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T
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p
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n
tr
ic
a
n
n
u
l
u
s
,
w
h
ile
S
h
eik
h
o
le
s
la
m
i
an
d
Go
r
j
i
-
B
an
d
p
y
[
1
5
]
ex
a
m
i
n
ed
th
e
n
u
m
er
ical
s
o
l
u
tio
n
f
o
r
f
r
ee
co
n
v
ec
tio
n
o
f
f
er
r
o
f
l
u
id
i
n
a
ca
v
i
t
y
h
ea
ted
f
r
o
m
b
elo
w
i
n
t
h
e
p
r
esen
ce
o
f
ex
ter
n
al
m
ag
n
etic
f
ie
ld
.
Sh
ei
k
h
o
le
s
la
m
i
e
t
al.
[
1
6
]
in
v
esti
g
ated
MH
D
n
atu
r
al
co
n
v
ec
ti
o
n
o
f
n
an
o
f
l
u
id
in
a
co
n
ce
n
tr
ic
an
n
u
l
u
s
b
et
w
ee
n
a
co
ld
o
u
ter
s
q
u
ar
e
cy
l
in
d
er
an
d
a
h
ea
ted
in
n
er
cir
cu
lar
.
I
n
o
th
er
w
o
r
k
,
Sin
g
h
et
al.
[
1
7
]
s
tu
d
ied
n
at
u
r
al
co
n
v
ec
tio
n
i
n
v
er
t
ical
co
n
ce
n
tr
ic
a
n
n
u
lu
s
u
n
d
er
a
r
ad
ia
l
m
ag
n
etic
f
ield
,
w
h
e
r
e
th
e
y
o
b
s
er
v
ed
th
at
b
o
th
v
elo
cit
y
a
n
d
te
m
p
er
atu
r
e
ar
e
m
o
r
e
in
ca
s
e
o
f
is
o
th
er
m
al
co
n
d
itio
n
co
m
p
ar
e
d
w
it
h
co
n
s
ta
n
t
h
ea
t
f
lu
x
ca
s
e
w
h
en
g
ap
b
et
w
ee
n
c
y
lin
d
er
s
is
less
o
r
eq
u
al
to
r
ad
iu
s
o
f
in
n
er
c
y
li
n
d
er
.
Nir
m
al
et
al.
[
1
8
]
s
tu
d
ied
an
e
x
ac
t
s
o
lu
tio
n
f
o
r
u
n
s
tead
y
m
a
g
n
eto
h
y
d
r
o
d
y
n
a
m
ic
f
r
ee
co
n
v
ec
tio
n
f
lo
w
w
i
th
co
n
s
ta
n
t
h
ea
t
f
l
u
x
an
d
co
n
cl
u
d
ed
t
h
at
t
h
e
m
a
g
n
etic
f
ield
h
a
s
a
r
etar
d
in
g
ef
f
ec
t
o
n
th
e
v
elo
cit
y
w
h
i
le
th
e
s
k
i
n
-
f
r
i
ctio
n
at
th
e
p
late
in
cr
ea
s
es
w
it
h
it.
Fu
r
t
h
er
m
o
r
e,
J
h
a
et
al.
[
1
9
]
c
o
n
s
id
er
ed
f
u
ll
y
d
ev
elo
p
ed
MH
D
n
at
u
r
al
co
n
v
ec
tio
n
f
lo
w
in
a
v
er
tical
m
icr
o
ch
an
n
el
w
i
th
th
e
e
f
f
ec
t o
f
th
e
t
r
an
s
v
er
s
e
m
ag
n
etic
f
ield
in
th
e
p
r
ese
n
ce
o
f
v
e
lo
cit
y
s
lip
a
n
d
te
m
p
er
atu
r
e
j
u
m
p
at
th
e
an
n
u
lar
m
icr
o
-
ch
a
n
n
el,
th
e
y
estab
li
s
h
ed
t
h
at
in
cr
ea
s
e
i
n
cu
r
v
atu
r
e
r
ad
iu
s
le
ad
s
to
an
in
cr
ea
s
e
i
n
th
e
m
as
s
f
lo
w
r
ate.
I
n
a
r
e
l
a
t
e
d
a
r
t
i
c
l
e
,
A
b
b
a
s
e
t
a
l
.
[
2
0
]
i
n
v
e
s
t
i
g
a
t
e
d
a
p
p
l
i
c
a
t
i
o
n
o
f
d
r
u
g
d
e
l
i
v
e
r
y
i
n
m
a
g
n
e
t
o
h
y
d
r
o
d
y
n
a
m
i
c
s
p
er
is
taltic
b
lo
o
d
f
lo
w
o
f
n
an
o
f
l
u
id
in
a
n
o
n
-
u
n
if
o
r
m
c
h
an
n
el,
w
h
i
le
B
h
atti
et
al.
[
2
1
]
co
n
s
id
er
e
d
co
m
b
i
n
ed
ef
f
ec
t o
f
m
a
g
n
eto
h
y
d
r
o
d
y
n
a
m
ics an
d
p
ar
tial s
lip
o
n
p
er
is
talt
ic
b
lo
o
d
f
lo
w
o
f
R
ee
-
E
y
r
i
n
g
with
w
al
l p
r
o
p
er
ties
.
An
a
n
al
y
s
i
s
to
i
n
v
e
s
ti
g
ate
t
h
e
co
m
b
in
ed
e
f
f
ec
ts
o
f
h
ea
t
an
d
m
a
s
s
tr
an
s
f
er
o
n
f
r
ee
c
o
n
v
ec
tio
n
u
n
s
tead
y
m
ag
n
eto
h
y
d
r
o
d
y
n
a
m
ic
s
(
MH
D)
f
lo
w
o
f
th
e
v
i
s
co
u
s
f
lu
id
em
b
ed
d
ed
in
a
p
o
r
o
u
s
m
ed
iu
m
w
a
s
later
p
r
ese
n
ted
b
y
A
li
et
al.
[
2
2
]
.
Mo
s
t
r
ec
en
tl
y
,
t
w
o
-
d
i
m
e
n
s
io
n
al
m
ag
n
eto
h
y
d
r
o
d
y
n
a
m
ic
f
lo
w
o
f
a
v
is
co
u
s
f
l
u
id
o
v
er
a
co
n
s
tan
t
w
ed
g
e
i
m
m
er
s
ed
in
a
p
o
r
o
u
s
m
ed
iu
m
w
as
s
t
u
d
i
ed
b
y
Ku
d
en
att
i
et
al.
[
2
3
]
,
th
e
y
f
o
u
n
d
th
at
MH
D
ef
f
ec
ts
o
n
th
e
b
o
u
n
d
ar
y
la
y
er
ar
e
ex
ac
tl
y
t
h
e
s
a
m
e
a
s
t
h
e
p
o
r
o
u
s
m
ed
i
u
m
in
w
h
ic
h
b
o
th
r
e
d
u
ce
th
e
b
o
u
n
d
ar
y
la
y
er
th
ick
n
es
s
.
T
h
is
p
ap
er
is
d
ev
o
ted
to
in
v
e
s
ti
g
ate
t
h
e
r
o
le
o
f
m
a
g
n
etic
f
ield
o
n
f
u
ll
y
d
ev
elo
p
ed
n
atu
r
al
co
n
v
ec
tio
n
f
lo
w
o
f
a
n
i
n
co
m
p
r
ess
ib
le
an
d
elec
tr
icall
y
co
n
d
u
ctin
g
f
l
u
id
f
i
lled
b
et
w
ee
n
t
w
o
v
er
tical
co
ax
ia
l
c
y
li
n
d
er
s
,
w
h
e
n
th
e
t
w
o
c
y
li
n
d
er
s
ar
e
s
u
b
j
ec
ted
to
co
n
s
tan
t
h
ea
ti
n
g
.
2.
RE
S
E
ARCH
M
E
T
H
O
D
C
o
n
s
id
er
a
tr
an
s
ien
t
la
m
i
n
ar
f
u
ll
y
d
e
v
elo
p
ed
n
atu
r
al
co
n
v
ec
tio
n
f
lo
w
o
f
a
n
in
co
m
p
r
ess
ib
l
e,
v
is
co
u
s
an
d
elec
tr
icall
y
co
n
d
u
cti
n
g
f
l
u
id
i
n
an
an
n
u
lu
s
o
f
i
n
f
i
n
ite
l
en
g
t
h
u
n
d
er
th
e
in
f
l
u
en
ce
o
f
t
r
an
s
v
er
s
e
m
ag
n
eti
c
f
ield
.
T
h
e
z
-
ax
is
is
tak
e
n
alo
n
g
t
h
e
ax
is
o
f
t
h
e
c
y
li
n
d
er
in
th
e
v
er
ticall
y
u
p
w
ar
d
d
ir
ec
tio
n
an
d
′
-
ax
i
s
is
i
n
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2252
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8792
I
n
t J
A
p
p
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o
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er
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n
g
,
Vo
l.
9
,
No
.
1
,
A
p
r
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0
2
0
:
5
8
–
66
60
th
e
r
ad
ial
d
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A
m
a
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ie
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ir
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p
er
p
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lar
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ter
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r
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o
f
th
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n
er
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y
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d
er
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d
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n
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er
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r
f
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e
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u
ter
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er
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h
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>
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as
p
r
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ted
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n
Fi
g
u
r
e
1
.
T
h
e
f
lo
w
is
ass
u
m
ed
to
b
e
f
u
ll
y
d
ev
elo
p
ed
b
o
th
th
er
m
all
y
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d
h
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d
r
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d
y
n
a
m
icall
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,
an
d
th
e
v
is
co
u
s
d
is
s
ip
atio
n
,
r
ad
iatio
n
,
an
d
co
m
p
r
ess
ib
ilit
y
ef
f
ec
t
s
ar
e
n
eg
lecte
d
.
Fo
llo
w
in
g
t
h
e
wo
r
k
o
f
J
h
a
et
al.
[
2
4
]
th
e
m
o
m
en
tu
m
an
d
e
n
er
g
y
eq
u
atio
n
s
g
o
v
er
n
in
g
t
h
e
p
r
esen
t p
h
y
s
ical
s
it
u
atio
n
i
s
g
i
v
e
n
b
y
(
1
)
an
d
(
2
)
.
Fig
u
r
e
1
.
Sch
e
m
atic
d
iag
r
a
m
o
f
th
e
p
r
o
b
le
m
′
′
=
[
2
′
′
2
+
1
′
′
′
]
−
0
2
′
+
(
′
−
0
)
(
1
)
′
′
=
[
2
′
′
2
+
1
′
′
′
]
(
2
)
T
h
e
r
elev
an
t d
i
m
e
n
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io
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al
i
n
iti
al
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d
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o
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ar
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n
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itio
n
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ar
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=
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o
r
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{
′
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2
(
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)
I
n
tr
o
d
u
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g
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e
f
o
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w
i
n
g
d
i
m
en
s
io
n
les
s
q
u
an
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es i
n
(
1
)
an
d
(
2
)
.
=
′
1
2
,
=
′
1
,
=
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E
q
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s
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d
(
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)
in
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o
llo
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s
:
=
2
2
+
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−
2
+
(
5
)
=
2
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+
1
(
6
)
T
h
e
in
itial a
n
d
b
o
u
n
d
ar
y
co
n
d
itio
n
s
i
n
d
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r
e:
≤
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r
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≤
(
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)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
A
p
p
l P
o
w
er
E
n
g
I
SS
N:
2252
-
8792
Tr
a
n
s
ien
t d
ev
elo
p
men
t o
f MHD
n
a
tu
r
a
l c
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n
tr
ic
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n
n
u
l
u
s
(
B
a
s
a
n
t K.
Jh
a
)
61
>
0
{
=
0
,
=
1
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=
1
=
0
,
=
1
at
=
(
8
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T
h
e
p
h
y
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ical
q
u
an
titi
es
u
s
ed
in
(
1
)
to
(
6
)
ar
e
d
ef
in
ed
in
th
e
n
o
m
e
n
clat
u
r
e
.
T
h
e
s
o
lu
tio
n
o
f
(
5
)
an
d
(
6
)
w
it
h
t
h
e
ass
o
ciate
d
in
itial
a
n
d
b
o
u
n
d
ar
y
co
n
d
itio
n
s
(
7
)
an
d
(
8
)
ca
n
b
e
o
b
tain
ed
b
y
u
s
i
n
g
t
h
e
L
ap
lace
tr
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s
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o
r
m
tec
h
n
iq
u
e.
Def
in
i
n
g
th
e
f
o
llo
w
i
n
g
tr
an
s
f
o
r
m
v
ar
ia
b
les
.
̅
(
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(
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2252
-
8792
I
n
t J
A
p
p
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9
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3
T
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.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
A
p
p
l P
o
w
er
E
n
g
I
SS
N:
2252
-
8792
Tr
a
n
s
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t d
ev
elo
p
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t o
f MHD
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(
B
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s
a
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t K.
Jh
a
)
63
T
ab
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1
.
Nu
m
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.
RE
F
E
R
E
NC
E
S
[1
]
P
.
Ch
a
n
d
ra
n
,
N
.
S
a
c
h
e
ti
,
a
n
d
A
.
K.
S
in
g
h
,
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A
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if
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ti
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f
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e
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o
n
v
e
c
ti
o
n
f
lo
w
,
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c
ien
ti
a
e
M
a
t
h
e
ma
ti
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a
e
J
a
p
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n
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a
e
,
v
o
l.
5
3
,
n
o
.
3
,
p
p
.
4
6
7
-
4
7
6
,
2
0
0
1
.
[2
]
D.
T
.
S
w
i
f
t
-
Ho
o
k
a
n
d
J.
K.
W
rig
h
t,
“
T
h
e
c
o
n
sta
n
t
-
M
a
c
h
-
n
u
m
b
e
r
M
HD
g
e
n
e
ra
to
r,
”
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o
u
rn
a
l
o
f
F
l
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id
M
e
c
h
a
n
ics
,
v
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l.
1
5
,
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o
.
1
,
p
p
.
9
7
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1
0
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9
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.
[3
]
A
.
A
.
Ka
u
fm
a
n
a
n
d
G
.
V
.
Ke
ll
e
r,
“
T
h
e
M
a
g
n
e
to
tellu
ric S
o
u
n
d
in
g
M
e
th
o
d
,
”
El
se
v
ier
,
Ne
w
Yo
rk
,
N
Y,
USA
,
1
9
8
1
.
[4
]
Y.
A
o
k
i,
T
.
S
e
id
o
u
,
a
n
d
N.
Oh
t
o
m
o
,
“
M
e
a
su
re
m
e
n
t
o
f
th
e
ti
m
e
d
e
p
e
n
d
e
n
t
tem
p
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ra
tu
re
v
a
riatio
n
o
f
c
o
m
b
u
stio
n
M
HD
p
las
m
a
,
”
J
a
p
a
n
e
se
J
o
u
rn
a
l
o
f
A
p
p
l
ied
P
h
y
sic
s
,
v
o
l.
2
3
,
n
o
.
1
2
,
p
p
.
1
6
2
8
-
1
6
3
3
,
1
9
8
4
.
[5
]
J.
E.
B
o
ro
v
sk
y
,
R.
C.
El
p
h
ic,
H.
O.
F
u
n
ste
n
,
a
n
d
M
.
F
.
T
h
o
m
se
n
,
“
T
h
e
Earth
’s
p
las
m
a
sh
e
e
t
a
s
a
la
b
o
ra
t
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ry
f
o
r
f
lo
w
tu
rb
u
len
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in
h
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h
-
β M
HD
M
HD
,
”
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o
u
rn
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l
o
f
Pl
a
sm
a
P
h
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sic
s
,
v
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l.
5
7
,
n
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.
1
,
p
p
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1
-
3
4
,
1
9
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.
[6
]
D.
M
.
Bu
sh
n
e
ll
a
n
d
C.
B.
M
c
G
in
ley
,
“
T
u
rb
u
len
c
e
c
o
n
tr
o
l
in
w
a
ll
f
lo
ws
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A
n
n
u
a
l
Re
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ie
w
o
f
F
lu
id
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e
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h
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,
v
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l.
2
1
,
p
p
.
1
–
2
0
,
1
9
8
9
.
[7
]
T
.
Ha
y
a
t
a
n
d
Z.
A
b
b
a
s,
“
He
a
t
tran
sf
e
r
a
n
a
l
y
sis
o
n
th
e
M
HD
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3
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:
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.
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D.
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,
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a
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B
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rb
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m
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h
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s
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m
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re
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ti
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tern
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ti
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a
l
jo
u
rn
a
ls.
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re
se
a
rc
h
in
tere
st
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s
f
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ro
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p
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ia,
m
a
g
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to
h
y
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ro
d
y
n
a
m
ics
,
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o
m
p
u
tatio
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a
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f
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id
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a
m
ics
,
a
n
d
h
e
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t
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m
a
s
s
tran
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r.
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s
u
f
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a
m
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l
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iw
o
w
a
s
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o
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in
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o
y
i,
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g
i
S
tate
,
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e
ria.
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is
a
lec
tu
re
r
a
n
d
c
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r
re
n
tl
y
a
P
h
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stu
d
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n
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De
p
a
rtm
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n
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th
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m
a
ti
c
s
A
h
m
a
d
u
Be
ll
o
Un
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rsity
(
A
BU),
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,
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e
ria.
He
h
a
d
h
is
B.
S
c
.
(
f
irst
c
las
s
h
o
n
o
u
rs
)
in
th
e
De
p
a
rtm
e
n
t
o
f
M
a
th
e
m
a
ti
c
s,
A
b
u
Zaria
(2
0
1
0
).
His
a
re
a
s
o
f
in
tere
st
in
c
l
u
d
e
f
lo
w
th
ro
u
g
h
p
o
r
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u
s
m
e
d
ia,
c
o
m
p
u
tatio
n
a
l
f
lu
id
d
y
n
a
m
ics
,
a
s we
ll
a
s h
e
a
t
a
n
d
m
a
ss
tran
s
f
e
r
.
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