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BULLETIN OF THE TRANSILVANIA UNIVERSITY OF BRAŞOV
A GENERAL METHOD FOR TREATING
A CERTAIN FLOW OF AN INVISCID FLUID
Part Two: An Extension to Some Special Cases
in Magneto-Plasma Dynamics
R. SELESCU*
Abstract: MHD models are extensively used in the analysis of magnetic fusion
devices, industrial processing plasmas, and ionospheric/astrophysical plasmas.
MHD is the extension of fluid dynamics to ionized gases, including the effects
of electric and magnetic fields. So, the general method presented in the part
one can be extended to some special (but usual) cases in magneto-plasma
dynamics, considering an adiabatic but non-isentropic flow (taking into
account the flow vorticity effects, as well as those of the associated Joule-Lenz
heat losses) in an external magnetic field, obtaining new first integrability
cases (similar to the “D. Bernoulli” ones): a) assuming a continuous medium;
b) making no distinction between the intensity of the magnetic field and the
magnetic induction of the medium (in the Gaussian system of units), since for
all conducting fluids the magnetic permeability is approximately equal to 1
(see [1 - 3]); c) assuming that the value of the electric conductivity of the fluid
medium is uniform and isotropic throughout and independent on the magnetic
field intensity. 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 (the highly
ionized plasma), these curves also are the isentropic lines of the flow, in both
cases enabling the treatment of any 3-D flow as a “quasi-potential” 2-D one.
Like the part one, this work also deals with the unsteady non-isentropic flows,
giving a first integral for the equation of motion in the general case.
Keywords and phrases: non-isentropic flow of a barotropic inviscid
electroconducting fluid in an external magnetic field, steady and unsteady
flows, Lorentz’ force, Maxwell’s equations, non-relativistic form of Ohm’s
law for a medium in motion, associated Joule-Lenz heat losses, second law of
thermodynamics, generalized Crocco’s equation for magneto-gasdynamics,
Selescu’s vector, Selescu’s space curves
1. Introduction, Nomenclature and the First Approach to the New Method
Analogously to the part one, in the magneto-plasma dynamics (by plasma
understanding a mixture of neutral and excited atoms, ions, electrons and photons), the
general form of the differential equation of motion for (not as usual) an adiabatic but nonisentropic flow of a barotropic (see the explanation in part one) inviscid
electroconducting fluid in an external magnetic field, considering the flow vorticity Ω , is
*
Dept. Of Aerodynamics, “Elie Carafoli” National Institute For Aerospace Research - Incas, Bucharest
348
A general method for treating a certain flow of an inviscid fluid
Part two: an extension to some special cases in magneto-plasma dynamics
(see
[1
10],
for
the
right-hand
side
only):
V
p
V
p ρe ~ 1
1 
1 
  V 2   Ω  V  f 
 feL , or
  V 2   Ω  V  f 
 E Hj
t
ρ
t
ρ
ρ
cρ
2 
2 
(for the left-hand side one usually writes: ∂V/∂t + (V)V ) , with:  - nabla (the Hamilton’s
operator) and t - the time, binding the acceleration and the force density terms of a fluid
particle, where (in case of quasi-neutral plasma, when the medium behaves itself in the
first approximation like an uniform conducting fluid):
ρ  na ma  n m  n m  na ma  nm  m   na  n ma - plasma density, with:
ma  m  m - the mass of a neutral atom; m - the mass of the positive ion (considering
a single species); m - the mass of the electron; na - plasma concentration in neutral atoms,
and respectively (for a three component neutral plasma): n  n  n  - the concentration
in positive and negative particles (a single species of ions and electrons; one can see that
in this neutrality case, the plasma density is the same as for the non-ionized fluid: ρ = ρa );
ρ , ρ , ρ - the densities of the components;
1
V  ρ a Va  ρ  V  ρ  V  - the velocity;  a  
ρ
Va , V , V - the velocitie s of the components;
Ω = V = 2 ω (with ω - the instantaneous velocity of rotation of the little fluid particle);
p  p a  p   p  - the pressure, resulted by totalizing of pressures of the three fluid components;
feL is the density of the volumic (as a rule pure electromagnetic) generalized Lorentz force, with:
ρ e - 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 local electric field in the proper (own) coordinate
system (E - given by the Maxwell’s equations: E  4πρ e ;   E   1 c  H t );
H - the intensity of the local magnetic field (as a rule variable in time; can be a periodical, even
alternating quantity, but assumed to have a constant direction, varying the modulus only);
j  λE - the density of the conduction electric current (the second Maxwell’s equation:
j  k  H , with k = c/4π , the terms jind  λ c  ( V  H) and jconv  ρ e V - the
densities of 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  ρ e V - the non-relativistic form of Ohm’s law for
a medium in motion, so having jtot = j ), where: λ - the electric conductivity of the fluid.
Searching for a steady motion solution we have V t  0 and for gases f = 0 (mass force),
H being considered an oscillating field, but reaching a damped state (constant in time).
In the non-relativistic theory (V << c), or for a neutral (ρe = 0), or an ideal (λ →
~
~
∞ ; E  0 ) gas, the term (ρ e ρ) E can be neglected (like the currents j ind and j conv ),
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 

with H  0 .
For a steady but isentropic motion (irrotational - Ω = 0 in an ideal fluid - λ → ∞)
see [5], pp. 65 - 67 and [6], pp. 94 - 96, giving a “D. Bernoulli” integral for some very
particular cases ( H  V ; H || V - along the H vector lines ; H·V = const. ); also see [10 20].
A complete approach to the electromagnetic theory and the continuum mechanics
presented from an unified point of view can be found, e.g., in [21].
Bulletin of the Transilvania University of Brasov Vol. 13(48) - 2006
349
2. Extension of the Method from the Part One to the Non-Isentropic Flows in MHD
Let multiply scalarly the last equation by a certain virtual elementary displacement
vector
dR
:
2
V 
dp 1
1
1
  (Ω  V)  dR   dp  (H  j)  dR  ; VdV  (Ω  V)  dR    ( j  H)  dR
d
ρ
c
ρ cρ

 2 
One can see that, besides the trivial cases of disappearance of both non-conservative
terms:
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);
2. Ω  V  ( j  H) cρ , meaning that all the vectors V, Ω, H and j are coplanar, also being
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:
3. dR coplanar with both the pairs (V and Ω) and (H and j); for all the considered special
(but usual) cases, there are some lines (space curves) along which an elementary vector dR
is coplanar with both the vectors V and Ω - contained in the tangent plane to the 0-work
sheet of (Ω  V) elementary 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 virtual
dR , disappearing the mixed products (Ω  V)·dR and (H  j)·dR , one gets successively
(considering a barotropic gas and assuming an alternating external electromagnetic field):
1
1 dp
1

1

d V 2    dp , or d V 2   
dρ ; 
ρ
ρ dρ
2

2

dρ
1 2
1
C
V   a 2 (ρ)
 i 0  W 2  H 02 ,
2
ρ
2
ρ


1
1 T
1 T 2
; ρ   ρ  (t)dt ; H 0  H ef 
H  (t)  E 2 (t ) dt ,

0
0
8π
T
T
where T is the period of H  (t) - the instantaneous value of the incident magnetic field H.
This first integral is very similar to the “D. Bernoulli” one in the compressible
steady aero-gasdynamics, the constant i0 being the flow total (fluido-electromagnetic)
specific energy.
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 vectors kx, ky, kz , 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 j x  k y j y  k z j z , 
with:
C
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:
A general method for treating a certain flow of an inviscid fluid
Part two: an extension to some special cases in magneto-plasma dynamics
350
kx
ky
Ω  V  Ωx
Ωy
Vx
Vy
k x Ω y Vz  Ω z Vy   k y Ω z Vx  Ω x Vz   k z Ω x Vy  Ω y Vx  ;
Ωz  
k x A1 ( x, y, z )  k y B1 ( x, y, z )  k z C1 ( x, y, z ) .
Vz 
kz
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  j z H y   k y  j z H x  j x H z   k z j x 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  B 2 ( x, y, z)  y  C 2 ( x, y, z)  z  D 2  0 .
The direction of the intersection straight line of the two current tangent planes
above (upon which is directed the searched elementary 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   B2 C 2
C2 A2
A 2 B2
$ ,

A 2 B2 C 2 k x A 3 ( x, y, z)  k y B3 ( x, y, z )  k z C 3 ( x, y, z )  k x $ x  k y $ y  k z $ z 
where: A 3  B1C 2  B2 C1  $ x ; B3  C1A 2  A1C 2  $ y ; C3  A1 B2  A 2 B  $z (all ≠ 0)
(with A1  Ω y Vz  Ωz Vy
; B1  Ωz Vx  Ωx Vz
; C1  Ωx Vy  Ω y Vx
;
A2  jy H z  jz H y ; B2  jz H x  jx H z ; C2  jx H y  jy H x , generally  0 )
are the director parameters of the searched direction, dR being therefore parallel to the vector
(Selescu) $ = (Ω  V)  (j  H) - the vector product of two accelerations (elementary
forces), multiplied by cρ , the first one being similar to the Coriolis acceleration and the
second one being given by the Lorentz force, $ representing a new physical quantity.
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 || $,
so that:
dR  (V, Ω) and also dR  (H, j)  dR  (V, Ω)  (H, j) || $ ,
(the $ vector lines) are therefore:
dx dy dz


,
$x $y $ z
(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’s” rigid
surfaces and lying on these ones.
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).
Bulletin of the Transilvania University of Brasov Vol. 13(48) - 2006
351
In the right-hand side the first term is the dissipation heat due to the flow vorticity Ω
=2ω.
The second term is the associated Joule-Lenz heat loss (dt being an elementary time)
with:
~ ~
~ ~ ~
~ ~
j  λE  ρe V  λE; E  E  ( V  H) c ;  j  E  λE2  λ E2  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:
~ ~
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 elementary 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 third 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  1 ρ | j | | E | dt  0 ( j || E ) , then dS ≠ 0.
Let now consider the particular case when the electroconducting fluid is an ideal
gas, e.g. the highly ionized plasma (having an infinite electric conductivity, λ → ∞) - [2],
p. 363, [6], pp. 91 - 92, 94 - 96.
From the non-relativistic form of Ohm’s law for a medium in motion, it results
~
that the intensity of the electric field, E , 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
second 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 magneto-hydrodynamics) the density of the
induction (displacement) electric current (the term k c  E 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:
352
ρe 
A general method for treating a certain flow of an inviscid fluid
Part two: an extension to some special cases in magneto-plasma dynamics
1
k
k
div E  E   2 (V  H) .
4π
c
c
From this last formula one can see that ρ e is little for usual conditions.
In magneto-gas(hydro)dynamics, one generally neglects in the vector equation of
~
motion the term ρe E  ρe E  ( V  H) c (now just zero), which is small with respect to
1 c  ( j  H) .
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
d H H 
 c  rot E  c    E    ( V  H) , or
       V ,
H  0 ;
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.
~ ~
~
Due to the fact that E  0 rigorously, the associated Joule-Lenz heat losses ( 1 ρ  j  E dt )
are zero for the medium with an infinite electric conductivity (zero electric resistance).
For an adiabatic process dq add
ext  0 , resulting: TdS  (Ω  V)  dR , the increase in
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  ,
2
ρ
c




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 elementary
displacement along the intersection line of the isentropic (D. Bernoulli’s) surface
(containing the stream- and the vortex-lines 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:
γ
p  Kργ , with K  p 0 ρ 0  exp (S0  c) Cv  ,
the equation of motion becoming now:
1
1

1

d V 2    dp , or d V 2    γKρ 2dρ ,
2
ρ
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 3-D flow as a potential 2-D one, introducing a 2-D “velocity
quasi-potential” Φi (M), specific to any “i” isentropic (or “D. Bernoulli’s”) surface on which
these curves are lying.
Bulletin of the Transilvania University of Brasov Vol. 13(48) - 2006
353
But actually, the term ∂V/∂t in the left-hand side of the differential vector
equation of motion can not disappear, because it is originating in the time-dependent term
(H  j)/cρ in the right-hand side of this equation, this fact being the essentials of the
electromagnetic field (H may even have a time variable direction). Taking into account that
on any “i” stream and vortex (V, Ω) surface (even if this is not an isentropic one, due to
the Joule-Lenz heat losses) the velocity vector can be written as Vi = Φi - see the part
one (where, this time, the “velocity quasi-potential” Φi is a function depending not only on
ξ and η, but also on t), after scalarly multiplying this equation by an elementary dR
vector, integrating the term ∂(Φ)/∂tdR , one gets the term ∂Φ/∂t , so that along the
“Selescu’s curves” (these ones ceasing to be rigid, now becoming deformable in time) the
~ ~
first integral for an unsteady (∂V/∂t  0) and non-isentropic (Ω  0; 1 ρ  j  E dt  0)
motion of a barotropic gas in an external magnetic field becomes:

dρ
Φ 1 2
1
C
 V   a 2 (ρ)  i 0 (t)  W 2 (t) 
H 2 (t)  E 2 (t)
t 2
ρ
2
ρ  (t)

.
This first integral is very similar to the “D. Bernoulli-Lagrange” one in the
compressible unsteady isentropic aerogasdynamics, i0 (t) being the flow total (fluidoelectromagnetic) specific enthalpy.
It is to be noticed that the dR elementary displacement being not a real, but a
virtual one, the works of the (Ω  V) elementary force and of the Lorentz one, (H  j)/cρ ,
made with this dR , are also virtual and generally ≠ 0 (being 0 for dR  (V, Ω) and dR 
(H,
j)
only).
If in some points the vectors V, Ω, H and j are coplanar, the Selescu’s vector $
becomes 0 and the Selescu’s curves are locally interrupted - undefined direction in the (V, Ω,
H, j) plane.
3. Conclusions
Taking into account the flow vorticity Ω effects as well as those of the associated
Joule-Lenz heat losses and searching for a steady motion solution (∂V/∂t = 0),
considering a barotropic gas, there are some space curves (Selescu) very similar to the
“Bernoulli’s” rigid surfaces, curves along which the vector equation of motion admits in the
general case a first integral (very similar to the “D. Bernoulli” one in the compressible
steady aerogasdynamics), also making evident a new physical quantity - the Selescu’s
vector: $ = (Ω  V)  (j  H).
In the particular case of a fluid medium having an infinite electric conductivity
and therefore without Joule-Lenz heat (λ → ∞ - the highly ionized plasma), these space
curves also are the isentropic lines of the respective plasma flow, in both cases enabling
the treatment of any 3-D flow as a potential 2-D one, introducing a 2-D “velocity quasipotential” Φi (M), specific to any “i” isentropic surface on which these curves are lying.
~ ~
For an unsteady (∂V/∂t  0) and non-isentropic (Ω  0 and 1 ρ  j  E dt  0) flow,
writing Vi = Φi on any “i” stream and vortex ( V, Ω ) surface, (where, this time, the
“velocity quasi-potential” Φi is a scalar function depending on ξ, η and t), along the
“Selescu’s curves” the first integral of the vector equation of motion of a barotropic gas
in an external magnetic field is slightly different with respect to that for a steady and nonisentropic flow (appearing the additional term ∂Φ/∂t ), being very similar to the “D.
Bernoulli-Lagrange” integral in the compressible unsteady isentropic aerogasdynamics.
354
A general method for treating a certain flow of an inviscid fluid
Part two: an extension to some special cases in magneto-plasma dynamics
This theory is useful especially for the cases when the terms Ω  V and (H 
j)/cρ have comparable sizes, and can find many areas of interesting applications, among
which
we
cite:
1. in the high-speed aerogasdynamics, to study the V, H fields around a
hypersonic aircraft or a rocket subject to a strong magnetic field (with respect to the
Earth’s one) in the ionized surrounding air, due to the high temperatures downstream the
strong (detached) shock waves, these waves now becoming discontinuity surfaces also for
the γ ratio of gas specific heats, this varying as the temperature increases (by jump);
2. in astrophysics, to study the flow around a spacecraft moving in the proximity
of a star (including magnetar - a species of neutron star with an extremely strong magnetic
field) or a gas giant planet, like Jupiter (a magnetic field 14 times stronger than the Earth’s
one).
References
1. Landau, L., Lifshitz, E.: Electrodynamics of Continuous Media, Pergamon Press, 1960;
[also Landau, L., Lifchitz, E.: Physique théorique, tome VIII (Électrodynamique des
milieux continus), ch. VIII (Magnéto-hydrodynamique), pp. 280 - 313, Éditions Mir,
Moscou, 1969.]
2. 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.
3. 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.]
4. Cowling, T. G.: Magnetohydrodynamics, Interscience Publishers, New York, 1957.
5. Dragoş, L.: Magnetofluid Dynamics, ch. 1 (Electromagnetic field theory), pp. 17 - 40,
ch. 2 (Magnetofluid dynamics), pp. 41 - 78, Editura Academiei, Bucureşti – Abacus
Press, Tunbridge Wells, Kent, England, 1975.
6. Dragoş, L.: Magnetodinamica fluidelor, cap. III (Electrodinamica mediilor în mişcare),
pp. 50 - 59, cap. IV (Ecuaţiile de mişcare), pp. 61 - 102, Editura Academiei, Bucureşti, 1969.
7. Dinu, Marina: Teoria rotaţiei sistemelor fluide, cap. IV (Mişcări de rotaţie ale fluidelor
vâscoase conductoare), par. 4.3 (Straturi limită Eckman-Hartmann liniare), subpar.
4.3.1 (Stratul limită format la o placă plană izolată), pp. 433 - 441, Editura ştiinţifică şi
enciclopedică, Bucureşti, 1983.
8. Tauth, T.: secţiunea I (Elemente de fizica plasmei şi aplicaţiile ei). In the volume: “Probleme
actuale de fizică”, cap. 4 (Teoria magnetohidrodinamică a plasmei), subcap. 4.1 (Ecuaţiile
de bază ale magnetohidrodinamicii plasmei), pp. 50 - 52, subcap. 4.2 (Comportarea plasmei
în câmp electromagnetic), pp. 53 - 55, cap. 5 (Aplicaţiile descărcărilor în gaze), subcap. 5.2
(Plasma izotermă, caldă), par. a (Arzătoarele cu plasmă (plasmatronul)) şi par. b (Generatoare
magnetohidro-dinamice (MHD)), pp. 67 - 74, Editura didactică şi pedagogică, Bucureşti, 1969.
9. 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.
Bulletin of the Transilvania University of Brasov Vol. 13(48) - 2006
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Magnetoplasmodinamica), pp. 217 - 221, Editura didactică şi pedagogică, Bucureşti, 1970.
10. Mercheş, I., Burlacu, L.: Mecanică analitică şi a mediilor deformabile, cap. VI
(Mecanica mediilor continue deformabile, autor I. Mercheş), subcap. 5. (Studiul unor
modele de mediu continuu deformabil), par. 5.2 (Fluidul ideal), subpar. Magnetodinamica
fluidului ideal, pp. 228 - 229, Editura didactică şi pedagogică, Bucureşti, 1983.
11. Ionescu-Bujor, Cl.: Introducere în studiul intrinsec al clasei mişcărilor permanente
ale plasmei perfecte într-un cîmp magnetic staţionar. In: Stud. şi cerc. de mec. apl.,
XIV, 3, pp. 537 - 557, 1963.
12. Smith, P.: The Steady Magnetodynamics Flow of Perfectly Conducting Fluids. In: J.
Math. Mech., 12, pp. 505 - 520, 1963.
13. Germain, P.: Introduction à l’étude de l’aéromagnétodynamique. In: Cah. Phys., 13,
103, 1959.
14. Germain, P.: Sur certains écoulements d’un fluide parfaitement conducteur. In: La
Rech. Aéro., 74, pp. 13 - 22, 1960.
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Mod. Phys., 32, pp. 830 - 847, 1960.
16. Kato, Y., Taniuti, T.: Hydromagnetic Plane Steady Flow in Compressible Ionized
Gases. In: Progress Theor. Phys., 21, pp. 606 - 612, 1959.
17. Imai, I.: General Principles of Magneto-Fluid Dynamics. In: Progress Theor. Phys.
Suppl., 24, pp. 1 - 34, 1963.
18. Power, G., Walker, D.: Plane Gasdynamic Flow with Orthogonal Magnetic and
Velocity Field Distribution. In: ZAMP, 16, pp. 803 - 816, 1965.
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25, 235, 1958.
20. Sagdeev, R. Z., Moiseev, S. S., Tur, A. V. and Yanovskii, V. V.: Problems of the Theory
of Strong Turbulence and Topological Solitons. Article no. 6 in the volume: “Nonlinear
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6.4 (Invariant Properties of Hydrodynamic Models and Topological Solitons), subchapter
6.4.3 (Examples of Topological Solitons), pp. 137 - 182, Mir Publishers, Moscow, 1986.
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York, Berlin, 1990.
O metodă generală de tratare a unei curgeri oarecare
a unui fluid nevâscos
Partea a II-a: O extindere la unele cazuri speciale
din magnetoplasmodinamică
Rezumat: Modelele MHD sunt larg folosite în analiza instalaţiilor de topire
magnetică, a plasmelor de prelucrare industrială şi a plasmelor ionosferice/
astrofizice. MHD este o prelungire a dinamicii fluidelor la gazele ionizate, care
include efectele câmpurilor electric şi magnetic. Aşadar, metoda generală din
partea I poate fi extinsă la unele cazuri speciale (dar uzuale) din magnetoplasmodinamică, considerând o curgere adiabatică, dar neizentropică (ţinând seama
de efectele vorticităţii curgerii, precum şi de cele ale pierderilor de căldură JouleLenz asociate) într-un câmp magnetic exterior, obţinând noi cazuri de primă
integrabilitate (similare celor “D. Bernoulli”): a) presupunând un mediu
continuu; b) nefăcând nici o deosebire între intensitatea câmpului magnetic şi
inducţia magnetică a mediului (în sistemul de unităţi gaussian), întrucât pentru
toate fluidele conductoare permeabilitatea magnetică este aproximativ egală cu
356
A general method for treating a certain flow of an inviscid fluid
Part two: an extension to some special cases in magneto-plasma dynamics
1 (v. [1 – 3]); c) admiţând că valoarea conductivităţii electrice a mediului fluid
este egală şi izotropă peste tot şi independentă de intensitatea câmpului magnetic.
Există totdeauna nişte curbe în spaţiu (Şelescu) în lungul cărora ecuaţia
vectorială de mişcare admite o integrală primă în cazul general. În cazul
particular al unui fluid ce are conductivitate electrică infinită (plasmă puternic
ionizată), aceste curbe sunt şi liniile izentropice ale curgerii, în ambele cazuri
permiţând tratarea oricărei curgeri 3-D ca pe una 2-D “cvasi-potenţială”. Ca şi
partea I, această lucrare se ocupă şi de curgerile neizentropice nestaţionare,
dându-se o integrală primă a ecuaţiei de mişcare, în cazul general.
Cuvinte şi fraze cheie: curgerea neizentropică a unui fluid electroconductor
nevâscos barotrop într-un câmp magnetic exterior, curgeri staţionare şi
nestaţionare, forţa Lorentz, ecuaţiile lui Maxwell, forma nerelativistă a legii
lui Ohm pentru un mediu în mişcare, pierderile de căldură Joule-Lenz asociate,
al doilea principiu al termodinamicii, ecuaţia lui Crocco generalizată pentru
magnetogazodinamică, vectorul Şelescu, curbele în spaţiu ale lui Şelescu