Study of lack of thermal equilibrium and anisotropy effects on double-diffusive ferroconvection in porous medium

Study of lack of thermal equilibrium and anisotropy effects on double-diffusive ferroconvection in porous medium

Anu Sharma¹, Neeti Goel²

¹ Department of Mathematics, Rajdhani College, University of Delhi, Raja Garden,

Ring Road, New Delhi, 110015, India.

²Department of Mathematics, Acharya Narendra Dev College, University of Delhi,

Govindpuri, Kalka Ji, New Delhi, 110019, India.

ABSTRACT:

In this work, we study theoretically the effect of lack of local thermal equilibrium and anisotropy on double-diffusive ferro convection in aporous medium. Stability analysis is carried out for ferrofluid layer contained between two stress-free boundaries using normal mode method. The eigen value problem is solved using regular perturbation technique to obtain critical thermal Rayleigh number. The impact of anisotropy parameter of permeability of porous medium, heat transfer coefficient and salinity Rayleigh number on the onset of convection are examined.

1. INTRODUCTION

In double-diffusive convection problems, there are two destabilizing sources for density difference, the temperature field and salt field. Double diffusion was first explored because of its application to oceanic phenomenon. When a fluid permeates through a porous material the flow is analyzed by macroscopic law, called Darcy law as referred byLapwood [1].In literature majority of the studies on convection in porous mediumhave been dealt with local thermal equilibrium conditions. In many situations (Nield andBejan[2]), lackof local thermal equilibrium exists. In such cases local thermal nonequilibrium effects are to be considered by two field model energy equationrepresenting fluid and solid phases separately. An excellent review of research on local thermal non-equilibrium phenomenon in porous medium convection, primarily free and forced convection boundary layers and free convection within cavities is given by Rees and Pop [3].

Ferrofluids are electrically non-conducting colloidal suspensions of tiny magnetic particles

coated with surfactant immersed in a carrier fluid. These fluids behave as continuum and

exhibit a variety of interesting phenomenon. Ferrofluids are widely used in magnetic inkjet

printers, inertial dampers, switches, sensors, magnetic and nanofluidic devices, magnetic targeted drug delivery etc. Rosenweig[4] reviews several applications of heat transfer through ferrofluids. This heat transfer through ferrofluids is called ferroconvection. The study of convection in two component ferrofluids is referred as double-diffusive ferroconvection studied by Baines and Gill [5] and later on extensively studied by many authors [6-9].

Most of the studies have been concerned with isotropic media, but geological and pedagogical processes rarely form isotropic porous media.Vaidyanathan et al. [10] studied ferroconvection in an anisotropic densely packed porous medium.Suresh et al. [11] studied numerical analysis of ferroconvection with temperature dependent viscosity and anisotropic porous medium. In the present work, it is attempted to analyze the effect of lack of thermal equilibrium and anisotropy effect on double-diffusive convection in ferrofluid layer saturated porous medium and free boundaries are considered. An analytical formula is found for critical Rayleigh number to study effect of various parameters.

2  MATHEMATICAL FORMULATION

An infinite incompressible ferrofluid saturated an anisotropic porous layer of thickness ‘ ’ heated from below and soluted from below in the presence of uniform vertical magnetic field is considered. A constant temperature gradient and solute gradient is maintened across the layer. Darcy model for flow through porous medium with fluid and solid phases not in thermal equilibrium is considered. With these assumptions the basic goverening equations are