International Journal Of Scientific & Engineering Research Volume 3, Issue 7, July-2012 1
ISSN 2229-5518
Angular Hydromagnetic Stability of
Incompressible Dusty Fluid Flow between Two
Rotating Cylinders
(Gurpreet Kaur, Arun Kumar Tomer and Shivdeep Singh Patial)
Abstract: In this Paper we have examined the stability of inviscid, incompressible, dusty fluid between two co-axial rotating cylinders with different angular velocities in the presence of angular magnetic field. We found out the sufficient conditions for stability when DN0< 0 and DN0>0 and observe that the effect of magnetic field is to stabilize. Also we obtained modes for non oscillatory and observe that the oscillatory modes are stable.
Keywords: is density, μ is magnetic permeability, perturbation, τ is Relaxation time, k is Stoke’s resistance coefficient, μ is viscosity of clean fluid, π is pressure, N is number density of dust particle.
—————————— ——————————
The necessary and sufficient condition of stability on physical grounds had been given by Rayleigh in 1916 and later on in 1938, Synge [6] gave the analytical proof of Rayleigh's criterion of stability. The problem of linearized stability of a plane parallel flow of dusty gas had been studied by Saffman [5] in 1962. A number of physical situations associated with the motion of dusty fluid between two rotating cylinders have been discussed by Greenpan (1983), Ungarish [7] (1988) and the stability thereof by Reeta in 1991.
Many research workers have done the work in last few years on the stability problems related to Couette flow in different physical contexts. Liu et.al (2001) examined the stability of an azimuthal base flow to both axisymmetric and plane-polar distributions for an electrically conducting fluid confined between stationary, concentric, infinitely long circular cylinder. An electric potential difference exists between the two cylinder walls and drives a radial electric current and without a magnetic field , this flow remains stationary. However, if an axial magnetic field is applied, the interaction between the radial electric current and the magnetic field give rise to an azimuthal electromagnetic body force which drives an azimuthal velocity. Infinitesimal axisymmetric disturbances lead to the instability in the base flow.
Author Gurpreet Kaur is currently pursuing PHD in Mathematics from
Dravidian University Kuppam (A.P), India,
E –mail: dazy_782@yahoo.co.in
Infinitesimal plane-polar disturbances do not appear to destabilize the base flow until shear flow transition to turbulence.
The effect of a rotating magnetic field on the stability of a fluid contained in a cylindrical column and heated from below was investigated by Volz and Mazurug [8] (1999). Dris [1] (1998) examined both experimentally and theoretically the flow of an elastic fluid between eccentric cylinders. They investigated the way the characteristics of the base flow ultimately influence the flow stability.
In this paper, we wish to examine the stability of inviscid, incompressible, dusty fluid between two co-axial rotating cylinders with different angular velocities in the presence of a magnetic field which is applied in the angular direction.
Following assumptions have been made to simplify the equations of motion:-
(a) The gap between the cylinders is small as compared to their radii.
(b) The perturbations are axisymmetric in nature. (c) Relaxation time is small.
(d) Number density depends upon r
(e) Velocity of sedimentation is negligible.
The initial state of which we wish to examine the stability or instability is given by
Co-Author Shivdeep Singh Patial is currently pursuing PHD in
Mathematics from Dravidian University Kuppam (A.P), India, PH-
09814853959. E-mail: shivdeeppatial@gmail.com,shivdeeppatial@yahoo.co.in
Co-Author Arun Kumar Tomer is currently working as Associate Prof. in
SMDRSD College, Pathankot,India.
IJSER © 2012
International Journal Of Scientific & Engineering Research Volume 3, Issue 7, July-2012 2
ISSN 2229-5518
v
mN v
U U v
U
v
u , u
uz
0, U(r), 0
0 t
r r
r
r r
r θ,
h
K *N
u
v 1 H
h H
(8)
v , v , v
0 ,U(r) 0,
0
0 r
r r
r θ z
(1)
v
U v
0 z z
H , H
r θ
,H z
0, H(const ant), 0
t
r
h
K *N
u v 1 H z
(9)
ρ ρ
0
r , and π π0 r
v
0 z z
v v
v
0 r
r r 1
z 0
provided that the equations
r r
r z
N
(10)
N v 0 U N
U 2 H2
-
and
g r U 2 H
hold, where
t r r
0
r
(11)
r r
r mN
0 0
h h
r r
h
h
z 0
0 stands for magnetic permeability.
r r
r z
(12)
Let
, , N , u ,u , u , v , v , v and h , h , h
denote
h h u
r r r
r z
r z
r z
t r
0
θ r θ
(13)
the perturbation in density, pressure, number density of
dust particles, velocity of clear fluid, velocity of dust
h
U h
H u
h U U h H u 0
particle and the magnetic field respectively. Then the
t r
r
r r r r r r
(14)
linearized hydromagnetic perturbation equations of the
h
z
h
z
u
z
fluid particle layer are:
t r
0
r
(15)
u
0 r
u
r
Uu
2
U 2
Where k 6a is the stoke’s resistance coefficient, a is the radius of dust particles, is pressure and is viscosity of
t
r
r
r
h
2Hh
clean fluid.
To examine the stability of the stationary solution (1), we
k*N
v u 1 H
r
(2)
consider the infinitesimal axisymmetric perturbation of the
r
u
0 r r
U U u
0 r
U
r
form
u
u
0 t
r r
r
r r
f rexpipt kz
(16)
h
where k is real wave number in the axial direction and p is
1 k*N
0
v
u 1 H
h H
r
(3)
complex frequency. Further if the fluid motion takes place
r
0 r
r
between two coaxial vertical cylinders at r = R1
and R2 (>
u
U u
R1), then µr must vanish on the walls of the cylinders. The
z z
0 t
r
h
boundary conditions of the problem are thus given by
R and R
k*N
v u 1 H z
(4)
µr = 0 at r = 1 2
z 0 z
z 0 r
after simplifying, we get a dispersion relation
u u
r r
1 u
u
z 0
(5)
pDDu
u
k r u
r r
r z
r
r p r
u 0 U 0
t r r
r
(6)
2
k DN g
2 2
- k 2 p1 0 N u
A u 0
(17)
v
U v U
DN
2 r 2 r
mN r
r 2
v
p p i 0 g p pr
0 t
r
r
h
Hh
0 N
0
mN g K *N
u v 1 H
r 2
(7)
2 2 D 2
0 r r
r r
wh ere
2 H k
, N2 0 U
0
A 00
0 r
IJSER © 2012
International Journal Of Scientific & Engineering Research Volume 3, Issue 7, July-2012 3
ISSN 2229-5518
d
Clearly, condition (19 a) is satisfied if either
and
r 2 dr r is the Rayleigh discriminant.
(a)
r > 0; recovery of Rayleigh’s criterion for
(17) is to be considered along with the boundary DN
conditions or
2
ur 0 at
r R
1
and
R R
(b) If r < 0 then for r + N
or
(r ) >0
22
(c) If r + N 2 (r ) < 0 then for r + N 2 (r) A > 0
k 2r 2
condition (b) clearly ensures the stability of the system
even when r <0 and thus a stabilizing role of N 2 .
Multiplying (17) by
ru
r
and neglecting over the range
Condition (a) ensures the stability of the system even when
of r and taking the imaginary part and then we get
r + N 2 (r ) < 0 in the presence of a magnetic field when
p I I
I I
I 0
(18)
2 22
i 1 2 3 4 5
the condition r + N
(r) A > 0
where
I r
1
du u
r r
dr r
2
dr,
k 2r 2
This establishes a stabilizing role of magnetic field.
I2 k2
r ur
2
dr,
Equation (18) can also written as
2
2 N
r
22 2
A u
p I I
i 1 2
k 2 r
I3 k
r 2
p
p 2 k 2
p 2 r 2 r
dr,
22
p2 p2 kg DN u 2
N (r )
A
r
i 0
r d I
k 2r 2
DN
2
p 2 5
rkDN
g u 2
p2 p 0 g
I k 2 0 r dr
r i N
0
4 DN
0
p i 0 g
0 N
0
If the condition
(20)
and
rk DN
2 g 2 u 2
and
DN > 0 (21)
I k 2 0 r dr
5 2
DN
N00 p i
0 g
N0
2 2
p2 p2 kg DN
N 2 (r )
A r i 0 0
Clearly
I , I2 and I5
are positive definite integrals and
k 2r 2
DN
2
p2 p 0
I3 an d I 4 are also positive definite if
r
i N
g
0
φ r N 2
2Ω 2
A 0
and
DN 0
(19a,b)
0
(22)
k 2 0
r
hold in the range of r then
i 0
necessarily which
Then under the above conditions, it follows from (18)
that pi 0 , implying thereby, the stability of the system. First condition of equation (19) is quite important in nature and it illustrates the role of various physical quantities.
ensures the stability of the system. Since for DN0> 0, we have
IJSER © 2012
International Journal Of Scientific & Engineering Research Volume 3, Issue 7, July-2012 4
ISSN 2229-5518
p2 p2
r i 1
DN
The existence of oscillatory modes is not ruled out. In fact,
p2 p 0 g
r i N
0
therefore; condition (22) can be replaced by a stronger condition, namely
if either any one of conditions (24 a, b) or both are not
satisfied, then oscillatory modes may exist. If this is the case, then we prove below the stability of such oscillatory modes.
Equation (17) can now be rewritten as
2 2
kg DN
N 2 (r)
A 0 0
du u
2 2
p 2 D r r k 2 r u r
k r
0
R
r R .
(23)
dr
r
everywhere in the range of r i.e. 1 2
kDN
g
2 2 2
- k 2 p2 1 0 N u
A u
(25)
Thus for DN0> 0, the system is stable if the modified
DN
2 r 2 r 0
Rayleigh discriminant (23) is positive in the region
p p i 0 g p r
R r R
1 2
0 N0
Multiplying this equation by
ru and integrating over the
range of r and then the integrating part is
p p I
k 2I
Multiplying (17) by
ru and integrating over the range of
r
r 2 i 1
2 0
r. The real part of the equation is
Where
2
du u
p I I
I I
0
1 r r
D r
k r r
dr,
r 1 2 3 4
I
dr r
2
u
2 u 2
where
I r
1
du u
r r
dr r
2
dr,
and
kDN
2 g 2 2 2
I 0 r u
2 DN
0
dr,
I2 k
r ur
dr,
0N0 p i N g
0
I3 k2
r
r N 2
22 u
A
2 2
2
2 dr
Since
I1 and
I 2 are definite positive integrals and
p
pr 0
k r
p
(for oscillatory modes); it follows that
i 0.
and
kDN
g u 2
Thus the oscillatory modes, if exist, are stable.
I k 2 0 r dr
4 2
DN
I Gurpreet Kaur wish to thank Arun Kumar Tomer and
Shivdeep Singh Patial. This work was supported in part by
p i p 0 g
0 r i N
0
a grant from Arun Kumar Tomer and Shivdeep Singh
Patial.
since
I1, I2
are positive definite integrals and the
integrals I3 and
I4 are also positive definite if
[1] Dris, I and Shaqfeh,E.S.G(1998). The flow of an elastic fluid
r N 2
2 2
A 0
2 2
and
DN 0 0
(24 a,b)
between eccentric cylinders. J. non-newtonian fluid mech.,80(1),59).
k r
hold in the range of the integration, it follows that
p r = 0 so that the modes are non-oscillatory.
[2] Greenspan,H.P.(1983). The motion of dusty fluid between two rotating cylinders. J.fluid mech, 127, 91.
[3] Liu.L. and Talmage, G. (2001).J.fluid engg.,121,31.
[4] Rayleigh. (1916). On convection currents in a horizontal layer of
IJSER © 2012
International Journal Of Scientific & Engineering Research Volume 3, Issue 7, July-2012 5
ISSN 222S-5518
fluid when temperature is on the underside, Phil.Mag.32, 529.
[5) Saffman, P.G (1962). Linearized stabihty of a plane parallel flow of dusty gas. J.fluid mech., 13,120.
(6) Synge,J.L(1938). Proc.Roy.Soc.A,167,250
(7) Ungarish,M.(1988). Int.J.multiphase flow, 14,729.
(8) Volz, M.P. and Mazurug, K.(1999). Int. J.Heat Mass Transfer,
42(16), 1037.
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