Brinkman convection induced by purely internal heating in a rotating porous medium layer saturated by a nanofluid

Highlights

• Brinkman nanofluid convection via purely internal heating with rotation is studied.

• Rigid–rigid boundary surfaces are taken with two sets of thermal boundary conditions.

• Higher order Galerkin method is used.

• The relevant parameter is an internal Rayleigh number, based on heat source strength.

Abstract

The onset of Brinkman convection induced by purely internal heating in a rotating porous medium layer saturated by a nanofluid is investigated for rigid–rigid boundary surfaces with two sets of thermal boundary conditions namely, Case (i) – both boundaries isothermal – and Case (ii) – lower insulated and upper isothermal. The zero nanoparticle flux condition under the thermophoretic effects is taken at the boundaries. The model used for nanofluid combines the effects of Brownian motion and thermophoresis, while for porous medium Brinkman model is considered. The effect of rotation parameter, porosity parameter, Darcy number, Lewis number, nanoparticle Rayleigh number, modified diffusivity ratio and modified specific heat increment on the onset of convection is shown graphically and analyzed in detail. It is found that the Lewis number, the nanoparticle Rayleigh number and the modified diffusivity ratio have destabilizing effect on the onset of convection, while modified specific heat increment has no significant effect on the stability of the system. The rotation parameter, the porosity parameter and the Darcy number delay the onset of convection. The system is found to be more stable when both boundaries are isothermal.

Introduction

In recent years, nanofluid flow in porous medium has increased into a substantial area of research due to their numerous applications in different fields such as chemical industry, solar collectors, microbial fuel cells, materials processing, biological propulsion, nuclear waste disposals, and geothermal energy systems, where nanofluid injection results in greater thermal efficiency [1], [2], [3], [4], [5]. The term “nanofluid” was first coined by Choi [6] to show engineered colloids which are composed of nanoparticles dispersed in a base fluid. Materials commonly utilized for nanoparticles include metals such as copper and gold, and oxides such as alumina, silica and copper oxide. Popular base fluids consist of water, oil and organic fluids. Relative to the base fluid, it has been detected consistently by many researchers that the nanofluids have abnormal thermal conductivity, viscosity and single-phase convective heat transfer coefficient. These fluids are considered to offer vital advantages over conventional heat transfer fluids [7], [8], [9], [10].

In review of its importance, many authors have studied the flow of nanofluid in a porous medium between two parallel plates. Nield and Kuznetsov [11] were the first to study the onset of convection in the horizontal porous layer saturated by a nanofluid with Darcy model on the basis of the transport equations developed by Buongiorno [12]. Later, Yadav et al. [13], [14], [15], [16], [17], [18], [19], Nield and Kuznetsov [20], [21], [22], Kuznetsov and Nield [23], [24], [25], Umavathi et al. [26], Shivakumara et al. [27], Hosseini et al. [28] and Hatami et al. [29] extended this problem with varying assumptions. The effect of magnetic field on the nanofluid flow and heat transfer characteristic was studied by Yadav et al. [30], [31], [32], [33], Jalilpour et al. [34], Sheikholeslami et al. [35], Ghasemi et al. [36] and Hatami et al. [37]. The common finding of those studies was that the fluid experienced a Lorentz force. This force, in turn, affects the buoyant flow field and the heat transfer rate.

It is known that there are large numbers of practical situation in which convection is driven by its own source of heat. This gives a different way in which a convective flow can be set up through the internal heat generation within the porous medium layer. Such a situation can happen through nuclear reaction, radioactive decay or a rather weak exothermic reaction which can take place within the porous material. Therefore, the role of internal heat generation becomes very important in several applications including chemical engineering, nuclear heat cores, nuclear energy, nuclear waste disposals, oil extractions and crystal growth. But, there are only few studies available in which the convection induced by purely internal heating has been investigated. Gasser and Kazimi [38] were the first to study the convection in a porous medium which is induced by both internal heat generation and heating from below. They introduced two Rayleigh number; one corresponding to internal heating, other to the external temperature gradient. The Brinkman model for natural convection in a porous layer with nonuniform thermal gradient was analyzed by Vasseur and Robillard [39]. They obtained a stability problem similar to that of Gasser and Kazimi [38] except that the disturbance temperatures satisfy Neumann rather than Dirichlet boundary conditions. Later, Nouri-Borujerdi et al. [40] analysed the influence of Darcy number on the onset of convection in a porous layer induced by purely internal heating for rigid–rigid boundaries. They obtained a smooth monotonic variation in the critical internal Rayleigh number accurately. Recently, Ghasemi et al. [41] studied thermal analysis of convective fin with temperature-dependent thermal conductivity and heat generation. They used a simple and highly accurate semi-analytical method called the Differential Transformation Method (DTM) for solving the nonlinear temperature distribution equation.

The study related to internal heat source in porous medium saturated by nanofluid was first presented by Yadav et al. [42]. They completed their analysis in terms of a standard external Rayleigh–Darcy number and used the same boundary conditions on the nanofluid fraction as those used by Nield and Kuznetsov [11]. These boundary conditions were of limited practical validity. Very recently, Nield and Kuznetsov [43] considered the case of purely internal heating for Darcy porous medium and re-examine the boundary conditions for volumetric fraction of nanoparticles that the nanofluid particle fraction was allowed to adapt to the temperature profile induced by the internal heating. They obtained that the presence of the nanofluid particles leads to increased instability of the system.

In all of the above studies, the convection via purely internal heating in rotating porous medium saturated by a nanofluid was not studied. However, it is well known the convection via purely internal heating in rotating porous medium saturated by a nanofluid is of great interest in the microwave heating of liquids. Microwave heating finds application in the chemical engineering, particle deposition rate in nuclear reactors, electronic chips, semiconductor and wafers food industry such as in pasteurization, sterilization etc. Therefore, in this paper, we analyze the stability criteria of a horizontal nanofluid layer undergoing rotation, irradiated by purely internal heating, such as that produced by microwave heating or chemical reaction, instead of bottom heating for two sets of thermal boundary conditions.