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,,Caius Iacob” Conference on Fluid Mechanics&Technical Applications Bucharest, Romania, November 2005 A General Method for Treating the Flow of a Barotropic Inviscid Fluid; A Possible Extension to Some Special Cases in Magnetoplasmadynamics (A Case of First Integrability of the Vectorial Differential Equation of Motion for the Non-Isentropic Flow of a Barotropic Inviscid Electroconducting Fluid by an Adiabatic Evolution in an External Magnetic Field, Considering the Flow Vorticity and the Associated Joule-Lenz Heat Losses) by Richard SELESCU1 Abstract In this work a model of a certain flow of a barotropic inviscid fluid is introduced, in order to establish a new (simpler) form of general partial differential equation of the velocity “quasipotential” for these flows. This model consists mainly in using a new three-orthogonal curvilinear coordinate system (one of them being tied to the local specific entropy value and another being an intrinsic one). The choice of this new orthogonal curvilinear coordinate system (with two coordinate curves lying on the isentropic surfaces) presents the advantage of enabling the treatment of any three-dimensional flow (even rotational) of velocity field V as a “quasipotential” two-dimensional one, on the respective isentropic rigid surfaces (a particular case of “D. Bernoulli” ones for an isoenergetic flow, namely those having different constant values S 0i of the specific entropy S ), introducing a two-dimensional velocity “quasi-potential”, specific to any isentropic surface. On these surfaces the streamlines are orthogonal paths of a family of lines of equal velocity “quasi-potential” (equi”quasi-potential” lines). The here presented method can be generalized for the true “D. Bernoulli” surfaces B = B 0i = const. (the steady flow case). This general method can be applied to the conical supersonic flows, as well extended to some special (but usual) cases in magnetoplasmadynamics, considering (not as usual) an adiabatic, but non-isentropic (rotational and with Joule-Lenz heat losses) flow in an external magnetic field H , obtaining a new first integrability case (similar to the “D. Bernoulli” one): - assuming a continuous medium; - making no distinction between the intensity of the magnetic field and the magnetic induction of the medium, since for all conducting fluids the value of the magnetic permeability is equal to 1 (see [22 - 24]); - also assuming that the real part of the dielectric permittivity of the medium is constant; - assuming that the value of the electric conductivity λ of the fluid medium is uniform and isotropic throughout and independent of the magnetic field intensity, as well being sufficiently large so that (e/4π)·(ω/λ) << 1, where e is the magnitude of the charge of the electron and ω is the frequency of processes occurring in the medium, for instance, that of electromagnetic waves propagated in plasma; - in the non-relativistic theory, neglecting: “Elie Carafoli” National Institute for Aerospace Research – INCAS Department of Aerodynamics E-mail address: [email protected] 1 Flow of a Barotropic Inviscid Fluid Richard SELESCU 1. in the general differential equation of motion for a non-isentropic inviscid electroconducting fluid flow in an external magnetic field the term due to the electric charges in the considered fluid medium, thus retaining the Lorentz force only and 2. in the expression of the total electric current density the terms given by the densities of the induction (displacement) and respectively convection electric currents. Taking into consideration the flow vorticity effects, there always are some space curves (Selescu) along which the vector equation of motion admits a first integral in the general case. In the particular case of a fluid having an infinite electric conductivity (λ → ∞, like for the highly ionized plasma), these space curves also are the isentropic lines (of different values S0i of the specific entropy) of the flow, in both cases enabling the treatment of any threedimensional flow as a potential two-dimensional one, introducing a two-dimensional “velocity quasi-potential”, specific to any isentropic (or “D. Bernoulli”) surface on which these curves are lying. This theory can find some useful applications: to study the V, H fields around a hypersonic (including stealth) aircraft subject to a strong electromagnetic field (radar waves) in the Earth atmosphere, as well in astrophysics, to study the flow around a spacecraft, in the proximity of a star or a gas giant planet. Keywords and phrases: rotational flow of a barotropic inviscid fluid, isentropic “D.Bernoulli” surfaces, the general nonlinear PDE of the velocity “quasi-potential” for the flows of barotropic inviscid fluids, non-isentropic flow of a barotropic inviscid electroconducting fluid in an external magnetic field (magnetoplasmadynamics), Lorentz force, second law of thermodynamics, generalized Crocco’s equation for magnetogasdynamics, Maxwell’s equations, non-relativistic form of Ohm’s law for a medium in motion, associated Joule-Lenz heat losses, Selescu’s curves 2000 Mathematics Subject Classification: 70 - Mechanics of particles and systems; 76 - Fluid mechanics; 78 - Optics, electromagnetic theory; 80 - Classical thermodynamics, heat transfer; 85 - Astronomy and astrophysics Part one - A general method for treating the problem of a barotropic inviscid fluid flow Let start from the vectorial general form of the equation of motion for an inviscid fluid flow: V 1 1 V 2 Ω V f p ; t ρ 2 (the Gromeko-Lamb form, binding the acceleration and the force density terms of a little fluid particle, where Ω = V = 2 ω is the vortex (curl V ), with ω - the instantaneous velocity of rotation of the respective little particle, f being the massic force density (conservative): f = (–gz) = –(gz) . For this inviscid fluid flow the momentum equation yields the Crocco-Vazsonyi form: V 1 Ω V TS i 0 gz , with i 0 i V 2 t 2 - the total (stagnation) specific enthalpy, where i = U + p/ρ is the specific enthalpy, U - the specific internal energy; S - the specific entropy; T - the static temperature (absolute). For a steady motion and respectively for gases, we have V t 0 and f = 0 , thus remaining: 1 1 1 1 V 2 Ω V p , or V 2 TS p , ρ ρ 2 2 using the Crocco’s equation for the isoenergetic non-isentropic flow ( Ω V = TS ). 2 CIC-F.Mech.&Tech.Appl., Bucharest, November, 2005 Let introduce the following vectorial quantities (theorems due to Helmholtz and Hadamard): V dR dt Vp(ot) Vr(ot) , with Vp Φ and Vr Ψ , or Vr Φ1Φ 2 (see [1]) ; Vp Φ ( Vp 0) - a potential (irrotational) field ; Φ - scalar potential of the vector field V; Vr Ψ - a solenoidal (rotational) field (Vr 0) ; Ψ - vector potential of the vector field V; or Vr Φ1Φ 2 - a two-scalar field; Φ1 , Φ 2 - independent scalar functions (Φ1 C1, Φ2 C2). Vi eight scalar equations, namely : Vp : Vp Vp Φ 2 Φ Φ ; then Φ Vp ; ( Vp 0 - 3 equations ; ) ( Ψ) (Ψ) Ψ (Ψ) 2 Ψ (Ψ) Ψ Ψ ; then Vr : Vr Ω Ψ Vr Ω (assuming that Ψ - also a solenoidal field : Ψ 0) ; or (Vr 0 - 1 equation ) Φ1 Φ 2 , with Vr Vr Vr Ω 0 ; Ω Vr (in this case) . Let write now the first (left) form of the previously framed equation of motion as successively follows: 1 1 2 Vp Vr Ω Vp Vr p , ρ 2 1 1 Vp2 2Vp Vr Vr2 Vr Vp Vr p . ρ 2 Let separate the above equation of motion in the following complementary two: 1 1 1 1 Vp2 p , or d Vp2 dp , ρ ρ 2 2 (1) that is 3 scalar equations, describing the potential (irrotational) part of the motion (left) and 1 1 2Vp Vr Vr2 Ω Vp Vr 0 , or Vr 2Vp Vr Vr Vp Vr 0 , 2 2 (2) that means other 3 scalar equations, describing the rotational part of the motion respectively. Joining to the above system the continuity and the energy equations and expressing p as a function of ρ - the case of a general polytropic transformation (introducing a - the speed of sound in the respective gas - a barotropic fluid), meaning other 3 scalar equations, we get: (ρV ) [ρ(Vp Vr )] ρVp (Vp Vr )ρ 0 ; 1 1 2 1 (Vp Vr ) 2 a i0 W2 ; 2 γ 1 2 W - the maximum possible velocity in the gas, correspond ing to an expansion into vacuum (for a static pressure p 0) p a 2 dρ ; p f (ρ) . 3 Flow of a Barotropic Inviscid Fluid Richard SELESCU Solving this 9/9 system one can determine the entire flow field, even if this is achieved only in a particular case, representing a first approach in obtaining the problem solution. Let return to the first (initial) framed equation of motion and multiply scalarly this equation by a certain elemental displacement vector dR , thus obtaining in the case of an isoenergetic flow 1 1 1 1 d V 2 (Ω V ) dR dp , or d V 2 TdS dp , ρ ρ 2 2 for a non-isentropic (rotational) flow: dS ≠ 0. One can see that, besides the trivial cases: 1. V = 0 - equilibrium; 2. Ω = 0 - an irrotational (potential) motion; and 3. Ω = cV - a helicoidal motion, there also is an important case for which the second term (Ω V)·dR = TdS becomes zero, namely dR = c1V + c2 Ω , with two main particular subcases: 4. c2 = 0 (dR = c1V) , and 5. c1 = 0 (dR = c2Ω) , this meaning an elemental displacement along a stream- (and a vortex-)line, for which one has dS = 0 . In both last cases the specific entropy S remains constant (S0) on the surface containing the stream- and vortex-line passing through the considered point of the flow, that is on the whole above surface one can write for the physical equation: γ p Kρ γ , with K p0 ρ0 exp(S0 c) Cv (the isentropic constant of the streamand vortex-line, and more, the isentropic constant of the respective surface), the equation of motion becoming now: 1 1 1 d V 2 dp , or d V 2 γKρ 2 dρ , 2 ρ 2 presenting itself like the previous equation of the potential motion (even if Ω ≠ 0 on the whole stream- and vortex-surface) and admitting a first integral (the energy equation): 1 2 γK 1 1 V ρ i0 W2 2 γ 1 2 1 i0 i V2 2 , with , the “D. Bernoulli integral” for isoenergetic (adiabatic and steady, with constant total i specific enthalpy 0 ) non-isentropic (rotational) flows, where: K - the isentropic constant, different from a stream- and vortex-surface to another; i0 , W - invariants for the whole flow (due to general flow steadiness and to the inviscid incident flow parallelism and uniformity), even if there are discontinuity surfaces (shock waves). There are two main differences between the last framed differential equation of the potential motion and the previously written one: 4 CIC-F.Mech.&Tech.Appl., Bucharest, November, 2005 1. the last differential equation was derived rigorously for the case of isoenergetic flows, from the isentropicity condition along the flow stream- and vortex-lines (on some surfaces - a particular case of the “D. Bernoulli” ones [2 - 21], namely those having different constant values of specific entropy); 2. it contains the square of the global (total) velocity vector V and not only that of the potential part of this vector, Vp . The last framed equation of motion, joined to the continuity and the physical ones and taking into consideration the local speed of sound a definition, enables the determining of the total velocity vector V from the Steichen’s equation - usually a PDE (improperly called now the “velocity potential equation”, taking into account that there is a vector Ω ≠ 0 ; the flow being rotational, more appropriate would be the term “velocity quasi-potential equation”): V Φ 1 2 ( 2) V 2 , with V Φ , but Ω 0 ; ΔΦ 2 Φ , or ΔΦ 2 Φ Φ , 2 2a 2a a ( 2) 2 2 2 Φ Φ Φ Φ Φ Φ ( 2) Φ Φ Φ where : ΔΦ 2 2 2 and x y z x x y y z z 2 2 2 Φ 2 Φ 2 Φ 2 Φ Φ 2 Φ Φ 2 Φ Φ 2 2 2 2 Φ , y y 2 2 2 x y xy y z yz z x zx z z x x V in the vector equation above, the local speed of sound a being given by the energy equation: 2 2 2 γ 1 2 Φ Φ Φ 2 a W , 2 x y z all the points M(x, y, z) in which is satisfied the previous PDE of the velocity potential Φ belonging to a certain isentropic surface. The new equation is identical to the velocity potential equation, the vector V having now two components only (like the vector Ω, both lying in the tangent plane to an isentropic sheet). In a three-orthogonal curvilinear coordinate system Oξηζ tied to the stream- and vortex-surface, Ω 0; the vortex component normal to the isentropic sheet ζ ζ 0 , must be zero: ζ h ξ k ξ h ηk η h ζ k ζ 1 Ω V k ξ Ωξ k ηΩ η k ζ Ωζ , h ξ h η h ζ ξ η ζ h ξ Vξ h η Vη h ζ Vζ with Vξ h ξ ξ ; Vη h η η ; V h ζ 0 ; ζ ζ 1 (h η Vη ) (h ξ Vξ ) 0, h ξ h η ξ η coefficients. (h η Vη ) Ωζ ξ or 5 (h ξ Vξ ) η 0; hi - the Lamé’s Flow of a Barotropic Inviscid Fluid Richard SELESCU Let introduce a scalar function Φi (ξ, η, ζ0i), called by the author “quasi-potential”, whose partial derivatives along the directions of the elemental orthogonal arcs h ξdξ and hηdη on the (i) isentropic surface ζ = ζ0i are just the components Vξi and Vηi of the velocity vector Vi (with Vζi = 0). Let still define λ and μ as being two orthogonal arcs lengths, so that: dλ = hξdξ and dμ = hηdη , the elemental arc length ds on this surface being given by: ds 2 dx 2 dy 2 dz 2 Edξ 2 2Fdξdη Gdη2 h ξ2 dξ 2 h 2η dη2 dλ 2 dμ 2 , with x x y y z z E h ξ2 ; G h 2η ; F 0 ξ η ξ η ξ η - due to the orthogonality, x j h i j1 q i 3 where: so having 2 2 x y z h ξ ξ ξ ξ 2 2 2 2 x y z h ζ ζ ζ ζ x x, y, z ; (i 1, 3) ; j q i ξ, η, ζ 0 f 1 ξ, η, ζ 0i ζ ζ 0i 2 2 x j x, y, z q i λ, μ, ν 0 and 2 x y z ; h η η η η 2 , f 2 ξ, η, ζ 0i ; ζ ζ 0i f 3 ξ, η, ζ 0i , ζ ζ 0i thus resulting on the (i) isentropic surface: Φ i Φ i 1 Φ i Φ i 1 Φ i Φ i 1 Φ i Vξi ; Vηi ; Vζi 0 ; h ξ Vξi ; h η Vηi ; h ξ ξ λ h η η μ h ζ ζ ξ η (h η Vηi ) (h ξ Vξi ) Φ i 2 Φ i Φ i 2 Φ i ; ; ξ ξ η ξη η η ξ ηξ Ω 0 Therefore the ζi relation leads to: 2Φi 2Φi 0 , ξη ηξ a true relation - the Schwarz’ theorem for the functions of two variables (the so-called theorem of “the equality of the mixed derivatives of the second order”, they differing as to the order of differentiation). This relation proves that Ωζi = 0 and the existence of a “quasi-potential” function Φi so that Φ Φ Vξi Vλi i ; Vηi Vμi i , λ μ the entropy gradient vector Si being normal to the above introduced isentropic surfaces ζ = ζ0i. In a three-orthogonal coordinate system Oξηζ tied to these surfaces (or Oλμν, with λ, μ, ν - lengths of orthogonal arcs) the Laplace’s operator Δ is given by the general expression below: 6 CIC-F.Mech.&Tech.Appl., Bucharest, November, 2005 h η h ζ Φ i h ζ h ξ η h ξ h ξ η ξ 1 h η h ζ Φ i h ζ h ξ ΔΦ i h ξ h η h ζ ξ h ξ ξ η h η ΔΦ i 1 h ξ h ηh ζ h ξ h η Φ i ; but ζ h ζ ζ Φ i , η ζ ζ 0i Φ i η Vζi 0 e.g.: the λ arcs taken along the streamlines and the μ arcs - along the equipotential lines. Analogously, in the Steichen’s equation - a nonlinear PDE of the 2nd order in three variables - ξ, η and ζ (written for a rotational flow - Ω ≠ 0 , but in the (i) isentropic surface ζ = ζ0i ) disappear all the terms containing the partial differential about ζ of the potential function Φ i , ( Φi ζ ) and also its derivatives with respect to ξ, η and ζ, thus being obtained a nonlinear PDE of the 2nd order in only two variables - ξ and η (the “velocity quasi-potential equation”). Conclusion for part one The choice of a new orthogonal curvilinear coordinate system (with two coordinate curves lying on the isentropic surfaces) presents the advantage of enabling the treatment of any three-dimensional flow (even rotational) as a potential two-dimensional one, on the respective isentropic (“D. Bernoulli”) rigid (Vζi = 0) surfaces. On these surfaces the streamlines (of arcs λ) are orthogonal paths of a family of lines of equal velocity potential (of arcs μ). Applying this method to the conical supersonic flows, it was obtained an ODE for the usually written (approximate) PDE of the velocity potential function Φ . As a main disadvantage of the new treating method, we mention that for the effective expression of the mathematical relations between the new three-orthogonal curvilinear coordinates and the Cartesian ones, it must be given (known) the equations of a pair of flow stream- and vortex-lines. This method can be generalized for the true “D. Bernoulli” surfaces B = const. (B = V2/2 + ∫dp/ρ + gz ; B = V Ω). Part two - A possible extension to some special cases in the magnetogasdynamics of plasma Analogously, in the magnetoplasmadynamics, the vectorial general form of the differential equation of motion for an adiabatic but non-isentropic flow of a barotropic inviscid electroconducting fluid in an external magnetic field, considering the flow vorticity Ω , is (see [22 - 26] for the right-hand side): V2 V2 V p V p ρ e ~ 1 Ω V f Ω V f f eL , or E H j , t ρ t ρ ρ cρ 2 2 binding the acceleration terms of a fluid particle, where feL is the density of the volumic (as a rule pure electromagnetic) generalized Lorentz force, in the expression of which appear: ρe - the density of electric charges in the considered fluid medium; c - the light speed in vacuum; ~ E E (V H) c - the intensity of the electric field in the proper (own) coordinate system ( E - given by the Maxwell’s equations: Poincaré-Steklov problem); 7 E 4πρ e ; E 1 c H t , then a Φ i 0 ; ζ Flow of a Barotropic Inviscid Fluid Richard SELESCU H - the intensity of the magnetic field; j λE - the density of the conduction electric current (the 2nd Maxwell’s equation: j k H , with k = c/4π , the terms jind λ c (V H) and jconv ρe V - densities of the induction (displacement) and respectively convection electric currents being neglected in the expression of the total current density: jtot j jind jconv λE (V H) c ρeV - the non-relativistic form of Ohm’s law for a medium in motion, so having jtot = j ), where: λ - the electric conductivity of the considered fluid. For a steady motion and respectively for gases, we have V t 0 and f = 0 . In the ~ non-relativistic theory (V << c) the term (ρe ρ) E , due to electric charges, may be also neglected, so remaining: 1 1 1 1 1 1 V 2 Ω V p (H j) , or V 2 Ω V p (H j) . ρ cρ ρ c 2 2 Let multiply scalarly this equation by a certain elemental displacement vector dR , so obtaining: 1 1 1 d V 2 (Ω V ) dR dp (H j) dR , ρ c 2 with Ω = 2 ω (ω - the instantaneous velocity of rotation of the little fluid particle). One can see that, besides the trivial cases: 1. Ω V j H 0 , meaning either Ω = j = 0 - an irrotational (potential) fluid field V, as well magnetic field H , or Ω || V and j || H - ( Ω = c1V and j = c2H - helicoidal fields); ΩV 2. being 1 ( j H) , cρ meaning that all the vectors V, Ω, H and j are coplanar, also satisfied the sense and modulus conditions for the vector products – a very particular case; there also is an important case for which (Ω V) dR 1 (cρ) (H j) dR becomes zero, namely: 3. dR coplanar with both the pairs (V and Ω) and (H and j); for all the considered special (but usual) cases, there always are some lines (space curves) along which an elemental vector dR is coplanar with both the vectors V and Ω - contained in the tangent plane to the 0-work sheet of (Ω V ) elemental force, and the vectors H and j - contained in the tangent plane to the 0-work sheet of the Lorentz force, being directed upon the intersection straight line of the two tangent planes above. Multiplying scalarly the last framed equation by such a dR, disappearing the mixed products (Ω V ) dR and (H j) dR , one gets successively (considering a barotropic gas): 1 1 dp 1 1 d V 2 dp , or d V 2 dρ ; ρ ρ dρ 2 2 8 CIC-F.Mech.&Tech.Appl., Bucharest, November, 2005 and 1 2 dρ 1 C V a 2 (ρ) i 0 W 2 H 02 2 ρ 2 ρ . This first integral is similar to the “D. Bernoulli” one in the compressible aerogasdynamics. Let determine the analytic expression of the vector dR in order to satisfy the two coplanarity conditions for the vectors V, Ω, dR and H, j, dR respectively, and the differential equations of the intersection curves of the 0-work sheets for the (Ω V) force and for the Lorentz one. In a given three-orthogonal coordinate system, say the Cartesian one, having the axes unit k ,k ,k vectors x y z , the analytic expressions of the vectors V, Ω, H and j are as follows: V k x Vx k y Vy k z Vz ; Ω rotV k x Ω x k y Ω y k z Ω z ; H k x H x k y H y k z H z ; j k rotH k x jx k y jy k z jz , assuming all these quantities as being known (given) functions of the variables x, y and z. The direction of the normal N1 to the 0-work sheet for (Ω V = 2 ω V) is given by this vector (an acceleration similar to the Coriolis complementary one), having the analytic expression: kx ky kz k x Ω y Vz Ω z Vy k y Ω z Vx Ω x Vz k z Ω x Vy Ω y Vx ; Ω V Ωx Ωy Ωz k x A1 ( x, y, z) k y B1 ( x, y, z) k z C1 ( x, y, z) . Vx Vy Vz Analogously, the direction of the normal N2 to the 0-work sheet for the Lorentz force is given by the vector (j H) - acceleration of the Lorentz force, having the analytic expression: kx ky kz k x j y H z jz H y k y jz H x jx H z k z jx H y j y H x ; j H jx jy jz k x A 2 ( x, y, z) k y B 2 ( x, y, z) k z C 2 ( x, y, z) , Hx Hy Hz the equations of the current tangent planes to the 0-work surfaces for the force similar to the Coriolis complementary one and respectively for the Lorentz force being thus: A1 (x, y, z)x B1 (x, y, z) y C1 (x, y, z)z D1 0 ; A 2 (x, y, z)x B2 (x, y, z) y C2 (x, y, z)z D2 0 . The direction of the intersection straight line of the two current tangent planes above (upon which is directed the searched elemental vector dR ) is given by the vector (N1 N2) : k x k y k z B1 C1 C A1 A B1 ky 1 kz 1 ; k x N1 N 2 A1 B1 C1 B 2 C 2 C2 A 2 A 2 B2 $ , A 2 B 2 C 2 k x A 3 ( x, y, z) k y B 3 ( x, y, z) k z C 3 ( x, y, z) ; where: A3 B1C2 B2C1 $x ; B3 C1A 2 A1C2 $ y ; C3 A1B2 A 2 B $z ≠0) (all A1 Ω y Vz Ω z Vy ; B1 Ω z Vx Ω x Vz ; C1 Ω x Vy Ω y Vx ; (with A 2 j y H z jz H y ; B 2 jz H x j x H z ; C 2 j x H y j y H x , generally 0 ) are the director parameters of the searched direction, dR being parallel to the vector (Selescu) $ = (Ω V) (j H) - the vector product of two accelerations (elemental forces, the first being similar to the Coriolis complementary one and the second being the Lorentz one). 9 Flow of a Barotropic Inviscid Fluid Richard SELESCU The differential equations of the intersection curves of the 0-work sheets of (Ω V) force with the 0-work sheets of Lorentz force, lines on which lies the searched dR || $, such that dR (V, Ω) and also dR (H, j) dR (V, Ω) (H, j) || $ , dx dy dz , $x $y $z are therefore: (the “Selescu’s curves”, along which the vector equation of motion for an adiabatic, but non-isentropic flow of a barotropic inviscid electroconducting fluid in an external magnetic field admits a first integral), these rigid lines being very similar to the “D. Bernoulli” rigid surfaces and lying on these. Applying the second law of thermodynamics, one may write the heat transport equation: ~ ~ TdS dq int dq ext (Ω V) dR 1 ρ j E dt dq add ext (the generalized Crocco’s equation for MHD). In the right hand side: the 1st term is the dissipation heat due to the flow vorticity Ω = 2 ω . The 2nd term is the associated Joule-Lenz heat loss (dt being an elemental time) with: ~ ~ ~ ~ ~ ~ ~ j λE ρ e V λE ; E E (V H) c ; j E λE 2 λ E 2 2E (V H) c (V H) 2 c 2 the first relation expressing the non-relativistic form of Ohm’s law for a medium in ~ ~ motion, j and E being the values of j and E in the proper (own) coordinate system. Taking into account that in the non-relativistic theory V << c, one can neglect the last term, so having: ~ ~ j E dt λEE 2(V H) c dt jEdt 2 (Vdt H) c j Edt 2 j (dR H) c , ~ ~ or: j E dt jE dt 2dR ( j H) c jE dt 2 c ( j H) dR , with dR Vdt - an elemental vector directed parallel to the velocity V . The term containing dR also may be neglected. It is interesting to remark that though H is an external field, inducing E and j , the ~ ~ associated Joule-Lenz heat ( 1 ρ j E dt ) is an internal one, generated and preserved inside the system (an adiabatic evolution). The 3rd term represents the additional heat brought (introduced or extracted) to the fluid mass unit from the outside, due to thermal conductivity, radiation etc. and must be equal to zero (an adiabatic process). So: TdS (Ω V) dR 1 ρ jE dt 2 (c ρ) ( j H) dR (Ω V) dR 1 ρ jE dt 0 , even if dR = Vdt , so that (Ω V ) dR 0 , resulting TdS 1 ρ j Edt 0 (j || E), then dS ≠ 0 . This means that along the Selescu’s curves it is conserved an other physical quantity, being respected the equation of the adiabatic (but non-isentropic) transformation. Let now consider the particular case when the electroconducting fluid is an ideal gas, e.g. the highly ionized plasma, this having an infinite electric conductivity ( λ → ∞ ) - see [23], p. 363. 10 , CIC-F.Mech.&Tech.Appl., Bucharest, November, 2005 From the non-relativistic form of Ohm’s law for a medium in motion, it results that the ~ ~ E intensity of the electric field, , must tend to zero, for having a finite density of the current, j ~ ~ (both j and E in the proper coordinate system). So, in the arbitrary inertial system, the electric field E is determined by V and H : E (V H) c . Let resume in detail the system formed by the equation of motion, the Maxwell’s ones, the physical one and the generalized Crocco’s (heat transport) one. According to the 2nd Maxwell’s equation, the electric current density j has the expression: j 1 E 1 c rot H H 2 (V H ) , with k . k c t 4π c t Neglecting (as it is often made in magnetohydrodynamics) the density of the induction k E (displacement) electric current (the term c t ), one may write as previously: j k H . The density of electric charges ρe is also expressed by V and H , as it follows: 1 k k div E E 2 (V H) . 4π c c From this last formula one can see that ρe is little for usual conditions. In magnetogas(hydro)dynamics, one generally neglects in the vector equation of motion ~ the term ρeE ρe E (V H) c (now just zero), which is small with respect to 1 c ( j H) ρe . In a medium (fluid) with an infinite electric conductivity one deducts from the Maxwell’s equations for the magnetic field of intensity H , the two following relations: H 0 , and H d H H V , c rot E c E (V H) , or t d t ρ ρ to which is reduced, in this case ( λ → ∞ ), all the system of electrodynamic equations; The second one of these two Maxwell’s equations express the law of conservation of the magnetic flux through any surface that moves together with the fluid, enabling the conception of magnetic lines of force “frozen in” the fluid to be introduced. ~ ~ ~ 1 ρ j E dt ) E 0 Due to the fact that rigorously, the associated Joule-Lenz heat losses ( are zero for the medium with an infinite electric conductivity (zero electric resistance). dq add TdS (Ω V ) dR , the increase in ext 0 , resulting For an adiabatic process, entropy being given by the vorticity Ω only, so that the vector equation of motion was reduced to 1 1 1 d V 2 TdS dp (H j) dR , ρ c 2 for an adiabatic non-isentropic (rotational) flow: dS ≠ 0. In the particular case when dR is chosen to be parallel to the Selescu’s vector (tangent to the Selescu’s curves) $ = (Ω V) (j H) , this meaning an elemental displacement along the intersection line of the isentropic (D. Bernoulli) surface (containing the stream- and the vortexlines passing through the considered point of the flow) with the 0-work sheet of the Lorentz force (containing the field lines of the vectors H and j passing through the same point), the entropy S remaining this time constant (S0) on this line (dS = 0), that is on the whole above rigid space curve one can write for the physical equation: 11 Flow of a Barotropic Inviscid Fluid Richard SELESCU p Kρ γ , K p 0 ρ 0 exp(S0 c) Cv , γ with the equation of motion becoming now: 1 1 1 d V 2 dp , or d V 2 γKρ 2 dρ , ρ 2 2 presenting itself like the previous equation of the potential motion (even if Ω ≠ 0 on the whole isentropic - Selescu’s curve) and also admitting a first integral (the energy equation): 1 2 γK 1 1 C V ρ i 0 W 2 H 02 2 γ 1 2 ρ , the “D. Bernoulli” integral for isoenergetic (adiabatic and steady) non-isentropic (rotational) flows, for an ideal gas (highly ionized plasma, λ → ∞) in MHD. In addition to this, one has to remark that in both the general and the particular λ → ∞ cases, the dR || $ choice above enables the treatment of any three-dimensional flow as a potential two-dimensional one, introducing a two-dimensional “velocity quasi-potential”, specific to any isentropic (or “D. Bernoulli”) surface on which these curves are lying. Conclusion for part two Taking into account the flow vorticity Ω effects, there always are some space curves (Selescu) very similar to the “D. Bernoulli” rigid surfaces, curves along which the vector equation of motion admits a first integral in the general case. In the particular case of a fluid medium having an infinite electric conductivity (λ → ∞ - the highly ionized plasma), these space curves also are the isentropic lines of the flow, in both cases enabling the treatment of any three-dimensional flow as a potential two-dimensional one, introducing a two-dimensional “velocity quasipotential”, specific to any isentropic (or Bernoulli) surface on which these curves are lying. This theory can find some useful applications: 1. to study the V, H fields around a hypersonic (including stealth) aircraft or a rocket subject to a strong electromagnetic field (radar waves) in the ionized surrounding air, due to the high temperatures downstream the strong shock waves, these waves now becoming discontinuity surfaces also for the ratio γ of gas specific heats, this ratio first decreasing as the temperature increases (by jump), passing from the value 7/5 (5 degrees of freedom: 3 translations + 2 rotations) for diatomic gas molecules upstream the wave to the value 9/7 (7 degrees of freedom: the previous 5 + 2 oscillations) for hot gas diatomic molecules downstream the wave and finally here rising to the value 5/3 (3 degrees of freedom: 3 translations only) for monatomic (diatomic completely dissociated: free atoms) gas molecules, or for completely ionized gas (plasma: neutral and excited atoms, ions, electrons and photons) particles; 2. as well in astrophysics, to study the flow around a spacecraft moving in the proximity of a star or a gas giant planet, like Jupiter (having a magnetic field about 20,000 times stronger than that of the Earth and with the same distribution law: strongest at the poles and weakest in the equator plane, but much less distorted by the solar wind) and Saturn (a field about 30 times weaker than that of Jupiter). 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