Effect of Magnetic Field on Thermal Instability of Non-newtonian Fluids in a Rotating Medium

Effect of Magnetic Field on Thermal Instability of Non-newtonian 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 Non-Newtonian 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 visco-elastic fluid layer in Hydromagnetics

2.Mathematical Formulation

We consider an infinite horizontal layer of electrically conducting WaltersB elastic-viscous 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 × (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

1. (19), in non-dimensional 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

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

   2. 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.

   3. 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.391-399.

   4. 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 71-283.

   5. 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.

   6. Sharma.R.C ,Thermal instability in a visoelastic fluid in hydromagnetics -Acta Physics Hungarica,Vol.38,pp.293-298 (1975).

   7. 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).

   8. Sharma R.C., Sunil and Chand

      .S., Thermosolutal instability of Walters rotating fluid (model B) in porous medium Arch.Mech., Vol.51,pp.181-191 (1999).

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

J.Math.Anal.Appl.Vol-41,pp- 271.(1973).