 Open Access
 Total Downloads : 485
 Authors : Mrs. G.Soudjada, Dr.S.Subbulakshmi
 Paper ID : IJERTV1IS7503
 Volume & Issue : Volume 01, Issue 07 (September 2012)
 Published (First Online): 26092012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Effect of Magnetic Field on Thermal Instability of Nonnewtonian Fluids in a Rotating Medium
Mrs. G.Soudjada1 Dr.S.Subbulakshmi2
1Assistant Professor, Department of Mathematics, Arignar Anna Government Arts & Science College, Karaikal, Puducherry, India.
2Assistant Professor, Department of Mathematics, D.G.G.A College(W), Mayiladuthurai, Nagapattinam Dt, Tamil Nadu.
Abstract
Effect of magnetic field on the thermal instability of Walters B Viscoelastic fluid in a rotating medium is considered. For stationary convection ,Walters B viscoelastic fluid behaves like a Newtonian fluid. The rotation and magnetic field have a stabilizing effect where as the suspended particles has destabilizing effect on the system. Numerical computations are made and illustrated graphically.
Key Words : Magnetic field , Thermal instability, Viscoelasticity, Suspended particle , Rotating medium.
1.Introduction
The effect of rotation and magnetic field on thermal instability of Non Newtonian fluid is considered. The
importance of NonNewtonian fluids in modern technology and industries is ever increasing and the investigations on such fluids are desirable. One such class of Non Newtonian fluids is WaltersB fluid. Here
,the effect of suspended particles, rotation and magnetic field on the Walters B viscoelastic fluid heated from below is considered.
A detailed account of the theoretical and experimental results of the onset of thermal instability in a fluid layer under varying assumptions of hydrodynamics has been given by Chandrasekhar(1981). Bhatia and Steiner(1972) have studied the problem of thermal instability of a Maxwellian viscoelastic fluid in the presence of rotation . Sharma and Aggarwal(2006) have studied the effect compressibility and suspended particles on thermal convection in a Walters B elastic viscous fluid in hydromagnetics . Bhatia and Steiner(1973) analyzed the thermal instability of a Maxwellian viscoelastic fluid in the presence of a magnetic field . Aggarwal and Prakash(2009) have discussed the effect of suspended particles and rotation on thermal instability of ferromagnetic fluids .Sharma (1975) studied the stability of a layer of an electrically conducting Oldroydian fluid in the presence of a magnetic field
.Sharma(1977) analyzed the thermal instability in compressible fluid in the presence of rotation and a magnetic fluid. Sharma(1999) et al have considered the
thermosolutal instability of Walters B rotating in porous medium . Bhatia and Steiner(1973) have studied the problem of thermal instability in a viscoelastic fluid layer in Hydromagnetics
2.Mathematical Formulation
We consider an infinite horizontal layer of electrically conducting WaltersB elasticviscous fluid layer of thickness d permeated with suspended particles, bounded by the planes z=0 and z=d in the presence of rotation. This layer is heated from below so that , the temperature and density at the bottom surface z=0 are and at the upper surface z= d are respectively and that
a uniform adverse temperature is maintained. A uniform magnetic field
= (0,0,H) and gravity field (0,0,g) pervades the system. The equations of motion and continuity for WaltersB viscoelastic fluid in the presence of suspended particles and magnetic field with rotation are
of motion and continuity for the particles, under the above assumptions are
+2 Ã— (1)
(2)
where p, , T (u,v,w), t) , t), , denote fluid pressure, density, temperature, fluid velocity, suspended particles velocity, suspended particles number density, kinematic viscosity and kinematic viscoelasticity respectively. Here
(3)
(4)
If denote the heat
capacity of fluid at constant volume, heat capacity of the particle, temperature and effective thermal conductivity of the pure fluid respectively. The equation of heat conduction gives
(0,0,g) is acceleration due to gravity, (x,y,z) and , being particle
radius, is the stokes drag coefficient.
= (5)
Since the force exerted by the fluid on the particle is equal and opposite to that exerted by the particle on the fluid, there must be an extra force term, equal in magnitude but opposite in sign, in the equation of motion for the particles. The effect due to pressure, gravity, Darcys force and magnetic field on the particles are small
The Maxwells equation yield
= (6)
=0 (7)
The equation of state for the fluid is
and so are ignored. If mN is the mass of particles per unit volume, then the equations
(8)
where is the coefficient of thermal expansion and the subscript zero refers to values at the reference level z=0.The kinematic viscosity v, kinematic viscoelasticity electrical resistivity and coefficient of thermal expansion are all assumed to be constants.
3.Perturbation equations
The basic motionless solution is
= (0,0,0), = (0,0,0), T = –
Then the linearized perturbation equations of WaltersB viscoelastic fluid become
= – +
+ (
+ +2( )
(11)
(1+
(9)
= 0 (12)
m =K ) (13)
Assume small perturbations around the basic solution and let
( denote
respectively the perturbations in fluid pressure P, density , temperature T and magnetic field . The change in density
caused mainly by the perturbation in
temperature is given by
(10)
. (14)
(1+ = s)+
(15)
= 0 (16)
where stands for electrical resistivity ,K = and
=
Eliminating in equation (11) with the help of equation (13) ,writing the scalar components of resulting equations and eliminating and between them by using equation (12) and equation (16), we obtain
2 2 4 0 2 + 2
+ =
' 4 (17)
(18)
(19)
4.Dispersion relation
We now analyze the disturbances into normal modes, assuming that the perturbation quantities are of the form
+ (20)
where are wave numbers along x and y directions respectively,
k(= ) is the resultant wave number of the disturbances and n is the growth rate. Using expression (20) equations

(19), in nondimensional form, become
2+ 1+ 1 W 4 0
+
(21)
1 1 = 1+ 1
(22)
(23)
1+ 1+ 1 = 4 0
(24)
(25)
+Q =0 (26)
where R= is the Rayleigh
where we have expressed the co ordinate x, y and z in the new unit of length
d, time t in the new unit of length and put a=kd , , J= , ,
M= , is the prandtl number, is the magnetic prandtl
number, F= is the dimensionless kinematic viscoelasticity, and
D= eliminating from
equation (21) to (25), we obtain
W
number and Q= is the Chandrasekhar number.
Free Free boundary conditions are
W= =0, DZ=0, =0 at Z=0,1 and DX=0,K=0 (27)
Using the above boundary conditions given in (27), it can be shown with the help of equations (21) (25) that all the even order derivatives of W must vanish for z=0 and z=1 hence the proper solution W characterizing the lowest mode is
W= (28)
where is a constant, substituting the proper solution (28) in equation (26), we obtain the dispersion relation,
To study the effects of magnetic field, suspended particles and rotation, we
examined the natures of , .
Equation (7.30) yields,
+
(29)
where x= , i = ,
1+ 1+ 1+1+ 2
(32)
, , , = F .
5.Stationary Convection
(33)
It is clear from equatio (31) – (33)
When the instability sets in as stationary convection, the marginal state will be characterized by =0. Putting =o, the dispersion relation (7.29) reduces to
that for stationary convection the magnetic field and rotation postpone the onset of convection where as the suspended particles hasten the onset of convection.
(30)
TABLE 1
Variation of with for a fixed value of = 50 and 20.
Sl.No
x=6
x=8
x=10
x=12
1
100
3.4935
4.0947
4.8839
5.8582
2
200
5.6361
6.3369
7.0770
8.0225
3
300
8.1410
8.5830
9.2734
10.1893
4
400
10.4689
10.8300
11.4712
12.3572
5
500
12.7976
13.0794
13.6697
14.5257
6
600
15.1267
15.3284
15.8687
16.6946
x=6 x=8 x=10 x=12
18
16
Rayleigh Number (R)
14
12
10
8
6
4
2
100 200 300 400 500 600
1
Magnetic Field (Q )
Fig.1 : Variation of with for a fixed value of = 50 and 20.
TABLE 2
Variation of with for a fixed value of =20 and =60.
Sl. No
x=2
x=4
x=6
x=8
x=10
1
100
5.5010
6.0658
8.9755
11.1626
13.6318
2
200
5.8271
7.3816
11.5963
14.3940
17.3086
3
300
6.1533
8.6973
14.2171
17.6253
20.9853
4
400
6.4793
10.0131
16.8380
20.8560
24.6621
5
500
6.8054
11.3289
19.4588
24.0881
28.3389
6
600
7.1315
12.6447
22.0796
27.3195
32.0157
x=2 x=4 x=6 x=8 x=10
35
30
Rayleigh Number (R)
25
20
15
10
5
100 200 300 400 500 600
A
Rotation (T )
Fig .2 : Variation of with for a fixed value of =20 and =60.
TABLE: 3
Variation of R1 with H1 for a fixed value , and
Sl
No
x=2
x=4
x=6
x=8
x=10
1
100
0.9000
0.95417
1.1715
1.4892
1.8952
2
200
0.4501
0.4779
0.5858
0.7446
0.9475
3
300
0.3008
0.3180
0.3905
0.4964
0.6318
4
400
0.2250
0.2385
0.2929
0.3723
0.4738
5
500
0.1800
0.1908
0.2343
0.2978
0.3790
6
600
0.1500
0.1590
0.1952
0.2482
0.3159
2.0
1.8
1.6
Rayieigh Number (R)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
x= 2
x= 4
x= 6
x= 8 x=10
100 200 300 400 500 600
1
Suspended Particles ( H )
Fig 3 : Variation of R1 with H1 for a fixed value , and
6.Conclusion
The Walters B fluid is one such important Non Newtonian fluid. For stationary convection, Walters B viscoelastic fluid behaves like a Newtonian fluid. It is also found that, rotation and magnetic field postpones the onset of convection whereas the suspended particles hasten the onset of convection. The rotation and magnetic field have stabilizing effect whereas the suspended particles has destabilizing effect on the system.
7. References

Chandrasekhar .S., Hydrodynamic and Hydromagnetics Stability, New York, Dover Publication, 1981.

Bhatia P.K and Steiner J.M, Convective instability in a rotating viscoelastic fluid layer, Z,Angew. Math.Mech.,Vol.52,1972,pp.321 324.

Sharma R.C and Aggarwal A.K., Effect of compressibility and suspended particles on thermal convection in a WaltersB elastic viscous fluid in hydromagnetics, Int. J.of App.Mech. and Engg., Vol.11,No.2,2006,pp.391399.

Bhatia,P.K. and Steiner, J.M., Thermal instability in a
viscoelstic fluid layer in hydromagnetics, J.Math.Appl.,Vol.41,2,1973,pp.2 71283.

Aggarwal, A.K and Prakash, K.,
Effect of suspended particles and rotation on thermal instability of ferro fluids, Int.J.Of App.Mech. and Engg., Vol.14,No.1,2009,pp 55 66.

Sharma.R.C ,Thermal instability in a visoelastic fluid in hydromagnetics Acta Physics Hungarica,Vol.38,pp.293298 (1975).

Sharma.R.C ,Thermal instability in a compressible fluid in the presence of rotation and magnetic field
J.Math.Anal.Appl.,vol.60,pp.227
235 (1977).

Sharma R.C., Sunil and Chand
.S., Thermosolutal instability of Walters rotating fluid (model B) in porous medium Arch.Mech., Vol.51,pp.181191 (1999).

Bhatia.P.K and Steiner, J.M., Thermal instability in a visco elastic fluid layer in hydromagnetics ,

J.Math.Anal.Appl.Vol41,pp 271.(1973).