997904 119860119889119905 (31)

indicates the time coordinate on hypersurface Σ andΣ997904997904997904

means the estimation is applied on Σ of both sidesIn a similar manner we prescribed the exterior line-

element119872+ on 3119863 hypersurface Σ takes the form

119896+ (] ) = minus Σ (]) = 0 (32)

Thus the exterior spacetime manifold119872+ on hypersurface isgiven by

1198891199042+ Σ997904997904997904 minus2(minus (]) + 119889Σ (])119889] )119889]2

+ 2 (1198891206012 + 12057421198891199112) (33)

The following conditions due to Darmois [50] are to besatisfied

(i) The continuity of the first fundamental form on Σbecomes

1198891199042Σ = (1198891199042minus)Σ = (1198891199042+)Σ (34)

(ii) The continuity of the second fundamental form overboundary three-surface is as follows

[119870120572120573] = 119870+120572120573 minus 119870minus

120572120573 = 0 (35)

where 119870120572120573 is the extrinsic curvature over the hyper-surface Σ and given into

119870plusmn120572120573 = minus119899plusmn120583 ( 1205972119909120583plusmn120597120588120572120597120588120573 + Γ120583119886119887

120597119909119886plusmn120597119909119887plusmn120597120588120572120597120588120573 ) (120583 119886 119887 = 0 1 2 3)

(36)

From the above equation of the extrinsic curvature theChristoffel symbols are calculated for the interior and exte-rior manifolds 119872minus and 119872+ accordingly and we labeled(1205880 1205882 1205883)=(120591 120601 119911) for the intrinsic coordinates on bound-ary surface Σ and 119899plusmn120583 are the components of the outward unitnormal over Σ in the coordinates 119909120583plusmn thus we have119899minus120583 Σ997904997904997904 (0 119861 0 0) 119899+120583 Σ997904997904997904 120582(minus119889119889] 1 0 0)

120582 = 12 (minus (]) + 119889Σ (]) 119889])12

(37)

The following expressions are taken from the continuity of thefirst fundamental form

119862 (119905 119903Σ) Σ997904997904997904 Σ (]) 119863 (119905 119903Σ) Σ997904997904997904 120574Σ (])

119889119905119889120591 = 1119860 119889]119889120591 = 120582

(38)

Hence the nonzero components of extrinsic curvature 119870plusmn120572120573

are

119870minus00 = (minus 1198601015840

119860119861)Σ

(39)

119870+00 = [(119889]119889120591)

minus1 (1198892]1198891205912) minus 1198772 (

119889]119889120591)]Σ

(40)

119870minus22 = (1198621015840119862119861 )

Σ

(41)

119870minus33 = (1198631015840119863119861 )

Σ

(42)

119870+22 = [(119889119889120591 ) minus 2 (119889]119889120591)]

Σ

= 120574minus2119870+33 (43)

Thus continuity of the extrinsic curvature is organized with(38) we obtain the following results over hypersurface Σ [1651]

(]) Σ997904997904997904 2 [(

119860)2 minus (1015840119861 )

2] (44)

119864 Σ997904997904997904 1198978 + 120574 (]) (45)

119902 Σ997904997904997904 (119875 + Π + 2Ω3 ) + 11987911986301119860119861 + 119879119863111198612 (46)

Here 11987911986301 and 11987911986311 are the dark source components given inAppendices (A1) and (A2) respectively

Therefore (44) shows the total mass of the interiorboundary surface Σ whereas (45) gives the smooth associ-ation between the 119862 energy for the cylindrical symmetricgeometry and the active mass on hypersurface Σ Moreover(46) describes the linear association among the fluid quan-tities (119875 ΠΩ 119902) over the boundary three-surface Σ Thuseffective pressure is in usual nonvanishing over Σ due todissipative nature of the dark source and this dissipationthrough viscosity parameters in the form of heat flow andundergoes free radiation streaming on the boundary surfaceThe extra components on the right hand side defines the darksource and appeared due to higher curvature of the geometryIt should be very interesting to mention that if the fluid has


no dark source components then effective pressure and heatflux are equal on the boundary surface Σ

Furthermore other important results of the collapsingfluid are performed on the bounding three-surface Σ for thematching conditions of the Ricci invariant 119877 and its normalderivativeThe detailed description ofmatching conditions in119891(119877) gravity theory has been given in [50 52ndash56]We requirethematching conditions for the continuity of both spacetimes119872minus and119872+ over the hypersurface Σ the following matchingconditions 119891(119877) theory are given by

119877|plusmn = 0119891119877119877 [120597]1198771003816100381610038161003816plusmn = 0

119891119877119877 = 0(47)

The above expressions are required for the continuity of Ricciinvariant 119877 over the boundary surface Σ in the cylindricalcollapsing fluid

32 Dynamical Equations Theproper time and radial deriva-tives are given we use Misner and Sharp technique [2]

119863119879 = 1119860 120597120597119905 119863119862 = 11198621015840 120597120597119903

(48)

where 119862 is the areal radius of a spherical surface inside thelimit The velocity of the collapsing matter is described by theproper time derivative of 119862 and 119863 [17] ie

119880 = 119863119879119862 = 119860119881 = 119863119879119863 = 119860

(49)

which is always negative Applying these results and (27) itturns out into

119864 equiv 1198621015840119861= [(119881119862 + 119880119863119863 )2 minus 8119863 (119898 (119903 119905) minus 1198978)]

12 minus 1198621198631015840

119861119863 (50)

The rate of change of mass is wrt proper time in (27) we use(21) (22) and (23)-(25) as follows

119863119879119898 (119903 119905) = 119862119863119865 [minus4120587 (120583 + 2120598 minus 119901 minus Π + 4Ω3 )119880+ 119864 (119902 + 120598)

minus 119880 119864211986210158402 (11986510158401198621015840119862 + 11986510158401198631015840

119863 minus 311986411986510158401198631198621198612 )+ 11986422 (119863119862119860119863119879119865 + 31198651015840101584011986210158402 )+ 1198642119865119863 (119863119862119862119863 + (119863119862119863)24119863 + 1198631198621198634119862 )+ 1198642119865119863119862119862119862119862 minus 11986431198651198631198621198611198621015840 ( 1119862 + 119863119862119863119863 )minus 119865119861 (119881119863119879119861119863 + 119863119879119879119861) minus 1198654 (119863119879119879119862119862 + 119863119879119879119863119863 )+ 119877119865 (119877)2 minus 119891 (119877)2 minus 1198803 (1198631198791198602 minus 1119862)minus 22 +

1198651198631198791198604119862 + 1198802 1198642 (311986311987911986121198641198621015840 minus 119865101584011986311986211986021198621015840 ) + 119881

119863+ 119865119863119879119861119861 ( 1119862 minus 119863119879119862 )minus 1198642

119862 (119863119862119862119860 minus 1198641198631198621198601198631198621198611198621015840 + 1198631198621198602119862 )+ 1198811198652119863 ( 34119862 minus 119863119879119862 ) minus 11988121198652119863 (1198642119863119862119860 minus 1198802119863)minus 119881311986541198632

(119862119863 + 119862119863119879119860 ) + 1198811198654119863 119862119863119879119879119863119863 + 119863119879119879119862

+ 1198642 (119862 (119863119862119863)221198632minus 12119862)

minus 119864221198621015840 119864119863119879119861(119863119862119865 + 1198652119862 minus 119862 (119863119862119863)2 11986521198632)

minus 119865(11986311987911986210158402119862 + 1198631198791198631015840

2119863 )+ 1198651198631198621198632119863 (1198621198631198791198631015840

119863 + 1198631198791198621015840) minus 1198631198791198651015840] (51)

The overhead result leads to telling us that the variation rateof the total energy interior of the cylinder of radius C and theRHS of (51) (120583+2120598minus119901minusΠ+4Ω3)119880 reflects that the energyis increasing in the interior surface of radius 119862 (in case ofcollapsing situation (119880 lt 0)) through the rate of work beingdone by the active radial pressure the energy density 120583 andthe radiation pressure 120598 The usual thermodynamical result is120587120572120573 = minus2120578120590120572120573 used in the relaxation phase The second value119864(119902+ 120598) of the right-hand side illustrates the matter energy ofthe fluid which is leaving the cylindrical surface Moreover


all other extra terms play the natural role of the dark energydissipative source under the formation of modified theory ofgravity and decreasing the pressure in the core of the star dueto continuous collapsing phase of the dark energy Similarlywe can calculate

119863119862119898 (119903 119905) = 1198621198632119865 [120581 120583 + 2120598 + 119901 + Π + 2Ω3 + 119864 (119902

+ 120598) + 11988031198651198631198621198602119862 + 1198802 119863119879119865119863119862119860 + 119865119862 (1198631198621198634119863minus 119863119879119862 ) minus 119880119864119863119862119865(119864119863119862119860 minus 1198631198791198611198621015840 )+ 119863119879119865119863119879119860+ 119865( 11198621015840 (11986311987911986210158402119862 + 1198631198791198631015840

2119863 minus 119864119863119879119861119863119862119863119863 )+ 1198642 (119863119862119860119862 + 119863119862119860119863119862119863119863 ))

minus 119881211986521198632119862(119880119863119862119860 + 1198631198621198632119863 ) minus 1

+ 11988111986521198621015840119863 (1198621198631198791198631015840

119863 + 1198631198791198621015840) + 119880119881119865119863 ( 12119862minus 119863119879119862 ) minus 119864211986511986311986211986321198632

(119862119863119862119862119863 + 119863119862119863

minus 119862 (119863119862119863)22119863 minus 1198641198621198631198621198611198631198621198631198621015840 ) minus 12119862 (119863119862119862119862

minus 31198631198621198632119863 minus 1198641198631198621198611198621015840 ) + 12119863 (119863119862119863119863119862119862119862

minus 119863119862119862119863) minus 1198641198621015840 (119863119879119865119863119879119861 + 1198642119863119862119865119863119862119861

+ 119865119863119879119861119863119879119863119863 ) + 119863119879119879119865 + 11986510158401015840119864211986210158402 minus 11988011986311987911986510158401198621015840+ 119865119863119879119879119862119862 + 119865119863119879119879119863119863 ]

(52)

This solution of (52) explains how variational quantities affectthe matter distribution between the neighboring exteriors inthe object of radius C The picture of the first two quantitieson the RHS of (52) is 120583 + 2120598 related with energy density ofthe fluid and plus the addiction of the null fluid conferringdissipation in the outgoing streaming approximation Thelast factor (119880119864)(119902 + 120598) plays negative role in the collapsingsituation and estimates the dissipation in terms of heat flowand radiation streaming The extra values signify the inputof the dark energy (DE) due to its higher order of thecurvature matter We Apply integration on (52) with 119862 weget

119898(119903 119905) = 12 int119862

0

1198621198632119865 [120581 120583 + 2120598 + 119901 + Π + 2Ω3 + 119864 (119902 + 120598) + 11988031198651198631198621198602119862 + 1198802 119863119879119865119863119862119860 + 119865119862 (1198631198621198634119863 minus 119863119879119862 )

minus 119880119864119863119862119865(119864119863119862119860 minus 1198631198791198611198621015840 ) + 119863119879119865119863119879A+ 119865( 11198621015840 (11986311987911986210158402119862 + 1198631198791198631015840

2119863 minus 119864119863119879119861119863119862119863119863 ) + 1198642 (119863119862119860119862 + 119863119862119860119863119862119863119863 )) minus 119881211986521198632

119862(119880119863119862119860 + 1198631198621198632119863 ) minus 1

+ 11988111986521198621015840119863 (1198621198631198791198631015840

119863 + 1198631198791198621015840) + 119880119881119865119863 ( 12119862 minus 119863119879119862 )

minus 119864211986511986311986211986321198632(119862119863119862119862119863 + 119863119862119863 minus 119862 (119863119862119863)22119863 minus 1198641198621198631198621198611198631198621198631198621015840 ) minus 12119862 (119863119862119862119862 minus 31198631198621198632119863 minus 1198641198631198621198611198621015840 )

+ 12119863 (119863119862119863119863119862119862119862 minus 119863119862119862119863) minus 1198641198621015840 (119863119879119865119863119879119861 + 1198642119863119862119865119863119862119861 + 119865119863119879119861119863119879119863119863 ) + 119863119879119879119865 + 11986510158401015840119864211986210158402 minus 11988011986311987911986510158401198621015840

+ 119865119863119879119879119862119862 + 119865119863119879119879119863119863 ]119889119862

(53)


The above equation gives the total mass in terms of C-energyinside the cylinder with the contribution of 119891(119877) dark sourceterm Toobserve the dynamical equation by assumingMisnerand Sharp approach[2 57] the contracted Bianchi identitiestake the form

(119879(119898)120572120573 + 119879(119863)120572120573 )120573 119881120572 = 0(119879(119898)120572120573 + 119879(119863)120572120573 )120573 120594120572 = 0

(54)

which produce

1119860 [ (120583 + 120598) + (120583 + 119901 + Π + 2120598 + 2Ω3 ) 119861+ (120583 + 119901 + Π + 120598 minus Ω3 )(119862 + 119863)]+ 1119861 [(119902 + 120598)1015840 + (119902 + 120598) (2119860

1015840

119860 + (119862119863)1015840119862119863 )] minus 1198631

= 0

(55)

1119860 [ (119902 + 120598) + (119902 + 120598) (2119861 + 119862 + 119863)]+ 1119861 [(119901 + Π + 120598 + 2Ω3 )1015840

+ (120583 + 119901 + Π + 2120598 + 2Ω3 ) 1198601015840

119860 + (120598 + Ω) (119862119863)1015840119862119863 ]+ 1198632 = 0

(56)

Here 1198631 and 1198632 are the dark source components shown inAppendix (A3) and (A4) respectively

From (23) and (48)-(50) it follows that

119863119879 (120583 + 120598) + 13 (3120583 + 3119901 + 3Π + 4120598)Θ+ 13 (120598 + Ω)( 2119860119861 minus 119880119862 minus 119881119863) + 119864119863119862 (119902 + 120598)+ (119902 + 120598) (21198861119861 + 119864119862 + 1198631015840

119861119863) minus 1198631 = 0(57)

119863119879 (119902 + 120598) + (119902 + 120598) (Θ + 119863119879119861119861 )+ 119864119863119862 (119901 + Π + 120598 + 2Ω3 )+ (120583 + 119901 + Π + 2120598 + 2Ω3 ) 1198861119861 + (120598 + Ω) (119862119863)1015840119861119862119863+ 1198632 = 0

(58)

The acceleration 119863119879119880 of the dissipative viscous collapsingsource is attained by using (23) and (48)-(50) it becomes

119863119879119880 = minus119898 (119903 119905)119862119863 + 1198978119862119863 minus 4120587119862119865 (119901 + Π + 120598 + 2Ω3 )+ (1198642 + 11986310158401198622119861119863) 1198861119861 minus 119862119865 [1198631198791198791198652+ 1198801198631198791198652 ( 1119862 minus 119863119879119862 ) + 1198811198631198791198652119863 minus 11986411986310158401198631198621198652119861119863minus 11986421198631198621198652 (119880119863119862119860 + 1119862)] + 1198631198791198791198622 minus 1198621198631198791198791198632119863minus 1198631198791198602119860 (119880 minus 119862119881119863 ) minus 14119863 (119880119881 minus 1198641198631015840

119861 )+ 181198632

(1198621198812 minus 11986211986310158402

1198612 ) + 18119862 (1198802 minus 1198642)minus 119862 (119891 (119877) minus 119877119865 (119877))

4119865

(59)

After inserting 1198861119861 from (59) into (58) we get

(120583 + 119901 + Π + 2120598 + 2Ω3 )119863119879119880 = minus(120583 + 119901 + Π + 2120598+ 2Ω3 )[119898 (119903 119905)119862119863 minus 1198978119862119863+ 4120587119862119865 (119901 + Π + 120598 + 2Ω3 ) + 119862 (119891 (119877) minus 119877119865 (119877))

4119865+ 1198621198631198791198791198652119865 + ( 1119862 minus 119863119879119862 ) 1198801198621198631198791198652119865 + 1198811198621198631198791198652119863119865minus 11986411986211986310158401198631198621198652119861119863119865 minus 11986421198621198631198621198652119865 ( 1119862 + 119880119863119862119860 )minus 1198631198791198791198622 + 1198621198631198791198791198632119863 + 1198631198791198602119860 (119880 minus 119881119862119863 )+ 14119863 (119880119881 minus 1198641198631015840

119861 ) minus 181198632(1198812119862 minus 11986211986310158402

1198612 )minus 18119862 (1198802 minus 1198642)] minus (1198642 + 11986310158401198622119861119863)sdot [119864119863119862 (119901 + Π + 120598 + 2Ω3 ) + 119864119862 (120598 + Ω)+ 1198631015840

119861119863 (120598 + Ω)] minus (1198642 + 11986310158401198622119861119863)[119863119879 (119902 + 120598)+ (119902 + 120598) (Θ + 119863119879119861119861 ) + 1198632]

(60)

In this interpretation to analyze the factor (120583 + 119901 + Π +2120598 + 2Ω3) that comes on the LHS and on the RHS this


is effective inertial mass and defines through equivalenceprinciple It can also be recognized as passive gravitationalmass On the RHS the first term of square bracket illus-trates the impacts of the collapsing variables on the activegravitational mass for the cylindrical collapsing object withthe dissipative dark source in f(R) metric theory this datumwas early pointed by Herrera et al [35] in the context ofGR Moreover in the second square bracket there is thegradient of the whole active pressure which is influenced bycollapsing variables and radiating density The final bracketincludes unalike quantities that define the collapsing natureof the source The second value of this bracket (119902 + 120598) ispositive concluding that dissipation is continuous in the formof heat flow and free streaming radiation and finally decreasesthe total energy of the system which reduces the degreeof collapse The last term of this bracket expresses the darkenergy source of the dissipative gravitating collapse

4 Transport Equations

The purpose of this profile is to discuss the full causaltechnique for the viscous dissipative gravitating collapse inself-gravitating system accompanied by heat transferenceThis study tells us that all collapsing variables should satisfythe transportation equations attained from causal thermody-namics Therefore we take the transportation equations forheat bulk and shear viscosity from 119872119897119897119890119903-Israel-Stewartformalism [41ndash43] for dissipative source Herrera et al [35]discussed transportation equations for heat bulk and shearviscosity The entropy flux is given by

119878120583 = 119878119899119881120583 + 119902120583119879 minus (1205730Π2 + 1205731119902]119902] + 1205732120587]120581120587]120581) 119881120583

2119879+ 1205720Π119902120583119879 + 1205721120587120583]119902]119879

(61)

where 1205731 and 1205732 are thermodynamic factors for unalikeadditions to entropy density 1205720 and 1205721 are thermodynamicsheat coupling factors and 119879 is temperature

Furthermore from the Gibbs equation and Bianchi iden-tities it follows that

119879119878120572120572 = minusΠ[119881120572120572 minus 1205720119902120572120572 + 1205730Π120572119881120572

+ 1198792 (1205730119879 119881120572)120572

Π] minus 119902120572 [ℎ120583120572 (ln119879)120583 (1 + 1205720Π)+ 119881120572120583119881120583 minus 1205720Π120572 minus 1205721120587120583120572120583 + 1205721120587120583120572ℎ120573120583 (ln119879)120573+ 1205731119902120572120583119881120583 + 1198792 (1205731119879 119881120583)

120583

119902120572] minus 120587120572120583 [120590120572120583minus 1205721119902120583120572 + 1205732120587120572120583]119881] + 1198792 (1205732119879 119881])

]120587120572120583]

(62)

Finally by the standard procedure the constitutive transportequations follow from the requirement 119878120572120572 ge 0

1205910Π120572119881120572 + Π = minus120577120579 + 1205720120577119902120572120572 minus 12120577119879( 1205910120577119879119881120572)120572

Π (63)

1205911ℎ120573120572119902120573120583119881120583 + 119902120572 = minus120581 [ℎ120573120572119879120573 (1 + 1205720Π) + 1205721120587120583120572ℎ120573120583119879120573+ 119879 (119886120572 minus 1205720Π120572 minus 1205721120587120583120572120583)] minus 121205811198792 ( 12059111205811198792119881120573)

120573

sdot 119902120572(64)

1205912ℎ120583120572ℎ]120573120587120583]120588119881120588 + 120587120572120573 = minus2120578120590120572120573 + 21205781205721119902⟨120573120572⟩minus 120578119879( 12059122120578119879119881])

]120587120572120573 (65)

with

119902⟨120573120572⟩ = ℎ120583120573ℎ]120572 (12 (119902120583] + 119902]120583) minus 13119902120590120581ℎ120590120581ℎ120583]) (66)

and where the relaxational times are given by

1205910 = 12057712057301205911 = 12058111987912057311205912 = 21205781205732

(67)

where 120577 and 120578 are the factors of bulk and shear viscosity Weuse the interior metric of the cylindrical structure from (63)-(65) to get the following set of equations

1205910Π = minus (120577 + 12059102 Π)119860Θ + 1198601198611205720120577 [1199021015840 + 119902(1198601015840

119860 + 1198621015840119862+ 1198631015840

119863 )] minus Π[1205771198792( 1205910120577119879) + 119860]

(68)

1205911 119902 = minus119860119861 1205811198791015840 (1 + 1205720Π + 231205721Ω) + 119879[1198601015840

119860 minus 1205720Π1015840

minus 231205721 (Ω1015840 + 1198601015840

119860 Ω + 32 (1198621015840

119862 + 1198631015840

119863 )Ω)]minus 119902[12058111987922

( 12059111205811198792 ) + 12059112 119860Θ + 119860] (69)

1205912Ω = minus120578(2119861 minus 119862 minus 119863) + 212057812057211198601198611199021015840

minus Ω[120578119879 ( 12059122120578119879) + 12059122 119860Θ + 119860] (70)


wherewe have to analyze the effect on several dissipative vari-ables for the cylindrical interior surface For this persistencewe use (69) and (60) and get

(120583 + 119901 + Π + 2120598 + 23Ω) (1 minus Λ)119863119879119880 = (1 minus Λ) 119865119892119903119886V+ 119865ℎ119910119889 + 119864(1198642 + 11986310158401198622119861119863)sdot 1205811205911 119863119862119879(1 + 1205720Π + 231205721Ω)minus 119879[1205720119863119862Π + 231205721 (119863119862Ω + 3 (119862119863)101584021198621198631198621015840 Ω)]+ (1198642 + 11986310158401198622119861119863)[120581119879

211990221205911 119863119879 ( 12059111205811198792 ) minus 119863119879120598] minus (1198642+ 11986310158401198622119861119863)[(1199022 + 2120598)Θ minus 1199021205911 + (119902 + 120598)

119863119879119861119861 + 1198632]

(71)

Here 119865119892119903119886V and 119865ℎ119910119889 are introduced into the form

119865119892119903119886V = minus(120583 + 119901 + Π + 2120598 + 2Ω3 ) [119898 (119903 119905) minus 1198978+ 4120587 (119901 + Π + 120598 + 2Ω3 ) 1198622119863119865+ 1198622119863(119891 (119877) minus 119877119865 (119877))

4119865 + 11986221198631198631198791198791198652119865+ 11988011986221198631198631198791198652119865 ( 1119862 minus 119863119879119862 ) + 11988111986221198631198791198652119865minus 119864119862211986310158401198631198621198652119861119865 minus 119864211986221198631198631198621198652119865 ( 1119862 + 119880119863119862119860 )minus 1198621198631198631198791198791198622 + 11986221198631198791198791198632 + 1198621198631198631198791198602119860 (119880 minus 119881119862119863 )+ 1198624 (119880119881 minus 1198641198631015840

119861 ) minus 1198628119863 (1198812119862 minus 11986211986310158402

1198612 )

+ minus1198638 (1198802 minus 1198642)] 1119862119863119865ℎ119910119889 = minus(1198642 + 11986310158401198622119861119863)[119864119863119862 (119901 + Π + 120598 + 2Ω3 )

+ 119864119862 (120598 + Ω) + 1198631015840

119861119863 (120598 + Ω)]

(72)

and Λ is defined as

Λ = 1205811198791205911 (120583 + 119901 + Π + 2120598 + 23Ω)minus1 (1 minus 231205721Ω) (73)

From (68) and (71) subsequently it becomes

(120583 + 119901 + Π + 2120598 + 23Ω) (1 minus Λ + Δ)119863119879119880 = (1 minus Λ+ Δ) 119865119892119903119886V + 119865ℎ119910119889 + 1198641205811205911 (

1198642 + 11986310158401198622119861119863)sdot 119863119862119879(1 + 1205720Π + 231205721Ω)minus 119879[1205720119863119862Π + 231205721 (119863119862Ω + 3 (119862119863)101584021198621198631198621015840 Ω)]minus 119864(1198642 + 11986310158401198622119861119863)(120583 + 119901 + Π + 2120598 + 23Ω)sdot Δ(119863119862119902119902 + (119862119863)10158401198621198631198621015840) + (1198642 + 11986310158401198622119861119863)sdot [120581119879211990221205911 119863119879 ( 12059111205811198792 ) minus 119863119879120598] + (1198642 + 11986310158401198622119861119863)[ 1199021205911minus (119902 + 120598) 119863119879119861119861 minus 1198632] + (1198642 + 11986310158401198622119861119863) Δ1205720120577119902 (120583+ 119901 + Π + 2120598 + 23Ω)[1 + 1205771198792 119863119879 ( 1205910120577119879)]Π+ 1205910119863119879Π

(74)

where Δ is recognized as

Δ = 1205720120577119902 (120583 + 119901 + Π + 2120598 + 23Ω)minus1 ( 119902 + 21205982120577 + 1205910Π) (75)

In this account once we have made the interpretation of thecausal transportation equations and then coupled with thedynamical equation we found the factor 1minusΛ+Δ the roll ofthis factor reduced the inertial energy density and effectivegravitational mass density that appears in the system Thisconsequence agrees with the findings of Herrera et al [35]

5 Conclusions

The beginning of the 20th century after presenting generaltheory of relativity Weyl [58] and Levi-Civita [59] discussedthe current issues with static cylindrical symmetric struc-tures In the starting scenario astrophysicists were focusedto find the self-gravitating models that are axially symmetricThe relativistic fluids having dissipative nature are very usefulin the formulation of stellar models Therefore one cannot disagree with the effects of dissipation during the self-gravitational collapse

In this format we have arranged the dynamical equationwhich plays the meaningful role in progressive stages ofthe viscous dissipative gravitational collapse We notice theimpacts of 119891(119877) terms in the dynamical progression forthe dissipative case this dissipation is undergone in theform of heat flow and free streaming approximation Wehave considered the convenient formation of the collapsing


variables involved in the transportation equations in terms ofheat flux bulk and shear viscosity ensuing from causal ther-modynamical approachMoreover the dynamical equation iscoupled with full causal transportation equations for gettingthemeaningful results that appear in the form of heat dissipa-tion bulk and shear viscosity In extensive point of view themain debate of the general thermodynamical approach is tonotice the relaxation phase whose direction either less thanor equal to radiating time It is interesting to mention thatthe hyperbolic concept of the collapse is much consistent andhaving rare problems than parabolic concept [60ndash62]

A full causal technique has been implemented in [35] toevaluate the impacts on collapsing variables with sphericallysymmetric collapse geometry this study gives the expressiveand momentous contribution in relativistic physics Theapplication of this work is obtained by some astral objects andrelativistic structures These massive systems have possessivenature in dissipative collapsing situation in which dissipationis undergone in the formof heat flow and radiation streamingThe system was weakened due to leaving the energy fromthe system and continues collapse Furthermore in thisconnection it is worth mentioning that thermodynamicalviscousheat coupling factors have nonterminating possibilitydue to dissipative gravitational collapse this framework givesthe momentous foundation in the modeling of relativisticstructures In this respect we see another previous work givenby [63] a partial technique is used to discuss the impactsof bulk dissipative gravitating collapse in cylindrical surfaceof the star with leaving the thermodynamical heat radiatingcoupling factors in the transport equations

Finally in this evaluation we analyze the full causaltechnique with viscous dissipative gravitational collapse andget the dynamical equation (74) which explain how passivegravitational mass and heat coupling factors influence thegravitational collapse of radiating source The passive grav-itational mass density is affected by bulk and shear viscosityand these factors are responsible for decreasing the internalradial pressure of the collapsing source These factors alsodecrease the force of gravity in a system this lessening dueto huge dissipation in the form of thermal conduction ofthe gravitational collapse Consequently the outflow of heatreduces the total energy of the collapsing cylindrical objectHerrera et al [64] investigated the bouncing action predictedin mathematical framework for the current arithmetic calcu-lation The present work is reproduced in future with othermodified theories such as Gauss-Bonnet and 119891(119877 119879) gravitytheories This work has been done for spherical symmetrywith and without electromagnetic field [65]

Appendix

11987911986301 = 1120581 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 ) (A1)

11987911986311 = 1198612120581 (119891 (119877) minus 119877119865 (119877)2 ) + 1198602+ 1198602

(119862 + 119863minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 ) (A2)

1198631 = 119860120581 [ 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )1

+ 119891 (119877) minus 119877119865 (119877)21198602minus 1198651015840101584011986021198612

+ 1198604(119861 + C119862 + 119863)

minus 119865101584011986021198612 (1198621015840

119862 + 1198631015840

119863 minus 1198611015840119861 )0

+ 21198603119891 (119877) minus 119877119865 (119877)2 minus 119865101584010158401198612

+ 1198602(119861 + 119862 + 119863) minus 11986510158401198612 (119862

1015840

119862 + 1198631015840

119863 minus 1198611015840119861 )+ 1198602119861 minus 1198602

minus 119865101584010158401198612 + 1198602(119860 + 119861)

+ 11986510158401198612 (1198601015840

119860 + 1198611015840119861 ) + 1198602119862 minus 1198602+ 1198602

119860+ 11986510158401198612 119860

1015840

119860 + 1198602119863 minus 1198602+ 1198602

119860 + 11986510158401198612 1198601015840

119860 + 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )(31198601015840

119860 + 1198611015840119861 + 1198621015840119862+ 1198631015840

119863 )]

(A3)

1198632 = 119861120581 [minus 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )0

+ 119891 (119877) minus 119877119865 (119877)21198612 + 11986021198612+ 11986021198612 (119862 + 119863 minus 119860)minus 11986510158401198614 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 )1

+ 1198601015840

1198601198612 1198602+ 119865101584010158401198612

minus 1198602(119860 + 119861) minus 11986510158401198612 (119860

1015840

119860 + 1198611015840119861 )+ 211986110158401198613 119891 (119877) minus 119877119865 (119877)2 + 1198602

+ 1198602(119862 + 119863 minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 )+ 11986210158401198612119862 119865101584010158401198612 minus 1198602

119861 minus 11986510158401198612 1198611015840

119861 + 1198631015840

1198612119863 119865101584010158401198612minus 1198602

119861 minus 11986510158401198612 1198611015840

119861 minus 111986021198612 (119860 + 3119861 + 119862 + 119863)sdot ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )]

(A4)


Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

One of the authors G Abbas appreciates the financial sup-port from HEC Islamabad Pakistan under NRPU projectwith Grant no 20-4059NRPUR amp DHEC141217

References

[1] J R Oppenheimer and H Snyder ldquoOn continued gravitationalcontractionrdquo Physical Review A Atomic Molecular and OpticalPhysics vol 56 no 5 pp 455ndash459 1939

[2] C W Misner and D H Sharp ldquoRelativistic equations for adi-abatic spherically symmetric gravitational collapserdquo PhysicalReview A Atomic Molecular and Optical Physics vol 136 no2 pp B571ndashB576 1964

[3] P C Vaidya ldquoThe gravitational field of a radiating starrdquoProceedings of the Indian Academy of Science vol A33 p 2641951

[4] S Capozziello ldquoCurvature quintessencerdquo International Journalof Modern Physics D vol 11 pp 483ndash492 2002

[5] M Sharif and Z Yousaf ldquoStability of a Class of Non-Static AxialSelf-Gravitating Systems in f(R)GravityrdquoAstrophysics and SpaceScience vol 352 p 943 2014

[6] M K Mak T Harko and R Soc ldquoAnisotropic Stars in GeneralRelativityrdquo Proceedings of the Royal Society vol A459 p 3932003

[7] L Herrera andNO Santos ldquoJeansmass for anisotropicmatterrdquoThe Astrophysical Journal vol 438 p 308 1995

[8] F Weber Pulsars as Astrophysical Laboratories for Nuclearand Particle Physics vol BS1 Bristol UK Institute of PhysicsPublishing Dirac House Temple Back 1999

[9] S Chakraborty S Chakraborty and U Debnath ldquoRole of Pres-sure in Quasi-Spherical Gravitational Collapserdquo InternationalJournal of Modern Physics vol D14 p 1707 2005

[10] R Garattini ldquoNaked Singularity in a Modified Gravity TheoryrdquoJournal of Physics Conference Series Article ID 012066 2009

[11] M Sharif and M Zaeem Ul Haq Bhatti ldquoStructure scalarsfor charged cylindrically symmetric relativistic fluidsrdquo GeneralRelativity and Gravitation vol 44 no 11 pp 2811ndash2823 2012

[12] M Sharif and M Z Bhatti ldquoStructure scalars in charged planesymmetryrdquo Modern Physics Letters A vol 27 no 27 p 12501412012

[13] M Sharif and Z Yousaf ldquoEvolution of expansion-free self-gravitating fluids and plane symmetryrdquo International Journal ofModern Physics vol D21 Article ID 1250095 2012

[14] M Sharif and Z Yousaf ldquoExpansion-free Cylindrically Sym-metric Modelsrdquo Canadian Journal of Physics vol 90 p 8652012

[15] M Sharif and H R Kausar ldquoEffects of f(R) model on thedynamical instability of expansionfree gravitational collapserdquoJournal of Cosmology and Astroparticle Physics vol 7 p 0222011

[16] M Sharif and M Azam ldquoEffects of Electromagnetic Field onthe Dynamical Instability of Cylindrical Collapserdquo Journal ofCosmology and Astroparticle Physics vol 02 p 043 2012

[17] M Sharif and S Fatima ldquoCharged cylindrical collapse ofanisotropic fluidrdquoGeneral Relativity andGravitation vol 43 no1 pp 127ndash142 2011

[18] M Sharif and G Abbas ldquoDynamics of Non-adiabatic ChargedCylindrical Gravitational Collapserdquo Astrophysics and SpaceScience vol 335 p 515 2011

[19] G Cognola E Elizalde S Nojiri S D Odintsov and SZerbini ldquoOne-loop f(R) gravity in de Sitter universerdquo Journalof Cosmology and Astroparticle Physics vol 02 Article ID 010p 010 2005

[20] S Capozziello and M de Laurentis ldquoExtended Theories ofGravityrdquo Physics Reports vol 509 no 4-5 pp 167ndash321 2011

[21] E J Copeland M Sami and S Tsujikawa ldquoDynamics of darkenergyrdquo International Journal of Modern Physics D vol 15 no11 pp 1753ndash1936 2006

[22] L Amendola R Gannouji D Polarski and S TsujikawaldquoConditions for the cosmological viability of f (R) dark energymodelsrdquo Physical Review D Particles Fields Gravitation andCosmology vol 75 no 8 Article ID 083504 2007

[23] S Chandrasekhar ldquoThedynamical instability of gaseousmassesapproaching the Schwarzschild limit in general relativityrdquo TheAstrophysical Journal vol 140 pp 417ndash433 1964

[24] L Herrera G le Denmat and N O Santos ldquoDynamical insta-bility for nonadiabatic spherical collapserdquo Royal AstronomicalSociety Monthly Notices vol 237 no 2 pp 257ndash268 1989

[25] G Abbas S Nazeer andM A Meraj ldquoCylindrically symmetricmodels of anisotropic compact starsrdquo Astrophysics and SpaceScience vol 354 p 449 2014

[26] M K Mak and T Harko ldquoQuark stars admitting a one-parameter group of conformal motionsrdquo International Journalof Modern Physics vol D13 p 149 2004

[27] F Rahaman R Sharma S Ray R Maulick and I KararldquoStrange stars in Krori-Barua space-timerdquoThe European Physi-cal Journal C vol 72 p 2071 2012

[28] L Herrera ldquoCracking of self-gravitating compact objectsrdquoPhysics Letters A vol 165 no 3 pp 206ndash210 1992

[29] L Herrera D Prisco and A Ibanez ldquoOn the Role of ElectricCharge and Cosmological Constant in Structure Scalarsrdquo Phys-ical Review A vol D84 p 107501 2011

[30] L Herrera J Ospino and A D Prisco ldquoAll static sphericallysymmetric anisotropic solutions of Einsteinrsquos equationsrdquo Physi-cal Review vol D77 p 027502 2008

[31] L Herrera N O Santos and A Wang ldquoShearing Expansion-free Spherical Anisotropic Fluid Evolutionrdquo Physical Review pD78 2008

[32] L Herrera and N O Santos ldquoDynamics of dissipative gravita-tional collapserdquo Physical Review vol D70 p 084004 2004

[33] L Herrera G Le Denmat and N O Santos ldquoDynamicalinstability and the expansion-free conditionrdquo General Relativityand Gravitation vol 44 no 5 pp 1143ndash1162 2012

[34] A D Prisco L Herrera J Ospino N O Santos and V MVina-Cervantes ldquoExpansion-Free Cavity Evolution Some exactAnalyticalModelsrdquo International Journal ofModern Physics volD20 p D20 2011

[35] L Herrera A D Prisco E Fuenmayor and O TroconisldquoDynamics of viscous dissipative gravitational collapse A fullcausal approachrdquo International Journal of Modern Physics volD18 p 129 2009


[36] A Di Prisco L Herrera G Le Denmat M A H MacCallumandNO Santos ldquoNonadiabatic charged spherical gravitationalcollapserdquo Physical Review D Particles Fields Gravitation andCosmology vol 76 Article ID 064017 2007

[37] L Herrera M A H Maccallum and N O Santos ldquoOn theMatchingConditions for theCollapsingCylinderrdquoClassical andQuantum Gravity vol 24 p 1033 2007

[38] A Di Prisco L Herrera M A MacCallum and N O SantosldquoShearfree cylindrical gravitational collapserdquo Physical ReviewD Particles Fields Gravitation and Cosmology vol 80 no 6Article ID 064031 10 pages 2009

[39] B C Nolan ldquoNaked singularities in cylindrical collapse ofcounterrotating dust shellsrdquo Physical Review vol D65 p104006 2002

[40] S A Hayward ldquoGravitational waves black holes and cosmicstrings in cylindrical symmetryrdquo Classical and Quantum Grav-ity vol 17 p 1749 2000

[41] I Muller ldquoZum Paradoxon der WarmeleitungstheorierdquoZeitschrift fur Physik vol 198 no 4 pp 329ndash344 1967

[42] W Israel ldquoNonstationary irreversible thermodynamics a causalrelativistic theoryrdquo Annals of Physics vol 100 no 1-2 pp 310ndash331 1976

[43] W Israel and J M Stewart ldquoTransient relativistic thermody-namics and kinetic theoryrdquo Annals of Physics vol 118 no 2 pp341ndash372 1979

[44] R Maartens ldquoCausal Thermodynamics in Relativityrdquo Astro-physics Article ID 9609119

[45] R Chan ldquoRadiating gravitational collapse with shear viscosityrdquoMonthly Notices of the Royal Astronomical Society vol 316 p588 2000

[46] K SThorne ldquoEnergy of infinitely long cylindrically symmetricsystems in general relativityrdquo Physical Review B CondensedMatter and Materials Physics vol 138 p B251 1965

[47] S M Shah and G Abbas ldquoDynamics of charged viscous dissi-pative cylindrical collapse with full causal approachrdquo EuropeanPhysical Journal vol A53 p 53 2017

[48] T Chiba ldquoCylindrical dust collapse in general relativity towardhigher dimensional collapserdquo Progress of Theoretical and Exper-imental Physics vol 95 no 2 pp 321ndash338 1996

[49] H Chao-Guang ldquoCharged static cylindrical black holerdquo ActaPhysica Sinica vol 4 no 8 pp 617ndash630 1995

[50] G Darmois Memorial des Sciences Mathematiques vol 25Gautheir-Viuars Paris France 1927

[51] S Guha and R Banerji ldquoDissipative cylindrical collapse ofcharged anisotropic fluidrdquo International Journal of TheoreticalPhysics vol 53 no 7 pp 2332ndash2348 2014

[52] J M M Senovilla ldquoJunction conditions for F(R)-gravity andtheir consequencesrdquo Physical Review vol 88 p 064015 2013

[53] T Clifton P K S Dunsby R Goswami and A M Nzioki ldquoOnthe absence of the usual weak-field limit and the impossibilityof embedding some known solutions for isolated masses incosmologies with f(R) dark energyrdquo Physical Review vol D87p 063517 2013

[54] N Deruelle M Sasaki and Y Sendouda ldquoJunction Conditionsin f(R) Theories of Gravityrdquo Progress of Theoretical and Experi-mental Physics vol 119 p 237 2008

[55] A Ganguly R Gannouji R Goswami and S Ray ldquoNeutronstars in the Starobinsky modelrdquo Physical Review D ParticlesFields Gravitation and Cosmology vol 89 no 6 2014

[56] R Goswami A M Nzioki S D Maharaj and S G GhoshldquoCollapsing spherical stars in f(R) gravityrdquo Physical Review Dvol 90 no 8 2014

[57] CWMisner and D Sharp ldquoRelativistic equations for sphericalgravitational collapsewith escaping neutrinosrdquoPhysical Reviewvol B137 p 1360 1965

[58] H Weyl ldquoOn the Theory of Gravitationrdquo Annalen der Physik(Leipzig) vol 54 p 117 1917

[59] T Levi-Civita ldquod2 einsteiniani in campi newtoniani IXLrsquoanalogo del potenziale logaritmicordquo Atti Della RealeAccademia Dei Lincei Rendiconti vol 28 p 101 1919

[60] C Cattaneo ldquoSulla conduzione del calorerdquo Atti del SeminarioMatematico e Fisico dellrsquo Universita di Modena e Reggio Emiliavol 3 p 3 1948

[61] A Anile D Pavon and V Romano ldquoThe Case for HyperbolicTheories of Dissipation in Relativistic Fluidsrdquo httpsarxivorgabsgr-qc9810014

[62] L Herrera and D Pavon ldquoHyperbolic theories of dissipationWhy and when do we need themrdquo Physica vol A307 p 1212002

[63] S M Shah and G Abbas ldquoDynamics of charged bulk viscouscollapsing cylindrical source with heat fluxrdquo Journal of Physicsvol C77 p 251 2017

[64] L Herrera A Di Prisco and W Barreto ldquoThermoinertialbouncing of a relativistic collapsing sphere A numericalmodelrdquo Physical Review D Particles Fields Gravitation andCosmology vol 73 no 2 2006

[65] G Abbas and H Nazar Dissipative and Viscous GravitationalCollapse in f(R) Gravity With Full Causal Approach (submittedfor publication 2018)

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Webber [8] examined the solutions of pressure anisotropyin dense models with electromagnetic field Chakraborty etal [9] analyzed the gears of tangential pressure and alongwith radial pressure in stellar structures by quasisphericalparadigms in the form of collapse Garattini [10] evaluatedthe formation of exposed singularity throughmass limitationin metric 119891(119877) theory of gravity Sharif and his coresearchers[11ndash18] studied the various consequences of energy densityinhomogeneity and Weyl tensor with gravitational collapsein dense objects with General relativity theory as well as inmodified theories of gravity Cognola et al [19] introducedthe results of spherical imploding object for the black hole in119891(119877) context by the parameter of positive Ricci invariant 119877Capozziello et al [20] examined the modified Lane-Emdenequation in context of metric 119891(119877) gravity theory and studiedthe hydrostatic periods of astral models Copeland et al[21] analyzed the several concepts for the rushing growthof universe Amendola et al [22] discussed the feasibleparadigms of metric 119891(119877) gravity in the context of bothEinstein and Jordan frames

Chandrasekhar [23] introduced the impacts on the con-stancy of dynamics with perfect fluid collapsing geometryHerrera et al [24] discussed the results of dynamical dise-quilibrium with spherically symmetric nonadiabatic collapseand found the heat flow out from the gravitating source in theform of radial heat flux Abbas et al [25] presented the resultsof compact star models in anisotropic fluid distributionwith cylindrical symmetric static spacetime geometry Makand Harko [26] studied the kind of exact solutions of fieldequations by considering the spherical symmetric structureand also conferred the energy density along with tangentialand radial pressure that are limited and increased in the coreof the imperfect fluid object Rahaman et al [27] prolongedthe Krori-Barua solution with spherical symmetric staticspacetime for the investigation of charge imperfect fluiddistribution Herrera and his collaborators [28ndash33] did workin different directions such as self-gravitating dens modelsdiscussed with local anisotropic matter distribution in thecracking spherical symmetric structure in context of equa-tion of state with perturbed approach They established thevarious results in gravitational collapse for the dissipative caseby using Misner and Sharp approach such gravitating sourceundergoes in form of free radiation streaming and diffusionapproximations They investigated the expansion free con-ditions for the spherically symmetric anisotropic dissipativecollapsing matter Further Herrera and his collaborators [34ndash38] have evaluated the Lake algorithm in static sphericallysymmetric anisotropic configuration for the determination ofnew formalism of two functions (instead of one) to generateall possible results They studied the results of structurescalars 119884119879119865 for the charged dissipative anisotropic collapsinggeometry in presence of cosmological constant They havealso done detailed work on the dynamical disequilibriumwith spherically symmetric locally anisotropic fluid whichcollapses adiabatically under the expansion-free conditionIn few investigations they have used the noncasual as wellas casual approaches to discuss the dynamics of gravitatingsource Nolan [39] examined the effects on gravitationalcollapse for counter rotating dust cloud with cylindrical

paradigms and found the naked singularity Hayward [40]produced the results of cylindrical geometry for black holesgravitational waves and cosmic strings

In this paper we analyze the self-gravitational collapsewith cylindrical symmetric geometry in context of metric119891(119877) gravity theory The main persistence of this study is toexamine the viscous dissipative collapse in the form of radialheat flow and free radiation streaming in cylindrical nonstaticspacetime with full causal approach

The proposal of this work is as follows In Section 2we express the cylindrical symmetric isotropic collapsingmatter distribution and its related variables in context of119891(119877)theory of gravity Section 3 consists of Einstein modified fieldequations matching conditions are given for the interior andexterior manifolds119872minus and119872+ and the dynamical equationthat was accompanied with physical parameters such as heatflux bulk and shear viscosity Section 4 is devoted to thetransportation equations attained in frame of119872119897119897119890119903-Israel-Stewart formalism [41ndash43] then transportation equations areassociated with dynamical equation Finally the conclusionsof the study have been discussed in the last section

2 Self-Gravitational Collapsing Geometry andRelated Conventions

In this section we consider a locally isotropic collapsingfluid with full causal description in context of Misner-Sharp approach formed by cylindrically symmetric nonstaticinterior and exterior geometry of the dissipative fluid whichundergoes dissipation in the form of heat flow and freestreaming radiation In general relativity the Einstein-Hilbert(EH) action can be expressed as follows

119878119864119867 = 12120581 int 1198894119909radicminus119892119877 (1)

The extended formulation of the Einstein-Hilbert (EH) in119891(119877) context is119878119898119900119889119894119891 = 12120581 int1198894119909radicminus119892 (119891 (119877) + 119871 (119898119886119905119905119890119903)) (2)

where 119891(119877) is arbitrary function of Ricci scalar 119877 120581 is cou-pling constant and 119871 (119898119886119905119905119890119903) the Lagrangian matter density ofthe action The following modified 119891(119877) field equations areobtained by the variation of (2) with respect to 119892120572120573119865 (119877)119877120572120573 minus 12119891 (119877) 119892120572120573 minus nabla120572nabla120573119865 (119877) + 119892120572120573nabla120572nabla120572119865 (119877)

= 120581119879120572120573(3)

where 119865(119877) = 119889119891(119877)119889119877 nabla120572nabla120572 is the Drsquo Alembert oper-ator nabla120572 denotes the covariant derivative and 119879120572120573 is theisotropic energy-momentum tensorThe above equations canbe expressed in Einstein tensor as given below

119866120572120573 = 120581119865 (119879119898120572120573 + 119879119863120572120573) (4)


where


(5)


119889119904minus2 = minus11986021198891199052 + 11986121198891199032 + 11986221198891206012 + 11986321198891199112 (6)



+ 120598119897120572119897120573 + 120587120572120573 (7)


119881120572119881120572 = minus1119881120572119902120572 = 0119897120572119881120572 = minus1119897120572119897120572 = 0

(8)

120587120583]119881] = 0120587[120583]] = 0120587120572120572 = 0

(9)


120587120572120573 = minus2120578120590120572120573Π = minus120577Θ (10)



120590120572120573 = 119881(120572120573) + 119886(120572119881120573) minus 13Θℎ120572120573 (11)


119886120572 = 119881120572120573119881120573Θ = 119881120572

120572 (12)



(13)


1205870120572 = 012058711 = minus212058722 = minus212058733 (14)


120587120572120573 = Ω(120594120572120594120573 minus 13ℎ120572120573) (15)


120594120572120594120572 = 1120594120572119881120572 = 0120594120572 = 119861minus11205751205721

(16)



12059011 = 11986123119860 [Σ1 minus Σ3] 12059022 = 11986223119860 [Σ2 minus Σ1] 12059033 = 1198632

3119860 [Σ3 minus Σ2] (17)

where

120590120572120573120590120572120573 = 1205902 = 131198602[Σ21 + Σ22 + Σ23] (18)

where


(19)


1198861 = 119860 Θ = 1119860 (119861 + 119862 + 119863)

(20)




(119860119861)2 [minus11986210158401015840119862 minus 11986310158401015840

119863 + 1198611015840119861 (1198621015840119862 + 1198631015840

119863 ) minus 11986210158401198631015840

119862119863 ]

+ (119861119862 + 119861119863 + 119862119863) = 120581119865 [(120583 + 120598)1198602

+ 1198602

120581 minus(119891 (119877) minus 119877119865 (119877)2 )

minus 1198602(119861 + 119862 + 119863) + 119865101584010158401198612

+ 11986510158401198612 (1198621015840

119862 + 1198631015840

119863 minus 1198611015840119861 )]

(21)

minus ( 1198621015840119862 + 1015840

119863 minus 1198621015840119861119862 minus 1198631015840

119861119863 minus 1198601015840

119860119862 minus 1198601015840

119860119863 )



10158401198631015840

119862119863+ 11986010158401198621015840119860119862 + 11986010158401198631015840

119860119863 ) = 120581119865 [(119901 + Π + 120598 + 2Ω3 )1198612

+ 1198612120581 (119891 (119877) minus 119877119865 (119877)2 ) + 1198602

+ 1198602(119862 + 119863 minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 )]

(23)

minus (119862119860)2 [119861 + 119863 minus 119860 (119861 + 119863) + 119861119863] + (119862119861)

2

sdot [11986010158401015840

119860 + 11986310158401015840

119863 minus 1198601015840

119860 (1198611015840119861 minus 1198631015840

119863 ) minus 11986110158401198631015840

119861119863 ]

= 120581119865 [(119901 + Π minus Ω3 )1198622



+ 11986510158401198612 (1198601015840

119860 minus 1198611015840119861 + 1198621015840119862 + 1198631015840

119863 )]

(24)

minus (119863119860)2 [119861 + 119862 minus 119860 (119861 + 119862) + 119861119862] + (119863119861 )

2

sdot [11986010158401015840

119860 + 11986210158401015840119862 minus 1198601015840


minus 1198632



+ 11986510158401198612 (1198601015840

119860 minus 1198611015840119861 + 1198621015840119862 + 1198631015840

119863 )]

(25)





119898(119903 119905) = 119864119897= 1198978+ 18119863 [ 11198602

(119862 + 119863)2 minus 11198612 (1198621198631015840 + 1198621015840119863)2] (27)



+ 2 (1198891206012 + 12057421198891198852) (28)



119896minus (119905 119903) = 119903 minus 119903Σ = 0 (29)


1198891199042minus Σ997904997904997904 minus1198891205912 + 11986221198891206012 + 11986321198891199112 (30)


where

119889120591 Σ997904997904997904 119860119889119905 (31)




119896+ (] ) = minus Σ (]) = 0 (32)


1198891199042+ Σ997904997904997904 minus2(minus (]) + 119889Σ (])119889] )119889]2

+ 2 (1198891206012 + 12057421198891199112) (33)



1198891199042Σ = (1198891199042minus)Σ = (1198891199042+)Σ (34)


[119870120572120573] = 119870+120572120573 minus 119870minus

120572120573 = 0 (35)



120597119909119886plusmn120597119909119887plusmn120597120588120572120597120588120573 ) (120583 119886 119887 = 0 1 2 3)

(36)


120582 = 12 (minus (]) + 119889Σ (]) 119889])12

(37)


119862 (119905 119903Σ) Σ997904997904997904 Σ (]) 119863 (119905 119903Σ) Σ997904997904997904 120574Σ (])

119889119905119889120591 = 1119860 119889]119889120591 = 120582

(38)


are

119870minus00 = (minus 1198601015840

119860119861)Σ

(39)

119870+00 = [(119889]119889120591)

minus1 (1198892]1198891205912) minus 1198772 (

119889]119889120591)]Σ

(40)

119870minus22 = (1198621015840119862119861 )

Σ

(41)

119870minus33 = (1198631015840119863119861 )

Σ

(42)

119870+22 = [(119889119889120591 ) minus 2 (119889]119889120591)]

Σ

= 120574minus2119870+33 (43)


(]) Σ997904997904997904 2 [(

119860)2 minus (1015840119861 )

2] (44)

119864 Σ997904997904997904 1198978 + 120574 (]) (45)

119902 Σ997904997904997904 (119875 + Π + 2Ω3 ) + 11987911986301119860119861 + 119879119863111198612 (46)






119877|plusmn = 0119891119877119877 [120597]1198771003816100381610038161003816plusmn = 0

119891119877119877 = 0(47)



119863119879 = 1119860 120597120597119905 119863119862 = 11198621015840 120597120597119903

(48)


119880 = 119863119879119862 = 119860119881 = 119863119879119863 = 119860

(49)



12 minus 1198621198631015840

119861119863 (50)



minus 119880 119864211986210158402 (11986510158401198621015840119862 + 11986510158401198631015840


1198651198631198791198604119862 + 1198802 1198642 (311986311987911986121198641198621015840 minus 119865101584011986311986211986021198621015840 ) + 119881

119863+ 119865119863119879119861119861 ( 1119862 minus 119863119879119862 )minus 1198642


(119862119863 + 119862119863119879119860 ) + 1198811198654119863 119862119863119879119879119863119863 + 119863119879119879119862

+ 1198642 (119862 (119863119862119863)221198632minus 12119862)

minus 119864221198621015840 119864119863119879119861(119863119862119865 + 1198652119862 minus 119862 (119863119862119863)2 11986521198632)

minus 119865(11986311987911986210158402119862 + 1198631198791198631015840

2119863 )+ 1198651198631198621198632119863 (1198621198631198791198631015840

119863 + 1198631198791198621015840) minus 1198631198791198651015840] (51)




119863119862119898 (119903 119905) = 1198621198632119865 [120581 120583 + 2120598 + 119901 + Π + 2Ω3 + 119864 (119902


2119863 minus 119864119863119879119861119863119862119863119863 )+ 1198642 (119863119862119860119862 + 119863119862119860119863119862119863119863 ))

minus 119881211986521198632119862(119880119863119862119860 + 1198631198621198632119863 ) minus 1

+ 11988111986521198621015840119863 (1198621198631198791198631015840

119863 + 1198631198791198621015840) + 119880119881119865119863 ( 12119862minus 119863119879119862 ) minus 119864211986511986311986211986321198632

(119862119863119862119862119863 + 119863119862119863


minus 31198631198621198632119863 minus 1198641198631198621198611198621015840 ) + 12119863 (119863119862119863119863119862119862119862

minus 119863119862119862119863) minus 1198641198621015840 (119863119879119865119863119879119861 + 1198642119863119862119865119863119862119861

+ 119865119863119879119861119863119879119863119863 ) + 119863119879119879119865 + 11986510158401015840119864211986210158402 minus 11988011986311987911986510158401198621015840+ 119865119863119879119879119862119862 + 119865119863119879119879119863119863 ]

(52)


119898(119903 119905) = 12 int119862

0

1198621198632119865 [120581 120583 + 2120598 + 119901 + Π + 2Ω3 + 119864 (119902 + 120598) + 11988031198651198631198621198602119862 + 1198802 119863119879119865119863119862119860 + 119865119862 (1198631198621198634119863 minus 119863119879119862 )

minus 119880119864119863119862119865(119864119863119862119860 minus 1198631198791198611198621015840 ) + 119863119879119865119863119879A+ 119865( 11198621015840 (11986311987911986210158402119862 + 1198631198791198631015840

2119863 minus 119864119863119879119861119863119862119863119863 ) + 1198642 (119863119862119860119862 + 119863119862119860119863119862119863119863 )) minus 119881211986521198632

119862(119880119863119862119860 + 1198631198621198632119863 ) minus 1

+ 11988111986521198621015840119863 (1198621198631198791198631015840

119863 + 1198631198791198621015840) + 119880119881119865119863 ( 12119862 minus 119863119879119862 )



+ 119865119863119879119879119862119862 + 119865119863119879119879119863119863 ]119889119862

(53)



(119879(119898)120572120573 + 119879(119863)120572120573 )120573 119881120572 = 0(119879(119898)120572120573 + 119879(119863)120572120573 )120573 120594120572 = 0

(54)

which produce

1119860 [ (120583 + 120598) + (120583 + 119901 + Π + 2120598 + 2Ω3 ) 119861+ (120583 + 119901 + Π + 120598 minus Ω3 )(119862 + 119863)]+ 1119861 [(119902 + 120598)1015840 + (119902 + 120598) (2119860

1015840

119860 + (119862119863)1015840119862119863 )] minus 1198631

= 0

(55)

1119860 [ (119902 + 120598) + (119902 + 120598) (2119861 + 119862 + 119863)]+ 1119861 [(119901 + Π + 120598 + 2Ω3 )1015840

+ (120583 + 119901 + Π + 2120598 + 2Ω3 ) 1198601015840

119860 + (120598 + Ω) (119862119863)1015840119862119863 ]+ 1198632 = 0

(56)



119863119879 (120583 + 120598) + 13 (3120583 + 3119901 + 3Π + 4120598)Θ+ 13 (120598 + Ω)( 2119860119861 minus 119880119862 minus 119881119863) + 119864119863119862 (119902 + 120598)+ (119902 + 120598) (21198861119861 + 119864119862 + 1198631015840

119861119863) minus 1198631 = 0(57)

119863119879 (119902 + 120598) + (119902 + 120598) (Θ + 119863119879119861119861 )+ 119864119863119862 (119901 + Π + 120598 + 2Ω3 )+ (120583 + 119901 + Π + 2120598 + 2Ω3 ) 1198861119861 + (120598 + Ω) (119862119863)1015840119861119862119863+ 1198632 = 0

(58)



119861 )+ 181198632

(1198621198812 minus 11986211986310158402


4119865

(59)




119861 ) minus 181198632(1198812119862 minus 11986211986310158402


119861119863 (120598 + Ω)] minus (1198642 + 11986310158401198622119861119863)[119863119879 (119902 + 120598)+ (119902 + 120598) (Θ + 119863119879119861119861 ) + 1198632]

(60)






119878120583 = 119878119899119881120583 + 119902120583119879 minus (1205730Π2 + 1205731119902]119902] + 1205732120587]120581120587]120581) 119881120583

2119879+ 1205720Π119902120583119879 + 1205721120587120583]119902]119879

(61)



119879119878120572120572 = minusΠ[119881120572120572 minus 1205720119902120572120572 + 1205730Π120572119881120572

+ 1198792 (1205730119879 119881120572)120572


120583

119902120572] minus 120587120572120583 [120590120572120583minus 1205721119902120583120572 + 1205732120587120572120583]119881] + 1198792 (1205732119879 119881])

]120587120572120583]

(62)


1205910Π120572119881120572 + Π = minus120577120579 + 1205720120577119902120572120572 minus 12120577119879( 1205910120577119879119881120572)120572

Π (63)


120573

sdot 119902120572(64)

1205912ℎ120583120572ℎ]120573120587120583]120588119881120588 + 120587120572120573 = minus2120578120590120572120573 + 21205781205721119902⟨120573120572⟩minus 120578119879( 12059122120578119879119881])

]120587120572120573 (65)

with

119902⟨120573120572⟩ = ℎ120583120573ℎ]120572 (12 (119902120583] + 119902]120583) minus 13119902120590120581ℎ120590120581ℎ120583]) (66)


1205910 = 12057712057301205911 = 12058111987912057311205912 = 21205781205732

(67)


1205910Π = minus (120577 + 12059102 Π)119860Θ + 1198601198611205720120577 [1199021015840 + 119902(1198601015840

119860 + 1198621015840119862+ 1198631015840

119863 )] minus Π[1205771198792( 1205910120577119879) + 119860]

(68)

1205911 119902 = minus119860119861 1205811198791015840 (1 + 1205720Π + 231205721Ω) + 119879[1198601015840

119860 minus 1205720Π1015840

minus 231205721 (Ω1015840 + 1198601015840

119860 Ω + 32 (1198621015840

119862 + 1198631015840

119863 )Ω)]minus 119902[12058111987922

( 12059111205811198792 ) + 12059112 119860Θ + 119860] (69)


minus Ω[120578119879 ( 12059122120578119879) + 12059122 119860Θ + 119860] (70)





119863119879119861119861 + 1198632]

(71)




119861 ) minus 1198628119863 (1198812119862 minus 11986211986310158402

1198612 )


+ 119864119862 (120598 + Ω) + 1198631015840

119861119863 (120598 + Ω)]

(72)


Λ = 1205811198791205911 (120583 + 119901 + Π + 2120598 + 23Ω)minus1 (1 minus 231205721Ω) (73)




(74)


Δ = 1205720120577119902 (120583 + 119901 + Π + 2120598 + 23Ω)minus1 ( 119902 + 21205982120577 + 1205910Π) (75)


5 Conclusions







Appendix

11987911986301 = 1120581 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 ) (A1)

11987911986311 = 1198612120581 (119891 (119877) minus 119877119865 (119877)2 ) + 1198602+ 1198602

(119862 + 119863minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 ) (A2)

1198631 = 119860120581 [ 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )1

+ 119891 (119877) minus 119877119865 (119877)21198602minus 1198651015840101584011986021198612

+ 1198604(119861 + C119862 + 119863)

minus 119865101584011986021198612 (1198621015840

119862 + 1198631015840

119863 minus 1198611015840119861 )0

+ 21198603119891 (119877) minus 119877119865 (119877)2 minus 119865101584010158401198612

+ 1198602(119861 + 119862 + 119863) minus 11986510158401198612 (119862

1015840

119862 + 1198631015840

119863 minus 1198611015840119861 )+ 1198602119861 minus 1198602

minus 119865101584010158401198612 + 1198602(119860 + 119861)

+ 11986510158401198612 (1198601015840

119860 + 1198611015840119861 ) + 1198602119862 minus 1198602+ 1198602

119860+ 11986510158401198612 119860

1015840

119860 + 1198602119863 minus 1198602+ 1198602

119860 + 11986510158401198612 1198601015840

119860 + 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )(31198601015840

119860 + 1198611015840119861 + 1198621015840119862+ 1198631015840

119863 )]

(A3)



1015840

119860 + 1198621015840119862 + 1198631015840

119863 )1

+ 1198601015840

1198601198612 1198602+ 119865101584010158401198612

minus 1198602(119860 + 119861) minus 11986510158401198612 (119860

1015840

119860 + 1198611015840119861 )+ 211986110158401198613 119891 (119877) minus 119877119865 (119877)2 + 1198602

+ 1198602(119862 + 119863 minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 )+ 11986210158401198612119862 119865101584010158401198612 minus 1198602

119861 minus 11986510158401198612 1198611015840

119861 + 1198631015840

1198612119863 119865101584010158401198612minus 1198602

119861 minus 11986510158401198612 1198611015840


(A4)


Data Availability




Acknowledgments


References






































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






where


(5)


119889119904minus2 = minus11986021198891199052 + 11986121198891199032 + 11986221198891206012 + 11986321198891199112 (6)



+ 120598119897120572119897120573 + 120587120572120573 (7)


119881120572119881120572 = minus1119881120572119902120572 = 0119897120572119881120572 = minus1119897120572119897120572 = 0

(8)

120587120583]119881] = 0120587[120583]] = 0120587120572120572 = 0

(9)


120587120572120573 = minus2120578120590120572120573Π = minus120577Θ (10)



120590120572120573 = 119881(120572120573) + 119886(120572119881120573) minus 13Θℎ120572120573 (11)


119886120572 = 119881120572120573119881120573Θ = 119881120572

120572 (12)



(13)


1205870120572 = 012058711 = minus212058722 = minus212058733 (14)


120587120572120573 = Ω(120594120572120594120573 minus 13ℎ120572120573) (15)


120594120572120594120572 = 1120594120572119881120572 = 0120594120572 = 119861minus11205751205721

(16)



12059011 = 11986123119860 [Σ1 minus Σ3] 12059022 = 11986223119860 [Σ2 minus Σ1] 12059033 = 1198632

3119860 [Σ3 minus Σ2] (17)

where

120590120572120573120590120572120573 = 1205902 = 131198602[Σ21 + Σ22 + Σ23] (18)

where


(19)


1198861 = 119860 Θ = 1119860 (119861 + 119862 + 119863)

(20)




(119860119861)2 [minus11986210158401015840119862 minus 11986310158401015840

119863 + 1198611015840119861 (1198621015840119862 + 1198631015840

119863 ) minus 11986210158401198631015840

119862119863 ]

+ (119861119862 + 119861119863 + 119862119863) = 120581119865 [(120583 + 120598)1198602

+ 1198602

120581 minus(119891 (119877) minus 119877119865 (119877)2 )

minus 1198602(119861 + 119862 + 119863) + 119865101584010158401198612

+ 11986510158401198612 (1198621015840

119862 + 1198631015840

119863 minus 1198611015840119861 )]

(21)

minus ( 1198621015840119862 + 1015840

119863 minus 1198621015840119861119862 minus 1198631015840

119861119863 minus 1198601015840

119860119862 minus 1198601015840

119860119863 )



10158401198631015840

119862119863+ 11986010158401198621015840119860119862 + 11986010158401198631015840

119860119863 ) = 120581119865 [(119901 + Π + 120598 + 2Ω3 )1198612

+ 1198612120581 (119891 (119877) minus 119877119865 (119877)2 ) + 1198602

+ 1198602(119862 + 119863 minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 )]

(23)

minus (119862119860)2 [119861 + 119863 minus 119860 (119861 + 119863) + 119861119863] + (119862119861)

2

sdot [11986010158401015840

119860 + 11986310158401015840

119863 minus 1198601015840

119860 (1198611015840119861 minus 1198631015840

119863 ) minus 11986110158401198631015840

119861119863 ]

= 120581119865 [(119901 + Π minus Ω3 )1198622



+ 11986510158401198612 (1198601015840

119860 minus 1198611015840119861 + 1198621015840119862 + 1198631015840

119863 )]

(24)

minus (119863119860)2 [119861 + 119862 minus 119860 (119861 + 119862) + 119861119862] + (119863119861 )

2

sdot [11986010158401015840

119860 + 11986210158401015840119862 minus 1198601015840


minus 1198632



+ 11986510158401198612 (1198601015840

119860 minus 1198611015840119861 + 1198621015840119862 + 1198631015840

119863 )]

(25)





119898(119903 119905) = 119864119897= 1198978+ 18119863 [ 11198602

(119862 + 119863)2 minus 11198612 (1198621198631015840 + 1198621015840119863)2] (27)



+ 2 (1198891206012 + 12057421198891198852) (28)



119896minus (119905 119903) = 119903 minus 119903Σ = 0 (29)


1198891199042minus Σ997904997904997904 minus1198891205912 + 11986221198891206012 + 11986321198891199112 (30)


where

119889120591 Σ997904997904997904 119860119889119905 (31)




119896+ (] ) = minus Σ (]) = 0 (32)


1198891199042+ Σ997904997904997904 minus2(minus (]) + 119889Σ (])119889] )119889]2

+ 2 (1198891206012 + 12057421198891199112) (33)



1198891199042Σ = (1198891199042minus)Σ = (1198891199042+)Σ (34)


[119870120572120573] = 119870+120572120573 minus 119870minus

120572120573 = 0 (35)



120597119909119886plusmn120597119909119887plusmn120597120588120572120597120588120573 ) (120583 119886 119887 = 0 1 2 3)

(36)


120582 = 12 (minus (]) + 119889Σ (]) 119889])12

(37)


119862 (119905 119903Σ) Σ997904997904997904 Σ (]) 119863 (119905 119903Σ) Σ997904997904997904 120574Σ (])

119889119905119889120591 = 1119860 119889]119889120591 = 120582

(38)


are

119870minus00 = (minus 1198601015840

119860119861)Σ

(39)

119870+00 = [(119889]119889120591)

minus1 (1198892]1198891205912) minus 1198772 (

119889]119889120591)]Σ

(40)

119870minus22 = (1198621015840119862119861 )

Σ

(41)

119870minus33 = (1198631015840119863119861 )

Σ

(42)

119870+22 = [(119889119889120591 ) minus 2 (119889]119889120591)]

Σ

= 120574minus2119870+33 (43)


(]) Σ997904997904997904 2 [(

119860)2 minus (1015840119861 )

2] (44)

119864 Σ997904997904997904 1198978 + 120574 (]) (45)

119902 Σ997904997904997904 (119875 + Π + 2Ω3 ) + 11987911986301119860119861 + 119879119863111198612 (46)






119877|plusmn = 0119891119877119877 [120597]1198771003816100381610038161003816plusmn = 0

119891119877119877 = 0(47)



119863119879 = 1119860 120597120597119905 119863119862 = 11198621015840 120597120597119903

(48)


119880 = 119863119879119862 = 119860119881 = 119863119879119863 = 119860

(49)



12 minus 1198621198631015840

119861119863 (50)



minus 119880 119864211986210158402 (11986510158401198621015840119862 + 11986510158401198631015840


1198651198631198791198604119862 + 1198802 1198642 (311986311987911986121198641198621015840 minus 119865101584011986311986211986021198621015840 ) + 119881

119863+ 119865119863119879119861119861 ( 1119862 minus 119863119879119862 )minus 1198642


(119862119863 + 119862119863119879119860 ) + 1198811198654119863 119862119863119879119879119863119863 + 119863119879119879119862

+ 1198642 (119862 (119863119862119863)221198632minus 12119862)

minus 119864221198621015840 119864119863119879119861(119863119862119865 + 1198652119862 minus 119862 (119863119862119863)2 11986521198632)

minus 119865(11986311987911986210158402119862 + 1198631198791198631015840

2119863 )+ 1198651198631198621198632119863 (1198621198631198791198631015840

119863 + 1198631198791198621015840) minus 1198631198791198651015840] (51)




119863119862119898 (119903 119905) = 1198621198632119865 [120581 120583 + 2120598 + 119901 + Π + 2Ω3 + 119864 (119902


2119863 minus 119864119863119879119861119863119862119863119863 )+ 1198642 (119863119862119860119862 + 119863119862119860119863119862119863119863 ))

minus 119881211986521198632119862(119880119863119862119860 + 1198631198621198632119863 ) minus 1

+ 11988111986521198621015840119863 (1198621198631198791198631015840

119863 + 1198631198791198621015840) + 119880119881119865119863 ( 12119862minus 119863119879119862 ) minus 119864211986511986311986211986321198632

(119862119863119862119862119863 + 119863119862119863


minus 31198631198621198632119863 minus 1198641198631198621198611198621015840 ) + 12119863 (119863119862119863119863119862119862119862

minus 119863119862119862119863) minus 1198641198621015840 (119863119879119865119863119879119861 + 1198642119863119862119865119863119862119861

+ 119865119863119879119861119863119879119863119863 ) + 119863119879119879119865 + 11986510158401015840119864211986210158402 minus 11988011986311987911986510158401198621015840+ 119865119863119879119879119862119862 + 119865119863119879119879119863119863 ]

(52)


119898(119903 119905) = 12 int119862

0

1198621198632119865 [120581 120583 + 2120598 + 119901 + Π + 2Ω3 + 119864 (119902 + 120598) + 11988031198651198631198621198602119862 + 1198802 119863119879119865119863119862119860 + 119865119862 (1198631198621198634119863 minus 119863119879119862 )

minus 119880119864119863119862119865(119864119863119862119860 minus 1198631198791198611198621015840 ) + 119863119879119865119863119879A+ 119865( 11198621015840 (11986311987911986210158402119862 + 1198631198791198631015840

2119863 minus 119864119863119879119861119863119862119863119863 ) + 1198642 (119863119862119860119862 + 119863119862119860119863119862119863119863 )) minus 119881211986521198632

119862(119880119863119862119860 + 1198631198621198632119863 ) minus 1

+ 11988111986521198621015840119863 (1198621198631198791198631015840

119863 + 1198631198791198621015840) + 119880119881119865119863 ( 12119862 minus 119863119879119862 )



+ 119865119863119879119879119862119862 + 119865119863119879119879119863119863 ]119889119862

(53)



(119879(119898)120572120573 + 119879(119863)120572120573 )120573 119881120572 = 0(119879(119898)120572120573 + 119879(119863)120572120573 )120573 120594120572 = 0

(54)

which produce

1119860 [ (120583 + 120598) + (120583 + 119901 + Π + 2120598 + 2Ω3 ) 119861+ (120583 + 119901 + Π + 120598 minus Ω3 )(119862 + 119863)]+ 1119861 [(119902 + 120598)1015840 + (119902 + 120598) (2119860

1015840

119860 + (119862119863)1015840119862119863 )] minus 1198631

= 0

(55)

1119860 [ (119902 + 120598) + (119902 + 120598) (2119861 + 119862 + 119863)]+ 1119861 [(119901 + Π + 120598 + 2Ω3 )1015840

+ (120583 + 119901 + Π + 2120598 + 2Ω3 ) 1198601015840

119860 + (120598 + Ω) (119862119863)1015840119862119863 ]+ 1198632 = 0

(56)



119863119879 (120583 + 120598) + 13 (3120583 + 3119901 + 3Π + 4120598)Θ+ 13 (120598 + Ω)( 2119860119861 minus 119880119862 minus 119881119863) + 119864119863119862 (119902 + 120598)+ (119902 + 120598) (21198861119861 + 119864119862 + 1198631015840

119861119863) minus 1198631 = 0(57)

119863119879 (119902 + 120598) + (119902 + 120598) (Θ + 119863119879119861119861 )+ 119864119863119862 (119901 + Π + 120598 + 2Ω3 )+ (120583 + 119901 + Π + 2120598 + 2Ω3 ) 1198861119861 + (120598 + Ω) (119862119863)1015840119861119862119863+ 1198632 = 0

(58)



119861 )+ 181198632

(1198621198812 minus 11986211986310158402


4119865

(59)




119861 ) minus 181198632(1198812119862 minus 11986211986310158402


119861119863 (120598 + Ω)] minus (1198642 + 11986310158401198622119861119863)[119863119879 (119902 + 120598)+ (119902 + 120598) (Θ + 119863119879119861119861 ) + 1198632]

(60)






119878120583 = 119878119899119881120583 + 119902120583119879 minus (1205730Π2 + 1205731119902]119902] + 1205732120587]120581120587]120581) 119881120583

2119879+ 1205720Π119902120583119879 + 1205721120587120583]119902]119879

(61)



119879119878120572120572 = minusΠ[119881120572120572 minus 1205720119902120572120572 + 1205730Π120572119881120572

+ 1198792 (1205730119879 119881120572)120572


120583

119902120572] minus 120587120572120583 [120590120572120583minus 1205721119902120583120572 + 1205732120587120572120583]119881] + 1198792 (1205732119879 119881])

]120587120572120583]

(62)


1205910Π120572119881120572 + Π = minus120577120579 + 1205720120577119902120572120572 minus 12120577119879( 1205910120577119879119881120572)120572

Π (63)


120573

sdot 119902120572(64)

1205912ℎ120583120572ℎ]120573120587120583]120588119881120588 + 120587120572120573 = minus2120578120590120572120573 + 21205781205721119902⟨120573120572⟩minus 120578119879( 12059122120578119879119881])

]120587120572120573 (65)

with

119902⟨120573120572⟩ = ℎ120583120573ℎ]120572 (12 (119902120583] + 119902]120583) minus 13119902120590120581ℎ120590120581ℎ120583]) (66)


1205910 = 12057712057301205911 = 12058111987912057311205912 = 21205781205732

(67)


1205910Π = minus (120577 + 12059102 Π)119860Θ + 1198601198611205720120577 [1199021015840 + 119902(1198601015840

119860 + 1198621015840119862+ 1198631015840

119863 )] minus Π[1205771198792( 1205910120577119879) + 119860]

(68)

1205911 119902 = minus119860119861 1205811198791015840 (1 + 1205720Π + 231205721Ω) + 119879[1198601015840

119860 minus 1205720Π1015840

minus 231205721 (Ω1015840 + 1198601015840

119860 Ω + 32 (1198621015840

119862 + 1198631015840

119863 )Ω)]minus 119902[12058111987922

( 12059111205811198792 ) + 12059112 119860Θ + 119860] (69)


minus Ω[120578119879 ( 12059122120578119879) + 12059122 119860Θ + 119860] (70)





119863119879119861119861 + 1198632]

(71)




119861 ) minus 1198628119863 (1198812119862 minus 11986211986310158402

1198612 )


+ 119864119862 (120598 + Ω) + 1198631015840

119861119863 (120598 + Ω)]

(72)


Λ = 1205811198791205911 (120583 + 119901 + Π + 2120598 + 23Ω)minus1 (1 minus 231205721Ω) (73)




(74)


Δ = 1205720120577119902 (120583 + 119901 + Π + 2120598 + 23Ω)minus1 ( 119902 + 21205982120577 + 1205910Π) (75)


5 Conclusions







Appendix

11987911986301 = 1120581 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 ) (A1)

11987911986311 = 1198612120581 (119891 (119877) minus 119877119865 (119877)2 ) + 1198602+ 1198602

(119862 + 119863minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 ) (A2)

1198631 = 119860120581 [ 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )1

+ 119891 (119877) minus 119877119865 (119877)21198602minus 1198651015840101584011986021198612

+ 1198604(119861 + C119862 + 119863)

minus 119865101584011986021198612 (1198621015840

119862 + 1198631015840

119863 minus 1198611015840119861 )0

+ 21198603119891 (119877) minus 119877119865 (119877)2 minus 119865101584010158401198612

+ 1198602(119861 + 119862 + 119863) minus 11986510158401198612 (119862

1015840

119862 + 1198631015840

119863 minus 1198611015840119861 )+ 1198602119861 minus 1198602

minus 119865101584010158401198612 + 1198602(119860 + 119861)

+ 11986510158401198612 (1198601015840

119860 + 1198611015840119861 ) + 1198602119862 minus 1198602+ 1198602

119860+ 11986510158401198612 119860

1015840

119860 + 1198602119863 minus 1198602+ 1198602

119860 + 11986510158401198612 1198601015840

119860 + 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )(31198601015840

119860 + 1198611015840119861 + 1198621015840119862+ 1198631015840

119863 )]

(A3)



1015840

119860 + 1198621015840119862 + 1198631015840

119863 )1

+ 1198601015840

1198601198612 1198602+ 119865101584010158401198612

minus 1198602(119860 + 119861) minus 11986510158401198612 (119860

1015840

119860 + 1198611015840119861 )+ 211986110158401198613 119891 (119877) minus 119877119865 (119877)2 + 1198602

+ 1198602(119862 + 119863 minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 )+ 11986210158401198612119862 119865101584010158401198612 minus 1198602

119861 minus 11986510158401198612 1198611015840

119861 + 1198631015840

1198612119863 119865101584010158401198612minus 1198602

119861 minus 11986510158401198612 1198611015840


(A4)


Data Availability




Acknowledgments


References






































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







(119860119861)2 [minus11986210158401015840119862 minus 11986310158401015840

119863 + 1198611015840119861 (1198621015840119862 + 1198631015840

119863 ) minus 11986210158401198631015840

119862119863 ]

+ (119861119862 + 119861119863 + 119862119863) = 120581119865 [(120583 + 120598)1198602

+ 1198602

120581 minus(119891 (119877) minus 119877119865 (119877)2 )

minus 1198602(119861 + 119862 + 119863) + 119865101584010158401198612

+ 11986510158401198612 (1198621015840

119862 + 1198631015840

119863 minus 1198611015840119861 )]

(21)

minus ( 1198621015840119862 + 1015840

119863 minus 1198621015840119861119862 minus 1198631015840

119861119863 minus 1198601015840

119860119862 minus 1198601015840

119860119863 )



10158401198631015840

119862119863+ 11986010158401198621015840119860119862 + 11986010158401198631015840

119860119863 ) = 120581119865 [(119901 + Π + 120598 + 2Ω3 )1198612

+ 1198612120581 (119891 (119877) minus 119877119865 (119877)2 ) + 1198602

+ 1198602(119862 + 119863 minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 )]

(23)

minus (119862119860)2 [119861 + 119863 minus 119860 (119861 + 119863) + 119861119863] + (119862119861)

2

sdot [11986010158401015840

119860 + 11986310158401015840

119863 minus 1198601015840

119860 (1198611015840119861 minus 1198631015840

119863 ) minus 11986110158401198631015840

119861119863 ]

= 120581119865 [(119901 + Π minus Ω3 )1198622



+ 11986510158401198612 (1198601015840

119860 minus 1198611015840119861 + 1198621015840119862 + 1198631015840

119863 )]

(24)

minus (119863119860)2 [119861 + 119862 minus 119860 (119861 + 119862) + 119861119862] + (119863119861 )

2

sdot [11986010158401015840

119860 + 11986210158401015840119862 minus 1198601015840


minus 1198632



+ 11986510158401198612 (1198601015840

119860 minus 1198611015840119861 + 1198621015840119862 + 1198631015840

119863 )]

(25)





119898(119903 119905) = 119864119897= 1198978+ 18119863 [ 11198602

(119862 + 119863)2 minus 11198612 (1198621198631015840 + 1198621015840119863)2] (27)



+ 2 (1198891206012 + 12057421198891198852) (28)



119896minus (119905 119903) = 119903 minus 119903Σ = 0 (29)


1198891199042minus Σ997904997904997904 minus1198891205912 + 11986221198891206012 + 11986321198891199112 (30)


where

119889120591 Σ997904997904997904 119860119889119905 (31)




119896+ (] ) = minus Σ (]) = 0 (32)


1198891199042+ Σ997904997904997904 minus2(minus (]) + 119889Σ (])119889] )119889]2

+ 2 (1198891206012 + 12057421198891199112) (33)



1198891199042Σ = (1198891199042minus)Σ = (1198891199042+)Σ (34)


[119870120572120573] = 119870+120572120573 minus 119870minus

120572120573 = 0 (35)



120597119909119886plusmn120597119909119887plusmn120597120588120572120597120588120573 ) (120583 119886 119887 = 0 1 2 3)

(36)


120582 = 12 (minus (]) + 119889Σ (]) 119889])12

(37)


119862 (119905 119903Σ) Σ997904997904997904 Σ (]) 119863 (119905 119903Σ) Σ997904997904997904 120574Σ (])

119889119905119889120591 = 1119860 119889]119889120591 = 120582

(38)


are

119870minus00 = (minus 1198601015840

119860119861)Σ

(39)

119870+00 = [(119889]119889120591)

minus1 (1198892]1198891205912) minus 1198772 (

119889]119889120591)]Σ

(40)

119870minus22 = (1198621015840119862119861 )

Σ

(41)

119870minus33 = (1198631015840119863119861 )

Σ

(42)

119870+22 = [(119889119889120591 ) minus 2 (119889]119889120591)]

Σ

= 120574minus2119870+33 (43)


(]) Σ997904997904997904 2 [(

119860)2 minus (1015840119861 )

2] (44)

119864 Σ997904997904997904 1198978 + 120574 (]) (45)

119902 Σ997904997904997904 (119875 + Π + 2Ω3 ) + 11987911986301119860119861 + 119879119863111198612 (46)






119877|plusmn = 0119891119877119877 [120597]1198771003816100381610038161003816plusmn = 0

119891119877119877 = 0(47)



119863119879 = 1119860 120597120597119905 119863119862 = 11198621015840 120597120597119903

(48)


119880 = 119863119879119862 = 119860119881 = 119863119879119863 = 119860

(49)



12 minus 1198621198631015840

119861119863 (50)



minus 119880 119864211986210158402 (11986510158401198621015840119862 + 11986510158401198631015840


1198651198631198791198604119862 + 1198802 1198642 (311986311987911986121198641198621015840 minus 119865101584011986311986211986021198621015840 ) + 119881

119863+ 119865119863119879119861119861 ( 1119862 minus 119863119879119862 )minus 1198642


(119862119863 + 119862119863119879119860 ) + 1198811198654119863 119862119863119879119879119863119863 + 119863119879119879119862

+ 1198642 (119862 (119863119862119863)221198632minus 12119862)

minus 119864221198621015840 119864119863119879119861(119863119862119865 + 1198652119862 minus 119862 (119863119862119863)2 11986521198632)

minus 119865(11986311987911986210158402119862 + 1198631198791198631015840

2119863 )+ 1198651198631198621198632119863 (1198621198631198791198631015840

119863 + 1198631198791198621015840) minus 1198631198791198651015840] (51)




119863119862119898 (119903 119905) = 1198621198632119865 [120581 120583 + 2120598 + 119901 + Π + 2Ω3 + 119864 (119902


2119863 minus 119864119863119879119861119863119862119863119863 )+ 1198642 (119863119862119860119862 + 119863119862119860119863119862119863119863 ))

minus 119881211986521198632119862(119880119863119862119860 + 1198631198621198632119863 ) minus 1

+ 11988111986521198621015840119863 (1198621198631198791198631015840

119863 + 1198631198791198621015840) + 119880119881119865119863 ( 12119862minus 119863119879119862 ) minus 119864211986511986311986211986321198632

(119862119863119862119862119863 + 119863119862119863


minus 31198631198621198632119863 minus 1198641198631198621198611198621015840 ) + 12119863 (119863119862119863119863119862119862119862

minus 119863119862119862119863) minus 1198641198621015840 (119863119879119865119863119879119861 + 1198642119863119862119865119863119862119861

+ 119865119863119879119861119863119879119863119863 ) + 119863119879119879119865 + 11986510158401015840119864211986210158402 minus 11988011986311987911986510158401198621015840+ 119865119863119879119879119862119862 + 119865119863119879119879119863119863 ]

(52)


119898(119903 119905) = 12 int119862

0

1198621198632119865 [120581 120583 + 2120598 + 119901 + Π + 2Ω3 + 119864 (119902 + 120598) + 11988031198651198631198621198602119862 + 1198802 119863119879119865119863119862119860 + 119865119862 (1198631198621198634119863 minus 119863119879119862 )

minus 119880119864119863119862119865(119864119863119862119860 minus 1198631198791198611198621015840 ) + 119863119879119865119863119879A+ 119865( 11198621015840 (11986311987911986210158402119862 + 1198631198791198631015840

2119863 minus 119864119863119879119861119863119862119863119863 ) + 1198642 (119863119862119860119862 + 119863119862119860119863119862119863119863 )) minus 119881211986521198632

119862(119880119863119862119860 + 1198631198621198632119863 ) minus 1

+ 11988111986521198621015840119863 (1198621198631198791198631015840

119863 + 1198631198791198621015840) + 119880119881119865119863 ( 12119862 minus 119863119879119862 )



+ 119865119863119879119879119862119862 + 119865119863119879119879119863119863 ]119889119862

(53)



(119879(119898)120572120573 + 119879(119863)120572120573 )120573 119881120572 = 0(119879(119898)120572120573 + 119879(119863)120572120573 )120573 120594120572 = 0

(54)

which produce

1119860 [ (120583 + 120598) + (120583 + 119901 + Π + 2120598 + 2Ω3 ) 119861+ (120583 + 119901 + Π + 120598 minus Ω3 )(119862 + 119863)]+ 1119861 [(119902 + 120598)1015840 + (119902 + 120598) (2119860

1015840

119860 + (119862119863)1015840119862119863 )] minus 1198631

= 0

(55)

1119860 [ (119902 + 120598) + (119902 + 120598) (2119861 + 119862 + 119863)]+ 1119861 [(119901 + Π + 120598 + 2Ω3 )1015840

+ (120583 + 119901 + Π + 2120598 + 2Ω3 ) 1198601015840

119860 + (120598 + Ω) (119862119863)1015840119862119863 ]+ 1198632 = 0

(56)



119863119879 (120583 + 120598) + 13 (3120583 + 3119901 + 3Π + 4120598)Θ+ 13 (120598 + Ω)( 2119860119861 minus 119880119862 minus 119881119863) + 119864119863119862 (119902 + 120598)+ (119902 + 120598) (21198861119861 + 119864119862 + 1198631015840

119861119863) minus 1198631 = 0(57)

119863119879 (119902 + 120598) + (119902 + 120598) (Θ + 119863119879119861119861 )+ 119864119863119862 (119901 + Π + 120598 + 2Ω3 )+ (120583 + 119901 + Π + 2120598 + 2Ω3 ) 1198861119861 + (120598 + Ω) (119862119863)1015840119861119862119863+ 1198632 = 0

(58)



119861 )+ 181198632

(1198621198812 minus 11986211986310158402


4119865

(59)




119861 ) minus 181198632(1198812119862 minus 11986211986310158402


119861119863 (120598 + Ω)] minus (1198642 + 11986310158401198622119861119863)[119863119879 (119902 + 120598)+ (119902 + 120598) (Θ + 119863119879119861119861 ) + 1198632]

(60)






119878120583 = 119878119899119881120583 + 119902120583119879 minus (1205730Π2 + 1205731119902]119902] + 1205732120587]120581120587]120581) 119881120583

2119879+ 1205720Π119902120583119879 + 1205721120587120583]119902]119879

(61)



119879119878120572120572 = minusΠ[119881120572120572 minus 1205720119902120572120572 + 1205730Π120572119881120572

+ 1198792 (1205730119879 119881120572)120572


120583

119902120572] minus 120587120572120583 [120590120572120583minus 1205721119902120583120572 + 1205732120587120572120583]119881] + 1198792 (1205732119879 119881])

]120587120572120583]

(62)


1205910Π120572119881120572 + Π = minus120577120579 + 1205720120577119902120572120572 minus 12120577119879( 1205910120577119879119881120572)120572

Π (63)


120573

sdot 119902120572(64)

1205912ℎ120583120572ℎ]120573120587120583]120588119881120588 + 120587120572120573 = minus2120578120590120572120573 + 21205781205721119902⟨120573120572⟩minus 120578119879( 12059122120578119879119881])

]120587120572120573 (65)

with

119902⟨120573120572⟩ = ℎ120583120573ℎ]120572 (12 (119902120583] + 119902]120583) minus 13119902120590120581ℎ120590120581ℎ120583]) (66)


1205910 = 12057712057301205911 = 12058111987912057311205912 = 21205781205732

(67)


1205910Π = minus (120577 + 12059102 Π)119860Θ + 1198601198611205720120577 [1199021015840 + 119902(1198601015840

119860 + 1198621015840119862+ 1198631015840

119863 )] minus Π[1205771198792( 1205910120577119879) + 119860]

(68)

1205911 119902 = minus119860119861 1205811198791015840 (1 + 1205720Π + 231205721Ω) + 119879[1198601015840

119860 minus 1205720Π1015840

minus 231205721 (Ω1015840 + 1198601015840

119860 Ω + 32 (1198621015840

119862 + 1198631015840

119863 )Ω)]minus 119902[12058111987922

( 12059111205811198792 ) + 12059112 119860Θ + 119860] (69)


minus Ω[120578119879 ( 12059122120578119879) + 12059122 119860Θ + 119860] (70)





119863119879119861119861 + 1198632]

(71)




119861 ) minus 1198628119863 (1198812119862 minus 11986211986310158402

1198612 )


+ 119864119862 (120598 + Ω) + 1198631015840

119861119863 (120598 + Ω)]

(72)


Λ = 1205811198791205911 (120583 + 119901 + Π + 2120598 + 23Ω)minus1 (1 minus 231205721Ω) (73)




(74)


Δ = 1205720120577119902 (120583 + 119901 + Π + 2120598 + 23Ω)minus1 ( 119902 + 21205982120577 + 1205910Π) (75)


5 Conclusions







Appendix

11987911986301 = 1120581 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 ) (A1)

11987911986311 = 1198612120581 (119891 (119877) minus 119877119865 (119877)2 ) + 1198602+ 1198602

(119862 + 119863minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 ) (A2)

1198631 = 119860120581 [ 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )1

+ 119891 (119877) minus 119877119865 (119877)21198602minus 1198651015840101584011986021198612

+ 1198604(119861 + C119862 + 119863)

minus 119865101584011986021198612 (1198621015840

119862 + 1198631015840

119863 minus 1198611015840119861 )0

+ 21198603119891 (119877) minus 119877119865 (119877)2 minus 119865101584010158401198612

+ 1198602(119861 + 119862 + 119863) minus 11986510158401198612 (119862

1015840

119862 + 1198631015840

119863 minus 1198611015840119861 )+ 1198602119861 minus 1198602

minus 119865101584010158401198612 + 1198602(119860 + 119861)

+ 11986510158401198612 (1198601015840

119860 + 1198611015840119861 ) + 1198602119862 minus 1198602+ 1198602

119860+ 11986510158401198612 119860

1015840

119860 + 1198602119863 minus 1198602+ 1198602

119860 + 11986510158401198612 1198601015840

119860 + 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )(31198601015840

119860 + 1198611015840119861 + 1198621015840119862+ 1198631015840

119863 )]

(A3)



1015840

119860 + 1198621015840119862 + 1198631015840

119863 )1

+ 1198601015840

1198601198612 1198602+ 119865101584010158401198612

minus 1198602(119860 + 119861) minus 11986510158401198612 (119860

1015840

119860 + 1198611015840119861 )+ 211986110158401198613 119891 (119877) minus 119877119865 (119877)2 + 1198602

+ 1198602(119862 + 119863 minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 )+ 11986210158401198612119862 119865101584010158401198612 minus 1198602

119861 minus 11986510158401198612 1198611015840

119861 + 1198631015840

1198612119863 119865101584010158401198612minus 1198602

119861 minus 11986510158401198612 1198611015840


(A4)


Data Availability




Acknowledgments


References






































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






where

119889120591 Σ997904997904997904 119860119889119905 (31)




119896+ (] ) = minus Σ (]) = 0 (32)


1198891199042+ Σ997904997904997904 minus2(minus (]) + 119889Σ (])119889] )119889]2

+ 2 (1198891206012 + 12057421198891199112) (33)



1198891199042Σ = (1198891199042minus)Σ = (1198891199042+)Σ (34)


[119870120572120573] = 119870+120572120573 minus 119870minus

120572120573 = 0 (35)



120597119909119886plusmn120597119909119887plusmn120597120588120572120597120588120573 ) (120583 119886 119887 = 0 1 2 3)

(36)


120582 = 12 (minus (]) + 119889Σ (]) 119889])12

(37)


119862 (119905 119903Σ) Σ997904997904997904 Σ (]) 119863 (119905 119903Σ) Σ997904997904997904 120574Σ (])

119889119905119889120591 = 1119860 119889]119889120591 = 120582

(38)


are

119870minus00 = (minus 1198601015840

119860119861)Σ

(39)

119870+00 = [(119889]119889120591)

minus1 (1198892]1198891205912) minus 1198772 (

119889]119889120591)]Σ

(40)

119870minus22 = (1198621015840119862119861 )

Σ

(41)

119870minus33 = (1198631015840119863119861 )

Σ

(42)

119870+22 = [(119889119889120591 ) minus 2 (119889]119889120591)]

Σ

= 120574minus2119870+33 (43)


(]) Σ997904997904997904 2 [(

119860)2 minus (1015840119861 )

2] (44)

119864 Σ997904997904997904 1198978 + 120574 (]) (45)

119902 Σ997904997904997904 (119875 + Π + 2Ω3 ) + 11987911986301119860119861 + 119879119863111198612 (46)






119877|plusmn = 0119891119877119877 [120597]1198771003816100381610038161003816plusmn = 0

119891119877119877 = 0(47)



119863119879 = 1119860 120597120597119905 119863119862 = 11198621015840 120597120597119903

(48)


119880 = 119863119879119862 = 119860119881 = 119863119879119863 = 119860

(49)



12 minus 1198621198631015840

119861119863 (50)



minus 119880 119864211986210158402 (11986510158401198621015840119862 + 11986510158401198631015840


1198651198631198791198604119862 + 1198802 1198642 (311986311987911986121198641198621015840 minus 119865101584011986311986211986021198621015840 ) + 119881

119863+ 119865119863119879119861119861 ( 1119862 minus 119863119879119862 )minus 1198642


(119862119863 + 119862119863119879119860 ) + 1198811198654119863 119862119863119879119879119863119863 + 119863119879119879119862

+ 1198642 (119862 (119863119862119863)221198632minus 12119862)

minus 119864221198621015840 119864119863119879119861(119863119862119865 + 1198652119862 minus 119862 (119863119862119863)2 11986521198632)

minus 119865(11986311987911986210158402119862 + 1198631198791198631015840

2119863 )+ 1198651198631198621198632119863 (1198621198631198791198631015840

119863 + 1198631198791198621015840) minus 1198631198791198651015840] (51)




119863119862119898 (119903 119905) = 1198621198632119865 [120581 120583 + 2120598 + 119901 + Π + 2Ω3 + 119864 (119902


2119863 minus 119864119863119879119861119863119862119863119863 )+ 1198642 (119863119862119860119862 + 119863119862119860119863119862119863119863 ))

minus 119881211986521198632119862(119880119863119862119860 + 1198631198621198632119863 ) minus 1

+ 11988111986521198621015840119863 (1198621198631198791198631015840

119863 + 1198631198791198621015840) + 119880119881119865119863 ( 12119862minus 119863119879119862 ) minus 119864211986511986311986211986321198632

(119862119863119862119862119863 + 119863119862119863


minus 31198631198621198632119863 minus 1198641198631198621198611198621015840 ) + 12119863 (119863119862119863119863119862119862119862

minus 119863119862119862119863) minus 1198641198621015840 (119863119879119865119863119879119861 + 1198642119863119862119865119863119862119861

+ 119865119863119879119861119863119879119863119863 ) + 119863119879119879119865 + 11986510158401015840119864211986210158402 minus 11988011986311987911986510158401198621015840+ 119865119863119879119879119862119862 + 119865119863119879119879119863119863 ]

(52)


119898(119903 119905) = 12 int119862

0

1198621198632119865 [120581 120583 + 2120598 + 119901 + Π + 2Ω3 + 119864 (119902 + 120598) + 11988031198651198631198621198602119862 + 1198802 119863119879119865119863119862119860 + 119865119862 (1198631198621198634119863 minus 119863119879119862 )

minus 119880119864119863119862119865(119864119863119862119860 minus 1198631198791198611198621015840 ) + 119863119879119865119863119879A+ 119865( 11198621015840 (11986311987911986210158402119862 + 1198631198791198631015840

2119863 minus 119864119863119879119861119863119862119863119863 ) + 1198642 (119863119862119860119862 + 119863119862119860119863119862119863119863 )) minus 119881211986521198632

119862(119880119863119862119860 + 1198631198621198632119863 ) minus 1

+ 11988111986521198621015840119863 (1198621198631198791198631015840

119863 + 1198631198791198621015840) + 119880119881119865119863 ( 12119862 minus 119863119879119862 )



+ 119865119863119879119879119862119862 + 119865119863119879119879119863119863 ]119889119862

(53)



(119879(119898)120572120573 + 119879(119863)120572120573 )120573 119881120572 = 0(119879(119898)120572120573 + 119879(119863)120572120573 )120573 120594120572 = 0

(54)

which produce

1119860 [ (120583 + 120598) + (120583 + 119901 + Π + 2120598 + 2Ω3 ) 119861+ (120583 + 119901 + Π + 120598 minus Ω3 )(119862 + 119863)]+ 1119861 [(119902 + 120598)1015840 + (119902 + 120598) (2119860

1015840

119860 + (119862119863)1015840119862119863 )] minus 1198631

= 0

(55)

1119860 [ (119902 + 120598) + (119902 + 120598) (2119861 + 119862 + 119863)]+ 1119861 [(119901 + Π + 120598 + 2Ω3 )1015840

+ (120583 + 119901 + Π + 2120598 + 2Ω3 ) 1198601015840

119860 + (120598 + Ω) (119862119863)1015840119862119863 ]+ 1198632 = 0

(56)



119863119879 (120583 + 120598) + 13 (3120583 + 3119901 + 3Π + 4120598)Θ+ 13 (120598 + Ω)( 2119860119861 minus 119880119862 minus 119881119863) + 119864119863119862 (119902 + 120598)+ (119902 + 120598) (21198861119861 + 119864119862 + 1198631015840

119861119863) minus 1198631 = 0(57)

119863119879 (119902 + 120598) + (119902 + 120598) (Θ + 119863119879119861119861 )+ 119864119863119862 (119901 + Π + 120598 + 2Ω3 )+ (120583 + 119901 + Π + 2120598 + 2Ω3 ) 1198861119861 + (120598 + Ω) (119862119863)1015840119861119862119863+ 1198632 = 0

(58)



119861 )+ 181198632

(1198621198812 minus 11986211986310158402


4119865

(59)




119861 ) minus 181198632(1198812119862 minus 11986211986310158402


119861119863 (120598 + Ω)] minus (1198642 + 11986310158401198622119861119863)[119863119879 (119902 + 120598)+ (119902 + 120598) (Θ + 119863119879119861119861 ) + 1198632]

(60)






119878120583 = 119878119899119881120583 + 119902120583119879 minus (1205730Π2 + 1205731119902]119902] + 1205732120587]120581120587]120581) 119881120583

2119879+ 1205720Π119902120583119879 + 1205721120587120583]119902]119879

(61)



119879119878120572120572 = minusΠ[119881120572120572 minus 1205720119902120572120572 + 1205730Π120572119881120572

+ 1198792 (1205730119879 119881120572)120572


120583

119902120572] minus 120587120572120583 [120590120572120583minus 1205721119902120583120572 + 1205732120587120572120583]119881] + 1198792 (1205732119879 119881])

]120587120572120583]

(62)


1205910Π120572119881120572 + Π = minus120577120579 + 1205720120577119902120572120572 minus 12120577119879( 1205910120577119879119881120572)120572

Π (63)


120573

sdot 119902120572(64)

1205912ℎ120583120572ℎ]120573120587120583]120588119881120588 + 120587120572120573 = minus2120578120590120572120573 + 21205781205721119902⟨120573120572⟩minus 120578119879( 12059122120578119879119881])

]120587120572120573 (65)

with

119902⟨120573120572⟩ = ℎ120583120573ℎ]120572 (12 (119902120583] + 119902]120583) minus 13119902120590120581ℎ120590120581ℎ120583]) (66)


1205910 = 12057712057301205911 = 12058111987912057311205912 = 21205781205732

(67)


1205910Π = minus (120577 + 12059102 Π)119860Θ + 1198601198611205720120577 [1199021015840 + 119902(1198601015840

119860 + 1198621015840119862+ 1198631015840

119863 )] minus Π[1205771198792( 1205910120577119879) + 119860]

(68)

1205911 119902 = minus119860119861 1205811198791015840 (1 + 1205720Π + 231205721Ω) + 119879[1198601015840

119860 minus 1205720Π1015840

minus 231205721 (Ω1015840 + 1198601015840

119860 Ω + 32 (1198621015840

119862 + 1198631015840

119863 )Ω)]minus 119902[12058111987922

( 12059111205811198792 ) + 12059112 119860Θ + 119860] (69)


minus Ω[120578119879 ( 12059122120578119879) + 12059122 119860Θ + 119860] (70)





119863119879119861119861 + 1198632]

(71)




119861 ) minus 1198628119863 (1198812119862 minus 11986211986310158402

1198612 )


+ 119864119862 (120598 + Ω) + 1198631015840

119861119863 (120598 + Ω)]

(72)


Λ = 1205811198791205911 (120583 + 119901 + Π + 2120598 + 23Ω)minus1 (1 minus 231205721Ω) (73)




(74)


Δ = 1205720120577119902 (120583 + 119901 + Π + 2120598 + 23Ω)minus1 ( 119902 + 21205982120577 + 1205910Π) (75)


5 Conclusions







Appendix

11987911986301 = 1120581 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 ) (A1)

11987911986311 = 1198612120581 (119891 (119877) minus 119877119865 (119877)2 ) + 1198602+ 1198602

(119862 + 119863minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 ) (A2)

1198631 = 119860120581 [ 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )1

+ 119891 (119877) minus 119877119865 (119877)21198602minus 1198651015840101584011986021198612

+ 1198604(119861 + C119862 + 119863)

minus 119865101584011986021198612 (1198621015840

119862 + 1198631015840

119863 minus 1198611015840119861 )0

+ 21198603119891 (119877) minus 119877119865 (119877)2 minus 119865101584010158401198612

+ 1198602(119861 + 119862 + 119863) minus 11986510158401198612 (119862

1015840

119862 + 1198631015840

119863 minus 1198611015840119861 )+ 1198602119861 minus 1198602

minus 119865101584010158401198612 + 1198602(119860 + 119861)

+ 11986510158401198612 (1198601015840

119860 + 1198611015840119861 ) + 1198602119862 minus 1198602+ 1198602

119860+ 11986510158401198612 119860

1015840

119860 + 1198602119863 minus 1198602+ 1198602

119860 + 11986510158401198612 1198601015840

119860 + 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )(31198601015840

119860 + 1198611015840119861 + 1198621015840119862+ 1198631015840

119863 )]

(A3)



1015840

119860 + 1198621015840119862 + 1198631015840

119863 )1

+ 1198601015840

1198601198612 1198602+ 119865101584010158401198612

minus 1198602(119860 + 119861) minus 11986510158401198612 (119860

1015840

119860 + 1198611015840119861 )+ 211986110158401198613 119891 (119877) minus 119877119865 (119877)2 + 1198602

+ 1198602(119862 + 119863 minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 )+ 11986210158401198612119862 119865101584010158401198612 minus 1198602

119861 minus 11986510158401198612 1198611015840

119861 + 1198631015840

1198612119863 119865101584010158401198612minus 1198602

119861 minus 11986510158401198612 1198611015840


(A4)


Data Availability




Acknowledgments


References






































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery








119877|plusmn = 0119891119877119877 [120597]1198771003816100381610038161003816plusmn = 0

119891119877119877 = 0(47)



119863119879 = 1119860 120597120597119905 119863119862 = 11198621015840 120597120597119903

(48)


119880 = 119863119879119862 = 119860119881 = 119863119879119863 = 119860

(49)



12 minus 1198621198631015840

119861119863 (50)



minus 119880 119864211986210158402 (11986510158401198621015840119862 + 11986510158401198631015840


1198651198631198791198604119862 + 1198802 1198642 (311986311987911986121198641198621015840 minus 119865101584011986311986211986021198621015840 ) + 119881

119863+ 119865119863119879119861119861 ( 1119862 minus 119863119879119862 )minus 1198642


(119862119863 + 119862119863119879119860 ) + 1198811198654119863 119862119863119879119879119863119863 + 119863119879119879119862

+ 1198642 (119862 (119863119862119863)221198632minus 12119862)

minus 119864221198621015840 119864119863119879119861(119863119862119865 + 1198652119862 minus 119862 (119863119862119863)2 11986521198632)

minus 119865(11986311987911986210158402119862 + 1198631198791198631015840

2119863 )+ 1198651198631198621198632119863 (1198621198631198791198631015840

119863 + 1198631198791198621015840) minus 1198631198791198651015840] (51)




119863119862119898 (119903 119905) = 1198621198632119865 [120581 120583 + 2120598 + 119901 + Π + 2Ω3 + 119864 (119902


2119863 minus 119864119863119879119861119863119862119863119863 )+ 1198642 (119863119862119860119862 + 119863119862119860119863119862119863119863 ))

minus 119881211986521198632119862(119880119863119862119860 + 1198631198621198632119863 ) minus 1

+ 11988111986521198621015840119863 (1198621198631198791198631015840

119863 + 1198631198791198621015840) + 119880119881119865119863 ( 12119862minus 119863119879119862 ) minus 119864211986511986311986211986321198632

(119862119863119862119862119863 + 119863119862119863


minus 31198631198621198632119863 minus 1198641198631198621198611198621015840 ) + 12119863 (119863119862119863119863119862119862119862

minus 119863119862119862119863) minus 1198641198621015840 (119863119879119865119863119879119861 + 1198642119863119862119865119863119862119861

+ 119865119863119879119861119863119879119863119863 ) + 119863119879119879119865 + 11986510158401015840119864211986210158402 minus 11988011986311987911986510158401198621015840+ 119865119863119879119879119862119862 + 119865119863119879119879119863119863 ]

(52)


119898(119903 119905) = 12 int119862

0

1198621198632119865 [120581 120583 + 2120598 + 119901 + Π + 2Ω3 + 119864 (119902 + 120598) + 11988031198651198631198621198602119862 + 1198802 119863119879119865119863119862119860 + 119865119862 (1198631198621198634119863 minus 119863119879119862 )

minus 119880119864119863119862119865(119864119863119862119860 minus 1198631198791198611198621015840 ) + 119863119879119865119863119879A+ 119865( 11198621015840 (11986311987911986210158402119862 + 1198631198791198631015840

2119863 minus 119864119863119879119861119863119862119863119863 ) + 1198642 (119863119862119860119862 + 119863119862119860119863119862119863119863 )) minus 119881211986521198632

119862(119880119863119862119860 + 1198631198621198632119863 ) minus 1

+ 11988111986521198621015840119863 (1198621198631198791198631015840

119863 + 1198631198791198621015840) + 119880119881119865119863 ( 12119862 minus 119863119879119862 )



+ 119865119863119879119879119862119862 + 119865119863119879119879119863119863 ]119889119862

(53)



(119879(119898)120572120573 + 119879(119863)120572120573 )120573 119881120572 = 0(119879(119898)120572120573 + 119879(119863)120572120573 )120573 120594120572 = 0

(54)

which produce

1119860 [ (120583 + 120598) + (120583 + 119901 + Π + 2120598 + 2Ω3 ) 119861+ (120583 + 119901 + Π + 120598 minus Ω3 )(119862 + 119863)]+ 1119861 [(119902 + 120598)1015840 + (119902 + 120598) (2119860

1015840

119860 + (119862119863)1015840119862119863 )] minus 1198631

= 0

(55)

1119860 [ (119902 + 120598) + (119902 + 120598) (2119861 + 119862 + 119863)]+ 1119861 [(119901 + Π + 120598 + 2Ω3 )1015840

+ (120583 + 119901 + Π + 2120598 + 2Ω3 ) 1198601015840

119860 + (120598 + Ω) (119862119863)1015840119862119863 ]+ 1198632 = 0

(56)



119863119879 (120583 + 120598) + 13 (3120583 + 3119901 + 3Π + 4120598)Θ+ 13 (120598 + Ω)( 2119860119861 minus 119880119862 minus 119881119863) + 119864119863119862 (119902 + 120598)+ (119902 + 120598) (21198861119861 + 119864119862 + 1198631015840

119861119863) minus 1198631 = 0(57)

119863119879 (119902 + 120598) + (119902 + 120598) (Θ + 119863119879119861119861 )+ 119864119863119862 (119901 + Π + 120598 + 2Ω3 )+ (120583 + 119901 + Π + 2120598 + 2Ω3 ) 1198861119861 + (120598 + Ω) (119862119863)1015840119861119862119863+ 1198632 = 0

(58)



119861 )+ 181198632

(1198621198812 minus 11986211986310158402


4119865

(59)




119861 ) minus 181198632(1198812119862 minus 11986211986310158402


119861119863 (120598 + Ω)] minus (1198642 + 11986310158401198622119861119863)[119863119879 (119902 + 120598)+ (119902 + 120598) (Θ + 119863119879119861119861 ) + 1198632]

(60)






119878120583 = 119878119899119881120583 + 119902120583119879 minus (1205730Π2 + 1205731119902]119902] + 1205732120587]120581120587]120581) 119881120583

2119879+ 1205720Π119902120583119879 + 1205721120587120583]119902]119879

(61)



119879119878120572120572 = minusΠ[119881120572120572 minus 1205720119902120572120572 + 1205730Π120572119881120572

+ 1198792 (1205730119879 119881120572)120572


120583

119902120572] minus 120587120572120583 [120590120572120583minus 1205721119902120583120572 + 1205732120587120572120583]119881] + 1198792 (1205732119879 119881])

]120587120572120583]

(62)


1205910Π120572119881120572 + Π = minus120577120579 + 1205720120577119902120572120572 minus 12120577119879( 1205910120577119879119881120572)120572

Π (63)


120573

sdot 119902120572(64)

1205912ℎ120583120572ℎ]120573120587120583]120588119881120588 + 120587120572120573 = minus2120578120590120572120573 + 21205781205721119902⟨120573120572⟩minus 120578119879( 12059122120578119879119881])

]120587120572120573 (65)

with

119902⟨120573120572⟩ = ℎ120583120573ℎ]120572 (12 (119902120583] + 119902]120583) minus 13119902120590120581ℎ120590120581ℎ120583]) (66)


1205910 = 12057712057301205911 = 12058111987912057311205912 = 21205781205732

(67)


1205910Π = minus (120577 + 12059102 Π)119860Θ + 1198601198611205720120577 [1199021015840 + 119902(1198601015840

119860 + 1198621015840119862+ 1198631015840

119863 )] minus Π[1205771198792( 1205910120577119879) + 119860]

(68)

1205911 119902 = minus119860119861 1205811198791015840 (1 + 1205720Π + 231205721Ω) + 119879[1198601015840

119860 minus 1205720Π1015840

minus 231205721 (Ω1015840 + 1198601015840

119860 Ω + 32 (1198621015840

119862 + 1198631015840

119863 )Ω)]minus 119902[12058111987922

( 12059111205811198792 ) + 12059112 119860Θ + 119860] (69)


minus Ω[120578119879 ( 12059122120578119879) + 12059122 119860Θ + 119860] (70)





119863119879119861119861 + 1198632]

(71)




119861 ) minus 1198628119863 (1198812119862 minus 11986211986310158402

1198612 )


+ 119864119862 (120598 + Ω) + 1198631015840

119861119863 (120598 + Ω)]

(72)


Λ = 1205811198791205911 (120583 + 119901 + Π + 2120598 + 23Ω)minus1 (1 minus 231205721Ω) (73)




(74)


Δ = 1205720120577119902 (120583 + 119901 + Π + 2120598 + 23Ω)minus1 ( 119902 + 21205982120577 + 1205910Π) (75)


5 Conclusions







Appendix

11987911986301 = 1120581 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 ) (A1)

11987911986311 = 1198612120581 (119891 (119877) minus 119877119865 (119877)2 ) + 1198602+ 1198602

(119862 + 119863minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 ) (A2)

1198631 = 119860120581 [ 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )1

+ 119891 (119877) minus 119877119865 (119877)21198602minus 1198651015840101584011986021198612

+ 1198604(119861 + C119862 + 119863)

minus 119865101584011986021198612 (1198621015840

119862 + 1198631015840

119863 minus 1198611015840119861 )0

+ 21198603119891 (119877) minus 119877119865 (119877)2 minus 119865101584010158401198612

+ 1198602(119861 + 119862 + 119863) minus 11986510158401198612 (119862

1015840

119862 + 1198631015840

119863 minus 1198611015840119861 )+ 1198602119861 minus 1198602

minus 119865101584010158401198612 + 1198602(119860 + 119861)

+ 11986510158401198612 (1198601015840

119860 + 1198611015840119861 ) + 1198602119862 minus 1198602+ 1198602

119860+ 11986510158401198612 119860

1015840

119860 + 1198602119863 minus 1198602+ 1198602

119860 + 11986510158401198612 1198601015840

119860 + 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )(31198601015840

119860 + 1198611015840119861 + 1198621015840119862+ 1198631015840

119863 )]

(A3)



1015840

119860 + 1198621015840119862 + 1198631015840

119863 )1

+ 1198601015840

1198601198612 1198602+ 119865101584010158401198612

minus 1198602(119860 + 119861) minus 11986510158401198612 (119860

1015840

119860 + 1198611015840119861 )+ 211986110158401198613 119891 (119877) minus 119877119865 (119877)2 + 1198602

+ 1198602(119862 + 119863 minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 )+ 11986210158401198612119862 119865101584010158401198612 minus 1198602

119861 minus 11986510158401198612 1198611015840

119861 + 1198631015840

1198612119863 119865101584010158401198612minus 1198602

119861 minus 11986510158401198612 1198611015840


(A4)


Data Availability




Acknowledgments


References






































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







119863119862119898 (119903 119905) = 1198621198632119865 [120581 120583 + 2120598 + 119901 + Π + 2Ω3 + 119864 (119902


2119863 minus 119864119863119879119861119863119862119863119863 )+ 1198642 (119863119862119860119862 + 119863119862119860119863119862119863119863 ))

minus 119881211986521198632119862(119880119863119862119860 + 1198631198621198632119863 ) minus 1

+ 11988111986521198621015840119863 (1198621198631198791198631015840

119863 + 1198631198791198621015840) + 119880119881119865119863 ( 12119862minus 119863119879119862 ) minus 119864211986511986311986211986321198632

(119862119863119862119862119863 + 119863119862119863


minus 31198631198621198632119863 minus 1198641198631198621198611198621015840 ) + 12119863 (119863119862119863119863119862119862119862

minus 119863119862119862119863) minus 1198641198621015840 (119863119879119865119863119879119861 + 1198642119863119862119865119863119862119861

+ 119865119863119879119861119863119879119863119863 ) + 119863119879119879119865 + 11986510158401015840119864211986210158402 minus 11988011986311987911986510158401198621015840+ 119865119863119879119879119862119862 + 119865119863119879119879119863119863 ]

(52)


119898(119903 119905) = 12 int119862

0

1198621198632119865 [120581 120583 + 2120598 + 119901 + Π + 2Ω3 + 119864 (119902 + 120598) + 11988031198651198631198621198602119862 + 1198802 119863119879119865119863119862119860 + 119865119862 (1198631198621198634119863 minus 119863119879119862 )

minus 119880119864119863119862119865(119864119863119862119860 minus 1198631198791198611198621015840 ) + 119863119879119865119863119879A+ 119865( 11198621015840 (11986311987911986210158402119862 + 1198631198791198631015840

2119863 minus 119864119863119879119861119863119862119863119863 ) + 1198642 (119863119862119860119862 + 119863119862119860119863119862119863119863 )) minus 119881211986521198632

119862(119880119863119862119860 + 1198631198621198632119863 ) minus 1

+ 11988111986521198621015840119863 (1198621198631198791198631015840

119863 + 1198631198791198621015840) + 119880119881119865119863 ( 12119862 minus 119863119879119862 )



+ 119865119863119879119879119862119862 + 119865119863119879119879119863119863 ]119889119862

(53)



(119879(119898)120572120573 + 119879(119863)120572120573 )120573 119881120572 = 0(119879(119898)120572120573 + 119879(119863)120572120573 )120573 120594120572 = 0

(54)

which produce

1119860 [ (120583 + 120598) + (120583 + 119901 + Π + 2120598 + 2Ω3 ) 119861+ (120583 + 119901 + Π + 120598 minus Ω3 )(119862 + 119863)]+ 1119861 [(119902 + 120598)1015840 + (119902 + 120598) (2119860

1015840

119860 + (119862119863)1015840119862119863 )] minus 1198631

= 0

(55)

1119860 [ (119902 + 120598) + (119902 + 120598) (2119861 + 119862 + 119863)]+ 1119861 [(119901 + Π + 120598 + 2Ω3 )1015840

+ (120583 + 119901 + Π + 2120598 + 2Ω3 ) 1198601015840

119860 + (120598 + Ω) (119862119863)1015840119862119863 ]+ 1198632 = 0

(56)



119863119879 (120583 + 120598) + 13 (3120583 + 3119901 + 3Π + 4120598)Θ+ 13 (120598 + Ω)( 2119860119861 minus 119880119862 minus 119881119863) + 119864119863119862 (119902 + 120598)+ (119902 + 120598) (21198861119861 + 119864119862 + 1198631015840

119861119863) minus 1198631 = 0(57)

119863119879 (119902 + 120598) + (119902 + 120598) (Θ + 119863119879119861119861 )+ 119864119863119862 (119901 + Π + 120598 + 2Ω3 )+ (120583 + 119901 + Π + 2120598 + 2Ω3 ) 1198861119861 + (120598 + Ω) (119862119863)1015840119861119862119863+ 1198632 = 0

(58)



119861 )+ 181198632

(1198621198812 minus 11986211986310158402


4119865

(59)




119861 ) minus 181198632(1198812119862 minus 11986211986310158402


119861119863 (120598 + Ω)] minus (1198642 + 11986310158401198622119861119863)[119863119879 (119902 + 120598)+ (119902 + 120598) (Θ + 119863119879119861119861 ) + 1198632]

(60)






119878120583 = 119878119899119881120583 + 119902120583119879 minus (1205730Π2 + 1205731119902]119902] + 1205732120587]120581120587]120581) 119881120583

2119879+ 1205720Π119902120583119879 + 1205721120587120583]119902]119879

(61)



119879119878120572120572 = minusΠ[119881120572120572 minus 1205720119902120572120572 + 1205730Π120572119881120572

+ 1198792 (1205730119879 119881120572)120572


120583

119902120572] minus 120587120572120583 [120590120572120583minus 1205721119902120583120572 + 1205732120587120572120583]119881] + 1198792 (1205732119879 119881])

]120587120572120583]

(62)


1205910Π120572119881120572 + Π = minus120577120579 + 1205720120577119902120572120572 minus 12120577119879( 1205910120577119879119881120572)120572

Π (63)


120573

sdot 119902120572(64)

1205912ℎ120583120572ℎ]120573120587120583]120588119881120588 + 120587120572120573 = minus2120578120590120572120573 + 21205781205721119902⟨120573120572⟩minus 120578119879( 12059122120578119879119881])

]120587120572120573 (65)

with

119902⟨120573120572⟩ = ℎ120583120573ℎ]120572 (12 (119902120583] + 119902]120583) minus 13119902120590120581ℎ120590120581ℎ120583]) (66)


1205910 = 12057712057301205911 = 12058111987912057311205912 = 21205781205732

(67)


1205910Π = minus (120577 + 12059102 Π)119860Θ + 1198601198611205720120577 [1199021015840 + 119902(1198601015840

119860 + 1198621015840119862+ 1198631015840

119863 )] minus Π[1205771198792( 1205910120577119879) + 119860]

(68)

1205911 119902 = minus119860119861 1205811198791015840 (1 + 1205720Π + 231205721Ω) + 119879[1198601015840

119860 minus 1205720Π1015840

minus 231205721 (Ω1015840 + 1198601015840

119860 Ω + 32 (1198621015840

119862 + 1198631015840

119863 )Ω)]minus 119902[12058111987922

( 12059111205811198792 ) + 12059112 119860Θ + 119860] (69)


minus Ω[120578119879 ( 12059122120578119879) + 12059122 119860Θ + 119860] (70)





119863119879119861119861 + 1198632]

(71)




119861 ) minus 1198628119863 (1198812119862 minus 11986211986310158402

1198612 )


+ 119864119862 (120598 + Ω) + 1198631015840

119861119863 (120598 + Ω)]

(72)


Λ = 1205811198791205911 (120583 + 119901 + Π + 2120598 + 23Ω)minus1 (1 minus 231205721Ω) (73)




(74)


Δ = 1205720120577119902 (120583 + 119901 + Π + 2120598 + 23Ω)minus1 ( 119902 + 21205982120577 + 1205910Π) (75)


5 Conclusions







Appendix

11987911986301 = 1120581 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 ) (A1)

11987911986311 = 1198612120581 (119891 (119877) minus 119877119865 (119877)2 ) + 1198602+ 1198602

(119862 + 119863minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 ) (A2)

1198631 = 119860120581 [ 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )1

+ 119891 (119877) minus 119877119865 (119877)21198602minus 1198651015840101584011986021198612

+ 1198604(119861 + C119862 + 119863)

minus 119865101584011986021198612 (1198621015840

119862 + 1198631015840

119863 minus 1198611015840119861 )0

+ 21198603119891 (119877) minus 119877119865 (119877)2 minus 119865101584010158401198612

+ 1198602(119861 + 119862 + 119863) minus 11986510158401198612 (119862

1015840

119862 + 1198631015840

119863 minus 1198611015840119861 )+ 1198602119861 minus 1198602

minus 119865101584010158401198612 + 1198602(119860 + 119861)

+ 11986510158401198612 (1198601015840

119860 + 1198611015840119861 ) + 1198602119862 minus 1198602+ 1198602

119860+ 11986510158401198612 119860

1015840

119860 + 1198602119863 minus 1198602+ 1198602

119860 + 11986510158401198612 1198601015840

119860 + 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )(31198601015840

119860 + 1198611015840119861 + 1198621015840119862+ 1198631015840

119863 )]

(A3)



1015840

119860 + 1198621015840119862 + 1198631015840

119863 )1

+ 1198601015840

1198601198612 1198602+ 119865101584010158401198612

minus 1198602(119860 + 119861) minus 11986510158401198612 (119860

1015840

119860 + 1198611015840119861 )+ 211986110158401198613 119891 (119877) minus 119877119865 (119877)2 + 1198602

+ 1198602(119862 + 119863 minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 )+ 11986210158401198612119862 119865101584010158401198612 minus 1198602

119861 minus 11986510158401198612 1198611015840

119861 + 1198631015840

1198612119863 119865101584010158401198612minus 1198602

119861 minus 11986510158401198612 1198611015840


(A4)


Data Availability




Acknowledgments


References






































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







(119879(119898)120572120573 + 119879(119863)120572120573 )120573 119881120572 = 0(119879(119898)120572120573 + 119879(119863)120572120573 )120573 120594120572 = 0

(54)

which produce

1119860 [ (120583 + 120598) + (120583 + 119901 + Π + 2120598 + 2Ω3 ) 119861+ (120583 + 119901 + Π + 120598 minus Ω3 )(119862 + 119863)]+ 1119861 [(119902 + 120598)1015840 + (119902 + 120598) (2119860

1015840

119860 + (119862119863)1015840119862119863 )] minus 1198631

= 0

(55)

1119860 [ (119902 + 120598) + (119902 + 120598) (2119861 + 119862 + 119863)]+ 1119861 [(119901 + Π + 120598 + 2Ω3 )1015840

+ (120583 + 119901 + Π + 2120598 + 2Ω3 ) 1198601015840

119860 + (120598 + Ω) (119862119863)1015840119862119863 ]+ 1198632 = 0

(56)



119863119879 (120583 + 120598) + 13 (3120583 + 3119901 + 3Π + 4120598)Θ+ 13 (120598 + Ω)( 2119860119861 minus 119880119862 minus 119881119863) + 119864119863119862 (119902 + 120598)+ (119902 + 120598) (21198861119861 + 119864119862 + 1198631015840

119861119863) minus 1198631 = 0(57)

119863119879 (119902 + 120598) + (119902 + 120598) (Θ + 119863119879119861119861 )+ 119864119863119862 (119901 + Π + 120598 + 2Ω3 )+ (120583 + 119901 + Π + 2120598 + 2Ω3 ) 1198861119861 + (120598 + Ω) (119862119863)1015840119861119862119863+ 1198632 = 0

(58)



119861 )+ 181198632

(1198621198812 minus 11986211986310158402


4119865

(59)




119861 ) minus 181198632(1198812119862 minus 11986211986310158402


119861119863 (120598 + Ω)] minus (1198642 + 11986310158401198622119861119863)[119863119879 (119902 + 120598)+ (119902 + 120598) (Θ + 119863119879119861119861 ) + 1198632]

(60)






119878120583 = 119878119899119881120583 + 119902120583119879 minus (1205730Π2 + 1205731119902]119902] + 1205732120587]120581120587]120581) 119881120583

2119879+ 1205720Π119902120583119879 + 1205721120587120583]119902]119879

(61)



119879119878120572120572 = minusΠ[119881120572120572 minus 1205720119902120572120572 + 1205730Π120572119881120572

+ 1198792 (1205730119879 119881120572)120572


120583

119902120572] minus 120587120572120583 [120590120572120583minus 1205721119902120583120572 + 1205732120587120572120583]119881] + 1198792 (1205732119879 119881])

]120587120572120583]

(62)


1205910Π120572119881120572 + Π = minus120577120579 + 1205720120577119902120572120572 minus 12120577119879( 1205910120577119879119881120572)120572

Π (63)


120573

sdot 119902120572(64)

1205912ℎ120583120572ℎ]120573120587120583]120588119881120588 + 120587120572120573 = minus2120578120590120572120573 + 21205781205721119902⟨120573120572⟩minus 120578119879( 12059122120578119879119881])

]120587120572120573 (65)

with

119902⟨120573120572⟩ = ℎ120583120573ℎ]120572 (12 (119902120583] + 119902]120583) minus 13119902120590120581ℎ120590120581ℎ120583]) (66)


1205910 = 12057712057301205911 = 12058111987912057311205912 = 21205781205732

(67)


1205910Π = minus (120577 + 12059102 Π)119860Θ + 1198601198611205720120577 [1199021015840 + 119902(1198601015840

119860 + 1198621015840119862+ 1198631015840

119863 )] minus Π[1205771198792( 1205910120577119879) + 119860]

(68)

1205911 119902 = minus119860119861 1205811198791015840 (1 + 1205720Π + 231205721Ω) + 119879[1198601015840

119860 minus 1205720Π1015840

minus 231205721 (Ω1015840 + 1198601015840

119860 Ω + 32 (1198621015840

119862 + 1198631015840

119863 )Ω)]minus 119902[12058111987922

( 12059111205811198792 ) + 12059112 119860Θ + 119860] (69)


minus Ω[120578119879 ( 12059122120578119879) + 12059122 119860Θ + 119860] (70)





119863119879119861119861 + 1198632]

(71)




119861 ) minus 1198628119863 (1198812119862 minus 11986211986310158402

1198612 )


+ 119864119862 (120598 + Ω) + 1198631015840

119861119863 (120598 + Ω)]

(72)


Λ = 1205811198791205911 (120583 + 119901 + Π + 2120598 + 23Ω)minus1 (1 minus 231205721Ω) (73)




(74)


Δ = 1205720120577119902 (120583 + 119901 + Π + 2120598 + 23Ω)minus1 ( 119902 + 21205982120577 + 1205910Π) (75)


5 Conclusions







Appendix

11987911986301 = 1120581 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 ) (A1)

11987911986311 = 1198612120581 (119891 (119877) minus 119877119865 (119877)2 ) + 1198602+ 1198602

(119862 + 119863minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 ) (A2)

1198631 = 119860120581 [ 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )1

+ 119891 (119877) minus 119877119865 (119877)21198602minus 1198651015840101584011986021198612

+ 1198604(119861 + C119862 + 119863)

minus 119865101584011986021198612 (1198621015840

119862 + 1198631015840

119863 minus 1198611015840119861 )0

+ 21198603119891 (119877) minus 119877119865 (119877)2 minus 119865101584010158401198612

+ 1198602(119861 + 119862 + 119863) minus 11986510158401198612 (119862

1015840

119862 + 1198631015840

119863 minus 1198611015840119861 )+ 1198602119861 minus 1198602

minus 119865101584010158401198612 + 1198602(119860 + 119861)

+ 11986510158401198612 (1198601015840

119860 + 1198611015840119861 ) + 1198602119862 minus 1198602+ 1198602

119860+ 11986510158401198612 119860

1015840

119860 + 1198602119863 minus 1198602+ 1198602

119860 + 11986510158401198612 1198601015840

119860 + 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )(31198601015840

119860 + 1198611015840119861 + 1198621015840119862+ 1198631015840

119863 )]

(A3)



1015840

119860 + 1198621015840119862 + 1198631015840

119863 )1

+ 1198601015840

1198601198612 1198602+ 119865101584010158401198612

minus 1198602(119860 + 119861) minus 11986510158401198612 (119860

1015840

119860 + 1198611015840119861 )+ 211986110158401198613 119891 (119877) minus 119877119865 (119877)2 + 1198602

+ 1198602(119862 + 119863 minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 )+ 11986210158401198612119862 119865101584010158401198612 minus 1198602

119861 minus 11986510158401198612 1198611015840

119861 + 1198631015840

1198612119863 119865101584010158401198612minus 1198602

119861 minus 11986510158401198612 1198611015840


(A4)


Data Availability




Acknowledgments


References






































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery









119878120583 = 119878119899119881120583 + 119902120583119879 minus (1205730Π2 + 1205731119902]119902] + 1205732120587]120581120587]120581) 119881120583

2119879+ 1205720Π119902120583119879 + 1205721120587120583]119902]119879

(61)



119879119878120572120572 = minusΠ[119881120572120572 minus 1205720119902120572120572 + 1205730Π120572119881120572

+ 1198792 (1205730119879 119881120572)120572


120583

119902120572] minus 120587120572120583 [120590120572120583minus 1205721119902120583120572 + 1205732120587120572120583]119881] + 1198792 (1205732119879 119881])

]120587120572120583]

(62)


1205910Π120572119881120572 + Π = minus120577120579 + 1205720120577119902120572120572 minus 12120577119879( 1205910120577119879119881120572)120572

Π (63)


120573

sdot 119902120572(64)

1205912ℎ120583120572ℎ]120573120587120583]120588119881120588 + 120587120572120573 = minus2120578120590120572120573 + 21205781205721119902⟨120573120572⟩minus 120578119879( 12059122120578119879119881])

]120587120572120573 (65)

with

119902⟨120573120572⟩ = ℎ120583120573ℎ]120572 (12 (119902120583] + 119902]120583) minus 13119902120590120581ℎ120590120581ℎ120583]) (66)


1205910 = 12057712057301205911 = 12058111987912057311205912 = 21205781205732

(67)


1205910Π = minus (120577 + 12059102 Π)119860Θ + 1198601198611205720120577 [1199021015840 + 119902(1198601015840

119860 + 1198621015840119862+ 1198631015840

119863 )] minus Π[1205771198792( 1205910120577119879) + 119860]

(68)

1205911 119902 = minus119860119861 1205811198791015840 (1 + 1205720Π + 231205721Ω) + 119879[1198601015840

119860 minus 1205720Π1015840

minus 231205721 (Ω1015840 + 1198601015840

119860 Ω + 32 (1198621015840

119862 + 1198631015840

119863 )Ω)]minus 119902[12058111987922

( 12059111205811198792 ) + 12059112 119860Θ + 119860] (69)


minus Ω[120578119879 ( 12059122120578119879) + 12059122 119860Θ + 119860] (70)





119863119879119861119861 + 1198632]

(71)




119861 ) minus 1198628119863 (1198812119862 minus 11986211986310158402

1198612 )


+ 119864119862 (120598 + Ω) + 1198631015840

119861119863 (120598 + Ω)]

(72)


Λ = 1205811198791205911 (120583 + 119901 + Π + 2120598 + 23Ω)minus1 (1 minus 231205721Ω) (73)




(74)


Δ = 1205720120577119902 (120583 + 119901 + Π + 2120598 + 23Ω)minus1 ( 119902 + 21205982120577 + 1205910Π) (75)


5 Conclusions







Appendix

11987911986301 = 1120581 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 ) (A1)

11987911986311 = 1198612120581 (119891 (119877) minus 119877119865 (119877)2 ) + 1198602+ 1198602

(119862 + 119863minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 ) (A2)

1198631 = 119860120581 [ 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )1

+ 119891 (119877) minus 119877119865 (119877)21198602minus 1198651015840101584011986021198612

+ 1198604(119861 + C119862 + 119863)

minus 119865101584011986021198612 (1198621015840

119862 + 1198631015840

119863 minus 1198611015840119861 )0

+ 21198603119891 (119877) minus 119877119865 (119877)2 minus 119865101584010158401198612

+ 1198602(119861 + 119862 + 119863) minus 11986510158401198612 (119862

1015840

119862 + 1198631015840

119863 minus 1198611015840119861 )+ 1198602119861 minus 1198602

minus 119865101584010158401198612 + 1198602(119860 + 119861)

+ 11986510158401198612 (1198601015840

119860 + 1198611015840119861 ) + 1198602119862 minus 1198602+ 1198602

119860+ 11986510158401198612 119860

1015840

119860 + 1198602119863 minus 1198602+ 1198602

119860 + 11986510158401198612 1198601015840

119860 + 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )(31198601015840

119860 + 1198611015840119861 + 1198621015840119862+ 1198631015840

119863 )]

(A3)



1015840

119860 + 1198621015840119862 + 1198631015840

119863 )1

+ 1198601015840

1198601198612 1198602+ 119865101584010158401198612

minus 1198602(119860 + 119861) minus 11986510158401198612 (119860

1015840

119860 + 1198611015840119861 )+ 211986110158401198613 119891 (119877) minus 119877119865 (119877)2 + 1198602

+ 1198602(119862 + 119863 minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 )+ 11986210158401198612119862 119865101584010158401198612 minus 1198602

119861 minus 11986510158401198612 1198611015840

119861 + 1198631015840

1198612119863 119865101584010158401198612minus 1198602

119861 minus 11986510158401198612 1198611015840


(A4)


Data Availability




Acknowledgments


References






































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery









119863119879119861119861 + 1198632]

(71)




119861 ) minus 1198628119863 (1198812119862 minus 11986211986310158402

1198612 )


+ 119864119862 (120598 + Ω) + 1198631015840

119861119863 (120598 + Ω)]

(72)


Λ = 1205811198791205911 (120583 + 119901 + Π + 2120598 + 23Ω)minus1 (1 minus 231205721Ω) (73)




(74)


Δ = 1205720120577119902 (120583 + 119901 + Π + 2120598 + 23Ω)minus1 ( 119902 + 21205982120577 + 1205910Π) (75)


5 Conclusions







Appendix

11987911986301 = 1120581 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 ) (A1)

11987911986311 = 1198612120581 (119891 (119877) minus 119877119865 (119877)2 ) + 1198602+ 1198602

(119862 + 119863minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 ) (A2)

1198631 = 119860120581 [ 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )1

+ 119891 (119877) minus 119877119865 (119877)21198602minus 1198651015840101584011986021198612

+ 1198604(119861 + C119862 + 119863)

minus 119865101584011986021198612 (1198621015840

119862 + 1198631015840

119863 minus 1198611015840119861 )0

+ 21198603119891 (119877) minus 119877119865 (119877)2 minus 119865101584010158401198612

+ 1198602(119861 + 119862 + 119863) minus 11986510158401198612 (119862

1015840

119862 + 1198631015840

119863 minus 1198611015840119861 )+ 1198602119861 minus 1198602

minus 119865101584010158401198612 + 1198602(119860 + 119861)

+ 11986510158401198612 (1198601015840

119860 + 1198611015840119861 ) + 1198602119862 minus 1198602+ 1198602

119860+ 11986510158401198612 119860

1015840

119860 + 1198602119863 minus 1198602+ 1198602

119860 + 11986510158401198612 1198601015840

119860 + 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )(31198601015840

119860 + 1198611015840119861 + 1198621015840119862+ 1198631015840

119863 )]

(A3)



1015840

119860 + 1198621015840119862 + 1198631015840

119863 )1

+ 1198601015840

1198601198612 1198602+ 119865101584010158401198612

minus 1198602(119860 + 119861) minus 11986510158401198612 (119860

1015840

119860 + 1198611015840119861 )+ 211986110158401198613 119891 (119877) minus 119877119865 (119877)2 + 1198602

+ 1198602(119862 + 119863 minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 )+ 11986210158401198612119862 119865101584010158401198612 minus 1198602

119861 minus 11986510158401198612 1198611015840

119861 + 1198631015840

1198612119863 119865101584010158401198612minus 1198602

119861 minus 11986510158401198612 1198611015840


(A4)


Data Availability




Acknowledgments


References






































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery









Appendix

11987911986301 = 1120581 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 ) (A1)

11987911986311 = 1198612120581 (119891 (119877) minus 119877119865 (119877)2 ) + 1198602+ 1198602

(119862 + 119863minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 ) (A2)

1198631 = 119860120581 [ 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )1

+ 119891 (119877) minus 119877119865 (119877)21198602minus 1198651015840101584011986021198612

+ 1198604(119861 + C119862 + 119863)

minus 119865101584011986021198612 (1198621015840

119862 + 1198631015840

119863 minus 1198611015840119861 )0

+ 21198603119891 (119877) minus 119877119865 (119877)2 minus 119865101584010158401198612

+ 1198602(119861 + 119862 + 119863) minus 11986510158401198612 (119862

1015840

119862 + 1198631015840

119863 minus 1198611015840119861 )+ 1198602119861 minus 1198602

minus 119865101584010158401198612 + 1198602(119860 + 119861)

+ 11986510158401198612 (1198601015840

119860 + 1198611015840119861 ) + 1198602119862 minus 1198602+ 1198602

119860+ 11986510158401198612 119860

1015840

119860 + 1198602119863 minus 1198602+ 1198602

119860 + 11986510158401198612 1198601015840

119860 + 111986021198612 ( 1198651015840 minus 1198601015840119860 minus 1198651015840119861 )(31198601015840

119860 + 1198611015840119861 + 1198621015840119862+ 1198631015840

119863 )]

(A3)



1015840

119860 + 1198621015840119862 + 1198631015840

119863 )1

+ 1198601015840

1198601198612 1198602+ 119865101584010158401198612

minus 1198602(119860 + 119861) minus 11986510158401198612 (119860

1015840

119860 + 1198611015840119861 )+ 211986110158401198613 119891 (119877) minus 119877119865 (119877)2 + 1198602

+ 1198602(119862 + 119863 minus 119860) minus 11986510158401198612 (119860

1015840

119860 + 1198621015840119862 + 1198631015840

119863 )+ 11986210158401198612119862 119865101584010158401198612 minus 1198602

119861 minus 11986510158401198612 1198611015840

119861 + 1198631015840

1198612119863 119865101584010158401198612minus 1198602

119861 minus 11986510158401198612 1198611015840


(A4)


Data Availability




Acknowledgments


References






































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery






Data Availability




Acknowledgments


References






































































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery







































Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery








Shock and Vibration





Volume 2018














Geophysics



Volume 2018




Volume 2018






Journal of

Chemistry




RotatingMachinery





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Dynamics of Dissipative Viscous Cylindrical Collapse with Full …downloads.hindawi.com/journals/ahep/2018/9250786.pdf · 2019-07-30 · the hydrostatic periods of astral models