I nternat. J. Math. & Math. Sci.
Vol. 5 No. 2 (1982) 385-392
385
SOME GEOMETRIC PROPERTIES OF MAGNETO-FLUID FLOWS
$.S.
GANGWAR and RAM BABU
Department of Mathematics
Faculty of Science
Banaras Hindu University
Varanasi 221005, INDIA
(Received November i, 1979)
ABSTRACT.
By employing an anholonomic description of the governing equations,
certain geometric results are obtained for a class of non-dissipatlve magneto-
fluid flows.
The stream lines are geodesics on a normal congruence of the sur-
faces which are the Maxwellian surfaces.
KEY WORDS AND PHRASES. Maxw Surface, Geodi, Developa.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES;
i.
76W05.
INTRODUCTION.
In recent years in order to study the kinematic and kinetic properties of
fluid flows, the attention of research investigators has been directed toward
the application of differential geometry to fluid dynamics [I]
[9].
The
characterization of a three-dimensional vector field in terms of anholonomic
coordinates was first introduced by Vranceanu
Marris and Passman
dynamics.
[5], [?
i0] and has been employed by
to describe some kinematic properties in hydro-
G. Purushotham and his group
6],
9] have also used anholonomic
geometric results in their study of fluid flows.
S.S. GANGWAR AND R. BABU
386
This paper presents a kinematic study of the steady and non-dissipative
magneto-fluid flows with the use of anholonomic geometric results.
It has been
shown that if at each point of the flow region the magnetic field line is in the
plane of the stream line and its binormal, there exists a normal congruence of
the surfaces which are the Maxwellian surfaces.
The stream lines are the geo-
desics and their binormals are parallels on these surfaces.
The speed and the
fluid pressure are constant along the binormals of the stream lines.
The
Bernoulli equation in MHD has also been derived in a special case.
2.
THE BASIC EQUATIONS
The basic equations governing the steady and non-dissipative magneto-fluid
ill are as follows:
flows given by Lundquist
(),j
PvJ
(v H
-*v(H2)’i’
(2.2)
(2.3)
0,
Q,j
j
vHj Hi,j
vi, j + P’i
j
v
(2.1)
0,
i
j
H
vi),j
(2.4)
0,
and
H
where
density,
,3
(2.5)
0,
is the constant magnetic permeability, p the fluid pressure, p
Q
the entropy of the fluid,
the fluid
@
the velocity field vector, and H
in
(2.2), (2.4), and (2.5), we get
j
the
magnetic field vector.
j
Using the substitution h
j
/H
respectively
pv3 v.
1,3
j
(v h
i
h
and
h
j
,3
0,
,
+
p i
j
v
i)
h
j
h.
(2.6)
(2.7)
GEOMETRIC PROPERTIES OF MAGNETO-FLUID FLOWS
where p*
+-h2
p
is the total pressure and h
2
i
h
h..
387
Equations (2.7) and
(2.8) imply
j
v
+
hi, j
J
h
hJ
v,j
i
O.
v.
1,3
(2.9)
ANHOLONOMIC GEOMETRIC RESULTS
3.
In this section we are going to state the results given by Marris and
Passman
If we consider the orthonormal basis {t
5].
the stream lines, where t
i
n
i
and b
i
i, n i, b i}
introduced along
are the unit vectors along the tangents,
the principal normals, and the binormals of the stream lines, we have the results
t
t
t..
k n.
ni, j
=-k t
j
(3.1)
i
+ bi
(3.2)
and
t
which are the Serret
torsion of t
J-lines.
rives along
n3-1ines
k,
, t’ n’
j
bi, j ---n i
Frenet formulae, where k and
j,
b3-1+/-nes
and
b,j, 8nt
i
nj
n
t,j
j
n
n
j
t
nJ
b
i
i
,j
bJ ti,j
j
n
,3
given by, in ters of the eight parameters
i
8
i
( + n
t
nt
i
t)t
( +
n
(b j
t
t
b
i
bi’
,j
)b
i
i
(3.4)
(3.5)
,3
( + n
( +
bj
8bt
ni’
’j
ent n
,3
b
are the curvature and
The latter results are augmented by the Intrinsic deriva-
J
n
(3.3)
n
)ni
+
i
+ (b j,j )n i
b t bi
(3.7)
)t
+ (k +
i
(k + nj .)n I
n
(3 6)
j)b
i
(3.8)
and
b
where
t
and
n
j
b
i
,j
8
bt
t
are abnormalities of
3
tJ-lines
and
nJ-lines
(3 9)
respectively given by
S.S. GANGWAR AND R. BABU
388
at
t
eijk(tk),j
i
(3.10)
and
n
n
e
i
ijk
(nk)
(3.11)
,j
It should be noted that here we employ the parameter
as employed by lrris and Passan;
instead of
bJ-lines,
the abnormality of
b
On
Ob-n
is expressed
in terms of other parameters by the relation
b
z-2 +
t-
0
(3.12)
n
The further results are
e
e
e
ijk
ijk
ijk
(tk),j
at
t
i
+ k
j
i
(r),j
(b
(bk),j
(k + nj
3)t,
,j
b
i
(3.13)
+ n
)ti
n
i
0bt
+
n
8
i
b
nt
+
i
fib
(3 14)
b
i
(3.15)
and
t3,j 0nt + 0bt
4.
(3.16)
INTRINSIC PROPERTIES OF THE FLOW.
Now we will derive some kinematic properties of magneto-fluid flows employing
the results of earlier section.
governing the flow.
First we will decompose the basic equations
j
j
The velocity vector v and magnetic field vector h can be expressed as
q t
j
(4.1)
and
=ht
t
j
+
h n
n
j
+
b
where q is the the magnitude of the velocity; and
of magnetic field vectors along t
j, nj,
b
j
j
(4.2)
ht,
are the components
respectively.
Using (4.1), (4.2) and the geometric results (3.1)
basic equations
hn,
(3.9), (3.16) in the
(2.1), (2.6), (2.8) and (2.9), we get the following decomposition
tJ (n0q),j + 0nt + 0bt
0
(4.3)
389
GEOMETRIC PROPERTIES OF MAGNETO-FLUID FLOWS
j
pq t q
+
j
j
t
p*.
2
h
0
n
+
pq
b
2
h
n
n
j
h
j
+
h
+
h
b
n
n
n
p*
j
+
h
b
h
k
h
n,j
+
h
(k+n
+
b
j
j)
j)
j
(4.4a)
ht (n
h
t
t
j
+ 2)
h
n,j
h
(4.4b)
n ,j’
ft
(k + n
n
(2 + 2n -ft
ht, j
0nt
b
(fin
bj
hb
+ hn h t
2
j
,j
h h
n t
,j
+
t,j
ht2
,3
+ hn
h k
n
t
2 0bt + ht tj ht,
nt
+ nj p*.
k
h
,3
h
+
h
2
t
t
b3 j
n
j
+
,j
ht ebt
+
h
n
n
j
,j
(4.4c)
t
j
h
n
+
ht, j
nJ
q
n
j
+
,j
h
n
hn,j
bj
hb
t
+ bj
j
,j
q t
q,j
q,j +
q
tJ
+
j
ht (0nt + 0bt) + hn nJ, j + hb bj,j
ht, j + q k hn q h t (ent + 0bt)
h
n,j
+
q
f
n
+
h
q(2 + f
h q
n
9b t"
0,
(4.5)
0,
(4.6a)
(4 6b)
0
and
tJ
q,j +
qtJ ,j + hb q 0nt
+
n
n
t
0
(4.6c)
Let us assume that the component of magnetic field vector in the direction
of the principal normal of the stream line vanishes at each point of the flow
region.
(4.6c) are reduced respectively to
Then the equations (4.4a)
tj
oq
pq
b
j
2
k
q
+
p*.
,3
t
j
ht,j
2
+ tj p,,j
J
p*.
,3
h
h
b
+
b
j
t
ht2
bt
,j
h
k
+ ht
+
h
t
j
t(0nt
+
0bt
t(fn
tj
t
ht, j
hb
0bt)
+
bj
bJ ht,j
+
2
+ 2
+
hb,j
+
h
3,j
,j
(4.7b)
(4.7c)
,j,
b
(k + n j
(4.7a)
0
(4.8)
S.S. GANGWAR AND R. BABU
390
h
b
b
j
q t
q,j
q
j
q h
ht, j
(ent
t
+
0bt)
0
(4.9a)
0
n
(4 9b)
and
t
j
q,j +
q t
j
,j
+
q
0.
0nt
(4.9c)
(4.9c) can be written as
t
j
(n q
),j
+
),j
+ bj
nt
0,
if
0
if
# 0.
(4.10)
# 0
(4.11)
(4.8) and (4.9a) imply
b
If q
j
(n q
h
b
0, the velocity vanishes; that means the fluid is stationary. [Since
the fluid is not stationary, q
# 0].
0
Therefore, (4.9b) implies
if
(4.12)
# 0"
that is, the abnormality of the principal normal vanishes which is the necessary
and sufficient condition for the existence of normal congruence of the surfaces
containing
b
j,
tJ-lines
and
bJ-lines.
Since the vector h
the magnetic field lines lie on these surfaces.
gruence of the surfaces is nothing but the
family
j
is in the plane of t
and
Hence, the normal conof the Maxwellian surfaces.
Since the principal normals of the stream lines are the normals of these
surfaces
Thus we have the theorem:
the stream lines are the geodesics.
THEOREM i.
j
If at each point of the flow region the component of the magnetic
field vector in the direction of the principal normal of the stream line
vanishes
there exists a normal congruence of the surfaces containing the stream lines and
their binormals which are the geodesics and parallels
respectively.
These
surfaces are the Maxwellian surfaces.
If the magnetic field lines coincide with the binormals of the stream lines
and the density does not vary along the stream lines, the equations (4.3)
(4.6c) will be put in the forms
tJ
q,j +
q
(nt
+
bt
0
(4.13)
GEOMETRIC PROPERTIES OF MAGNETO-FLUID FLOWS
tJ
0q
b
2
k
0q
2
(k +
eb t
nj
0,
0
,j
(4 14b)
b
j
0
,j
(4.15)
0,
q,j
0
(4.14a)
(4.14c)
+
h
j
n
h2
h2
0,
p,j
n
+ tj P*. +
,3
),j
+ nj p*. +
j
j
b
b
(1/2
391
(4.16a)
if
h# 0
(4.16b)
and
h t
j
q,j +
q t
j
h,j
+
h
8bt
h q
ent
(4.16c)
0.
Using (4.13) in (4.16c), we get
t
j
h,j
(4.17)
0.
From (4.14c) and (4.16a) we have the theorem:
THEOREM 2.
If the magnetic field lines coincide with the binormals of the
stream lines, the fluid pressure and magnitude of the velocity are constant a-
long the binormals.
Let us take the additional assumption that the magnetic intensity is constant along the stream lines, i.e.
t
j
8bt
that is the geodesic curvature of
h
,3
0, then (4.17) implies
(4.18)
0,
bJ-lines
vanishes.
Thus we see that the mag-
netic ield lines are the geodesics on the normal congruence of the
surfaces,
and hence we get the theorem:
THEOREM 3.
If the fluid density and the magnetic intensity do not vary
along the stream lines, the
congruence of developables.
family
of the Maxwellian surfaces is the normal
The stream lines and the magnetic field lines are
orthogonal geodesics on these developables.
Employing (4.18) in (4.14a), we get
392
S.S. GANGWAR AND R. BABU
-
that is
where
tJ
(1/2
0
q2
(1/2
q2
+
p
j
+ tj P*-
+ 1/2 h 2)
0
0,
is the intrinsic derivative along the stream lines.
we get
1/2
pq2
+ P + 1/2h 2
Integrating (4.19),
constant along the stream lines.
(4.20)
This is the Bernoulli equation.
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Truesdell, C. Intrinsic Equations for Spatial Gas Flows, Zeits. Angew.
Math. Mech., 40 (1960), 9-14.
[2]
Kanwal, R.P. Intrinsic Relations in Magnetohydrodynamics, Zeits. Angew.
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Suryanarayan, E.R.
On the Geometry of Stream Lines in Hydromagnetic Fluid
Flows When the Magnetic Field is Along a Fixed Direction, Proc. Amer.
Math. Soc., 16 (1965), 90-96.
4]
Wasserman, R.H. On the Geometry of Magnetohydrodynamic Flows, Quart. J.
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8]
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Ill]
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