Some basic concepts of fluid dynamics derived from ECE theory

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1 Some sic concepts of fluid dynmics 363 Journl of Foundtions of Physics nd Chemistry, 2, vol. (4) Some sic concepts of fluid dynmics derived from ECE theory M.W. Evns Alph Institute for Advnced Studies ( Bsic concepts of flow dynmics re derived from geometry in the context of Einstein Crtn Evns (ECE) theory. These include the concept of the Stokes connection, the continuity eqution, nd the equtions of inviscid nd viscous fluids. These concepts cn e derived from Crtn's differentil geometry using the minimum of hypothesis in ccordnce with Ockhm's Rzor. Keywords: ECE theory, flow dynmics, Stokes connection, continuity eqution, inviscid fluid, viscous fluid.. Introduction The equtions of flow dynmics re usully sed on the Nvier Stokes system of equtions, derived independently y Nvier nd Stokes in the nineteenth century []. These re well descried in mny textooks. In this pper the Einstein Crtn Evns theory (ECE theory) is pplied for the first time to flow dynmics to derive the sic concepts of flow dynmics from Crtn's differentil geometry, thus mking flow dynmics theory of generl reltivity from which its min equtions my e derived in well defined limits. In Section 2 the continuity eqution of the Nvier Stokes system is derived from the tetrd postulte of Crtn's differentil geometry [2 ]. It is shown tht the Stokes connection is limiting form of the spin connection of Crtn geometry. In Section 3 the eqution of the inviscid fluid is derived from the first Crtn structure eqution, nd in Section 4 the nlysis is extended to the eqution of the viscous fluid. Similrities with electrodynmics re pointed out within the context of ECE theory. 2. The continuity eqution of the Nvier Stokes system This eqution is the eqution of conservtion of mss density ρ, nd is nlogous to the eqution of continuity of chrge density in electrodynmics. The eqution e-mil: emyrone@ol.com

2 364 M.W. Evns cn e written s [] ( v) + ρ = () where v is liner velocity. Eq. () in tensor nottion is: U µ µ = (2) where: U µ ( cρ ρ) =, v (3) Here c is the vcuum speed of light nd the density nd velocity re: ρ = ρ (X, Y, Z, t) (4) nd ( X Y Z t) v = v,,, (5) Eqution () is: + ρ v + ρ v = (6) i.e.: Dρ + ρ v = Dt (7) where Dρ = + ρ v Dt (8) is the Stokes derivtive [[. This is mesure of the rte of chnge of density t point moving with the fluid. It is the derivtive long pth moving with velocity v. In this Section it is shown tht the Stokes derivtive is covrint derivtive of generl reltivity nd tht the continuity of the Nvier Stokes

3 system is tetrd postulte. Eqution (6) my e written s Some sic concepts of fluid dynmics ( v + v) ρ= (9) so the covrint derivtive is: D Dt = +( v + v) () The connection is therefore simple sclr connection: Γ= v + v () nd Eq. (9) is: + Γρ = t (2) which is n eqution of generl reltivity. The Stokes derivtive of velocity v is: Dv Dt = +( v ) v (3) nd this concept is used in the other equtions of the Nvier Stokes system. In order to derive Eq. (2) from unified field theory its sic geometricl structure hs to e found. A suitly self consistent geometry is Crtn's differentil geometry [2 ], whose well known tetrd postulte is: λ D q = q + ω q - Γ q = (4) µ v µ v µ v µ v λ Here q v is the Crtn tetrd, λ ω µ is the spin connection nd Γ µ v is the ntisymmetric Riemnnin connection: Γ λ = q λ Γ = q λ q +ω q (5) µ v µ v µ v µ v Introduce the mss density tetrd: ρ =ρ q (, ) = ρ ρ µ µ (6)

4 366 M.W. Evns where: = (), (), (2), (3) (7) is the complex circulr sis of spcetime [3 ]. The everydy sclr vlued mss density is limit of this more generl concept of mss density tetrd. The vector prt of the density tetrd is defined s: ρ ( ) ( ) ( ) ( ) = ρ i + ρ i + ρ k X Y Z (8) nd so on. Therefore: ( ) ( ) ( ) µ = ρ, ρ ρ (9) nd so on. The timelike prt of the mss density tetrd is defined s: ( ) ( ) ( ) ρ = ρ, (2) µ so ( ) ρ = (2) The complex circulr sis is defined in terms of the Crtesin unit vectors i, j, nd k s: 2 3 e = i ij, e = i+ ij, e = k (22) so: 2 2 ( ) ( ) ( ) ( 2) ( 2) ( 2) ( 3) ( 3) = ρx + ρ Y, =ρx + ρ Y, =ρz ρ i i ρ i i ρ k (23) re the currents of mss density. Equtions (8) to (23) re exmples of the fct tht the generl vector field my lwys e expressed [2] s: ( ) ( 2) ( 3) V = V + V + V (24) n extension of the Helmholtz theorem.

5 Some sic concepts of fluid dynmics 367 The sclr density my e defined now s the sum of three timelike components of the density tetrd: ( ) ( 2) ( 3) ρ = ρ + ρ + ρ (25) nd the tetrd postulte implies tht: D µ ρ = v (26) Therefore the continuity eqution (2) implies tht: ( ) ( 2) ( 3) ( ) ( 2) ( 3) ( ) ρ +ρ +ρ + Γ ρ +ρ +ρ = t (27) It is plusile to ssume tht: ( ) + Γ ρ = ( ) (28) ( 2) ( 2) + Γρ = (29) ( 3) + ( 3) Γ ρ = (3) so the indices in Eq. (26) re determined y: c ( ) +ω ( ) ( ) = Γ (3) c ( 2) +ω ( 2) Γ ( 2) = t (32) c ( 3) +ω ( 3) ( 3) Γ = (33) However:

6 368 M.W. Evns ( ) ( 2) ( 3) Γ =Γ = Γ = (34) so: ( 2 3 ) ( 2 3 c ) ρ +ρ +ρ + ω +ω +ω = t (35) i.e. ( ) ( 2) ( 3) ( ) + c ω +ω +ω = t (36) Compring Eqs. (2) nd (36): ( ) ( 2) ( 3) Γ= c ω +ω +ω (37) The connection of the Stokes derivtive is given y the comintion (37) of spin connections of Crtn's differentil geometry, Q.E.D. so the continuity eqution of the Nvier Stokes system is limiting eqution of generl reltivity. 3. Eqution of the inviscid fluid The inviscid fluid is well known textook ideliztion [] tht nevertheless serves to illustrte how the sics of fluid dynmics my e derived from Crtn's differentil geometry. The ECE engineering model [2 ] uses the first Crtn structure eqution nd one simple hypothesis to produce the following eqution for ccelertion in dynmics: g = c v ωcv + cv ω (38) nd lso generl expression for ngulr velocity: Ω = v ω v (39) The velocity vector field is expnded s n exmple of Eq. (24): ( ) ( 2) ( 3) v= v + v + v (4)

7 Some sic concepts of fluid dynmics 369 The spin connection in these equtions is defined s: (, ) ω µ = ω ω (4) nd the velocity tetrd s: = Φ (42) c v µ µ where: Φ =Φq µ µ (43) is the grvittionl potentil tetrd. It will e shown in this Section tht Eq. (38) gives the eqution of the inviscid fluid s prticulr limiting formt. In the non reltivistic limit [] ccelertion my e defined in liner pproximtion of McLurin series s: = ( v) v+ (44) t nd the vorticity [] defined s: Ω = v (45) The formt of Eq. (38) mens tht ccelertion in generl reltivity is: = +ω v + c v cvω (46) nd tht vorticity in generl reltivity is: Ω = v ω v (47) In the equtions the four velocity is: v = ( v, v ) = µ (, ) c Φ Φ (48)

8 37 M.W. Evns In n inviscid fluid [] the ccelertion is clculted s: = + φ m ( p ) (49) where p is the pressure, m is the mss, nd is the totl potentil energy per unit mss due to ll ody forces. So the textook description of n inviscid fluid is []: v + ( v) v = ( p + φ ) (5) m or: = + ( v) v+ ( p + φ ) = (5) m nd the net ccelertion is zero. In generl reltivity for zero net ccelertion Eq. (46) is: +ω v + c v cv ω = (52) This eqution my e simplified to: + vω + c v cφ ω = (53) where: ω =ω i+ ω j+ ω k (54) 2 3 nd Ω =ω i+ ω j+ ω k 2 3 (55) In this simplifiction we hve used: v ω =vω (56)

9 Some sic concepts of fluid dynmics 37 nd ω = v vω (57) For ech in Eq. (53) = + vω+ c v cvω (58) nd this reduces to Eq. (5) if: vω= v v (59) = c( v v ) ω (6) m ρ+ φ So we hve derived the textook inviscid fluid eqution from the Crtn structure eqution with the minimum use of hypothesis, Q.E.D. Note crefully tht this procedure uses the spin connection s n intrinsic prt of the fluid dynmics, so generl reltivity itself is extended considerly s suject. It is no longer smll correction to Newtonin dynmics ut n intrinsic prt of fluid dynmics. All equtions of fluid dynmics ecome equtions of generl reltivity nd some equtions of fluid dynmics my e corrected using the concept of spin connection. One of these is the vorticity eqution, which in generl reltivity is: Ω = v ω v ut which in the textooks is []: (6) Ω = v (62) Eq. (6) in the philosophy of generl reltivity is correction of Eq. (6). Accepting the textook eq. (6) for the ske of illustrtion only, the usul procedure [] is to use Eqs. (5) nd (6) to produce the textook eqution of motion of the inviscid fluid: Ω + ( Ω v) = (63) using the ssumption:

10 372 M.W. Evns v = (64) Generl reltivity shows tht there re terms missing in Eq. (63). These terms my led to effects which re experimentlly oservle. Accepting Eq. (63) gin just for the ske of illustrtion, it hs the sme structure s the textook Frdy lw of induction: B + E = (65) where B is the mgnetic flux density nd E is the electric field strength. Eqution (62) hs the structure: B= A (66) of electrodynmics in the stndrd model. In ECE theory Eq. (66) ecomes: B = A ω A (67) Eq. (64) is nlogous to: A = (68) in the stndrd model of electrodynmics. Using the miniml prescription: ρ = mv= ea (69) where m is mss nd e is chrge, the stndrd model of electrodynmics gives: e E= B A= B v m (7) which is the Lorentz force lw. In summry: Ω B + ( Ω v) = + E= Ω = v B= A v= A=

11 Some sic concepts of fluid dynmics 373 The quntity Ω x v is nlogous to the Coriolis ccelertion nd to the electric field strength E. The vorticity Ω plys the role of the mgnetic flux density B. The velocity v plys the role of the vector potentil A. Note crefully tht these concepts re extended nd self consistently generlized in ECE theory in which: = + cω v + c v cvω (7) Ω = v ω v (72) The expression for ccelertion used in Eq. (44) is: = ( v) v+ (73) nd is specil cse of: I = + cω v (74) Eq. (49) is specil cse of: ( v v ) = c ω II (75) nd the textook inviscid, incompressile fluid is: + = (76) I II Generl reltivity is no longer minor correction to Newtonin dynmics, ut ecomes in ECE theory n intrinsic prt of everydy dynmics, dding new terms to well known equtions, terms which my hve mesurle experimentl effect or which re given y well known effects interpreted in new wy. 4. Viscous fluid eqution In this cse there is viscous force f v, so Eq. (5) ecomes []: fv =ρ + ( v ) v + p +ρ φ (77)

12 374 M.W. Evns The most generl form of second derivtives tht cn occur in vector eqution is liner comintion of terms 2 v nd ( v), so the viscous force is expressed s: 2 v + f = µ v µ + µ v (78) where μ nd μ' re coefficients. So the viscous fluid eqution of motion is derived using the vorticity (62) nd is: Ω µ 2 + ( Ω v) = Ω (79) ρ In ECE theory nd generl reltivity there re gin new terms for the viscous fluid nd for other equtions of the Nvier Stokes system. These will e developed in lter ppers. References [] E.G. Milewski, Chief Ed., The Vector Anlysis Prolem Solver (Reserch nd Eduction Assocition, New York, 987). [2] S.P. Crroll, Spcetime nd Geometry: n Introduction to Generl Reltivity (Addison Wesley New York, 24). [3] M.W. Evns, Generlly Covrint Unified Field Theory (Armis Acdemic, New York, 25 onwrds), in six volumes to dte, voluem seven in prep. ( [4] L. Felker, The Evns Equtions of Unified Field Theory (Armis 27). [5] K. Pendergst, The Life of Myron Evns (Cmridge Interntionl Science Pulishing, 2). [6] M.W. Evns et l., nd [7] M.W. Evns, ed., Modern Non-Liner Optics (Wiley 2, second edition). [8] M. W. Evns nd S. Kielich (eds.), Modern Non-liner Optics (Wiley 992, 993, 997, first edition). [9] M.W. Evns nd L.B. Crowell, Clssicl nd Quntum Electrodynmics nd the B(3) Field (World Scientific, 2). [] M.W. Evns nd J.-P. Vigier, The Enigmtic Photon (Kluwer, Dordrecht, 994 to 22), in five volumes. [] M.W. Evns nd A. A. Hsnein, The Photomgneton in Quntum Field Theory (World Scientific, 994). [2] D. Reed in ref. (7).