Synthesis of bicyclic salicylates by [3+3] cyclization of 1,3-bis(silyl enol ethers) with cyclic 3-(silyloxy)alk-2-en-1-ones

Article abstract of DOI:10.1515/znb-2009-0616

Bicyclic salicylates were prepared by [3+3] cyclization of 1,3-bis(silyl enol ethers) with cyclic 3-(silyloxy)alk-2-en-1-ones.

Full text of DOI:10.1515/znb-2009-0616

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
effect of radiative processes on the gravitational insta-
bility in a static medium.
Jeans [1] considered the gravitational instability of
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
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,
several researchers have studied the velocity shear in-
stability in hydrodynamicsand plasmas under different
assumptions. Benjamin and Bridges [15] have shown
that the velocity shear instability problem in hydrody-
namics admits a canonical Hamiltonian formulation.
Allah [16] has examined the effects of heat and mass
transfer on the instability of streams. Luo et al. [17]
have investigated the effect of negatively charged dust
on the parallel velocity shear instability in a magne-
tized plasma. More recently Bhatia and Sharma [18]
have studied the effects of surface tension and perme-
ability of a porous medium on the stability of super-
posed viscous conducting streams. In all these studies
the streams are not gravitating and are under the action
of gravity.
up for perturbations of the wave number k less than
√
Jeans [1] wave number k j = Gρ , where ρ is the den-
c
sity, c the velocity of sound in the gas and G the grav-
itational constant. Chandrasekhar [2] examined the ef-
fects of an uniform magnetic field and uniform rotation
on the gravitational instability of the static medium
and found that both rotations and the magnetic field
inhibit the contraction and fragmentation of the in-
terstellar clouds. Since then several researchers have
studied this problem under varying assumptions. Tas-
soul [3] has investigated the gravitational instability
of a thermally conducting fluid, while Gliddon [4]
has studied this problem for an anisotropic plasma.
Mouschovias [5, 6] and Mestel and Paris [7] have
pointed out the importance of the gravitational insta-
bility of a self-gravitating static homogeneous plasma
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.
c
0932–0784 / 05 / 0700–0484 $ 06.00 ꢀ 2005 Verlag der Zeitschrift fu¨r Naturforschung, Tu¨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-
tion of the magnetic field we assume that all perturbed
quantities have the space and time dependence of the
form
2. Perturbation Equations
We consider two semi-infinitely ideally conducting
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, ρ1 and ρ2, and move with uniform speeds, V1
and V2. A uniform magnetic field is applied to the sys-
tem in the direction of the x-axis. The whole system
rotates about the y-axis with a small uniform angular
velocity Ω.
F(z)exp(ikxx+nt),
(7)
where F(z) is some function of z, kx is the wave num-
ber of the perturbation along the x-axis, and n (may be
complex) is the rate at which the system departs away
from equilibrium. The streaming velocity is also taken
ꢀ
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
dz
writing D ≡
:
ꢀ
ꢁ
ρsσsus = −ikxδ ps +ρsikxδφs −2ρswsΩ,
ρsσsvs = Hikx(hy)s,
(8)
(9)
∂
∂t
ꢀ
ρs
us +(Vs )ꢀus = − δ ps +ρs δφs
(1)
ꢀ
ꢀ
ꢀ
+2ρs(ꢀus ×Ω)+( ×hs)×H,
ρsσsws = −Dδ ps +ρsDδφs
(10)
∂
∂t
+2ρsusΩ −H[D(hx)s −ikx(hz)s],
ꢀ
δρs +(Vs )δρs +ρs( ꢀus) = 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(ꢀus ×H),
ꢀ
hs = 0,
(4)
(5)
(D2 −kx2)δφs = −Gδρs,
σsδ ps = C2σsδρs,
2δφs = −Gδρs,
ꢀ
ꢁ
∂
∂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
to the upper region z > 0 and s = 2 to the lower region
σs = n+ikxVs.
(16)
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
(D2 −kx2)(D2 −Ns2)δφs = 0,
(17)
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
above, we take here the horizontal magnetic field along
where
(Cs2kx2 +σs2 −Gρs)(σs2 +Ms2kx2)+4Ω2σs2
σs2(σs2 +Ms2)+Ms2Cs2kx2
ꢀ
Ns2 =
. (18)
the x-axis i. e. H = (H,0,0), and streams rotating about
an axis in the horizontal plane perpendicular to the di-
ꢀ
rection of the magnetic field, i. e. Ω = (0,Ω,0). We
2
H
ρ
Here Ms2 =
is the Alfven velocity. Equation (17)
investigate the stability problem for the two cases of
s
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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P. K. Bhatia and R. P. Mathur · Stability of Rotating Gravitating Streams
4. Solution of the Differential Equations
5. The Dispersion Relation
Now we seek the solutions of (17) which remain
bounded in the two regions. The appropriate solutions
for the two regions are therefore
For a non-trivial solution, the determinant of the ma-
trix of the coefficients of A1, A2, B1, and B2 in (21)
to (24) must vanish. This gives the dispersion relation
in the general form. Since the expressions for the Qi’s
and Ti’s are complex and quite complicated, an explicit
expression for the critical wave number k∗ (= kx) can-
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
δφ1 = A1e−k z +B1e−N z(z > 0)
(19)
x
1
and
δφ2 = A2ek z +B2eN z(z < 0),
(20)
x
2
where A1, A2, B1 and B2 are constants of integration. In
writing the solutions (19) and (20) for δφ it is assumed
that N1 and N2 are so defined that their real parts are
positive. The four boundary conditions to be satisfied
at the interface z = 0 are:
(i) Continuity of the perturbed gravitational poten-
tial, i. e. δφ1 = δφ2.
ρ1 = ρ2 = ρ, M12 = M22 = M2,
(27)
C12 = C22 = C2, V1 = V, V2 = −V.
(ii) Continuity of the normal derivative of the per-
turbed gravitational potential, i. e. D(δφ1) = D(δφ2).
(iii) Continuity of the total perturbed pressure, i. e.
δ p1 +H(hx)1 = δ p2 +H(hx)2.
The expressions for Q1 to Q4 and T1 to T4 are then
considerably simplified. Using the values of V1 and V2
given by (27) in (16), we find that σ12 = σ22 = σ2 (when
n = 0) and then N1 = N2 = N. For the above simple
configuration the dispersion relation becomes N = 0,
i. e.
(iv) The normal displacement at any point (fluid el-
w
w
σ
1
2
ement) is unique at z = 0, i. e.
=
.
σ
1
These conditions, on applying the2solutions (19)–
(20), lead to the four equations
(C2kx2 −Gρ +σ2)(σ2 +M2kx2)+4Ω2σ2 = 0. (28)
A1 +B1 −A2 −B2 = 0,
(21)
Now, using the value of σ 2 = −kx2V2 (when n = 0)
in (28), we find that the configuration of rotating grav-
itating streams is unstable for all wave numbers kx less
than the critical wave number kB∗ , where
kxA1 +N1B1 +kxA2 +N2B2 = 0,
Q1A1 +Q2B1 −Q3A2 −Q4B2 = 0,
T1A1 +T2B1 −T3A2 −T4B2 = 0,
(22)
(23)
(24)
ꢂ
where
GρM2 −GρV2 +4Ω2V2
kB∗ =
.
(29)
Q1 = ρ1σ12α12 +kx2ρ1(M12kx2 +σ12)
−(M12kx2 +σ12 +4Ω2)(C12kx2 +σ12)(α12 −kx2)/G (25)
+2ikxα1M12Ω[ρ1kx2 +(C12kx2 +σ12)(α12 −kx2)/G],
(C2 −V2)(M2 −V2)
When V = 0, i. e. when the streaming velocity van-
ishes, we obtain Jeans’ criterion.
T1 = ρ2σ2(C12kx2 +σ12)(σ2α1 +2ikxΩ)(α12 −kx2)/G
6. Discussion
+ρ1ρ2kx2(σ12α1 +2ikxΩ).
(26)
From (29) we see that in the present case the critical
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
The coefficient Q2 is obtained from Q1 by replacing
α1 by N1, Q3 is obtained from Q1 by replacing α1 by
α2, changing i to −i and interchanging the subscripts 1
and 2, and Q4 is obtained from Q3 by replacing α2 by
N2. Similaraly T2 to T4 are obtained from T1. Here the
values of α1 and α2 are the same in the two streams
and equal to kx, i. e. α1 = α2 = kx [see (17)].
ꢃ
Gρ
C2 −V2
ks∗ =
,
(30)
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P. K. Bhatia and R. P. Mathur · Stability of Rotating Gravitating Streams
487
Clearly ks∗ > kj. The streaming velocity has desta-
bilizing influence as it renders the wave number range
kj < k < ks∗ unstable. There is no effect of the magnetic
field in this case.
σsδ ps = Cs2σsδρs,
(40)
where in this case
σs = n+ikyVs.
(41)
When M = 0, i. e. when there is no magnetic field,
the critical wave number kΩ∗ is given by
Elimination of the variables leads to the differential
equation
ꢂ
Gρ −4Ω2
kΩ∗
=
.
(31)
(D2 −ky2)(D2 −Js2)δφs = 0,
(42)
C2 −V2
where
Rotation has stabilizing influence on the instability
of the configuration as the wave number range kΩ∗
<
(4Ω2 +σs2)(σs2 +Ms2ky2 +Cs2ky2 −Gρs)
σs2(Cs2 +Ms2)
Js2 =
. (43)
k < ks∗ is stabilized by rotation.
Considering now (29) we find that two cases can be
distinguished:
The solutions of the differential equation (42) for the
(a) M2 < V2, i. e. when the Alfven velocity is two regions are therefore
smaller than the streaming velocity, the effect of ro-
δφ1 = A1e−k z +B1e−J z (z > 0)
(44)
y
1
tation is stabilizing as ks∗ decreases on increasing Ω.
The magnetic field also has a stabilizing influence in
this case as kB∗ increases on increasing M.
and
(b) M2 > V2, both rotation and the magnetic field
have a destabilizing influence as kB∗ in this case in-
creases on increasing Ω or M, and in this case kB∗ > ks∗.
δφ2 = A2ek z +B2e−J z (z < 0).
(45)
y
2
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
Q1 = Gρ1α12 −[Gρ1ky2(4Ω2 +σ12)/σ12]
For propagation perpendicular to the magnetic field,
we assume that the perturbed quantities depend on the
space coordinates and time as
− (α12 −ky2)(4Ω2 +σ12)
(46)
· [σ12 +(M12 +C12)ky2]/σ12,
T1 = (Gα1ky2/σ12)
+α1(α12 −ky2)[σ12 +(M12 +C12)ky2]/ρ1σ12.
Again the coefficients Q2 to Q4 and T2 to T4 follow
from Q1 and T1, respectively, in exactly the same way
as for the other mode.
F(z)exp(ikyy+nt),
(32)
(47)
where F(z) and n are as explained above and ky is the
wave number of perturbation along the y-axis. Here we
ꢀ
take Vs = (0,Vs,0), i. e. the streaming velocity is along
the direction of propagation.
For the perturbations of the form (32), (1) to (6) give
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
(35)
(4Ω2 +σ2)(σ2 +M2ky2 +C2ky2 −Gρ) = 0. (48)
Using σ2 = −ky2V2 (when n = 0), in (48) we find that
the critical wave number k∗, below which the system is
unstable, is given by
+2ρsusΩ −HD(hx)s,
σsδρs = −ρs(ikyvs +Dws),
σs(hx,hy,hz)s = [−H(ikyvs +Dws),0,0], (37)
(36)
iky(hy)s +D(hz)s = 0,
(D2 −ky2)δφs = −Gδρs,
(38)
(39)
ꢃ
Gρ
k∗ = ky =
.
(49)
M2 +C2 −V2
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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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