Thermal Convection of Micropolar Fluid in the Presence of Suspended Particles and Rotation Saturating A Porous Medium
Veena Sharma1, Sumit Gupta2
1Department of Mathematics and Statistics, Himachal Pradesh University,Shimla-171 005, India.
2Department of Mathematics, Govt. Degree College Diggal Distt. Solan-173 218 India.
*Corresponding Author E-mail: sumitttgupta@gmail.com
ABSTRACT:
A study has been made of the
convection of micropolar fluids heated from below in the presence of suspended
particles (fine dust) and uniform vertical rotation
in a porous medium.
The effect of Coriolis forces on the stability is chosen along the direction of
the gravitational field. It is found that the presence of coupling between
thermal and micropolar effects, rotation parameter, medium permeability and
suspended particles may introduce overstability in the system. Using the
Boussinesq approximation, the linearized stability theory and normal mode
analysis, the exact solutions are obtained for the case of two free
boundaries. Graphs have been plotted by giving numerical values to the
parameters accounting for rotation
, medium
permeability, dynamic microrotation viscosity 
and coefficient of
angular viscosity 
to depict the
stability characteristics, for both the cases of stationary convection and
overstability. It is found that Rayleigh number for the case of overstability
and stationary convection increases with increase in rotation parametre and
decreases with increase in micropolar coefficients and medium permeability, for
a fixed wave number, showing thereby the stabilizing effect of rotation
parametre , destabilizing effect of micropolar coefficients and
medium permeability on the thermal convection of micropolar fluids.
KEYWORDS: Micropolar fluid; Rotation; Suspended particles (fine dust); Medium permeability; Microrotation; Coefficient of angular viscosity.
INTRODUCTION:
Micropolar theory was introduced by Eringen [1] in order to describe some physical systems which do not sastisfy the Navier Stokes equations. These fluids are able to describe the behaviour of colloidal solutions, liquid crystals; animal blood etc. The equations governing the flow of micropolar fluid theory involve a spin vector and a microinertia tensor in addition to velocity vector. A generalization of the theory including thermal effects has been developed by Kazakia and Ariman [2] and Eringen [3]. The stability investigations of the Bénard problem in the framework of various external force fields assume importance not only on account of being a meaningful mathematical extensions of the problem but also because of its importance in the problem of meteorology, oceanography and various other fields of practical importance. The effects of the action of a uniform vertical magnetic field and a uniform vertical rotation acting individually or simultaneously on the Bénard problem has been investigated by Chandrasekhar [4].
Micropolar fluid stabilities have become an important field of research these days. Ahmadi [5] and Pérez- Garcia et. al [6] have studied the effects of the microstructures on the thermal convection and have found that in the absence of coupling between thermal and micropolar effects, the principle of exchange of stabilities may not be fulfilled and consequently micropolar fluids introduce oscillatory motions. The existence of oscillatory motions in micropolar fluids has been depicted by Lekkerkerker in liquid crystal [7,8], Bradley in dielectric fluids [9] and Laidlaw in binary mixture [10]. In the study of problems of thermal convection, it is frequent practice to simplify the basic equations by introducing an approximation which is attributed to Boussinesq [11]. In geophysical situations, the fluid is often not pure but contains several suspended particles. Motivation for the study of certain effect of particles immersed in the fluid such as particle heat capacity, particle mass fraction and thermal force is due to the fact that the knowledge concerning fluid -particles mixture is not commensurate with their industrial and scientific importance. Saffman [12] has considered the stability of laminar flow of a dusty gas. Sharma et. al [13] have considered the effect of suspended particles on the onset of Bénard convection in hydromagnetics and found that the critical Rayleigh number is reduced because of the heat capacity of particles thereby destabilizing the system. On the other hand, multiphase fluid systems are concerned with the motion of liquid or gas containing immiscible inert identical particles of all multiphase fluid systems observed in nature, blood flow in arteries, flow in rocket tubes, dust-in-gas cooling system to enhance heat transfer processes, movement of inset solid particles in atmosphere, and sand or other particles on sea or ocean beaches are the most common examples of multiphase fluid systems. The Rayleigh instability in flow through a porous medium has been considered by Wooding [14]. Moreover, Saffman and Taylor [15] have shown that the motion in a Hele–Shaw cell is mathematically analogous to two dimensional flow in porous medium. In recent years, there has been a considerable interest in the study of breakdown of the stability of a layer of a fluid subjected to a vertical temperature gradient in a porous medium and also in the possibility of convective flow. Lapwood [16] has studied the convective flow in porous medium using linearized stability theory.
Generally, suspended particles number density has a destabilizing effect on the thermal convection on the fluids. From the physical point of view the effect of rotation on the micropolar fluids in the presence of suspended particles is interesting because there is a competition between the large enough stabilizing effect of rotation and destabilizing effect (to a smaller extent) of suspended particles. Moreover, rotation introduces Coriolis acceleration which plays an important role on the stability on the system and a centrifugal force which is neglected due to its small magnitude. The rotating fluid also finds its application in meteorphysics and oceanography. Sharma and Gupta [17] have studied the stability of micropolar fluids heated from below in the presence of suspended particles . Sharma and Gupta[18] have studied the effect of rotation on thermal convection in micropolar fluids in the presence of suspended particles.
Keeping in mind the importance and relevance of thermal convection with suspended particles in porous medium and rotation, the present paper therefore, deals with the thermal convection of micropolar fluid in the presence of suspended particles in rotation saturating porous medium.
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Received on 18.11.2016 Modified on 25.11.2016 Accepted on 04.12.2016 ©A&V Publications All right reserved DOI: 10.5958/2349-2988.2017.00037.7 Research J. Science and Tech. 2017; 9(1):62-70.
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