Onset of convection in a couple-stress fluid-saturated porous medium: effects of non-uniform temperature gradients

Abstract

Onset of convection in a layer of couple-stress fluid-saturated porous medium is investigated for different types of basic temperature gradients. The boundaries are considered to be adiabatically insulated to temperature perturbations. The eigenvalue equations of the perturbed state obtained from the normal mode analysis are solved analytically using a regular perturbation technique with wave number as a perturbation parameter and also numerically using the Galerkin technique. The critical stability parameters obtained from these two techniques are in excellent agreement and an increase in the value of couple-stress parameter is found to delay the onset of convection. The results also indicate that the piecewise linear temperature profile hastens the onset of convection when compared to linear, parabolic, and inverted parabolic temperature profiles. In addition, the influence of thermal depth on the critical conditions is assessed in the case of piecewise linear temperature profiles, and it is observed that the critical thermal depth decreases marginally with an increase in the couple-stress parameter.Keywords Convection · Couple-stress fluid · Non-uniform temperature gradient · Porous medium IntroductionDarcy-Benard convection in a layer of fluid-saturated porous medium has been studied extensively over the years because its relevance to variety of situations in many science and engineering problems. Applications include underground spread of pollutants, convection in the earth's crust, packed bed chemical reactors, highperformance thermal insulation development, geothermal energy extraction, ceramic processing, coal and grain storage, solid matrix heat exchangers, and nuclear waste disposal among others. The substantial part of theoretical and experimental works on thermal convective instability in porous media has been dealt with Newtonian fluids and the copious literature available on this topic is well documented by Ingham and Pop [1], Vafai [2,3] and Nield and Bejan [4].However, it is well known that fluids which are involved in many of the problems cited above are nonNewtonian and hence the traditional Newtonian fluids cannot precisely describe the characteristics of the fluid flow. Therefore, it is pertinent to investigate convective instability in a non-Newtonian fluid-saturated porous medium. Among the existence of different types of non-Newtonian fluids, couple-stress fluids have received wider attention in recent years. The study of couple-stress fluids has applications in a number of processes that occur in industry such as the extrusion of polymer fluids, solidification of liquid crystals, cooling of metallic plate in a bath, exotic lubricants, and colloidal solutions to mention a few. The couple-stress fluid has distinct features, such as polar effects whose microstructure is mechanically significant. The constitutive equations for couple-stress fluids are given by Stokes [5]. The theory proposed by Stokes is the simplest one for I. S. Shivakumara (B)

The Onset of Double Diffusive Convection in a Couple Stress Fluid Saturated Anisotropic Porous Layer

“…Malashetty et al (2009a) investigated the local thermal nonequilibrium effect on the onset of convection in a couple stress fluid saturated porous layer. Recently, Shivakumara (2009) have studied onset of convection in a couple stress fluid saturated porous medium with nonuniform temperature gradients. More recently, analyzed the double diffusive instability of a couple stress fluid saturated porous layer using linear and weakly nonlinear theories.…”

Effect of Internal Heating on Double Diffusive Convection in a Couple Stress Fluid Saturated Anisotropic Porous Medium

“…Malashetty et al [23] did an analytical study of linear and nonlinear double diffusive convection with Soret effect in couple stress liquids. Shivakumara [24] studied the effect of nonuniform temperature gradient on the onset of convection in a couple stress fluid saturated porous medium. Malashetty et al [25] performed linear and weakly nonlinear analyses of double diffusive convection in an isotropic porous medium saturated with couple stress fluid.…”

Linear and nonlinear stability of double diffusive convection in a couple stress fluid–saturated porous layer

“…Following the same procedure, the equation of motion for couple-stress fluid through porous media can be obtained from Eq. (3) and with the inclusion of inertial effect the equation reads as [17,18] …”

“…Gaikwad et al [16] have investigated linear and nonlinear double diffusive convection by considering both Soret and Dufour effects in a layer of couple-stress liquid. Recently, Shivakumara [17] has discussed the effect of various nonuniform basic temperature gradients on the onset of convection in a layer of couple stress fluid-saturated porous layer, while Shivakumara et al [18] have analyzed the effect of Coriolis force on both linear and weakly nonlinear thermal convection in a couple stress fluid-saturated rotating rigid porous layer.…”