1. International Journal of Advanced JOURNAL OF ADVANCED RESEARCH ISSN 0976 –
INTERNATIONAL Research in Engineering and Technology (IJARET), IN
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME
ENGINEERING AND TECHNOLOGY (IJARET)
ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
Volume 4, Issue 7, November - December 2013, pp. 92-100
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IJARET
©IAEME
SELF-GRAVITATING ELECTRODYNAMIC STABILITY OF
ACCELERATING STREAMING FLUID CYLINDERS IN
SELF-GRAVITATING TENUOUS MEDIUM
Hussain E. Hussain1,2 and Hossam A. Ghany1,3
1
Mathematics Department, Faculty of Science, Taif University, Hawia(888) Taif, Saudi Arabia
2
Mathematics Department, Faculty of Engineering, Ain Shams University, Cairo, Egypt
3
Mathematics Department, Faculty of Industrial Education, Helwan University, Cairo, Egypt
ABSTRACT
The stability of electrodynamic self-gravitational of accelerating streaming fluid cylinder
surrounded by tenuous medium has been studied for all axisymmetric and non-axisymmetric
perturbation modes. The dispersion relation is derived, discussed and some reported works are
recovered as limiting cases from it. The electro-dynamic forces interior and exterior the fluid
cylinder are destabilizing. The self-gravitating forces are destabilizing for small axisymmetric
domain while they are stabilizing in all other domains. The accelerating (periodic) streaming has a
stabilizing tendency. The self-gravitating electrodynamic stable domains are enlarged due to the
resonance of the periodic streaming and could overcome the instability character of the model, if it is
so strong.
1. INTRODUCTION
The electrodynamic stability of cylindrical fluids has been the attention of several scientists
(see Melcher et. al. (1971), and Nayyar et. al. (1970)) since the pioneering works of Kelly (1965).
The effect of the electrodynamic force on the capillary force and other parameters has been
examined by Radwan (1992), and Mestel (1994) & (1996).
In the last decades of 20th century interests have been devoted to the self-gravitating
instability of cylindrical models for their crucial applications. Chandrasekhar and Fermi (1953) were
the first to discuss the stability of the full fluid cylinder under the influence of the self-gravitating
force and did write about its importance with the oscillation of the spiral arm of galaxy, for more
studies see also Chandrasekhar (1981), Radwan (2007) and Radwan & Hussain(2009). Hasan (2011)
has discussed the stability of oscillating streaming fluid cylinder subject to combined effect of the
capillary, self gravitating and electrodynamic forces for all axisymmetric and non axisymmetric
perturbation modes. He (2012) studied the magnetodynamic stability of a fluid jet pervaded by
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2. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME
transverse varying magnetic field while its surrounding tenuous medium is penetrated by uniform
magnetic field.
In the present work we investigate the electrodynamic self-gravitational stability of oscillating
streaming fluid cylinder surrounded by self-gravitating tenuous medium of very slow motion.
2. FORMULATION OF THE PROBLEM
We consider a dielectric fluid cylinder of uniform cross-section of (radius R0 ) permittivity
ε
(1)
(2)
surrounded by tenuous medium of negligible motion of permittivity ε . The fluid model is
assumed to be incompressible, self-gravitating, inviscid and pervaded by uniform electric field
(1),(2)
E0
= (0,0, E0 )
(1)
where E0 is the intensity of the electric field in the fluid. The fluid streams with periodic streaming
at time t
u 0 = (0,0,U cos Ωt )
(2)
where U is streaming velocity at time t = 0 while Ω is the oscillation frequency of the velocity.
(1),(2)
The components of E 0
and u 0 are considered along the cylindrical coordinates (r ,ϕ , z ) with
the z -axis coinciding with the axis of the fluid cylinder. The cylindrical fluid model is acted by the
combined effect of the pressure gradient, electrodynamic and self-gravitating forces.
The required basic equations are the combination of the electro-dynamic Maxwell equations,
ordinary hydro-dynamic equations and Newtonian equation concerning the self-gravitating matter.
For the model under consideration, these equations are given as follows:
1
 ∂u

(1)
(1)
+ ( u.∇ ) u  = −∇p + ρ∇V (1) + ∇ε (1) E .E
2
 ∂t

(
ρ
∇.u = 0
∇ ∧ (ε E )
∇.E
,
(1),(2)
(1),(2)
)
=0
(3)
(4), (5)
=0
(6)
∇ 2V (1) = −4π G ρ (1)
(7)
∇2V (2) = 0
(8)
This system of equations is solved in the unperturbed state with
(1),(2)
E0
u 0 = (0,0,U ) ,
= (0,0, E0 ) , and after applying the required boundary conditions at r = R0 , we get
V0(1) = −π Gρ r 2
(9)
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3. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME
R 
V0(2) = 2π G ρ R02 ln  0  − π G ρ R02
 r 
(10)
z
Ucos t
r
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4. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME
3. PERTURBATION STATE
For small departures from the unperturbed state, every physical quantity
be expressed as
%
Q(r ,ϕ , z; t ) = Q0 (r ) + η (t )Q1 (r ,ϕ , z ) + L
Q(r ,ϕ , z; t ) may
(11)
Q stands for u, p,V (1) ,V (2) , E (1) , E (2) with Q0 (r ) is the value of Q in the unperturbed
state with Q1 is small increment of Q due to perturbation. Based on the expansion (11), the
where
deformation in the cylindrical interface is given by
%
r = R0 + η
(12)
%
η = η (t )exp ( i ( kz + mϕ ) )
(13)
with
is the elevation of the surface wave measured from the unperturbed level.
By using the expansion (11) in the fundamental equations (3)-(8), the relevant perturbation
equations are given as follows
(1)
(1)
ε
(1)
 ∂u

∇ ( 2E 0 .E 1 )
ρ  1 + (u 0 .∇ )u 1  = −∇p1 + ρ∇V 1 +
2
 ∂t

(1),( 2 )
∇.u1 = 0
∇ ∧ ( ε E1 )
∇.E1
,
(1),(2)
=0
(14)
(15), (16)
=0
(17)
(1)
∇ 2V1 = 0
(18)
∇2V1(2) = 0
(19)
Since the fluid is irrotational, we have
u 1 = −∇φ1
(20)
By inserting this in (16), the velocity potential
φ1
satisfies
∇ 2φ1 = 0
(21)
and based on the expansion (11)-(13), the solution of (21) give
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5. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME
(1)
φ1 = A B (t )I m (kr )exp i ( kz + mϕ ) 


(22)
with
u1r =
∂η
+ ikUη cos Ωt
∂t
(23)
From equation (14), we get
∂u
 ∂u 1r
+ U cos Ωt 1r
∂z
 ∂t
ρ
(
∂Π1

=−
∂r

(1)
(1)
π 1 = p1 − ρV1(1) − ε (1) E 0 .E1
(24)
)
(25)
So
ρ
∂  ∂η
∂Π

 ∂η

+ ikU η cos Ωt  + ikU ρ 
+ ikU η cos Ωt  = − 1

∂t  ∂t
∂r

 ∂t

Taking into account
∇2Π1 = 0 ,
so, we have
Π1 = C 1 (t )η (t )I m (kr )exp i ( kz + mϕ ) 


(26)
Hence
∂ 2η
∂η
ρ 2 + 2i ρ kU cos Ωt
− i ρ kUη sin Ωt − ρ k 2U 2 cos 2 Ωt
∂t
∂t
′
= −C1η kI m ( x)
(27)
and
Π1 =
− ρ I m (x )  ∂ 2η
∂η 
2
2
2
 2 + 2ikU cos Ωt
 − ikU η sin Ωt − k U cos Ωt (28)
′
k I m (x )  ∂t
∂t 
(1),(2)
Now, upon using equation (17) concerning E1
(1),(2)
, we have E1
= ∇ψ 1(1),(2) so,
∇2ψ 1(1),(2) = 0 , therefore
(1)
ψ 1 = C2 (t ) I m (kr )exp i ( kz + mϕ ) 


(30)
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6. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME
(2)
ψ 1 = C3 (t ) K m ( kr ) exp i ( kz + mϕ ) 


(31)
We have to apply the appropriate boundary conditions at r = R0 to get C2 (t ) and C3 (t )
:which are
ψ1 =ψ 2
and n0 .ε
(1)
(1)
(1)
(2)
(2)
E1 + n1.ε (1) E 0 = n0 .ε (2) E1 + n1.ε (2) E1
from which, we get
 K ( x) 
C2 (t ) = C3 (t )  m
,
 I m ( x) 
C3 (t ) =
x = kR0
(32)
−iE0 ( ε (1) − ε (2) ) I m ( x)
(ε
(1)
′
′
K m ( x) I m ( x ) − ε (2) K m ( x) I m ( x ) )
(33)
Therefore,
E
(1)
1
 −(ik )(−iE0 ) ( ε (1) − ε (2) ) K m ( x )

= ∇  (1)
I m ( kr )exp ( i (kz + mϕ ) )  (34)
(2)
′
′
 (ε K m ( x ) I m ( x) − ε K m ( x) I m ( x ) )



E
(2)
1
 −(ik )(−iE0 ) ( ε (1) − ε (2) ) I m ( x)

= ∇  (1)
K m ( kr ) exp ( i ( kz + mϕ ) )  (35)
(2)
′
′
 (ε K m ( x ) I m ( x) − ε K m ( x ) I m ( x) )



By solving equations (18) and (19), we get
V1(1) = B1 (t ) I m (kr )exp i(kz + mϕ )
(36)
V1(2) = B2 (t ) Km (kr )exp i(kz + mϕ )
(37)
The coefficients B1 (t ) and B2 (t ) are determined upon using the boundary conditions:



 at r = R0



∂V0(1)
∂V (2)
= V1(2) + η 0
∂r
∂r
(1)
2 (1)
(2)
∂V1
∂ V0
∂V1
∂ 2V0(2)
+η
=
+η
∂r
∂r 2
∂r
∂r 2
V1(1) + η
97
(38)

7. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME
and then upon utilizing the Wronskian relation ( cf.Abramowitz and Stegun)
′
′
I m ( x) Km ( x) − I m ( x) Km ( x) = − x −1
(39)
we get,
V1(1) = 4π Gρ R0η Km ( x) I m (kr )exp i(kz + mϕ )
(40)
V1(2) = 4π Gρ R0η I m ( x) Km (kr )exp i(kz + mϕ )
(41)
Now, we have to apply the continuity of the normal component of the total stress tensor at
r = R0 :
∂
ε (1)
ε (2)
(1)
(1)
(2)
(2)
(1)
Π 0 + ρV 0(1) ) +
2E 0 .E 1 =
2E 0 .E 1
(
∂r
2
2
with noting that Π 0 = const .
(
Π1 + ρV 1(1) + η
)
(
)
(42)
We finally obtain, the second order integro-differential equation
∂ 2η
∂η
= −2ikU cos Ωt
+ ikU Ωη sin Ωt + k 2U 2η cos Ωt
2
∂t
∂t
xI ′ ( x ) 
1
I m ( x ) K m ( x ) − η
+ 4π G ρ m
I m ( x) 
2

(43)
2
 ε (2) 
′
x 1 − (1)  I m ( x ) K m ( x )
 ε 
2
ε (1) E02
−
ρ R02 

ε (2)
′ ( x) K m ( x ) − (1) K m ( x) I m ( x) 
′
 Im
ε


η
4. DISCUSSION
Equation (43) is the second order integro-differential equation in the amplitude
η (t )
of the
perturbation. It relates η with the modified Bessel functions I m ( x) and K m ( x) , the wave numbers
x and m , the permittivity ε (1) and ε (2) of the fluid and tenuous medium and with the parameters
 ε E2 
ρ , R0 , E0 ,V and Ω of the problem. It contains  02 
 ρ R0 
(time)-1.
If we assume that η
yields
−
1
2
as well as ( 4π G ρ )
−
1
2
each of unit
exp(σ t ) where σ is the growth rate of the perturbation, equation (43),
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8. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME
σ 2 = ( −2ikU cos Ωt )σ + ikU Ω sin Ωt + k 2U 2 cos Ωt
+ 4π G ρ
′
xI m ( x ) 
1
 I m ( x) K m ( x) − 2 
I m ( x) 

(44)
2
 ε (2) 
′
x 1 − (1)  I m ( x) K m ( x)
 ε 
2
ε (1) E02
−
ρ R02 

ε (2)
′
′
I m ( x ) K m ( x) − (1) K m ( x ) I m ( x ) 

ε


As t = 0 and the fluid streams uniformly u 0 = (0,0,U ) equation (43) reduces to
(σ + ikU )
2
= 4π G ρ
′
xI m ( x) 
1
 I m ( x) K m ( x) − 2 
I m ( x) 

2
 ε (2) 
′
x 1 − (1)  I m ( x) K m ( x)
 ε 
2
(1)
−
2
0
2
0
ε E
ρR
(45)
ε (2)
′
′
I m ( x) K m ( x ) − (1) K m ( x ) I m ( x)
ε
The relation (45), as U = 0 is derived by Radwan (1992) in examining the effect of the
electric field on the self-gravitating fluid cylinder.
As u 0 = 0 and E 0 = 0 , the relation (45) gives the self-gravitating dispersion relation of a
fluid cylinder surrounded by self-gravitational tenuous medium, viz.,
σ 2 = 4π G ρ
′
xI m ( x) 
1
I m ( x) K m ( x) − 
I m ( x) 
2

(46)
This relation has been early derived by Chandrasekhar (1981).
We know that (cf. Abramowitz and Stegun (1970))
′
′
I m ( x) > 0, K m ( x) > 0, I m ( x) > 0, and K m ( x) < 0
(47)
In view of the inequalities (47), the analytical and numerical discussions of the relation (46)
reveal to the following results.
When m = 0 :
σ2
σ2
> 0 as 0 < x < 1.0667 while
≤ 0 as 1.0667 ≤ x < ∞ .
4πρG
4πρG
When m ≥ 0 : we have 0 < x < ∞ .
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9. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 7, November – December (2013), © IAEME
This means that the model is self-gravitating unstable in the axisymmetric perturbation
m = 0 in the domain 0 < x < 1.0667 . While it is stable in the neighboring axisymmetric domain
1.0667 ≤ x < ∞ and in the non-axisymmetric domain 0 < x < ∞ .
Therefore, we conclude that the fluid cylinder is self-gravitating unstable in small
axisymmetric domain and stable otherwise.
As U = 0 and we neglect the self-gravitating force effect, the relation (45), reduces to
2
 ε (2) 
′
x 1 − (1)  I m ( x) K m ( x)
 ε 
2
ε (1) E02
σ =
ρ R02 ε (2) ′
′
K ( x) I m ( x) − I m ( x ) K m ( x )
ε (1) m
2
(48)
The discussions of the relation (48), in view of the inequalities (47) show that the
electrodynamic forces acting on the cylindrical fluid and the surrounding tenuous medium have
strong stabilizing influence.
As G = 0 and E0 = 0 , the general relation (43), shows that the periodic oscillating stream
has stabilizing tendency. However, the uniform stream is destabilizing. Therefore, in our present case
the electrodynamic self-gravitational stable domains are enlarged and could suppress the unstable
domains and consequently the model will be completely stable as the oscillating stream of the fluid is
so strong.
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