Aromatization of Hantzsch 1,4-dihydropyridines with urea-hydrogen peroxide/maleic anhydride

Article abstract of DOI:10.1515/znb-2006-0314

A simple method for the oxidative aromatization of Hantzsch 1,4-dihydropyridines to the corresponding pyridines is reported using urea-hydrogen peroxide/maleic anhydride in acetonitrile.

Full text of DOI:10.1515/znb-2006-0314

Stability of Rotating Gravitating Streams
P. K. Bhatia and R. P. Mathur
Department of Mathematics and Statistics, Jai Narian Vyas University, Jodhpur, India
Reprint requests to Prof. P. K. B.; E-mail: pkbhatiamb@yahoo.com
Z. Naturforsch. 60a, 484 – 488 (2005); received October 25, 2004
This paper treats the stability of two superposed gravitating streams rotating about the axis trans-
verse to the horizontal magnetic field. The critical wave number for instability is found to be affected
by rotation for propagation perpendicular to the axis about which the system rotates. The critical
wave number for instability is not affected by rotation when waves propagate along the axis of rota-
tion. The critical wave number is affected by both the magnetic field and the streaming velocity in
both cases. Both the magnetic field and the rotation are stabilizing, while the streaming velocity is
destabilizing.
Key words: Stability; Gravitating Streams; Rotation; Magnetic Field.
1
. Introduction
Jeans [1] considered the gravitational instability of
effect of radiative processes on the gravitational insta-
bility in a static medium.
Singh and Khare [13] investigated the instability
of superposed gravitating streams in an uniform hor-
izontal magnetic field and rotation about the vertical.
Shrivastava and Vaghela [14] have studied the mag-
netogravitational instability of an interstellar medium
with variable streams and radiation. In recent years,
an infinite homogeneous self-gravitating medium in
context of the formation of astronomical bodies by the
fragmentation of interstellar matter. He has derived a
criterion that the medium becomes unstable and breaks
up for perturbations of the wave number k less than
√
Gρ
Jeans [1] wave number k j =
, where ρ is the den- several researchers have studied the velocity shear in-
c
sity, c the velocity of sound in the gas and G the grav- stability in hydrodynamicsand plasmas under different
itational constant. Chandrasekhar [2] examined the ef- assumptions. Benjamin and Bridges [15] have shown
fects of an uniform magnetic field and uniform rotation that the velocity shear instability problem in hydrody-
on the gravitational instability of the static medium namics admits a canonical Hamiltonian formulation.
and found that both rotations and the magnetic field Allah [16] has examined the effects of heat and mass
inhibit the contraction and fragmentation of the in- transfer on the instability of streams. Luo et al. [17]
terstellar clouds. Since then several researchers have have investigated the effect of negatively charged dust
studied this problem under varying assumptions. Tas- on the parallel velocity shear instability in a magne-
soul [3] has investigated the gravitational instability tized plasma. More recently Bhatia and Sharma [18]
of a thermally conducting fluid, while Gliddon [4] have studied the effects of surface tension and perme-
has studied this problem for an anisotropic plasma. ability of a porous medium on the stability of super-
Mouschovias [5, 6] and Mestel and Paris [7] have posed viscous conducting streams. In all these studies
pointed out the importance of the gravitational insta- the streams are not gravitating and are under the action
bility of a self-gravitating static homogeneous plasma of gravity.
in the context of fragmentation and collapse in mag-
In astrophysical situations the instability of the grav-
netic molecular clouds. Sengar [8, 9] and Radwan and itating rotating streams in a horizontal magnetic field
Elazab [10] examined the effect of variable streams on would be interesting when the system rotates about
the gravitational instability. Sorker and Sarazin [11] an axis perpendicular to the direction of the magnetic
have demonstrated the relevance of this problem in field, in the horizontal plane. This aspect forms the sub-
gravitational plasma filaments in cooling flows in clus- ject matter of this paper where we study the instability
ters and galaxies. Vranjes and Cadez [12] studied the of inviscid infinitely conducting gravitating streams.
0
932–0784 / 05 / 0700–0484 $ 06.00 ꢀc 2005 Verlag der Zeitschrift f u¨ r Naturforschung, T u¨ bingen · http://znaturforsch.com
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P. K. Bhatia and R. P. Mathur · Stability of Rotating Gravitating Streams
485
3
. The Governing Differential Equation for
Propagation along the Magnetic Field
We study the cases of propagation along and perpen-
dicular to the direction of the magnetic field.
For this mode of wave propagation along the direc-
2
. Perturbation Equations
tion of the magnetic field we assume that all perturbed
quantities have the space and time dependence of the
We consider two semi-infinitely ideally conducting form
homogeneous gravitating streaming fluids occupying
the regions z > 0 and z < 0 and separated by a plane
interface at z = 0. The streams possess uniform den-
sities, ρ and ρ , and move with uniform speeds, V
F(z)exp(ikxx+nt),
(7)
where F(z) is some function of z, kx is the wave num-
and V . A uniform magnetic field is applied to the sys- ber of the perturbation along the x-axis, and n (may be
1
2
1
2
tem in the direction of the x-axis. The whole system complex) is the rate at which the system departs away
rotates about the y-axis with a small uniform angular from equilibrium. The streaming velocity is also taken
ꢀ
velocity Ω.
along the x-axis in this mode, i. e. V = (V,0,0). Then
for the perturbations of the form (7), (1) to (6) give, on
The linearized perturbation equations relevant to the
problem are
d
writing D ≡
:
dz
ꢀ
ꢁ
ρsσsus = −ikxδ ps +ρsikxδφs −2ρswsΩ,
ρsσsvs = Hikx(hy)s,
(8)
(9)
∂
∂t
ꢀ
ρs
us +(Vs ) ꢀu s = − δ ps +ρs δφs
(
1)
ꢀ
ꢀ
ꢀ
+
2ρs( ꢀu s ×Ω)+( ×hs)×H,
ρsσsws = −Dδ ps +ρsDδφs
(
10)
∂
∂t
+2ρsusΩ −H[D(hx)s −ikx(hz)s],
ꢀ
δρs +(Vs )δρs +ρs( ꢀu s) = 0,
(2)
(3)
σsδρs = −ρs(ikyus +Dws),
σs(hx,hy,hz)s = [Dws,ikxvs,ikxws],
ikx(hx)s +D(hz)s = 0,
(11)
(12)
(13)
(14)
(15)
∂
∂t
ꢀ
ꢀ
ꢀ
ꢀ
hs +(Vs )hs = curl( ꢀu s ×H),
ꢀ
hs = 0,
(4)
(5)
2
2
2
(D −kx)δφs = −Gδρs,
δφs = −Gδρs,
2
ꢀ
ꢁ
σsδ ps = C σsδρs,
∂
∂t
∂
∂t
δρs +(Vs )δρs =Cs2
δρs +(Vs )δρs . (6)
ꢀ
ꢀ
where
The equations are same for both streams. The subscript
s’ distinguishes the two streams, s = 1 corresponding
σs = n+ikxVs.
(16)
‘
to the upper region z > 0 and s = 2 to the lower region
Eliminating the various quantities from the above
equations, we finally get the fourth order differential
equation in δφs
ꢀ
z < 0. In the above equations h = (hx,hy,hz), δφ, δ p
and δρ are the perturbations, respectively, in the mag-
ꢀ
netic field H, the gravitational potential φ, the pressure
2
2
x
2
2
s
p and density ρ due to a small disturbance of the sys-
tem which produces the velocity field ꢀu = (u,v,w) in
the system. Here C is the velocity of sound. As stated where
above, we take here the horizontal magnetic field along
(D −k )(D −N )δφ = 0,
(17)
s
2
2
2
2
2 2
2 2
ꢀ
2
s
(Cs kx +σs −Gρs)(σs +Ms kx)+4Ω σs
N = . (18)
2 2 2 2 2 2
the x-axis i. e. H = (H,0,0), and streams rotating about
σ (σ +M )+M C k
s x
an axis in the horizontal plane perpendicular to the di-
s
s
s
s
ꢀ
rection of the magnetic field, i. e. Ω = (0,Ω,0). We
2
investigate the stability problem for the two cases of Here Ms = H is the Alfven velocity. Equation (17)
2
ρs
propagation along and perpendicular to the axis about holds for both streams and must be solved subject to
which the streams rotate.
the appropriate boundary conditions.
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4
86
. Solution of the Differential Equations
P. K. Bhatia and R. P. Mathur · Stability of Rotating Gravitating Streams
4
5. The Dispersion Relation
Now we seek the solutions of (17) which remain
bounded in the two regions. The appropriate solutions trix of the coefficients of A , A , B , and B in (21)
For a non-trivial solution, the determinant of the ma-
1
2
1
2
for the two regions are therefore
to (24) must vanish. This gives the dispersion relation
in the general form. Since the expressions for the Qi’s
−
kxz
−N1z
+B1e
δφ = A e
(z > 0)
(19)
(20)
1
1
and T ’s are complex and quite complicated, an explicit
i
∗
expression for the critical wave number k (= kx) can-
and
not be obtained easily analytically. In order to get an
insight into the tendencies of the actual situations, we
consider now the case of two gravitating streams of the
same uniform densities, flowing past each other with
the same velocity in opposite directions, and with the
same magnetic field and the same velocity of sound in
the two streams. The same model has been considered
by Singh and Khare [13], we therefore set
δφ = A e +B2eN2z(z < 0),
kxz
2
2
where A , A , B and B are constants of integration. In
writing the solutions (19) and (20) for δφ it is assumed
that N and N are so defined that their real parts are
positive. The four boundary conditions to be satisfied
at the interface z = 0 are:
1
2
1
2
1
2
(
i) Continuity of the perturbed gravitational poten-
2
2
2
ρ = ρ = ρ, M = M = M ,
1
2
2
1
2
tial, i. e. δφ = δφ .
1
2
(27)
2
2
C = C = C , V1 = V, V2 = −V.
(ii) Continuity of the normal derivative of the per-
1
2
turbed gravitational potential, i. e. D(δφ ) = D(δφ ).
1
2
The expressions for Q to Q and T to T are then
1
4
1
4
(
iii) Continuity of the total perturbed pressure, i. e.
considerably simplified. Using the values of V and V
1
2
δ p +H(hx) = δ p +H(hx) .
1
1
2
2
2
1
2
2
2
given by (27) in (16), we find that σ = σ = σ (when
n = 0) and then N = N = N. For the above simple
(
iv) The normal displacement at any point (fluid el-
w1
w2
σ2
1 2
ement) is unique at z = 0, i. e. σ1
=
.
configuration the dispersion relation becomes N = 0,
i. e.
These conditions, on applying the solutions (19)–
(20), lead to the four equations
2
2
2
2
2 2
2
2
(
C k −Gρ +σ )(σ +M k )+4Ω σ = 0. (28)
A +B −A −B = 0,
(21)
x
x
1
1
2
2
2
2
2
Now, using the value of σ = −k V (when n = 0)
kxA +N B +kxA +N B = 0,
(22)
(23)
(24)
x
1
1
1
2
2 2
in (28), we find that the configuration of rotating grav-
Q A +Q B −Q A −Q B = 0,
1
1
2
1
3
2
4
2
itating streams is unstable for all wave numbers k less
than the critical wave number k , where
x
∗
T A +T B −T A −T B = 0,
B
1
1
2
1
3
2
4
2
ꢂ
where
2
2
2 2
GρM −GρV +4Ω V
∗
k =
.
(29)
2
2
2
2
2
2
1
2
1
2
1
B
2
2
2
2
Q = ρ σ α +k ρ (M k +σ )
(C −V )(M −V )
1
1
1
1
x
1
2
1
x
2
2
2
1
2
1
2 2
1
2
x
2
2
−
+
(M k +σ +4Ω )(C k +σ )(α −k )/G (25)
1
x
1
x
When V = 0, i. e. when the streaming velocity van-
ishes, we obtain Jeans’ criterion.
2
2
2
1
2
x
2ikxα M Ω[ρ k +(C k +σ )(α −k )/G],
1
1
1 x
x
2
2
2
2
1
2
x
T = ρ σ (C k +σ )(σ α +2ikxΩ)(α −k )/G
1
2
2
1
2
x
1
2
1
6. Discussion
2
+
ρ ρ k (σ α +2ikxΩ).
(26)
1
2 x
1
1
From (29) we see that in the present case the critical
The coefficient Q is obtained from Q by replacing
2
1
wave number depends on the rotation, magnetic field
and streaming velocity. When Ω = 0, i. e. when there
is no rotation, the critical wave number below which
the configuration is unstable is given by
α by N , Q is obtained from Q by replacing α by
1
1
3
1
1
α , changing i to −i and interchanging the subscripts 1
2
and 2, and Q is obtained from Q by replacing α by
4
3
2
N . Similaraly T to T are obtained from T . Here the
2
2
4
1
ꢃ
values of α and α are the same in the two streams
1
2
Gρ
C −V
∗
k =
,
2
(30)
and equal to kx, i. e. α = α = kx [see (17)].
s
2
1
2
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P. K. Bhatia and R. P. Mathur · Stability of Rotating Gravitating Streams
487
∗
2
Clearly k > kj. The streaming velocity has desta-
σsδ ps = C σsδρs,
(40)
s
s
bilizing influence as it renders the wave number range
kj < k < k unstable. There is no effect of the magnetic
field in this case.
When M = 0, i. e. when there is no magnetic field,
the critical wave number k is given by
where in this case
σs = n+ikyVs.
∗
s
(41)
∗
Elimination of the variables leads to the differential
equation
Ω
ꢂ
2
Gρ −4Ω
∗
2
2
2
2
k
=
.
(31)
(D −k )(D −J )δφs = 0,
(42)
Ω
2
2
y
s
C −V
where
Rotation has stabilizing influence on the instability
of the configuration as the wave number range k
k < k is stabilized by rotation.
∗
2
2
2
2 2
2 2
<
(4Ω +σ )(σ +M k +C k −Gρ )
Ω
2
s
s
s
s
y
s
y
s
∗
J =
. (43)
s
σ (C +M )
s s s
2
2
2
Considering now (29) we find that two cases can be
distinguished:
The solutions of the differential equation (42) for the
2
2
(a) M < V , i. e. when the Alfven velocity is two regions are therefore
smaller than the streaming velocity, the effect of ro-
−
1
kyz
−J1z
∗
δφ = A e
+B1e
(z > 0)
(44)
tation is stabilizing as k decreases on increasing Ω.
1
s
The magnetic field also has a stabilizing influence in
∗
and
this case as k increases on increasing M.
B
2
2
(
b) M > V , both rotation and the magnetic field
have a destabilizing influence as k in this case in-
creases on increasing Ω or M, and in this case k > k .
kyz
−J2z
δφ = A e +B2e
(z < 0).
(45)
2
2
∗
B
∗
∗
B
s
In this mode it is also assumed that J1 and J2 are so
defined that their real parts are positive. The boundary
conditions are the same and lead to the relations (21)
to (24), where in this case ky replaces kx in (22) and
7
. Stability of Streams for Propagation
Perpendicular to the Magnetic Field
2
2
2
2
2
For propagation perpendicular to the magnetic field,
we assume that the perturbed quantities depend on the
space coordinates and time as
Q = Gρ α −[Gρ k (4Ω +σ )/σ ]
1
1
1
1 y
1
1
2
2
2
2
−
(α −k )(4Ω +σ )
(46)
1
y
1
2
2
1
2
1
2
1
2
1
·
[σ +(M +C )k ]/σ ,
y
F(z)exp(ikyy+nt),
(32)
2
2
T = (Gα k /σ )
1
1 y
1
(47)
where F(z) and n are as explained above and ky is the
2
1
2
y
2
1
2
1
2
1
2
y
2
+
α (α −k )[σ +(M +C )k ]/ρ σ .
1
1
1
wave number of perturbation along the y-axis. Here we
ꢀ
take Vs = (0,Vs,0), i. e. the streaming velocity is along Again the coefficients Q2 to Q4 and T2 to T4 follow
the direction of propagation.
For the perturbations of the form (32), (1) to (6) give as for the other mode.
from Q1 and T1, respectively, in exactly the same way
Now in this mode also, we consider the same model
for the streams. Using therefore (27) and proceeding as
for the other mode, we find that the dispersion relation
for the considered mode is J = 0, i. e.
ρsσsus = −2ρswsΩ,
(33)
ρsσsvs = −ikyδ ps +ρsikyδφs −Hiky(hx)s, (34)
ρsσsws = −Dδ ps +ρsDδφs
2
2
2
2
2
y
2 2
y
(35)
(4Ω +σ )(σ +M k +C k −Gρ) = 0. (48)
+
2ρsusΩ −HD(hx)s,
σsδρs = −ρs(ikyvs +Dws),
σs(hx,hy,hz)s = [−H(ikyvs +Dws),0,0], (37)
2
2
2
Using σ = −k V (when n = 0), in (48) we find that
(36)
y
∗
the critical wave number k , below which the system is
unstable, is given by
iky(hy)s +D(hz)s = 0,
(38)
ꢃ
Gρ
M +C −V
∗
2
2
y
k = k =
.
2
(49)
y
(
D −k )δφs = −Gδρs,
(39)
2
2
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88
P. K. Bhatia and R. P. Mathur · Stability of Rotating Gravitating Streams
∗
We observe that k is independent of the Coriolis force. the horizontal plane, the critical wave number, below
∗
The magnetic field has stabilizing influence, as k de- which the system is unstable, is affected by rotation
creases on increasing the magnetic field. The stream- only for the mode of wave propagation perpendicular
∗
ing velocity is destabilizing, as k increases on increas- to the axis about which the system rotates. Rotation
ing V. The results obtained for this mode are the same and the magnetic field suppress the instability while
as for the mode of propagation along the magnetic the streaming velocity has destabilizing influence.
field.
Acknowledgements
P. K. B. is thankful to the University Grants Com-
8
. Conclusion
mission for the award of an Emeritus Fellowship to
We thus conclude, that when the streams rotate him during the tenure of which this work has been
about an axis perpendicular to the magnetic field in done.
[
[
[
[
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