Copyright © 2014 by American Scientific Publishers Journal of Nanofluids All rights reserved. Vol. 3, pp. 247–253, 2014 Printed in the United States of America (www.aspbs.com/jon) Hall Effect on the Thermal Instability in a Horizontal Layer of Nanofluid Ramesh Chand1, ∗ and G. C. Rana2 1Department of Mathematics, Government College, Dhaliara (Kangra) 177103, Himachal Pradesh, India 2Department of Mathematics, Government College, Nadaun (Hamirpur) 177033, Himachal Pradesh, India Hall effect on the thermal instability in a horizontal layer of nanofluid is investigated. The model used incor- porates the effect of Brownian diffusion, thermophoresis and magnetophoresis. The Eigen value problem is solved by employing the Galerkin weighted residuals method. A linear stability theory based upon normal mode analysis is used to find expressions for Rayleigh number for a layer of nanofluid confined between two free–free boundaries. The influence of Hall effect and other parameters on the stability is investigated both analytically and graphically. It is found that Hall effect destabilizes fluid layer for both cases of bottom-heavy distribution and top-heavy distribution of nanoparticles, while the magnetic field and Lewis number have stabilizing effect on system. KEYWORDS: Hall Effect, Nanofluid, Convection, Lewis Number, Chandrasekhar Number, Galerkin Method, Prandtl ARTICLE Number. 1. INTRODUCTION IP: 192.168.39.211 On: Sun,pointed 26 Sep out 2021 that 22:03:21 in the absence of turbulent effect, only The theory of nanofluid was first proposedCopyright: by American Choi1 ScientificBrownian Publishers diffusion; thermophoresis are important mecha- which deals with a fluid containing small amountDelivered of uni- bynisms Ingenta in nanofluids. Rainbow13 and Winslow14 found that formly dispersed and suspended nanometer-sized parti- it is possible to influence the viscosity and/or apply forces cles in base fluid. Nanoparticles used in nanofluid are inside non-magnetic fluids with magnetic fields. This was typically made of oxide ceramics (Al2O3, CuO), metal accomplished by dispersing ferromagnetic particles in the carbides (SiC), nitrides (AlN, SiN) or metals (Al, Cu) fluid. Since this discovery, ferrofluids and magnetorheolog- etc. Base fluids are water, ethylene or tri-ethylene-glycols ical fluids have been applied to a wide variety of mechani- and other coolants, oil and other lubricants, bio-fluids, cal and electromechanical applications in sensors, motors, polymer solutions, other common fluids. Typical dimen- dampers, seals, bearings, and brakes.15 It should be noted sion of the nanoparticles is in the range of a few that traditional magnetorheological fluids do not qualify as to about 100 nm. Philip and Shima2 studied the ther- nanofluids. mal properties while Keblinski et al.3 studied the ther- Magnetic fluids (ferromagnetic fluid) are kinds of mal conductivity of nanofluids and it was found that special nanofluids. They are stable colloidal suspen- nanofluid exhibit enhanced thermal properties. These novel sions of small magnetic particles such as magnetite properties of nanofluids make them potentially useful (Fe3O4). The properties of the magnetic nanoparticles, in many applications in heat transfer enhancement,4 in the magnetic component of magnetic nanofluids, may be cooling of micro-electronic components,5 in electronics tailored by varying their size and adapting their sur- cooling,6 in industrial cooling,7 smart material,8 in nuclear face coating in order to meet the requirements of col- system,9 in automatic transmissions10 and in Radiators.11 loidal stability of magnetic nanofluids with non-polar and Buongiorno12 studied the convective transport in nanofluid. polar carrier liquids. Recently, the study of magneto- He took seven slip mechanisms that show relative veloc- hydrodynamics (MHD) became important in engineering ity between the nanoparticles and the base fluid including applications, such as in designing cooling system with inertia, Brownian diffusion, thermophoresis, diffisiophore- liquid metals, MHD generator and other devices in the sis, the magnus effect, fluid drainage and gravity. He petroleum industry, materials processing, Plasma stud- ies, nuclear reactors, geophysics, astrophysics, aeronau- tics and aerodynamics.16 17 If an electric field is applied ∗Author to whom correspondence should be addressed. Email: [email protected] right angle to magnetic field, the whole current will not Received: 28 January 2014 flow along the electric field. The tendency of the elec- Accepted: 10 March 2014 tric current of flow across an electric field in the pres- J. Nanofluids 2014, Vol. 3, No. 3 2169-432X/2014/3/247/007 doi:10.1166/jon.2014.1100 247 Hall Effect on the Thermal Instability in a Horizontal Layer of Nanofluid Chand and Rana ence of magnetic field is called ‘Hall effect.’ Researchers Sherman and Sutton,18 Oberoi and Devanathan,19 Gupta,20 Sharma et al.,21 Sharma and Kumar22 have studied the Hall effect in thermal instability of different types of Newtonian and non-Newtonian fluids. The Hall effect is likely to be important in geophysical and astrophysical situation. The study of MHD flows with Hall currents has important engi- neering applications in MHD generators, Hall accelerators, refrigeration coils, electric transformers etc. The onset of convection in a horizontal nanofluid layer heated from below under various assumptions was stud- ied by Tzou,23 24 Vadasz,25 Alloui et al.,26 Kuznetsov and Fig. 1. Geometrical configuration of the problem. Nield,27 Nield and Kuznetsov,28–32 Kim et al.,33 Sheu,34 35 Chand and Rana,36–38 Chand et al.39 40 and Chand41 42 on the basis of the transport equations of Buongiorno.12 Xuan (iii) Nanoparticles are considered to be spherical in shape; and Li43 investigated convective heat transfer and flow (iv) No chemical reactions; features of Cu-water nanofluid. They observed that the (v) Nanofluid is incompressible, Newtonian and laminar; suspended nanoparticles remarkably enhance heat transfer (vi) Radiation heat transfer between the sides of wall is process and the nanofluid has larger heat transfer coeffi- negligible when compared with other modes of the heat cient than that of the original base liquid under the same transfer; Reynolds number. The heat transfer feature of a nanofluid (vii) Particle chain formation of the nanofluid fluid in increases with volume fraction of nanoparticles. Recently presence of external applied magnetic field. El-Aziz44 studied the effects of Hall current on the flow According to the works of Chandrasekhar,16 Tzou23 24 and heat transfer of a nanofluid and found that Hall effect and Sharma and Kumar22 the equations of continuity and enhances the heat transfer rate. Magnetic nanoparticles motion under the Boussinesq approximation are have recently got wide interest in many fields. Nakano et al.,45 Lai et al.,46 Singh andIP: Lillard, 192.168.39.21147 Zhang et On: al.48 Sun, 26 Sep 2021 22:03:21 · q = 0 (1) reported the biomedical applications ofCopyright: nanofluids American in drug Scientific Publishers Delivered by Ingentadq =− + + − − − delivery and anticancer drugs system and Mahendran and 0 p p 1 01 T T0g 49 dt Philip used nanofluid based optical sensor for rapid visual inspection of defects in structures such as rail tracks + 2q + e H · H (2) 4 and pipelines. ARTICLE Keeping in mind the various applications of nanofluid where d/dt = /t + q · is stands for convection and Hall effect as mentioned above, our aim in this paper derivative while q, p, , 0, , e and stands for fluid is to study effects the Hall effect on the onset of convec- velocity, hydrostatic pressure, density of nanofluid, density tion in a layer of nanofluid confined between two free-free of the nanofluid at reference temperature, viscosity mag- boundaries. netic permeability and the coefficient of thermal expansion respectively. 2. MATHEMATICAL FORMULATIONS AND The equation of energy is given as PERTURBATION EQUATIONS T c + q · T Consider an infinite horizontal nanofluid layer of thickness t ‘d’ which is bounded by planes z = 0 and z = d and heated D from below. The layer of fluid is acted upon by gravity = 2 + · + T · km T cpDB T T T (3) force g (0 0 −g) and a uniform vertical magnetic field H T1 (0 0H). The temperature (T ) and volumetric fraction () where c is the heat capacity of nanofluid, c is the of nano particles at z = 0 will be taken as T and while p 0 0 heat capacity of nano particles and k is thermal conduc- T and are taken at z = d, where (T >T ) as shown in m 1 1 0 1 tivity of the fluid. Figure 1. The reference temperature is taken to be T . The 1 The equation of continuity for the nanoparticles is mathematical equations describing the physical model are based upon the following assumptions: D + q · = D 2 + T 2T (4) (i) Themophysical properties expect for density in the t B T buoyancy force (Boussinesq Hypothesis) are considered to 1 be constant; where DB is the Brownian diffusion coefficient, given by (ii) The fluid phase and nanoparticles are in thermal equi- Einstein-Stokes equation and DT is the thermoporetic dif- librium state; fusion coefficient of the nanoparticles. 248 J. Nanofluids, 3, 247–253, 2014 Chand and Rana Hall Effect on the Thermal Instability in a Horizontal Layer of Nanofluid = − Maxwell equations are ratio, NB cp1 0/cf is modified particle- density increment, = / is ratio of Prandtl number to dH C 1 =H·q+ 2H − × ×H×H (5) magnetic Prandtl number. dt 4Ne The dimensionless boundary conditions are ·H=0 (6) = = = = w 0T T00 at z 0 and (14) where , C, N , e, stand for the electrical resitivity, speed =