International Journal of Thermodynamics, Vol. 18, No. 1, P (2015). Sergey G.

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1 International Journal of Therodynais Vol. 8 No. P. 3-4 (5). Four-diensional equation of otion for visous opressible and harged fluid with regard to the aeleration field pressure field and dissipation field Abstrat Sergey G. Fedosin PO box 6488 Sviazeva str. -79 Per Russia E-ail: intelli@list.ru Fro the priniple of least ation the equation of otion for visous opressible and harged fluid is derived. The visosity effet is desribed by the 4-potential of the energy dissipation field dissipation tensor and dissipation stress-energy tensor. In the weak field liit it is shown that the obtained equation is equivalent to the Navier-Stokes equation. The equation for the power of the kineti energy loss is provided the equation of otion is integrated and the dependene of the veloity agnitude is deterined. A oplete set of equations is presented whih suffies to solve the proble of otion of visous opressible and harged fluid in the gravitational and eletroagneti fields. Keywords: Navier-Stokes equation; dissipation field; aeleration field; pressure field; visosity.. Introdution Sine Navier-Stokes equations appeared in 87 (Navier 87) (Poisson 83) onstant attepts have been ade to derive these equations by various ethods. Stokes (Stokes 849) and Saint-Venant (Saint-Venant 843) in derivation of these equations relied on the fat that the deviatori stress tensor of noral and tangential stress is linearly related to the three-diensional deforation rate tensor and the fluid is isotropi. In book: (Khelnik ) it is onsidered that Navier- Stokes equations are the extreu onditions of soe funtional and a ethod of finding a solution of these equations is desribed whih onsists in the gradient otion to the extreu of this funtional. One of the variants of the four-diensional stressenergy tensor of visous stresses in the speial theory of relativity an be found in book: (Landau & Lifshitz 987). The divergene of this tensor gives the required visous ters in the Navier-Stokes equation. The phenoenologial derivation of this tensor is based on the assued ondition of entropy inreent during energy dissipation. As a onsequene in the o-oving referene frae the tie oponents of the tensor i.e. the dissipation energy density and its flux vanish. Therefore suh a tensor is not a universal tensor and annot serve for exaple as a basis for deterining the etri in the presene of visosity. In this artile our goal is to provide in general for the four-diensional stress-energy tensor of energy dissipation whih desribes in the urved spaetie the energy density and the stress and energy flux arising due to visous stresses. This tensor will be derived with the help of the priniple of least ation on the basis of a ovariant 4- potential of the dissipation field. Then we will apply these quantities in the equation of otion of the visous opressible and harged fluid and by seleting the salar potential of the dissipation field we will obtain the Navier- Stokes equation. The essential eleent of our alulations will be the use of the wave equation for the field potentials of the aeleration field. In onlusion we will present a oplete set of equations suffiient to desribe the otion of visous fluid. For sipliity we will assue that the various fields existing siultaneously would not produe any indued effets (urrents) due to oupling and interations between the fields are absent.. The ation funtion The starting point of our alulations is the ation funtion in the following for: k( R ) D J Φ Φ A j F F u J 6 G 4 S L g d u u J f f J h h () where L is the Lagrange funtion or Lagrangian R is the salar urvature is the osologial onstant J u is the 4-vetor of the ass (gravitational) urrent is the ass density in the referene frae assoiated with the fluid unit

2 dx u is the 4-veloity of a point partile is ds the speed of light D D is the 4-potential of the gravitational field desribed by the salar potential and the vetor potential D of this field G is the gravitational onstant Φ is the gravitational tensor A A is the 4-potential of the eletroagneti field whih is speified by the salar potential and the vetor potential A of this field j u is the 4-vetor of the eletroagneti q (harge) urrent is the harge density in the referene frae q assoiated with the fluid unit is the vauu perittivity F is the eletroagneti tensor u g u is the 4-veloity with the ovariant index expressed with the help of the etri tensor and the 4- veloity with the ontravariant index; the ovariant 4- veloity is the 4-potential of the aeleration field u U where and U denote the salar and vetor potentials respetively u is the aeleration tensor and are soe funtions of oordinates and tie p u Π is the 4-potential of the pressure field onsisting of the salar potential and the vetor potential Π p is the pressure in the referene p frae assoiated with the partile the ratio defines the equation of state of the fluid f is the pressure field tensor. The above-entioned quantities are desribed in detail in (Fedosin 4a). In addition to the we introdue the 4-potential of energy dissipation in the fluid: u Θ () where is the dissipation funtion and Θ are the salar and vetor dissipation potentials respetively. Using the 4-potential dissipation tensor: we onstrut the energy h. (3) The oeffiients and in order to siplify alulations we assue to be a onstants. The ter J in Eq. () reflets the fat that the energy of the fluid otion an be dissipated in the surrounding ediu and turn into the internal fluid energy while the syste s energy does not hange. The last ter in Eq. () is assoiated with the energy auulated by the syste due to the ation of the energy dissipation. The ethod of onstruting the dissipation 4-potential in Eq. () and the dissipation tensor h in Eq. (3) is fully idential to that whih was used earlier in (Fedosin 4a). Therefore we will not provide here the interediate results and will right away write the equations of otion of the fluid and field obtained as a result of the variation of the ation funtion in Eq. (). 3. Field equations The eletroagneti field equations have the standard for: F F F F j (4) where is the vauu pereability. The gravitational field equations are: Φ Φ Φ 4 G Φ J. (5) The aeleration field equations are: u u u 4 u J. (6) The pressure field equations are: f f f 4 f J. (7) The dissipation field equations are: h h h 4 h J. (8)

3 In order to obtain the Eqs. (4)-(8) variation by the orresponding 4-potential is arried out in the ation funtion in Eq. (). All the above-entioned fields have vetor harater. Eah field an be desribed by two three-diensional vetors inluded into the orresponding field tensor. One of these vetors is the strength of the orresponding field and the other solenoidal vetor desribes the field vortiity. For exaple the oponents of the eletri field strength E and the agneti field indution B are the oponents of the eletroagneti tensor F. The gravitational tensor Φ onsists of the oponents of the gravitational field strength Γ and the torsion field Ω. As an be seen fro Eqs. (4)-(8) the onstants and have the sae eaning as the onstants and G all these onstants reflet the relationship between the 4- urrent of dissipated fluid and the divergene of the orresponding field tensor. The properties of the dissipation field are provided in Appendix A. 4. Field gauge In order to siplify the for of equations we use the following field gauge: A A D D u u. (9) In Eq. (9) the gauge of eah field is arried out by equating the ovariant derivative of the orresponding 4- potential to zero. Sine the 4-potentials onsist of the salar and vetor potentials the gauge in Eq. (9) links the salar and vetor potential of eah field. As a result the divergene of the vetor potential of any field in a ertain volue is aopanied by hange in tie of the salar field potential in this volue and also depends on the tensor produt of the Christoffel sybols and the 4-potential that is on the degree of spaetie urvature. 5. Continuity equations In Eqs. (4)-(8) the divergenes of field tensors are assoiated with their soures i.e. with 4-urrents. The field tensors are defined by their 4-potentials siilarly to Eq. (3): F A A A A Φ D D D D () u u u u u f. If we substitute Eqs. (3) and () into Eqs. (4)-(8) and apply the ovariant derivative to all ters we obtain the following relations ontaining the Rii tensor: R F j 4 G R Φ J 4 R u J 4 R f J 4 R h J. () In the liit of speial theory of relativity the Rii tensor vanishes the ovariant derivative turns into the 4- gradient and then instead of Eq. () we an write: j J. () Eqs. () are the ordinary ontinuity equation of the harge and ass 4-urrents in the flat spaetie. 6. Equations of otion The variation of the ation funtion leads diretly to the equations desribing the otion of the fluid unit under the ation of the fields: u J Φ J F j f J h J. (3) The left side of the equality an be transfored taking into aount the expression J u for the 4-vetor of the ass urrent density and the definition in Eq. () for the aeleration tensor u u u : Du d u u J u u u u u u u a D d. (4) In Eq. (4) a denotes the 4-aeleration and we used D the operator of proper-tie-derivative u where D D is the sybol of 4-differential in the urved spaetie is the proper tie (Fedosin 9). If we substitute Eq. (4) into Eq. (3) we obtain the equation of otion in whih the 4-aeleration is expressed in ters of field tensors and 4-urrents: a Φ J F j f J h J. (5) Variation of the ation funtion allows us to find the for of stress-energy tensors of all the fields assoiated with the fluid: 4 W g F F g F F 3

4 U g Φ Φ g Φ Φ 4 G 4 B g u u g u u 4 4 P g f f g f f 4 4 Q g h h g h h 4 4. (6) One of the properties of these tensors is that their divergenes alongside with the field tensors speify the densities of 4-fores arising fro the influene of the orresponding field on the fluid: k ( f ) F j W e k k ( f ) Φ J U g k ( f ) a u J B k a k k ( f ) f J P p k k ( f ) h J Q. (7) d k The left side of Eq. (7) ontains the density of the orresponding 4-fore exluding ( whih up to sign f ) a denotes the density of the 4-fore ating fro the aelerated fluid on the rest four fields. Fro Eqs. (3)-(7) it follows that the equation of otion an be written only in ters of the divergenes of the stress-energy tensors of fields: ( W U B P Q ). (8) We have integrated equation Eq. (8) in (Fedosin 4b) in the weak field liit (exluding the stress-energy tensor of dissipation Q ) and this allowed us to explain the well-known 4/3 proble of the fields ass-energy inequality in the fixed and oving systes. The Eq. (8) will be used also in the Eq. () for the etri. 7. The syste s energy The ation funtion in Eq. () ontains the Lagrangian L. Applying to it the Legendre transforations for a syste of fluid units we an find the syste s Hailtonian. This Hailtonian is the relativisti energy of the syste written in an arbitrary referene frae. Sine the energy is dependent on the osologial onstant gauging of the osologial onstant should be done using the relation: k D J A j u J J J. (9) As a result we an write for the energy of the syste the following: 3 E q u g dx dx dx 3 F F Φ Φ u u f f h h g dx dx dx. 4 6 G () The energy of the syste in the for of a set of losely interating partiles and the related fields in the weak field liit was alulated in (Fedosin 4). The differene between the syste s ass and the gravitational ass was shown as well as the fat that the ass-energy of the proper eletroagneti field redues the gravitational ass of the syste. 8. Equation for the etri Aording to the logi of the ovariant theory of gravitation (Fedosin ) and the etri theory of relativity (Fedosin 4d) ontribution to the definition of the syste s etri is ade by the stress-energy tensors of all the fields inluding the gravitational field. The etri is a seondary funtion the derivative of the fields ating in the syste that define all the basi properties of the syste. The equation for the etri is obtained as follows: R R g ( W U B P Q ). () 4 k If we ultiply Eq. () by the etri tensor g and ontrat over all the indies the right and left sides of the equation vanish. It follows fro the properties of tensors in Eq. (). Outside the fluid liits with regard to the gauge the salar urvature R beoes equal to zero. If we take into aount the equation of otion in Eq. (8) the ovariant derivative of the right side of Eq. () is zero. The ovariant derivative of the left side of Eq. () is also zero sine R as a onsequene of the osologial onstant gauge and for the Einstein tensor the following equality holds: R R g. 9. The analysis of the equation of otion Eqs. (4)-(5) iply the onnetion between the ovariant 4-aeleration of a fluid unit and the densities of ating fores in the urved spaetie: 4

5 du a uu Φ J F j f J h J. d For the dissipation field we will use Eq. (A6) fro We will write this four-diensional equation separately for the tie and spae oponents given that Appendix A where in the general ase should be ds dx dx substituted instead of. Expressions for other fields an be J ( v ) ds ds ds found in (Fedosin 4a). This gives: dx as well as j q q ( v ). ds ds du q uu Γ v E v C v X v. d s ds dui i uu d s ds q Γ [ v Ω] E [ v B] C [ v I] X [ v Y ]. Here Γ E C and X are the vetors of strengths of gravitational and eletroagneti fields pressure field and dissipation field respetively. Notations Ω B I and Y refer to the torsion field the agneti field and the solenoidal vetors of the pressure and dissipation fields respetively. After redution by a fator we obtain: ds du ds uu q Γ v E v C v X v. () dui d i uu q q Γ [ v Ω] E [ v B] C [ v I] X [ v Y ]. (3) q q In Eq. (3) the su E [ vb ] is the ontribution of the eletroagneti Lorentz fore ating on the fluid unit into the total aeleration. The inus sign before this su appears beause u i is the spae oponent of the ovariant 4-veloity whih differs fro the ordinary i ontravariant spae oponent u in the for fator of the etri tensor. Siilarly the su Γ [ vω ] is the aeleration of the gravitational Lorentz fore. Gravitational and eletroagneti fores are the so-alled ass fores distributed over the entire volue where there is ass and harge of the fluid. 9. The equation of otion in Minkowski spae In order to siplify our analysis we will onsider Eqs. () and (3) in the fraework of the speial theory of relativity. The su of the last two ters in Eq. (3) taking into aount the Eq. (A7) fro Appendix A for X and Y gives the following: ( v) X [ vy] ( ) v[ v] t ( v) d( v) ( ) ( v )( v) ( v). t (4) In Eq. (4) v ds is the Lorentz fator. For the pressure field with regard to the definition of p the 4-potential in the for of u Π the pressure field tensor f fro Eq. () and the definition of the vetors C and I by the rule: f C fi j i j j i Ik i i i i we find the expression for the vetors: Π p pv C t t p v I Π. (5) Using Eq. (5) we alulate the su of two ters in Eq. (3): 5

6 p p v p v C [ vi] v t p p v p v p v p d p v ( ). v p t p (6) Substituting Eqs. (4) and (6) into Eq. (3) and taking into aount that in Minkowski spae the Christoffel sybols are zero and the spae oponent of the 4-veloity equals ui v we find: d p p v a. (7) In Eq. (7) we have introdued notation for the aeleration resulting fro the ation of ass fores: q q a Γ [ v Ω] E [ ] v B. Until now we have not defined the dissipation funtion. In this approxiation it is assoiated with a salar potential of the dissipation field by relation:. Let us assue that v ( v ) (8) that is d ( ) d v r v r. It eans that the salar potential of the dissipation field is proportional both to the veloity v of the onsidered fluid unit and the path traveled by it in the surrounding spae. Contribution to is also ade by the gradient of the veloity divergene with a ertain oeffiient. The oeffiient depends on the paraeters of interating fluid layers in a first approxiation it is inversely proportional to the square of the layers thikness. At the sae tie the oeffiient reflets the fluid properties and an be different in different fluids. Taking into aount Eq. (8) Eq. ( 7) is transfored as follows: d p p v ( ) a v v. (9) p Due to the presene in Eq. (9) of the gradient of the pressure to the ass density ratio there is aeleration direted against this gradient. The ter in Eq. (9) whih is proportional to the veloity v defines the rate of deeleration due to visosity. Sine the deeleration in Eq. (9) depends not on the absolute veloity but on the veloity of otion of soe fluid layers relative to the other layers the veloity v should be a relative veloity. We will use the freedo of hoosing the referene frae in order to ove fro absolute veloities to relative veloities. Suppose the referene frae is o-oving and it oves in the fluid with the ontrol volue of a sall size. Then in suh a referene frae the veloity v in Eq. (9) will be a relative veloity: soe layers will be ahead while others will lag behind and visous fores will appear. We will now write the equations for the aeleration field fro (Fedosin 4a): S 4 S 4 v t N u U equals the 4-veloity taken with the ovariant index. The aeleration tensor u is defined in Eq. () as a 4-url and it ontains the vetors S and N : ui ui i u S i u i j i u j j ui N k. In Minkowski spae we an ove fro the salar and vetor U potentials of the aeleration field to the 4- veloity oponents and express vetors S and N in ters of the: U ( v) S t t N U v. (3) Let us substitute Eq. (3) into the seond equation in Eq. (3): N N S. (3) t 4 v ( ) t t ( v) v. (3) The vetor S in Eq. (3) is the aeleration field strength and the vetor N is the solenoidal vetor of the aeleration field. The 4-potential of the aeleration field 6

7 The gauge ondition of the 4-potential of the aeleration field in Eq. (9) has the for: u. In Minkowski spae this relation is siplified: u t v. (33) t U or With regard to Eq. (33) we will transfor the left side of Eq. (3): ( v) ( v) ( v) ( v ). t Substituting this in Eq. (3) we obtain the wave equation: 4 v ( v) ( v ). (34) t Aording to Eq. (34) the veloity v of the fluid otion in the syste ust onfor to the wave equation that eans that the veloity is given by the syste s paraeters and hanges ontinuously in transition fro one ontrol volue to another. The wave equation for the Lorentz fator follows fro Eq. (3) and the first equation in Eq. (3) with regard to Eq. (33): t 4. We an express the veloity v fro Eq. (34) and substitute it in Eq. (9): d p ( ) p v v ( ) ( ) a v v. (35) 4 t Let us find out the physial eaning of the last ter in Eq. (35). The gauge ondition in Eq. (33) of the 4-potential of the aeleration field an be rewritten as follows: d dv v. 4 Hene provided we have: dv d ( v) v ( v ) v ( ) 4 v v. (36) Thus in Eq. (35) visosity is taken into aount not only due to the otion veloity but also due to the rate of aeleration hange of the fluid otion. The quantity v is the gradient of half of the squared veloity that is the gradient of the kineti energy per unit ass. This quantity is proportional to the aeleration arising due to the dissipation of the kineti energy of otion. The tie derivative of v leads to the rate of aeleration hange. Other ters in Eq. (36) also have the diension of the rate of aeleration hange. 9. Coparison with the Navier-Stokes equation The vetor Navier-Stokes equation in its lassial for is usually used for non-relativisti desription of the liquid otion and has the following for (Landau & Lifshitz 987): dv v a ( v ) v a p v ( v ) (37) t 3 where a and v are the veloity and aeleration of an arbitrary point unit of liquid is the ass density p is the pressure is the kineati visosity oeffiient is the volue (bulk or seond) visosity oeffiient a is the aeleration produed by ass fores in the liquid and it is assued that the oeffiients and are onstant in volue. In Eq. (37) the veloity v depends not only on the tie but also on the oordinates of the oving liquid unit. This allows us to expand the derivative d v into the su of two v partial derivatives: tie derivative and spae derivative t ( v) v that is to apply the aterial (substantial) derivative. Coparing Eqs. (35) and (37) for the ase of low veloities when tends to unity and at suffiiently low pressure and visosity we obtain the kineati visosity oeffiient:. (38) 4 7

8 Sine where is the dynai visosity oeffiient then we obtain:. 4 In this ratio depends priarily on the fluid properties and the oeffiients and also depend on the paraeters of the syste under onsideration. For exaple if we study the liquid flow between two losely loated plates the oeffiient is inversely proportional to the square of the distane between the plates. The equality of the last ters in Eqs. (35) and (37) iplies: 3 3. (39) 3 The presene of and in Eq. (37) iplies two auses of the rate of aeleration hange: one of the is due to the fluid density variation beause of the ediu resistane and the other is due to the oentu variation of the fluid oving in a visous ediu. 9.3 The energy power In Minkowski spae the tie oponent of the 4- veloity is equal to u the Christoffel sybols are zero and Eq. () an be written as follows: d d( ) v q Γ v E v C v X v. (4) We an also obtain Eq. (4) if in Minkowski spae we ultiply the equation of otion Eq. (3) by the veloity v. We substitute in Eq. (4) the vetor C fro Eq. (5) and the vetor X aording to Eq. (A7) fro Appendix A: d ( v) ( v) ( ) t t q p p v Γ v E v v v v v. The equivalent relation is obtained if Eq. (9) is ultiplied by the veloity v : d p p v v ( ) v a v v v v. (4) The left side of Eq. (4) ontains the rate of hange of the kineti energy per unit ass density (with ontribution fro the pressure and the dissipation funtion ) and the right side ontains the power of gravitational and eletroagneti fores va and the power of the pressure fore. The ter with the squared veloity in the right side of Eq. (4) is proportional to the kineti energy and the last ter desribes the power of the energy transfored during the fluid otion in a visous ediu with the effet of fluid opression and hange of its density. Another equivalent relation for the power of hange in the energy of oving fluid is obtained by ultiplying the veloity v by Eq. (35). In this ase in the right-hand side of Eq. (4) the Laplaian and the seond partial tie derivative appear. If we substitute the oeffiients and with Eqs. (38) and (39) we obtain the following: d p ( ) p v ( ) ( ). v v v a v v v v v (4) t Dependene of the veloity agnitude on the tie In this setion we will ake a onlusion about the nature of the kineti energy hange over tie. For onveniene we will onsider the o-oving referene frae in whih the veloities v are relative veloities of otion of the fluid layers relative to eah other. If we ultiply all the ters in Eq. (7) by and assue that the quantity p that is we neglet the ontribution of the pressure energy density and the dissipation funtion as opared to the energy density at rest we an write: v p d a. d d Assuing that the veloity v r d r ( v) dr d( v ) after ultipliation by dr and integration with regard to Eq. (8) for we obtain: p p v v d d d( ) a r v r v. (43) 8

9 Aording to Eq. (43) the kineti energy hanges when the work is arried out by the ass fores on the p fluid the fluid turns into a state with a different ratio in the fluid there is frition between the layers and the veloity divergene is non-zero. In Eq. (43) further siplifiation is possible if we assue that in the proess of integration the integrands hange insignifiantly and an be taken outside the integral sign. We will also use the ontinuity equation in the for: d v ( ). (44) All this with regard to Eqs. (38) and (39) gives: p p 4 d v v a r v r ( ). (45) 3 The left side of Eq. (45) ontains the hange of kineti energy per unit ass whih ours due to the veloity hange fro v to v.the kineti energy inreases if in the right side the projetion a of the ass fores aeleration on the displaeent vetor r has a positive sign. Meanwhile the seond third and fourth ters in the right side have a negative sign. This eans that the otion energy dissipation is proportional to the inrease in pressure during the fluid otion the veloity and the otion distane as well as to the inrease in the fluid density that prevents fro free otion. The salar produt vr vt where t denotes the tie of otion of one layer relative to another does not vanish during the urvilinear or rotational otion of the layers of fluid or liquid. Therefore in the oving fluid vorties and turbulene an easily our. This is ontributed by the fat that the ters in the right side of Eq. (45) an influene eah other. For exaple in the areas of high pressure the fluid strealines bent and the teperature hanges the density in the last ter in Eq. (45). Turbulene an be haraterized as a ethod of transferring the energy of linear otion of the fluid into the rotary otion of different sales. Equation Eq. (45) an be rewritten so as to ove all the ters depending on the veloity to the left side. Assuing a r a v t a v t os where is the angle between the veloity and the aeleration a we find a quadrati equation for the veloity as a funtion of the tie t and other paraeters: 4 p d v v t a v tos ( ) onst. 3 The onstant in the right side speifies the initial ondition of otion. The solution of this equation allows us to estiate the hange of the veloity agnitude over tie.. Conlusion For the ase of onstant oeffiients of visosity we showed that the Navier-Stokes equation of otion of the visous opressible liquid an be derived using the 4- potential of the energy dissipation field dissipation tensor and dissipation stress-energy tensor. First we wrote the equations of otion Eq. (5) in a general for then expressed the in Eqs. () and (3) through the strengths of the gravitational and eletroagneti fields the strengths of the pressure field and energy dissipation field. The 4- potential of the dissipation field inludes the dissipation funtion and the assoiated salar potential of the dissipation field. The quantity an be seleted so that in the Eq. (7) for the fluid aeleration the dependene on the veloity of the fluid otion appears assoiated with visosity when deeleration of the fluid is proportional to the relative veloity of its otion. We an also take into aount the dependene on the rate of aeleration hange over tie. This gives us the Eq. (9). Then we analyzed the wave equation for the veloity field in Eq. (34) and expressed the veloity fro to substitute it in Eq. (9). The resulting Eq. (35) oinides alost exatly with the Navier-Stokes Eq. (37). One differene is that in the aeleration fro pressure the ass p density in the expression is under the gradient sign and in Eq. (37) is taken outside the gradient sign. The seond differene is due to the fat that in Eq. (35) ( v ) there is an additional ter in the for. This ter is t proportional to the rate of aeleration hange over tie and desribes the phenoena in whih the hange of the ediu properties affeting visosity ours in a speified tie frae. In addition in Eq. (35) we took into aount the relativisti orretions of the Lorentz fator as well as the fluid aeleration dependene on the aeleration of the ass-energy of the pressure field and dissipation field (in square brakets in the left side of Eq. ( 35) ). The direted kineti energy of otion of the fluid in a visous ediu an dissipate into the rando otion of the partiles of the surrounding ediu and be onverted into heat. The inverse proess is also possible when heating of the ediu leads to a hange in the state of the fluid otion. In setion 9.3. we introdued the differential equations of the hange in the syste s kineti energy and its onversion into other energy fors inluding the dissipation field energy. These equations are not opletely independent sine they are obtained by salar 9

10 ultipliation of the equation of otion by the fluid veloity v. The dissipation stress-energy tensor Q is represented in Eq. (6) and its invariants are represented in Eq. (A8) in Appendix A. In setion 9.. the dissipation funtion is given by Eq. (8): d ( ) d v r v r. This funtion depends on the distane traveled by the fluid relative to the surrounding oving ediu and an be onsidered as a funtion of the tie of otion with respet to the referene frae whih is at the average o-oving with the fluid in this sall ontrol volue of the syste. With the help of the known quantity we an alulate aording the Eq. (A7) the vetors X and Y and therefore deterine the oponents of the tensor Q. In partiular the volue integral of the oponent Q of this tensor allows us to onsider all the energy that is transferred by the oving fluid to the surrounding ediu in the for of i dissipation field energy and the oponents Q define i the vetor Z Q as the energy flux density of the dissipation field. Under the assuptions ade the Navier-Stokes Eq. (37) redues to Eq. (7) wherein the aeleration depends besides the ass fores on the su of two gradients the p dissipation funtion and the quantity. Eq. (7) has suh a for that this equation should have sooth solutions if there are no disontinuities in the pressure or the dissipation funtion. If we onsider ondition in Eq. (8) and Eq. (36) as valid the gradient of will also be a sooth funtion. Instead of oving fro Eq. (7) to Eq. (35) whih is siilar to the Navier-Stokes Eq. (37) we an at in another way. Eq. (7) is an differential equation to deterine the veloity field v. In this equation there are at least three ore unknown funtions: the pressure field p the ass density and the dissipation funtion. Therefore it is neessary to add to Eq. (7) at least three equations in order to lose the syste of equations and ake it solvable in priniple. One of suh equations is the ontinuity Eq. () in the for of Eq. (44) whih relates the density and veloity. In order to deterine the dissipation funtion we have introdued the wave equation Eq. (A) in Appendix A. The pressure distribution in the syste an be found fro the wave equation Eq. (B4) in Appendix B. In Eq. (7) there is also aeleration a arising due to the ation of ass fores. This aeleration depends on the gravitational field strength Γ torsion field Ω eletri field strength E agneti field B and harge density : q q q a Γ [ v Ω] E [ ] v B. For eah of these quantities there are speial equations used to define the. For exaple the gravitational field equations (the Heaviside equations) an be represented aording to (Fedosin 999) as follows: Γ 4 G v Γ 4 G Ω t Ω Ω Γ. (46) t Equations (46) are derived in (Fedosin ) fro the priniple of least ation and are siilar in their for to Maxwell equations whih are used to alulate E and B. Finally the harge density an be related to the veloity by eans of the equation of the eletri harge ontinuity: d v ( q ). (47) q Thus the set of Eqs. (7) (44) (A) (B4) (46) and (47) together with Maxwell equations is a oplete set whih is suffiient to solve the proble of otion of visous opressible and harged fluid in the gravitational and eletroagneti fields. Noenlature A Vetor potential of eletroagneti field V s A 4-potential of eletroagneti field V s a Fluid aeleration s a Aeleration produed by ass fores s a 4-aeleration s B Magneti field T B Aeleration stress-energy tensor J 3 C Field strength of pressure field s D Vetor potential of gravitational field s D 4-potential of gravitational field s E Field strength of eletroagneti field V E Energy J Eletroagneti tensor T F f Pressure field tensor s f Density of 4-fore N 3 g g q Deterinant of etri tensor Metri tensor h Dissipation field tensor s I Solenoidal vetor of pressure field s J Mass 4-urrentl kg s j Eletroagneti (harge) 4-urrent C s L Lagrangian J N Solenoidal vetor of aeleration field s P Pressure stress-energy tensor J 3 p Pressure Pa Q Dissipation stress-energy tensor J 3 R Salar urvature R Rii tensor r Displaeent vetor S Field strength of aeleration field s S Ation funtion J s ds Spaetie interval

11 t Tie s U Vetor potential of aeleration field s U Gravitational stress-energy tensor J 3 u 4-veloity s u 4-potential of aeleration field s u Aeleration field tensor s v Fluid veloity s W Eletroagneti stress-energy tensor J 3 X Field strength of dissipation field s dx 4-displaeent 3 dx dx dx Produt of spae oordinate differentials 3 Y Solenoidal vetor of dissipation field s Z Energy-oentu flux vetor of dissipation field kg s 3 Dissipation funtion s Salar potential of dissipation field s Φ Gravitational tensor s Salar potential of eletroagneti field V Γ Field strength of gravitational field s Christoffel sybol Lorentz fator Metri tensor of Minkowski spae Cosologial onstant 4-potential of dissipation field s Π Vetor potential of pressure field s 4-potential of pressure field s Θ Vetor potential of dissipation field s Salar potential of aeleration field s Salar potential of pressure field s Mass density kg 3 Charge density C 3 q d Differential of 4-volue 4 Proper tie s Ω Gravitational torsion field s Salar potential of gravitational field s Subsript a Aeleration d Dissipation e Eletroagneti g Gravitational p Pressure Fedosin S. G. (4b). The Integral Energy-Moentu 4- Vetor and Analysis of 4/3 Proble Based on the Pressure Field and Aeleration Field. Aerian Journal of Modern Physis Fedosin S. G. (4). Relativisti Energy and Mass in the Weak Field Liit. vixra.org/abs/45.. Aepted by Jordan Journal of Physis. Fedosin S. G. (4d). The General Theory of Relativity Metri Theory of Relativity and Covariant Theory of Gravitation. Axioatization and Critial Analysis. International Journal of Theoretial and Applied Physis Khelnik S. I. (). Navier-Stokes equations. On the existene and the searh ethod for global solutions Bene-Ayish Israel: MiC - Matheatis in Coputer Cop. Landau L. D. Lifshitz E. M. (987). Fluid Mehanis Vol. 6 (nd ed.). Oxford: Butterworth-Heineann. Navier C. (87). Meoir sur Ies Iois du ouveent des fluids Me de L' A. Royale de s. de L' Institut de Frane 6I Poisson S. D. (83). Méoire sur les équations générales de l'équilibre et du ouveent des orps solides élastique et des fluids. J. Eole polytehn Saint-Venant (Barre) A. J. C. (843). Note à joindre au Méoire sur la dynaique des fluides Coptes rendus r. Aat. si Stokes G. G. (849). On the theories of the internal frition of fluids in otion and of the equilibriu and otion of elasti solids. Trans. of the Cabr. Philos. Soiety т. VIII Appendix A. Properties of the dissipation field The oponents of the antisyetri tensor of the dissipation field are obtained fro Eq. (3) using the Eq. (). We will introdue the following notations: h X hi j i j j i Yk (A) i i i i where the indies i j k for triplets of non-reurrent nubers of the for 3 or 3 or 3; the 3-vetors X and Y an be written by oponents: X X ( X X X ) ( X X X ) ; i 3 x y z Y Y ( Y Y Y ) ( Y Y Y ). i 3 x y z Using these notations the tensor h an be represented as follows: Referenes: Fedosin S. G. (999). Fizika i filosofiia podobiia ot preonov do etagalaktik. Per: Style-g. Fedosin S. G. (9). Fiziheskie teorii i beskonehnaia vlozhennost aterii. Per. Fedosin S. G. (). The Priniple of Least Ation in Covariant Theory of Gravitation. Hadroni Journal Fedosin S. G. (4a). About the osologial onstant aeleration field pressure field and energy. vixra.org/abs/43.3. h X X X x y z X X x y X z Yz Yy. (A) Yz Yx Yy Yx

12 The sae tensor with ontravariant indies equals: h g g h. In Minkowski spae the etri tensor does not depend on the oordinates in whih ase for the dissipation tensor it follows: h X X x y X z X x Yz Yy h. (A3) X y Yz Yx X z Yy Yx We an express the dissipation field equations Eq. (8) in Minkowski spae in ters of the vetors X and Y using J u v the 4-vetor of ass urrent: where derivatives v X 4 Y. Replaing in Eq. (8) the ovariant with the partial derivatives X 4 v t we find: Y Y X. (A4) t If we ultiply salarly the seond equation in Eq. (A4) by X and the fourth equation by Y and then su up the results we will obtain the following: ( X Y ) 4 v X [ X Y ]. (A5) t Eq. (A5) oprises the Poynting theore applied to the dissipation field. The eaning of this differential equation that if dissipation of the energy of oving fluid partiles takes plae in the syste then the divergene of the field dissipation flux is assoiated with the hange of the dissipation field energy over tie and the power of the dissipation energy density. Eq. (A5) in a ovariant for is written as the tie oponent of Eq. (7):. Q h J If we substitute Eq. (A) into Eq. (7) we an express the salar and vetor oponents of the 4-fore density of the dissipation field: ( f ) d h J X v ( f ) h J X [ vy ]. (A6) i d i The vetor X has the diension of an ordinary 3- aeleration and the diension of the vetor Y is the sae as that of the frequeny. Substituting the 4-potential of the dissipation field in Eq. () in the definition in Eq. (A) in Minkowski spae we find: Θ ( v) X ( ) t t Y Θ v. (A7) The vetor X is the dissipation field strength and the vetor Y is the solenoidal vetor of the dissipation field. Both vetors depend on the dissipation funtion whih in turn depends on the oordinates and tie. In real fluids there is always internal frition and the vetors X and Y are also not equal to zero. We an substitute the tensors in Eqs. (A) and (A3) in Eq. (6) and express the stress-energy tensor of the dissipation field Q in ters of the vetors X and Y. We will write here the expression for the tensor invariant h h and for the tie oponents of the tensor Q : ( ) h h X Y (A8) ( Q X Y ) 8 Q i [ XY ]. 4 The oponent Q defines the energy density of the dissipation field in the given volue and the vetor i Z Q [ X Y ] defines the energy flux density 4 of the dissipation field. If we substitute X fro Eq. (A7) into the first equation in Eq. (A4) and take into aount the gauge of the 4- potential in Eq. (9) as follows: Θ or t ( ) ( v ) (A9) t we will obtain the wave equation for the salar potential: 4 or t ( ) t ( ) 4. (A) Fro Eqs. (A7) (A9) and the seond equation in Eq. (A4) the wave equation follows for the vetor potential of the dissipation field:

13 4 v t Θ Θ or ( v) t ( v) 4 v. Appendix B. Pressure field equations Four vetor equations for the pressure field oponents within the speial theory of relativity were presented in ((Fedosin 4a)) as the onsequene of the ation funtion variation: The 4-potential gauge aording to Eq. (9) in the for in Minkowski spae is transfored into the expression. Substituting here the expression for the 4-potential of the pressure field we obtain: Π or t p p v. (B3) t C 4 I C 4 v t I I C. (B) t Substituting Eq. (B) into the first equation in Eq. (B) and using Eq. (B3) we obtain the wave equation for alulation of the salar potential of the pressure field: t 4 or The vetor of the pressure field strength C and the solenoidal vetor I are deterined with the 4-potential of p the pressure field u Π aording to the forulas: Π p pv C (B) t t p v I Π. p p 4. (B4) t The wave equation for the vetor potential of the pressure field follows fro Eqs. (B) (B3) and the seond equation in Eq. (B): Π t 4 Π v or t p v p v 4 v. 3