Effect of Hall Currents and Suspended Particles on Critical Thermal Rayleigh Number for Compressible Walters’ (Model B¢) Fluid

 

Urvashi Gupta*

University Institute of Chemical Engineering and Technology, Panjab University, Chandigarh-160014, India.

*Corresponding Author E-mail: dr_urvashi_gupta@yahoo.com

ABSTRACT:

A layer of a compressible, electrically conducting Walters’ (Model B¢) elastico-viscous fluid permeated with suspended particles heated from below in the presence of a magnetic field and Hall currents is considered. The perturbation equations are analyzed in terms of normal modes after linearizing the relevant hydromagnetic equations. A dispersion relation governing the effects of visco-elasticity, magnetic field, Hall currents and suspended particles has been obtained. For the case of stationary convection, Walters’ (Model B¢) fluid behaves like a  Newtonian fluid due to the vanishing of the visco-elastic parameter. The compressibility and magnetic field are found to have stabilizing effects on the system, that is, their effect is to postpone the onset of thermal instability whereas Hall currents and suspended particles are found to hasten the onset of thermal instability for permissible range of values of various parameters. The dispersion relation is also analyzed numerically. The critical Rayleigh numbers and the wave numbers of the associated disturbances for the onset of instability as stationary convection are obtained and the behaviour of various parameters on critical thermal Rayleigh numbers has been depicted graphically.

 

 

INTRODUCTION

A detailed account of the theoretical and experimental results of the onset of thermal instability (Bénard convection) in a fluid layer under varying assumptions of hydrodynamics and hydromagnetics is given by Chandrasekhar [1] in his celebrated monograph. Chandra [2] observed a contradiction between the theory and experiment for the onset of convection in fluids heated from below. Scanlon and Segel [3] investigated some of the continuum effects of particles on Bénard convection and found that a critical Rayleigh number was reduced solely because the heat capacity of the pure gas was supplemented by that of the particles. The effect of suspended particles was thus found to destabilize the layer. Palaniswamy and Purushotham [4] have considered the stability of shear flow of stratified fluids with fine dust and have found that the effect of fine dust is to increase the region of instability.

 

Sharma et al. [5] considered the effect of suspended particles on the onset of Bénard convection in hydromagnetics while Sharma and Gupta [6] investigated the effect of Hall currents and suspended particles on thermal instability of compressible fluids saturating a porous medium.

 

The tendency of the electric current to flow across an electric field in the presence of magnetic field is called Hall Effect. Sherman and Sutton [7] have considered the effect of Hall currents on the efficiency of a magneto-fluid dynamic (MHD) generator while Gupta [8] studied the effect of Hall currents on the thermal instability of electrically conducting fluid in the presence of uniform vertical magnetic field. Sharma and Gupta [9] investigated the effect of Hall currents on thermosolutal instability of a rotating plasma. For compressible fluids, the equations governing the system become quite complicated. Spiegel and Veronis [10] have simplified the set of equations governing the flow of compressible fluids assuming that the depth of the fluid layer is much smaller than the scale height as defined by them and the motions of infinitesimal amplitude are considered. Sharma [11] investigated the thermal instability of compressible fluids in the presence of rotation and magnetic field while Sharma and Gupta [12] studied the effect of finite Larmor radius on thermal instability of compressible rotating plasma. Thermal instability of compressible, finite Larmor radius Hall plasma has been studied by Sharma and Sunil [13] in a porous medium.

 

There is growing importance of non-Newtonian fluids in geophysical fluid dynamics, chemical technology and petroleum industry. Bhatia and Steiner [14] studied the problem of thermal instability of a Maxwellian visco-elastic fluid in the presence of rotation and found that rotation has a destabilizing influence in contrast to its stabilizing effect on a viscous Newtonian fluid. But Maxwell’s model does not describe all the characteristics of a visco-elastic fluid. Thermal instability of an Oldroydian visco-elastic fluid acted on by a uniform rotation has been studied by Sharma [15]. An experimental demonstration by Toms and Strawbridge [16] has revealed that a dilute solution of methyl methacrylate in n-butyl acetate agrees well with the theoretical model of Oldroyd [17]. There are many visco-elastic fluids which cannot be characterized by Maxwell’s constitutive relations or Oldroyd’s [17] constitutive relations. Two such classes of elastico-viscous fluids are Rivlin-Ericksen [18] and Walters’ (Model B¢) fluids. Walters [19] has proposed a theoretical model for such elastico-viscous fluids and reported [20] that the mixture of polymethyl methacrylate and pyridine at 25⁰C containing 30.5 g of polymer per liter behaves very nearly as the Walters’ (Model B¢) elastico-viscous fluid. Such and other polymers are used in agriculture, communication appliances and in biomedical applications. Sharma and Kumar [21] have studied the stability of two superposed Walters’ (Model B¢) liquids whereas thermosolutal convection problem in the presence of magnetic field for Walters’ (Model B¢) fluid in porous medium has been investigated by Sunil et al. [22]. Recently, Sharma et al. [23] considered the stability of stratified Walters’ (Model B¢) fluid in the presence of magnetic field and rotation in porous medium.         

 

Motivated by the fact that knowledge regarding fluid particle mixture is not commensurate with their industrial and scientific importance and the importance of flow of visco-elastic fluids in paper industry, petroleum industry, chemical technology and geophysical fluid dynamics; we set out to study the effect of suspended particles and Hall currents on thermal instability of a compressible Walters’ (Model B¢) fluid. Here, it is worthwhile to mention that Hall currents are important in many geophysical and astrophysical situations in addition to the flow of laboratory plasmas.

  

Let and denote, respectively, temperature, pressure, density, thermal coefficient of expansion, gravitational acceleration, electrical resistivity, magnetic permeability, electron number density, charge on an electron, kinematic viscosity, kinematic visco-elasticity, thermal diffusivity and fluid velocity.

 

This is to find out critical Rayleigh number R_c and the associated critical wavenumber x_c for various values of the parameters Q₁, M and H_d. The critical numbers are listed in Tables I - III and illustrated in Figures 2-4 are obtained in this fashion. It is clear from Figure 2 that the critical Rayleigh number R_c increases with the increase in magnetic field parameter Q₁ for fixed value of suspended particle parameter H_d. Also, the various curves for different values of H_d indicate the destabilizing influence of suspended particle parameter as R_c decreases with the increase in H_d. Thus, magnetic field has a stabilizing effect on the system as usual whereas the effect of suspended particles is destabilizing.

 

In Figure 3, R_c is plotted against H_d for various values of Q_1. For fixed Q_1, R_c decreases with increase in H_d and R_c increases with the increase in Q₁ (as is clear from curves for different values of Q₁). This confirms the above results and the results drawn analytically. In Figure 4, variation of R_c with the Hall current parameter M is investigated for different values of Q₁. The critical Rayleigh number R_c decreases with the increase in M confirming the usual destabilizing influence of Hall currents. The different curves for various values of Q₁ confirm the stabilizing effect of magnetic field as R_c increases with the increase in Q_1. Figures 3 and 4 show the decrease in critical Rayleigh number R_c with the increase in suspended particles parameter H_d and Hall current parameter M, respectively. Thus figs. 3 and 4 confirm the destabilizing influence of suspended particles and Hall currents, respectively.

 

Table I.  The critical Rayleigh numbers and the wavenumbers of the associated disturbances for the onset of instability as stationary convection for G =10, M = 10 and for various values of Q1 and Hd
	Hd =10		Hd =20		Hd =30		Hd =50
Q1	xc	Rc	xc	Rc	xc	Rc	xc	Rc
100	3.5	13.29	3.5	6.64	3.5	4.43	3.5	2.66
200	4.5	26.03	4.5	13.02	4.5	8.68	4.5	5.21
300	5.0	38.74	5.0	19.37	5.0	12.91	5.0	7.75
500	5.0	64.76	5.0	32.38	5.0	21.59	5.0	12.95
1000	5.0	130.83	5.0	65.42	5.0	43.61	5.0	26.17

 

 

 

 

Table II.  The critical Rayleigh numbers and the wavenumbers of the associated disturbances for the onset of instability as stationary convection for G =10, M = 10 and for various values of Q₁ and H_d                 

 

 

Table III. The critical Rayleigh numbers and the wavenumbers of the associated disturbances for the onset of instability as stationary convection for G =10, H_d = 10 and for various values of Q₁ and M

 

 

Figure 2. Variation of critical Rayleigh number R_c with magnetic field parameter Q₁ for fixed G=10, M=10 and for various values of H_d = 10, 20, 30 and 50

 

Figure 3. Variation of critical Rayleigh number R_c with suspended particle factor   H_d for fixed G=10, M=10 and for various values of Q₁ =100, 200, 300, 500 and 1000

 

Figure 4. Variation of critical Rayleigh number R_c with Hall current parameter M for fixed G=10, H_d =10 and for various values of Q₁ =100, 200, 300, 500 and 1000

 

 

 

CONCLUSIONS:

In the present paper, the combined effect of Hall currents, magnetic field and suspended particles on the stability of a compressible Walters’ (Model B¢) elastico-viscous fluid heated from below is considered. The effect of various parameters such as magnetic field, compressibility, Hall currents, and suspended particles has been investigated analytically as well as numerically. For the case of stationary convection, Walters’ (Model B¢) fluid behaves like a  Newtonian fluid due to the vanishing of the visco-elastic parameter. The expressions for ,  and are examined analytically and it has been found that the magnetic field has a stabilizing effect on the system whereas suspended particles and Hall currents have a destabilizing influence on the system. The reasons for stabilizing effect of magnetic field and destabilizing effect of suspended particles and Hall currents are accounted by Chandrasekhar [1], Scanlon and Segel [3] and Gupta [8] respectively. These are valid for second- order fluids as well. The effect of compressibility is to postpone the onset of instability, as is clear from eq. (27). The critical thermal Rayleigh numbers and the associated wavenumbers are found for stationary convection for various parameters involved and it has been found that it increases with the increase in magnetic field parameter and decreases with the increase in suspended particle factor and Hall current parameter thereby confirming the stabilizing role of magnetic field and destabilizing role of suspended particles and Hall currents.

 

NOMENCLATURE:

Cp	specific heat of the fluid at constant pressure, [J kg-1K-1]
Cpt	heat capacity of particles, [J kg-1K-1]
Cf	heat capacity of the fluid, [J kg-1K-1]
d	depth of fluid layer, [m]
e	charge of an electron, [C]
F	dimensionless kinematic visco-elasticity, [-]
	acceleration due to gravity, [ms-2]
	perturbation in magnetic field
	magnetic field vector having components , [G]
	wave number of the disturbance, [m-1]
	Stokes’ drag coefficient, [kgs-1]
	wavenumbers in x and y directions respectively, [m-1]
M	dimensionless Hall current parameter, [-]
n	growth rate of the disturbance, [s-1]
N0	particle number density, [m-3]
N	perturbation in suspended particle number density, [m-3]
	electron number density, [m-3]
P	fluid pressure, [Pa]
Pr1	thermal Prandtl number, [-]
Pr2	magnetic Prandtl number, [-]
q	effective thermal conductivity of the pure fluid, [Wm-1K-1]
Q1	dimensionless Chandrasekhar number, [-]
R1	dimensionless Rayleigh number, [-]
Rc	critical Rayleigh number, [-]
T	temperature, [K]
	fluid velocity vector having components , [ms-1]
	velocity of suspended particles, [ms-1]
(x, y, z)	x, y, z  directions
x	wavenumber, [m-1]
xc	critical wavenumber, [m-1]

Greek Letters

	thermal coefficient of expansion, [K-1]
	temperature gradient, [Km-1]
	curly operator, [-]
	del operator, [-]
	perturbation in the respective physical quantity, [-]
	particle radius, [m]
	resistivity, [m2s-1]
	perturbation in temperature, [K]
	thermal diffusivity, [m2s-1]
	viscosity of the fluid, [kg m-1s-1]
	visco-elasticity of the fluid, [kg m-1s-1]
	magnetic permeability, [H m-1]
	kinematic viscosity, [m2s-1]
	kinematic visco-elasticity, [m2s-1]
	density of the fluid, [kgm-3]

 

 

 

REFERENCES:

[1]     Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, Dover Publication, New  York, 1981

[2]     Chandra, K., Instability of Fluids Heated from Below, Proc. Roy. Soc., London.,      A164 (1938), pp. 231- 242

[3]     Scanlon, J.W., Segel, L.A., Some Effects of Suspended Particles on the Onset of Bénard Convection, Phys. Fluids, 16 (1973), pp. 1573-1578

[4]     Palaniswamy, V.A., Purushotham, C.M., Stability of Shear Flow of Stratified Fluids with Fine Dust, Phys.Fluids, 22 (1981), pp. 1224-1229

[5]     Sharma, R.C., Prakash, K., Dube, S.N., Effect of Suspended Particles on the Onset of Bénard Convection in Hydromagnetics, Acta Physica Hungarica, 40 (1976), pp. 3-10

[6]     Sharma, R.C., Gupta, U., Thermal Instability of Compressible Fluids with Hall Currents   and Suspended Particles in Porous Medium,  Int.J. Eng. Sci., 31(7) (1993), pp. 1053-1060

[7]     Sherman, A., Sutton, G.W., Magnetohydrodynamics, Northwestern University Press, Evanston, IL, 1962

[8]     Gupta, A.S., Hall Effects on Thermal Instability, Rev. Roum. Math. Pures Appl., 12     (1967), pp. 665-677

[9]     Sharma, K.C., Gupta, U., Hall Effects on Thermosolutal Instability of a Rotating Plasma, IL Nuovo   Cemento, 12D (12) (1990), pp. 1603-1610  

[10] Spiegal, E.A., Veronis, G., On the Boussinesq Approximation for a Compressible Fluid, Astrophys. J., 131 (1960), pp. 442

[11]   Sharma, R.C., Thermal Instability in Compressible Fluids in the Presence of Rotation and Magnetic Field, J. Math. Anal. Appl., 60 (1977), pp. 227-235

[12]   Sharma, K.C., Gupta, U., Finite Larmor Radius Effect on Thermal Instability of a Compressible Rotating Plasma, Astrophysics and Space Science, 167 (1990), pp. 235-243

[13]   Sharma, R.C., Sunil, Thermal Instability of Compressible Finite-Larmor-Radius Hall Plasma in Porous Medium, Journal of Plasma Physics, 55(1) (1996), pp. 35-45

[14] Bhatia, P.K., Steiner, J.M., Convective Instability in a Rotating Visco-elastic Fluid Layer, Z. Angew. Math. Mech., 52 (1972), pp. 321-327

[15]   Sharma, R.C., Effect of Rotation on Thermal Instability of a Visco-elastic Fluid, Acta Physica  Hungrica, 40 (1976), pp. 11-17

[16]   Toms, B.A., Strawbridge, D.J., Elastic and Viscous Properties of Dilute Solutions of Polymethyl Methacrylate in Organic Liquids, Trans. Faraday Soc., 49 (1953), pp. 1225-1232

[17]   Oldroyd, J.G., Non-Newtonian Effects in Steady Motion of Some Idealized Elastico–viscous Liquids,  Proc. Roy. Soc. (London), A245 (1958), pp. 278

[18]   Rivlin, R.S., Ericksen, J.L., Stress Deformation Relations for Isotropic Materials, J. Rat. Mech. Anal., 4 (1955), pp. 323-329

[19]   Walters, K., The Motion of an Elastico-Viscous Liquid Contained Between Coaxial Cylinders, Quart. J. Mech. Appl. Math., (13) (1960), pp. 444

[20]   Walters, K., Non-Newtonian Effects in Some Elastico-viscous Liquids Whose Behaviour at Some Rates of Shear is Characterized by a General Linear Equation of State, Quart. J. Mech. Appl. Math., 15 (1962), pp. 63- 76

[21]   Sharma, R.C., Kumar, P., Study of the Stability of Two Superposed Walters’ (Model B¢) Visco-Elastic Liquids, Czechoslovak Journal of Physics, 47 (1997), pp. 197-204

 

 

 

Received on 11.12.2011        Accepted on 16.01.2012        

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[22] Sunil, Sharma, R.C., Chand, S., Thermosolutal Convection in Walters’ (Model B¢) Fluid in Porous Medium in Hydromagnetics, Studia Geotechnica et Mechanica, 22(3-4) (2000), pp.3-14

[23]   Sharma, V., Sunil, Gupta, U., Stability of Stratified Elastico-viscous Walters’ (Model B¢) Fluid in the Presence of Horizontal Magnetic Field and Rotation in Porous Medium, Arch. Mech., 58(2) (2006), pp. 187-197 

[24]   Sharma, R.C., Aggarwal, A.K., Effect of Compressibility and Suspended Particles on Thermal Convection in a Walters’ Elastico-Viscous Fluid in Hydromagnetics, Int. J of Applied Mechanics and Engineering, 11(2) (2006), pp. 391-399

[25]   Sharma, R.C., Sunil, Chand, S., Hall Effect on Thermal Instability of  Rivlin-Ericksen Fluid, Indian J. of pure Appl. Math., 31 (1) (2000), pp. 49-59

[26]   Kumar, P., Singh, G.J., Lal, R., Thermal Instability of Walters B¢ Visco-elastic Fluid Permeated with Suspended Particles in Hydromagnetics in Porous Medium, Thermal Science, 8(1) (2004), pp. 51-61

[27]   Sunil, Sharma, R.C., Chand, S., Hall Effect on Thermal Instability of Walters’ (Model B¢)  Fluid, Indian Journal of Theoretical Physics, 48 (1) (2000), pp. 81-92