Four-dimensional equation of motion for viscous compressible substance with regard to the acceleration field, pressure field and dissipation field

Transcription

1 Four-dimensional equation of motion for visous ompressible substane with regard to the aeleration field, pressure field and dissipation field Sergey G. Fedosin PO box 6488, Sviazeva str. -79, Perm, Russia From the priniple of least ation the equation of motion for visous ompressible substane 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 limit 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 motion is integrated, and the dependene of the veloity magnitude is determined. A omplete set of equations is presented, whih suffies to solve the problem of motion of visous ompressible and harged substane in the gravitational and eletromagneti fields. Keywords: Navier-Stokes equation; dissipation field; aeleration field; pressure field; visosity.. Introdution Sine Navier-Stokes equations appeared in 87 [], [], onstant attempts have been made to derive these equations by various methods. Stokes [3] and Saint-Venant [4] in derivation of these equations relied on the fat that the deviatori stress tensor of normal and tangential stress is linearly related to the three-dimensional deformation rate tensor and the substane is isotropi. In book [5] it is onsidered that Navier-Stokes equations are the extremum onditions of some funtional, and a method of finding a solution of these equations is desribed, whih onsists in the gradient motion to the extremum of this funtional. One of the variants of the four-dimensional stress-energy tensor of visous stresses in the speial theory of relativity an be found in [6]. The divergene of this tensor gives the required visous terms in the Navier-Stokes equation. The phenomenologial derivation of this tensor is based on the assumed ondition of entropy inrement during energy dissipation. As a onsequene, in the o-moving referene frame the time omponents of the tensor, i.e. the dissipation energy density and its flux vanish. Therefore, suh a tensor is not a universal

2 tensor and annot serve, for example, as a basis for determining the metri in the presene of visosity. In this artile, our goal is to provide in general form the four-dimensional stress-energy tensor of energy dissipation, whih desribes in the urved spaetime 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 motion of the visous ompressible substane, and by seleting the salar potential of the dissipation field we will obtain the Navier-Stokes equation. The essential element 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 omplete set of equations suffiient to desribe the motion of visous substane.. The ation funtion The starting point of our alulations is the ation funtion in the following form: k( R ) D J Φ Φ A j F F u J 6 G 4 S L dt 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 osmologial onstant, J u is the 4-vetor of the mass (gravitational) urrent, is the mass density in the referene frame assoiated with the partile, dx u is the 4-veloity of a point partile, is the speed of light, ds 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,

3 Φ is the gravitational tensor, A, A is the 4-potential of the eletromagneti field, whih is speified by the salar potential and the vetor potential A of this field, j u is the 4-vetor of the eletromagneti (harge) urrent, q q is the harge density in the referene frame assoiated with the partile, is the vauum permittivity, F is the eletromagneti tensor, u g u is the 4-veloity with the ovariant index, expressed with the help of the metri 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, is some funtion of oordinates and time, p, Π is the 4-potential of the pressure field, onsisting of the salar u potential and the vetor potential Π, p is the pressure in the referene frame assoiated p with the partile, the ratio defines the equation of state of the substane, f is the pressure field tensor, is some funtion of oordinates and time. The above-mentioned quantities are desribed in detail in [7]. In addition to them, we introdue the 4-potential of energy dissipation in the medium: u, Θ, () where is the dissipation funtion, and Θ are the salar and vetor dissipation potentials, respetively. 3

4 Using the 4-potential we onstrut the energy dissipation tensor: h. (3) In () the oeffiient preeding the tensor invariant h h ontains the quantity, whih in order to simplify alulations we assume to be a onstant. The oeffiients and we will also assume to be onstants. The term J in () reflets the fat that the energy of the substane motion an be dissipated in the surrounding medium and turn into the internal substane energy, while the system s energy does not hange. The last term in () is assoiated with the energy, aumulated by the system due to the ation of the energy dissipation. The method of onstruting the dissipation 4-potential in () and the dissipation tensor h in (3) is fully idential to that, whih was used earlier in [7]. Therefore, we will not provide here the intermediate results from [7], and will right away write the equations of motion of the substane and field, obtained as a result of the variation of the ation funtion (). 3. Field equations The eletromagneti field equations have the standard form: F F F, F j, (4) where is the vauum permeability. The gravitational field equations are: 4 G Φ Φ Φ, Φ J. (5) 4

5 The aeleration field equations are: 4 u u u, u J. (6) The pressure field equations are: 4 f f f, f J. (7) The dissipation field equations are: 4 h h h, h J. (8) In order to obtain the equations (4-8), variation by the orresponding 4-potential is arried out in the ation funtion (). All the above-mentioned fields have vetor harater. Eah field an be desribed by two three-dimensional 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 example, the omponents of the eletri field strength E and the magneti field indution B are the omponents of the eletromagneti tensor F. The gravitational tensor Φ onsists of the omponents of the gravitational field strength Γ and the torsion field Ω. The summary of notation for all the fields is provided in Appendix A. The properties of the dissipation field are provided in Appendix B. 4. Field gauge In order to simplify the form of equations we use the following field gauge: A A, D D, (9) u u,,. 5

6 In (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 (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 volume is aompanied by hange in time of the salar field potential in this volume, and also depends on the tensor produt of the Christoffel symbols and the 4-potential, that is, on the degree of spaetime urvature. 5. Continuity equations In equations (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 similarly to (3): F A A A A, Φ D D D D, () u u u u u, f. If we substitute (3) and () into equations (4-8), and apply the ovariant derivative all terms, we obtain the following relations ontaining the Rii tensor: to R F j 4 G 4 R Φ J, R u J,, 4 R f J, 4 R h J. () In the limit of speial theory of relativity, the Rii tensor vanishes, the ovariant derivative turns into the 4-gradient, and then instead of () we an write: j, J. () Relations () are the ordinary ontinuity equation of the harge and mass 4-urrents in the flat spaetime. 6. Equations of motion 6

7 The variation of the ation funtion leads diretly to the equations desribing the motion of the substane unit under the ation of the fields: u J Φ J F j f J h J. (3) J The left side of the equality an be transformed, taking into aount the expression u for the 4-vetor of the mass urrent density and the definition () for the aeleration tensor u u u : Du d u u J u u u u u u u a D d. (4) u In (4) a denotes the 4-aeleration, and we used the operator of proper-time-derivative D, where D is the symbol of 4-differential in the urved spaetime, is the D proper time [8]. If we substitute (4) into (3), we obtain the equation of motion, in whih the 4-aeleration is expressed in terms 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 form of stress-energy tensors of all the fields assoiated with the substane: 4, U g Φ Φ g Φ Φ 4 G 4, W g F F g F F 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) 7

8 One of the properties of these tensors is that their divergenes alongside with the field tensors speify the densities of 4-fores, arising from the influene of the orresponding field on the substane: k k ( f ) F j W, ( f ) Φ J U, e k k a k g k k ( f ) a u J B, ( f ) f J P, p k k ( f ) h J Q. (7) d k The left side of (7) ontains the density of the orresponding 4-fore, exluding ( f ) a, whih up to sign denotes the density of the 4-fore, ating from the aelerated substane on the rest four fields. From (3-7) it follows that the equation of motion an be written only in terms of the divergenes of the stress-energy tensors of fields: ( W U B P Q ). (8) We have integrated equation (8) in [9] in the weak field limit (exluding the stressenergy tensor of dissipation Q ), and this allowed us to explain the well-known 4/3 problem. 7. The system s energy The ation funtion () ontains the Lagrangian L. Applying to it the Legendre transformations for a system of partiles, we an find the system s Hamiltonian. This Hamiltonian is the relativisti energy of the system, written in an arbitrary referene frame. Sine the energy is dependent on the osmologial onstant, gauging of the osmologial 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 system the following: 8

9 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 system in the form of a set of losely interating partiles and the related fields in the weak field limit was alulated in []. The differene between the system s mass and the gravitational mass was shown, as well as the fat that the mass-energy of the proper eletromagneti field redues the gravitational mass of the system. 8. Equation for the metri Aording to the logi of the ovariant theory of gravitation [] and the metri theory of relativity [], ontribution to the definition of the system s metri is made by the stressenergy tensors of all the fields, inluding the gravitational field. The metri is a seondary funtion, the derivative of the fields ating in the system that define all the basi properties of the system. The equation for the metri is obtained as follows: R R g ( W U B P Q ). () 4 k If we multiply () by the metri tensor g and ontrat over all the indies, the right and left sides of the equation vanish. It follows from the properties of tensors in (). Outside the substane limits, with regard to the gauge, the salar urvature R beomes equal to zero. If we take into aount the equation of motion (8), the ovariant derivative of the right side of () is zero. The ovariant derivative of the left side of () is also zero, sine R as a onsequene of the osmologial onstant gauge, and for the Einstein tensor the following equality holds: R R g. 9. The analysis of the equation of motion Equations (4-5) imply the onnetion between the ovariant 4-aeleration of a substane unit and the densities of ating fores in the urved spaetime: 9

10 du a uu Φ J F j f J h J. d We will write this four-dimensional equation separately for the time and spae omponents, given that dx dt dx dt J (, v ), ds ds dt ds dx dt as well as j q q (, v ). For the dissipation field will use relations (B6) ds ds from Appendix B, where in the general ase dt ds Expressions for other fields an be found in [7]. This gives: should be substituted instead of. du dt q uu Γ v E v C v X v. d s ds dui dt iuu 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 eletromagneti fields, pressure field and dissipation field, respetively. Notations Ω, B, I and Y refer to the torsion field, the magneti field and the solenoidal vetors of the pressure and dissipation fields, respetively. dt After redution by a fator we obtain: ds du dt ds Γ v E v C v X v. () q uu dt dui dt d iuu dt q q Γ [ v Ω] E [ v B] C [ v I] X [ v Y ]. (3)

11 In (3) the sum q q E [ vb ] is the ontribution of the eletromagneti Lorentz fore ating on the substane unit into the total aeleration. The minus sign before this sum appears beause u i is the spae omponent of the ovariant 4-veloity, whih differs from the ordinary ontravariant spae omponent i u in the form fator of the metri tensor. Similarly, the sum Γ [ vω ] is the aeleration of the gravitational Lorentz fore. Gravitational and eletromagneti fores are the so-alled mass fores distributed over the entire volume, where there is mass and harge of the substane. 9.. The equation of motion in Minkowski spae In order to simplify our analysis, we will onsider equations (-3) in the framework of the speial theory of relativity. The sum of the last two terms in (3), taking into aount the formulas (B7) from Appendix B, for X and Y gives the following: ( v) X [ vy] ( ) v[ v] t ( v) d( v) ( ) ( v)( v) ( v). t dt (4) In (4) v dt ds is the Lorentz fator. For the pressure field, with regard to the definition of the 4-potential in the form of p, Π, the pressure field tensor f from () and the definition of the u vetors C and I by the rule: f C, fi j i j ji Ik, i i i i we find the expression for the vetors:

12 Π p pv t t C, p v I Π. (5) Using (5) we alulate the sum of two terms in (3): p p v p v C [ vi] v t p p v p v p v p d p v ( ). v p t p dt (6) Substituting (4) and (6) into (3), and taking into aount that in Minkowski spae the Christoffel symbols are zero and the spae omponent of the 4-veloity equals ui find: v, we d p p v a m. (7) dt fores: In (7) we have introdued notation for the aeleration, resulting from the ation of mass q q a m Γ [ v Ω] E [ ] v B. Until now we have not defined the dissipation funtion. In this approximation, it is assoiated with a salar potential of the dissipation field by relation:. Let us assume that v ( v ), (8)

13 that is d ( ) d v r v r. It means that the salar potential of the dissipation field is proportional both to the veloity v of the onsidered substane unit and the path traveled by it in the surrounding spae. Contribution to is also made by the gradient of the veloity divergene with a ertain oeffiient. The oeffiient depends on the parameters of interating substane layers, in a first approximation it is inversely proportional to the square of the layers thikness. At the same time the oeffiient reflets the substane properties and an be different in different substanes. Taking into aount (8), equation ( 7) is transformed as follows: d p p v ( ) am v v. (9) dt p Due to the presene in (9) of the gradient of the pressure to the mass density ratio, there is aeleration direted against this gradient. The term in (9) whih is proportional to the veloity v, defines the rate of deeleration due to visosity. Sine the deeleration in (9) depends not on the absolute veloity but on the veloity of motion of some substane layers relative to the other layers, the veloity v should be a relative veloity. We will use the freedom of hoosing the referene frame in order to move from absolute veloities to relative veloities. Suppose the referene frame is o-moving and it moves in the substane with the ontrol volume of a small size. Then in suh a referene frame the veloity v in (9) will be a relative veloity: some layers will be ahead, while others will lag behind, and visous fores will appear. We will now write the equations for the aeleration field from [7]: S 4 v S 4, N, N, t N S. (3) t 3

14 The vetor S in (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 u, U equals the 4-veloity, taken with the ovariant index. The aeleration tensor u is defined in () as a 4-url and it ontains the vetors S and N : ui ui iu S i, u i j iu j j ui N k. In Minkowski spae we an move from the salar and vetor U potentials of the aeleration field to the 4-veloity omponents and express vetors S and N in terms of them: U ( v) t t S, N U v. (3) Let us substitute (3) into the seond equation in (3): 4 v ( ) t t ( v) v. (3) The gauge ondition of the 4-potential of the aeleration field (9) has the form: u. In Minkowski spae this relation is simplified: u t U, or v. (33) t With regard to (33) we will transform the left side of (3): ( v) ( v) ( v) ( v ). t Substituting this in (3), we obtain the wave equation: 4

15 4 v t ( v) ( v ). (34) Aording to (34) the veloity v of the substane motion in the system must onform to the wave equation, that means that the veloity is given by the system s parameters and hanges ontinuously in transition from one ontrol volume to another. The wave equation for the Lorentz fator follows from (3) and the first equation in (3) with regard to (33): t 4. We an express the veloity v from (34) and substitute it in (9): d p ( ) p v v ( ) ( ) am v v. dt 4 t (35) Let us find out the physial meaning of the last term in (35). The gauge ondition (33) of the 4-potential of the aeleration field an be rewritten as follows: d dv v. 4 dt dt Hene, provided we have: dv d ( v) v ( v ) v ( ) 4 v v. (36) dt dt The quantity v is the gradient of half of the squared veloity, that is the gradient of the kineti energy per unit mass. This quantity is proportional to the aeleration, arising due 5

16 to the dissipation of the kineti energy of motion. The time derivative of v leads to the rate of aeleration hange. Other terms in (36) also have the dimension of the rate of aeleration hange. Thus in (35) visosity is taken into aount not only due to the motion veloity, but also due to the rate of aeleration hange of the substane motion. 9.. Comparison with the Navier-Stokes equation The vetor Navier-Stokes equation in its lassial form is usually used for non-relativisti desription of the liquid motion and has the following form [6]: dv v a ( v) v a p v ( v ), (37) dt m t 3 where a and v are the veloity and aeleration of an arbitrary point unit of liquid, is the mass density, p is the pressure, is the kinemati visosity oeffiient, is the volume (bulk or seond) visosity oeffiient, a m is the aeleration of the mass fores in the liquid, and it is assumed that the oeffiients and are onstant in volume. In (37) the veloity v depends not only on the time but also on the oordinates of the moving liquid unit. This allows us to expand the derivative d v into the sum of two partial dt derivatives: time derivative (substantial) derivative. v and spae derivative ( v ) v, that is to apply the material t Comparing (35) and (37) for the ase of low veloities, when tends to unity, and at suffiiently low pressure and visosity, we obtain the kinemati visosity oeffiient:. (38) 4 Sine, where is the dynami visosity oeffiient, then we obtain: 6

17 . 4 In this ratio depends primarily on the substane properties, and the oeffiients and also depend on the parameters of the system under onsideration. For example, 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 terms in (35) and (37) implies:, 3 3. (39) 3 The presene of and in (37) implies two auses of the rate of aeleration hange: one of them is due to the substane density variation beause of the medium resistane and the other is due to the momentum variation of the substane moving in a visous medium The energy power In Minkowski spae the time omponent of the 4-veloity is equal to u Christoffel symbols are zero, and () an be written as follows:, the d d( v) q dt dt Γ v E v C v X v. (4) We an also obtain (4), if in Minkowski spae we multiply the equation of motion (3) by the veloity v. We substitute in (4) the vetor C from (5) and the vetor X aording to (B7) from Appendix B: d ( v) ( v) ( ) dt t t q p p v Γ v E v v v v v. The equivalent relation is obtained, if (9) is multiplied by the veloity v : 7

18 d p p v v ( ) v am v v v v. (4) dt The left side of (4) ontains the rate of hange of the kineti energy per unit mass density (with ontribution from the pressure and the dissipation funtion ), and the right side ontains the power of gravitational and eletromagneti fores va and the power of the m pressure fore. The term with the squared veloity in the right side of (4) is proportional to the kineti energy, and the last term desribes the power of the energy transformed during the substane motion in a visous medium with the effet of substane ompression and hange of its density. Another equivalent relation for the power of hange in the energy of moving substane is obtained by multiplying the veloity v by equation (35). In this ase, in the right-hand side of (4) the Laplaian and the seond partial time derivative appear. If we substitute the oeffiients and with (38) and (39), we obtain the following: d p v v dt p ( v) v am v v ( ) ( ). v v v t 3 (4) 9.4. Dependene of the veloity magnitude on the time In this setion we will make a onlusion about the nature of the kineti energy hange over time. For onveniene, we will onsider the o-moving referene frame in whih the veloities v are relative veloities of motion of the substane layers relative to eah other. If p we multiply all the terms in (7) by and assume that the quantity, that is we neglet the ontribution of the pressure energy density and the dissipation funtion as ompared to the energy density at rest, we an write: v p d a m. dt 8

19 d d Assuming that, the veloity v r, ( v) dr d( v ), after multipliation d r dt by dr and integration, with regard to (8) for, we obtain: p p v v m d d d( ) a r v r v. (43) Aording to (43), the kineti energy hanges, when the work is arried out by the mass p fores on the substane, the substane turns into a state with a different ratio, in the substane there is frition between the layers, and the veloity divergene is non-zero. In (43) further simplifiation is possible, if we assume 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 form: d v ( ). (44) dt All this with regard to (38-39) gives: p p 4 d v v am r v r ( ). 3 dt (45) The left side of (45) ontains the hange of kineti energy per unit mass, whih ours due to the veloity hange from v to v.the kineti energy inreases if in the right side the projetion a m of the mass fores aeleration on the displaement vetor r has a positive sign. Meanwhile the seond, third and fourth terms in the right side have a negative sign. This means that the motion energy dissipation is proportional to the inrease in pressure during the substane motion, the veloity and the motion distane, as well as to the inrease in the substane density that prevents from free motion. The salar produt vr vt, where t denotes the time of motion of one layer relative to another, does not vanish during the urvilinear or rotational motion of the layers of substane 9

20 or liquid. Therefore in the moving substane vorties and turbulene an easily our. This is ontributed by the fat that the terms in the right side of (45) an influene eah other. For example, in the areas of high pressure the substane streamlines bent, and the temperature hanges the density in the last term in (45). Turbulene an be haraterized as a method of transferring the energy of linear motion of the substane into the rotary motion of different sales. Equation (45) an be rewritten so as to move all the terms, depending on the veloity, to the left side. Assuming veloity and the aeleration of the time t and other parameters: a r a v t a v t os, where is the angle between the m m m a m, we find a quadrati equation for the veloity as a funtion 4 p d v v t a v tos ( ) onst. 3 dt m The onstant in the right side speifies the initial ondition of motion. The solution of this equation allows us to estimate the hange of the veloity magnitude over time.. Conlusion For the ase of onstant oeffiients of visosity we showed that the Navier-Stokes equation of motion of the visous ompressible 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 motion (5) in a general form, then expressed them in (-3) through the strengths of the gravitational and eletromagneti 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 equation (7) for the substane aeleration the dependene on the veloity of the substane motion appears, assoiated with visosity, when deeleration of the substane is proportional to the relative veloity of its motion. We an also take into aount the dependene on the rate of aeleration hange over time. This gives us the equation (9). Then we analyzed the wave equation for the veloity field (34) and expressed the veloity from to substitute it in (9). The resulting equation (35) oinides almost exatly with the Navier-Stokes equation (37). One differene is that in the aeleration from pressure the mass

21 p density in the expression is under the gradient sign, and in (37) is taken outside the gradient sign. The seond differene is due to the fat that in (35) there is an additional term in the form ( v ). This term is proportional to the rate of aeleration t hange over time and desribes the phenomena, in whih the hange of the medium properties affeting visosity ours in a speified time frame. In addition, in (35) we took into aount the relativisti orretions of the Lorentz fator, as well as the substane aeleration dependene on the aeleration of the mass-energy of the pressure field and dissipation field ( in square brakets in the left side of ( 35) ). The direted kineti energy of motion of the substane in a visous medium an dissipate into the random motion of the partiles of the surrounding medium and be onverted into heat. The inverse proess is also possible, when heating of the medium leads to a hange in the state of the substane motion. In setion 9.3. we introdued the differential equations of the hange in the system s kineti energy and its onversion into other energy forms, inluding the dissipation field energy. These equations are not ompletely independent, sine they are obtained by salar multipliation of the equation of motion by the substane veloity v. The dissipation stress-energy tensor Q is represented in (6) and its invariants are represented in (B8) in Appendix B. In setion 9.. the dissipation funtion is given by formula (8): d ( ) d v r v r. This funtion depends on the distane traveled by the substane relative to the surrounding moving substane, and an be onsidered as a funtion of the time of motion with respet to the referene frame, whih is at the average o-moving with the substane in this small ontrol volume of the system. With the help of the known quantity we an alulate aording the formulas (B7) the vetors X and Y and therefore determine the omponents of the tensor Q. In partiular, the volume integral of the omponent Q of this tensor allows us to onsider all the energy that is transferred by the moving substane to the surrounding medium during the observation, and the omponents i Q define the vetor i Z Q as the energy flux density of the dissipation field.

22 Under the assumptions made the Navier-Stokes equation (37) redues to equation (7), wherein the aeleration depends, besides the mass fores, on the sum of two gradients the p dissipation funtion and the quantity. Equation (7) has suh a form that this equation should have smooth solutions, if there are no disontinuities in the pressure or the dissipation funtion. If we onsider ondition (8) and formula (36) as valid, the gradient of will also be a smooth funtion. Instead of moving from equation (7) to equation (35), whih is similar to the Navier- Stokes equation (37), we an at in another way. Differential equation (7) is an equation to determine the veloity field v. In this equation, there are at least three more unknown funtions: the pressure field p, the mass density, and the dissipation funtion. Therefore, it is neessary to add to (7) at least three equations in order to lose the system of equations and make it solvable in priniple. One of suh equations is the ontinuity equation () in the form of (44), whih relates the density and veloity. In order to determine the dissipation funtion we have introdued the wave equation (B) in Appendix B. The pressure distribution in the system an be found from the wave equation (C4) in Appendix C. In equation (7) there is also aeleration a m, arising due to the ation of mass fores. This aeleration depends on the gravitational field strength Γ, torsion field Ω, eletri field strength E, magneti field B and harge density q : q q a m Γ [ v Ω] E [ ] v B. For eah of these quantities there are speial equations used to define them. For example, the gravitational field equations (the Heaviside equations) an be represented aording to [3] as follows: Γ 4 G v Γ 4 G, Ω, Ω, t Ω Γ. (46) t Equations (46) are derived in [] from the priniple of least ation and are similar in their form to Maxwell equations, whih are used to alulate E and B. Finally, the harge

23 density ontinuity: q an be related to the veloity by means of the equation of the eletri harge d v ( q). (47) dt q Thus, the set of equations (7), (44), (B), (C4), (46), (47) together with Maxwell equations is a omplete set, whih is suffiient to solve the problem of motion of visous ompressible and harged substane in the gravitational and eletromagneti fields.. Referenes. Navier C., Memoir sur Ies Iois du mouvement des fluids, Mem, de L' A. Royale de s. de L' Institut de Frane, 87, Vol. VI.. Poisson S. D., J. Eole polytehn., 83, Vol. 3, P Stokes G.G., On the theories of the internal frition of fluids in motion and of the equilibrium and motion of elasti solids, Trans, of the Cambr. Philos. Soiety, т. VIII , Vol. 8. P de Saint-Venant, C. Note à joindre au Mémoire sur la dynamique des fluides, présenté le 4 avril 834 ; Comptes rendus, r. Aat. si., 843, Vol. 7, No. 5. Khmelnik S. I. Navier-Stokes equations. On the existene and the searh method for global solutions., ISBN L.D. Landau, E.M. Lifshitz (987). Fluid Mehanis. Vol. 6 (nd ed.). Butterworth- Heinemann. ISBN Fedosin S.G. About the osmologial onstant, aeleration field, pressure field and energy. vixra.org, 5 Mar Fedosin S.G. Fiziheskie teorii i beskonehnaia vlozhennost materii. (Perm, 9). ISBN Fedosin S.G. The integral energy-momentum 4-vetor and analysis of 4/3 problem based on the pressure field and aeleration field. vixra.org, 3 Mar 4.. Fedosin S.G. Relativisti Energy and Mass in the Weak Field Limit. vixra.org, May 4.. Fedosin S.G. The Priniple of Least Ation in Covariant Theory of Gravitation. Hadroni Journal,, Vol. 35, No., P

24 . Fedosin S.G. The General Theory of Relativity, Metri Theory of Relativity and Covariant Theory of Gravitation. Axiomatization and Critial Analysis. International Journal of Theoretial and Applied Physis (IJTAP), ISSN: 5-634, Vol.4, No. I (4), pp Fedosin S. G. Fizika i filosofiia podobiia ot preonov do metagalaktik. (Perm, 999). ISBN Appendix A. Field notations Eletromagneti Gravitational Aeleration Pressure Dissipation field field field field field 4-potential A D u Salar potential Vetor potential Field strength Solenoidal vetor A D U Π Θ E Γ S C X B Ω N I Y Field tensor F Φ u f h Stress-energy tensor Energymomentum flux vetor W U B P Q P H K F Z In this Table P is the Poynting vetor, H is the Heaviside vetor. Appendix B. Properties of the dissipation field The omponents of the antisymmetri tensor of the dissipation field are obtained from relation (3) using the relation (). We will introdue the following notations: 4

25 h X, hi j i j j i Yk, (B) i i i i where the indies i, j, k form triplets of non-reurrent numbers of the form,,3 or 3,, or,3,; the 3-vetors X and Y an be written by omponents: X X i ( X, X, X 3) ( X x, X y, X z ) ; Yi Y Y Y3 Yx Yy Yz Y (,, ) (,, ). Using these notations the tensor h an be represented as follows: h X X X x y z X X x y X Y z Y y Y Y x z Y y z Y x. (B) The same tensor with ontravariant indies equals: h g g h. In Minkowski spae the metri tensor does not depend on the oordinates, in whih ase for the tensor dissipation it follows: h X X x y X z X x Yz Yy h. (B3) X y Yz Y x X z Yy Yx We an express the dissipation field equations (8) in Minkowski spae in terms of the vetors X and Y using the 4-vetor of mass urrent: J u v, where, 5

26 v. Replaing in (8) the ovariant derivatives with the partial derivatives we find: X 4 v X 4, Y, Y, t Y X. (B4) t If we multiply salarly the seond equation in (B4) by X and the fourth equation by Y and then sum up the results, we will obtain the following: ( X Y ) 4 vx [ XY ]. (B5) t Equation (B5) omprises the Poynting theorem applied to the dissipation field. The meaning of this differential equation that if dissipation of the energy of moving substane partiles takes plae in the system, then the divergene of the field dissipation flux is assoiated with the hange of the dissipation field energy over time and the power of the dissipation energy density. Relation (B5) in a ovariant form is written as the time omponent of equation (7):. Q h J If we substitute (B) into (7), we an express the salar and vetor omponents of the 4- fore density of the dissipation field: ( f ) d h J X v, ( f ) h J [ ] i d i X v Y. (B6) The vetor X has the dimension of an ordinary 3-aeleration, and the dimension of the vetor Y is the same as that of the frequeny. Substituting the 4-potential of the dissipation field () in the definition (B), in Minkowski spae we find: 6

27 Θ ( ) ( v) X, t t Y Θ v. (B7) 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 time. 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 (B) and (B3) in (6) and express the stress-energy tensor of the dissipation field Q in terms of the vetors X and Y. We will write here the expression for the tensor invariant h h and for the time omponents of the tensor Q : ( ) h h X Y, ( Q X Y ), 8 Q i [ XY ]. (B8) 4 The omponent Q defines the energy density of the dissipation field in the given volume, and the vetor Z [ X Y ] defines the energy flux density of the 4 i Q dissipation field. If we substitute X from (B7) into the first equation in (B4), and take into aount the gauge of the 4-potential (9) as follows: Θ, or t ( ) ( v ), (B9) t we will obtain the wave equation for the salar potential: 4, or t ( ) t ( ) 4. (B) From (B7), (B9) and the seond equation in (B4) the wave equation follows for the vetor potential of the dissipation field: 7

28 Θ 4 v ( v) Θ, or ( v) 4 v. t t Appendix C. Pressure field equations Four vetor equations for the pressure field omponents within the speial theory of relativity were presented in [7] as the onsequene of the ation funtion variation: C 4 v C 4, I, I, t I C. (C) t The vetor of the pressure field strength C and the solenoidal vetor I are determined p with the 4-potential of the pressure field u, Π aording to the formulas: Π p pv C, (C) t t p v I Π. The 4-potential gauge aording to (9) in the form in Minkowski spae is transformed into the expression. Substituting here the expression for the 4- potential of the pressure field, we obtain: p pv Π, or. (C3) t t Substituting (C) into the first equation in (C) and using (C3), we obtain the wave equation for alulation of the salar potential of the pressure field: t 4, or p p 4. (C4) t 8

29 The wave equation for the vetor potential of the pressure field follows from (C), (C3) and the seond equation in (C): Π t Π 4 v, или p v p v 4 t v. 9