Download Caius Iacob” Conference on

,,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  TS  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   TS   p ,
ρ
ρ
2

2

using the Crocco’s equation for the isoenergetic non-isentropic flow ( Ω  V = TS ).
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  Φ1Φ 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  Φ1Φ 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 )]  ρVp  (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 xy
y z yz
z x zx 
 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   
j1  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
ξη
η
η ξ
ηξ
Ω 0
Therefore the ζi
relation leads to:

 2Φi  2Φi

0 ,
ξη ηξ
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  λE  (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);
ΩV 
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  λEE  2(V  H) c dt  jEdt  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).
References
1. Hadamard, J. – Leçons sur la propagation des ondes et les équations de
l’hydrodynamique, Hermann, Paris, 1903;
(also at Chelsea Publishing Co., New York, 1949.)
12
CIC-F.Mech.&Tech.Appl., Bucharest, November, 2005
2. Poincaré, H. – Théorie des tourbillons, Paris, 1893.
3. Appell, P. É. – Traité de mécanique rationnelle, tome III, 3-ième éd., p. 410, GauthierVillars, Paris, 1921.
4. Segre, B. – Ann. di Mat., 1, pp. 31 – 55, 1924.
5. Caldonazzo, B. – Rend. della Accad. Naz. dei Lincei, 33, pp. 396 – 400, 1924.
6. Caldonazzo, B. – Rend. della Accad. Naz. dei Lincei, IV, pp. 124 – 126, 1926.
7. Caldonazzo, B. – Boll. dell Unione Mat. Italiana, IV, pp. 1 – 3, 1925.
8. Finzi, B. – Rend. della Accad. Naz. dei Lincei, VI, pp. 236 – 241, 1925.
9. Finzi, B. – Rend. del Circolo Mat. di Palermo, 51, pp. 1 – 24, 1927.
10. Cisotti, U. – Rend. della Accad. Naz. dei Lincei, VI, pp. 612 – 617, 1925.
11. Cisotti, U. – Boll. dell Unione Mat. Italiana, II, p. 125, 1923.
12. Masotti, A. – Rend. della Accad. Naz. dei Lincei, V, pp. 985 – 989, 1927.
13. Masotti, A. – Rend. della Accad. Naz. dei Lincei, VI, pp. 224 – 228, 1927.
14. Masotti, A. – Rend. del Circolo Mat. di Palermo, 52, pp. 313 – 330, 1928.
15. Vâlcovici, V. – Asupra mişcării turbionare a fluidelor barotrope, Bul. ştiinţ. Acad. R.
P. R., Secţiunea de ştiinţe mat. şi fiz., IV, 3, pp. 541 – 545, 1952.
16. Vâlcovici, V. – Liniile de curent şi liniile de vârtej în mişcarea permanentă a unui
fluid ideal, barotrop, Bul. ştiinţ. Acad. R. P. R., Secţiunea de ştiinţe mat. şi fiz., V, 1, pp. 147 –
154, 1953.
17. Vâlcovici, V. – Sur le mouvement des fluides barotropes, Rend. della Accad. Naz. dei
Lincei, XII, série VIII, 21, 5, pp. 288 – 296, 1956.
18. Vâlcovici, V. – Bernoulli’s surfaces, Rev. de méc. appl., VI, 1, pp. 5 – 32, 1961.
19. Dragoş, L. – Sur le mouvement d’un fluide barotrope, Rend. della Accad. Naz. dei
Lincei, XXIV, pp. 142 – 148, 1958.
20. Ionescu-Bujor, Cl. – Étude intrinsèque des écoulements permanents et rotationnels d’un
fluide parfait, Thèse présentée à la Faculté des Sciences de l’Université de Paris, 1961.
21. Cristea, I. I. – Studiul canonic al mişcărilor rotaţionale ale fluidelor perfecte, barotrope sau
incompresibile, Stud. şi cerc. mat., 26, nr. 5, 6, 7, 1974.
22. Landau, L., Lifchitz, E. – Physique théorique, tome VIII (Électrodynamique des
milieux continus), ch. VIII (Magnéto-hydrodynamique), pp. 280 – 313, Éditions Mir, Moscou,
1969.
23. Sedov, L. – Mécanique des milieux continus, tome I, ch. VI (Notions et équations
fondamentales de l’électrodynamique): par. 3 (Transformations de Lorentz et systèmes de
référence inertiaux), pp. 311 – 329; par. 4 (Interaction du champ électromagnétique avec les
conducteurs), pp. 329 – 338 et par. 6 (Hydrodynamique magnétique), pp. 361 – 364, Éditions
Mir, Moscou, 1975.
24. Yavorski, B., Detlaf, A. – Handbook of Physics, Part Four (Electricity and
Magnetism), Chapter 31 (Fundamentals of Magnetohydrodynamics), 2nd edition, pp. 525 – 533,
Mir Publishers, Moscow, 1975;
(also Aide-mémoire de physique, 4-ième partie (Électricité et magnétisme), ch. 12
(Éléments de l’hydrodynamique magnétique), 5-ième éd., pp. 534 – 543, Éditions Mir, Moscou,
1986.)
25. Dragoş, L. – Magnetofluid Dynamics, Editura Academiei, Bucureşti – Abacus Press,
Tunbridge, Wells, Kent, England, 1975;
(also
in
Romanian:
Magnetodinamica fluidelor, Editura Academiei, Bucureşti, 1969.)
26. Brătescu, G. – Fizica plasmei, cap. IV (Plasma), subcap. 4.2. (Metode teoretice
generale de studiu al plasmei), par. 4.2.8. (Ecuaţiile macroscopice globale ale plasmei.
Magnetoplasmodinamica), pp. 217 – 221, Editura didactică şi pedagogică, Bucureşti, 1970.
13