Comparative study of Rotation and Suspended Particles on Micropolar Fluid Heated and Soluted from Below Saturating Porous Medium

Comparative study of Rotation and Suspended Particles on Micropolar Fluid Heated and Soluted from Below Saturating Porous Medium

Sumit Gupta

Department of Mathematics, Rajiv Gandhi Govt. Degree College Kotshera, Shimla -17 1004, India

ABSTRACT:

This paper deals with the convection of micropolar fluids heated and soluted from below in the presence of suspended particles (fine dust) and uniform vertical rotation  in a porous medium and using the Boussinesq approximation, the linearized stability theory and normal mode analysis, the exact solutions are obtained for the case of two free boundaries.  It is found that the presence of the suspended particles number density, the rotation parameter, stable solute parameter and medium permeability bring oscillatory modes which were non–existent in their absence. It is found that the presence of coupling between thermal and micropolar effects, rotation parameter, solute parameter and suspended particles may introduce overstability in the system. Graphs have been plotted by giving numerical values to the parameters accounting for rotation , solute parameter, the 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 parameter, solute parameter and decreases with increase in micropolar coefficients and medium permeability, for a fixed wave number, implying thereby the stabilizing effect of rotation parameter, solute parameter and destabilizing effect of micropolar coefficients and medium permeability on the thermosolutal convection of micropolar fluids.

I. 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. Micropolar fluid stabilities have become an important field of research these days. A particular stability problem is the Rayleigh-Bénard instability in a horizontal thin layer of fluid heated from below.

A detailed account of thermal convection in a horizontal thin layer of Newtonian fluid heated from below has been given by Chandrasekhar ⁴. Ahmadi ⁵ and Pérez-Garcia et al⁶ 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 crystals^{7, 8}, Bradley in dielectric fluids⁹ and Laidlaw in binary mixture¹⁰. 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 suspended particles. Saffman ¹² has considered the stability of laminar flow of a dusty gas. Scanlon and Segel ¹³ have considered the effects of suspended particles on the onset of Bénard convection. The separate effects of suspended particles, rotation and solute gradient on thermal instability of fluids saturating a porous medium have been discussed by Sharma and Sharma ¹⁴. The suspended particles were thus found to destabilize the layer. Palaniswami and Purushotham ¹⁵ have studied the stability of shear flow of stratified fluids with the fine dust and found that the presence of dust particles increases the region of instability. On the other hand, multiphase fluid systems are concerned with the motion of liquid or gas containing immiscible inert identical particles. The theoretical and experimental results of the onset of thermal instability (Bénard convection) in a fluid layer under varying assumptions of hydromagnetics, has been depicted in a treatise by Chandrasekhar ⁴. Lapwood ¹⁶ has studied the convective flow in porous medium using linearized stability theory. The Rayleigh instability in flow through a porous medium has been considered by Wooding ¹⁷. The problem of thermal convection in a fluid in porous medium is of importance in geophysics, soil–science, ground–water, hydrology and astrophysics. The physical property of comets, meteororites and inter–planetary dust strongly suggests the importance of porosity in the astrophysical context McDonnel ¹⁸.