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 engineering 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 studied 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 = = = = w 0T T11 at z 1 of light, electron number density and charge of electron respectively. 2.1. Basic Solutions We assume the temperature and volumetric frac- The basic state of the nanofluid is assumed to be time tion of the nano particles are constant. Thus boundary independent and is described by conditions16 are quvw= 0p = pz H = H e T = T z = = = = z b w 0T T00 at z 0 (7) = z = = = = b and w 0T T11 at z d The approximate solution for the governing equation is We introduce non-dimensional variables as obtained as xyz u v w xyz = quvw = q d T = 1 − z and = z (15) d b b 2 These results are identical with the results obtained d − t = t p = p = 0 27 36–38 2 − by Kuznetsov and Nield, Chand and Rana and d 1 0 Nield and Kuznetsov.30 31 ARTICLE T − T H T = 1 H = − T0 T1 H0 2.2. Perturbation Solutions To study the stability of the system, we superimposed where = k /c is thermal diffusivity of the fluid. m infinitesimal perturbations on the basic state, which are of Equations (1)–(6) in a dimensionlessIP: 192.168.39.211 form can On: be Sun, 26 Sep 2021 22:03:21 Copyright: American Scientificthe forms Publishers written as Delivered by Ingenta q =0+qq =0+q T =T +T = + · q = 0 (8) d d b b p =p +p H=He +hh h h 1 dq b z x y z with =− + 2 − ˆ + ˆ p q Rmez RaT ez = − = Pr dt Tb 1 zb z (16) −Rneˆ + QH · H (9) z 1 There after dropping the dashes for simplicity. T N + q · T = 2T + B · T Using the Eq. (16) in the Eqs. (8)–(13), we obtain the t Le linearized perturbation (neglecting the product of the prime N N quantities) equations as: + A B T · T (10) Le · q = 0 (17) 1 N + · = 2 + A 2 q T (11) 1 q h t Le Le =− + 2 + ˆ − ˆ + p q RaT ez Rnez 1Q dH √ Pr t z = H · q + 2H − M (18) dt 1 1 T N T 2N N T × × H × H (12) − w = 2T + B − − A B (19) t Le z z Le z · H = 0 (13) 1 N + w = 2 + A 2T (20) where non-dimensional parameters are defined as fol- t Le Le = = h w √ h lows Pr / is Prandtl number, Le /DB = + 2h − M × (21) = 3 − 1 1 is Lewis number, Ra gd T0 T1/ is t z z = + − 3 Rayleigh number, Rm p0 1 0gd / · h = 0 (22) = − is basic-density Rayleigh number, Rn p − 3 1 0gd / is Nanoparticle Rayleigh number, The boundary conditions are: = 2 2 = Q eH d /40 is Chandrasekhar number, M = = = = 2 2 = w 0T 00atz 0 and CH0 /4Ne is Hall effect parameter, NA (23) − − = = = = DT T0 T1/DBT11 0 is modified diffusivity w 0T 00atz 1

J. Nanoﬂuids, 3, 247–253, 2014 249 Hall Effect on the Thermal Instability in a Horizontal Layer of Nanoﬂuid Chand and Rana

It will be noted that the parameter Rm is not involved in N 2N N N W + D2−a2+ B D− A B D − B D=0 (31) these and subsequent equations. It is just a measure of the Le Le Le basic static pressure gradient. N 1 By making use of Eqs. (17) and (22), Eqs. (18) and (21) W − A D2 −a2− D2 −a2−n =0 (32) give Le Le 1 where D = d/dz and a = k2 + k2 is dimensionless the 2 − 2 − − QD2 2w x y 1 t Pr t 1 resultant wave number. √ The boundary conditions of the problem in view of nor- + 2 − 2 − 2 − 2 2 2 = 1 Ra T Rn Q M D 0 mal mode analysis are t H H 1 (24) 2 W =0DW =0=0=0atz=01 (33) 1 − 2 − Q =0 (25) Pr t 1 z 4. METHOD OF SOLUTION √ hz The Galerkin weighted residuals method is used to obtain 2 − + − M2 =0 (26) 1 z z 1 z an approximate solution to the system of Eqs. (30)–(32) √ w with the corresponding boundary conditions (33). In this 2 − h − M + =0 (27) method, the test functions are the same as the base (trial) 1 t z 1 z z functions. Accordingly W , and are taken as = − = − where v/x u/y and hy/x hx/y stand n n n for the z-components of vorticity and current density = = = W ApWp Bpp Cpp (34) respectively. p=1 p=1 p=1 Eliminating , and h from Eqs. (24)–(27), we have z = Where Ap, Bp and Cp are unknown coefﬁcients, p 2 1 1 2 3N and the base functions Wp, p and p are − 2 2 − + Q 2 − D2 Pr t 1 t 1 1 t assumed in the following form IP: 192.168.39.211 On: Sun, 26 Sep 2021 22:03:21 1 Copyright: American ScientificW = Publisherspz = pz = pz + 2M − 2 2D2 2 − p sin p sin p sin (35) 1 Pr t 1 t Delivered by Ingenta such that W , and satisfy the corresponding bound- 1 p p p × 2 − − 2 + 3 2 4 2 W 1QD 1 MQ D w ary conditions. Using expression for , and in Pr t W Eqs. (30)–(32) and multiplying ﬁrst equation by p sec- 2 ond equation by p and third by p and integrating in

ARTICLE 1 + − 2 2 − + Q 2 − D2 Pr t 1 t 1 1 t the limits from zero to unity, we obtain a set of 3N linear homogeneous equations in 3N unknown A , B and C ; 1 p p p + 2 − 2 2 2 2 − p = 1 2 3N. For existing of non trivial solution, the 1M D 1 Pr t t vanishing of the determinant of coefficients produces the × 2 − 2 = characteristics equation of the system in term of Rayleigh RaH T RnH 0 (28) number Ra. 3. NORMAL MODES ANALYSIS Analyzing the disturbances into the normal modes and 5. LINEAR STABILITY ANALYSIS assuming that the perturbed quantities are of the form For one term Galerkin approximation gives the expression for Rayleigh number Ra as wT= W z z zexpik x + ik y (29) x y 1 Ra = 2 + a23 + 2 + a22Q where k , k are wave numbers in x and y directions a2 x y respectively MQ42 + a22 Using Eqs. (29), (28), (19) and (20) become − 2 + 2 2 + 2 + 2 + 2 2 Q 1M a a − D2 −a23 + 2QD2 −a2D2 − 2MD2 −a22D2 − + 1 1 1 NA LeRn (36) × D2 −a22− QD2+ 3MQD2 −a2D4 1 1 1 Equation (36) expresses the thermal Rayleigh number Ra 2 − 2 + − 2 − 2 3 + 2 2 − 2 2 D a W 1D a 1 QD a D as a function of dimensionless wave number a and the − 2 2 − 2 2 2 2 − 2 parameters Hall effect M, Chandrasekhar number Q (mag- 1 MD a D 1D a netic field), Lewis number Le, modified diffusivity NA, ×−a2Ra+a2Rn=0 (30) nanoparticles Rayleigh number Rn. It is also noted that

250 J. Nanoﬂuids, 3, 247–253, 2014 Chand and Rana Hall Effect on the Thermal Instability in a Horizontal Layer of Nanoﬂuid

5000 parameter NB does not appear in the equation, thus instability is purely phenomenon due to buoyancy coupled with Le = 500, Q = 50, N = 5, the conservation of nanoparticles. It is independent of the 4500 A Rn = –1, σ1 = 1.2 contributions of Brownian motion and thermophoresis to the thermal energy equation. The parameter NB drops out 4000 because of an orthogonal property of the first order trail functions and their first derivatives. 3500 In the absence of Hall effect and magnetic field the Rayleigh number Ra for steady onset is Rayleigh Number 3000 M = 0.2 M = 0.4 2 + a23 M = 0.6 Ra = − N + LeRn (37) a2 A 2500

This is the good agreement of the result as obtained by 2000 Tzou [23, 24] and Chand and Rana [36–38]. The inter- 123456 weaving behaviors’ of Brownian motion and thermopore- Wave Number sis of nanoparticles evidently does not change the critical Fig. 2. Variation of thermal Rayleigh number with wave number for size of the Bénard cell at the onset of instability. As such, different values of Hall effect parameter M. the critical size is not a function of any thermo physical properties of nanofluid. 6. RESULTS AND DISCUSSION In the absence of Hall effect magnetic field magnetic Equation (36) expresses the thermal Rayleigh number Ra field and nanoparticles i.e., for ordinary fluid, we have as a function of dimensionless wave number a and the ARTICLE parameters Hall effect M, Chandrasekhar number Q (mag- 2 + a23 Ra = netic field), Lewis number Le, modified diffusivity N 2 A a both cases of bottom-heavy distribution (Rn < 0) and top- Consequently critical Rayleigh numberIP: 192.168.39.211 is given by Ra On:= Sun,heavy 26 Sep distribution 2021 22:03:21 of nanoparticles (Rn > 0). Now we have 272/4. Copyright: American Scientificdiscussed Publishers the results graphically. The convection curves This is well known result derived by Chandrasekhar.Delivered16 byin Ingenta (Ra a) plane for various values of Hall effect param- To study the effect of Hall effect M, magnetic field Q, eter M and fixed values of other parameters is shown in Figure 2. It has been found that the Rayleigh number Lewis number Le, modified diffusivity ratio NA for both cases of bottom-heavy distribution (Rn < 0) and top-heavy decreases with increase in the value of Hall effect param- distribution of nanoparticles (Rn > 0), we examine the eter M, thus Hall effect has destabilizing effect on fluid layer for the case of both bottom-heavy distribution (neg- behaviour of Ra/M Ra/Q, Ra/Le and Ra/NA analytically. ative value of Rn) and top-heavy (positive value of Rn) Equation (36) yields distribution of nanoparticles. Figure 3 shows the variation of thermal Rayleigh number Ra 2 + a2Q26 + 2 + a23Q4 with wave number for different values of Chandrasekhar =− < 0 2 + 2 2 + 2 + 2 + 2 2 2 number Q. It is found that the thermal Rayliegh number M Q 1M a a

Thus Hall effect has a destabilizing effect on the layer of 14000 nanoﬂuid. Q = 100 12000 Le = 500, N = 10, Rn = –1, Equation (36) also yields A M = 0.01, σ1 = 1.2 Ra 10000 = 2 + 2 2 + 2 + 2 + 2 2 2 Q 1M a a Q 8000 Q = 50 − 2 2 + 2 2 + 2 + 2 2 2 + 2 1M a a M a 6000 2 2 2 2 2 2 2 2 Q = 20

× + + + + Rayleigh Number Q 1M a a 4000

Since M<<1, thus Ra/Q > 0 which imply that mag- 2000 netic field has stabilizing effect on the layer of nanofluid. 0 0123456 Also we have Ra/Le > 0 and Ra/NA > 0, thus Wave number Lewis number Le and modified diffusivity ratio NA have stabilizing on the layer of nanofluid in the presence of Hall Fig. 3. Variation of thermal Rayleigh number with wave number for Effect. different values of Chandrasekhar number Q.

J. Nanoﬂuids, 3, 247–253, 2014 251 Hall Effect on the Thermal Instability in a Horizontal Layer of Nanoﬂuid Chand and Rana

31000 of Rn) and top-heavy (positive value of Rn) distribution 29000 of nanoparticles.

27000

25000 7. CONCLUSIONS

23000 A linear stability analysis for a horizontal layer of nanoﬂuid in the presence of Hall effect for free-free 21000 boundaries is investigated. Galerkin residuals method is 19000

Rayleigh Number used for the stability analysis. Results has been depicted 17000 Le = 1500 both analytically graphically. The main conclusions of the Le = 1000 Q = 50, N = 5, Rn = –1 15000 Le = 500 A present work are summarized as follows: M = 0.01, σ1 = 1.2 (i) The critical cell size is not a function of any thermo 13000 1 1.5 2 2.5 3 3.5 4 4.5 physical properties of nanofluid. Wave number (ii) Instability is independent of the contributions of Brownian motion and is purely phenomenon due to buoy- Fig. 4. Variation of thermal Rayleigh number with wave number for ancy coupled with the conservation of nanoparticles. different values of Lewis number Le. (iii) The Hall effect has destabilizing effect while magnetic field, magnetic field, Lewis number and modified (Ra) increases as values of Chandrasekhar number Q diffusivity ratio have stabilizing effect on fluid layer for inceases. Therefore, magnetic field have stabilizing effect the case of both bottom-heavy distribution (negative value on fluid layer for the case of both bottom-heavy distribu- of Rn) and top-heavy (positive value of Rn) distribution of tion (negative value of Rn) and top-heavy (positive value nanoparticles. of Rn) distribution of nanoparticles. Figure 4 shows the variation of thermal Rayleigh num- NOMENCLATURE ber with wave number for different values of Lewis num- a Wave number ber Le. It is found that the thermal Rayliegh number (Ra) C Speed of light increases as values of Lewis number Le inceases. There- d Thickness of nanofluid layer fore, Lewis number Le have stabilizingIP: 192.168.39.211 effect on On: fluid Sun, 26 Sep 2021 22:03:21 D Diffusion coefficient (m2/s) layer for the case of both bottom-heavyCopyright: distribution American (neg- Scientific PublishersB Delivered by Ingenta D Thermophoretic diffusion coefficient ative value of Rn) and top-heavy (positive value of Rn) T e Charge of electron distribution of nanoparticles. g Acceleration due to gravity (m/s2) Figure 5 shows the variation of thermal Rayleigh num- H Magnetic field ber with wave number for different values of modified k Thermal conductivity (W/mK) ARTICLE diffusivity ratio (N ). It is found that the thermal Rayliegh m A Le Lewis number number (Ra) increases as values of modified diffusiv- N Electron number density ity ratio (N ) increases. Therefore, modified diffusivity A N Modified diffusivity ratio ratio (N ) have stabilizing effect on fluid layer for the A A N Modified particle-density increment case of both bottom-heavy distribution (negative value B M Hall effect parameter p Pressure

8400 Pr Prandtl number q Fluid velocity (m/s) 8200 Q Chandrasekhar number for nanoﬂuid

Le = 500, M = 0.01, Ra Thermal Rayleigh number 8000 σ Q = 50, Rn = –1, 1, = 1.2 Rac Critical Rayleigh number Rm Density Rayleigh number 7800 Rn Concentration Rayleigh number 7600 t Time

Rayleigh Number T Temperature 7400 (u v w) Fluid velocity components (m/s) (xyz) Space coordinates (m) 7200 NA = 10 N = 1 7000 A Greek Symbols 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 Thermal expansion coefﬁcient (1/K) Wave number e Magnetic permeability Fig. 5. Variation of thermal Rayleigh number with wave number for Viscosity (kg/ms) different values of modiﬁed diffusivity ratio NA. Volume fraction of the nanoparticles

252 J. Nanoﬂuids, 3, 247–253, 2014 Chand and Rana Hall Effect on the Thermal Instability in a Horizontal Layer of Nanoﬂuid

Density of the nanoﬂuid (kg/m3) 14. W. M. Winslow, J. Appl. Phys. 20, 1137 (1949). 15. J. L. Neuringer and R. E. Rosensweig, Physics of Fluids 7, 1927 0 Density of the nanoﬂuid at reference temperature 3 (1964). p Density of nanoparticles (kg/m ) 2 16. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Thermal diffusivity (m /s) Oxford University Press, Dover Publication, New York (1961). Dimensionless frequency 17. A. Kent, Phys. Fluids 9, 1286 (1966). Electrical resistivity 18. A. Sherman and G. W. Sutton, Magnetohydrodynamics, North- Western University Press, Evanston, Illinois (1962). 19. C. Oberoi and C. Devanathan, Proc. Summer Seminar in Magneto- Superscripts hydrodynamics, IIT Bangalore (1963). Non-dimensional variables 20. A. S. Gupta, Rev. Roumaine Math. Pures Appl. 12, 665 (1967). Perturbed quantity 21. R. Sharma, Sunil, and S. Chand, Indian J. Pure Appl. Math. 31, 49 (2000). Subscripts 22. R. Sharma and P. Kumar, Arch. Mech. 48, 199 (1996). p Particle 23. D. Y. Tzou, Int. J. Heat and Mass Transfer 51, 2967 (2008a). 24. D. Y. Tzou, ASME J. Heat Transfer 30, 372 (2008b). 0 Lower boundary 25. P. Vadasz, ASME J. Heat Transf. 128, 465 (2006). 1 Upper boundary 26. Z. Alloui, P. Vasseur, and M. Reggio, Int. J. Thermal Science 50, 385 H Horizontal plane. (2010). 27. A. V. Kuznetsov and D. A. Nield, Transp. Porous Medium 81, 409 (2010). Acknowledgments: The authors are grateful to the 28. D. A. Nield and A. V. Kuznetsov, Int. J. Heat Mass Transf. 52, 5796 reviewers for their valuable comments and suggestions to (2009). improve the present paper. 29. D. A. Nield and A. V. Kuznetsov, European Journal of Mechanics B/Fluid 29, 217 (2010a). 30. D. A. Nield and A. V. Kuznetsov, J. Heat Transfer 132, 052405

(2010b). ARTICLE References and Notes 31. D. A. Nield and A. V. Kuznetsov, J. Heat Transfer 132, 074503 1. S. Choi, Enhancing Thermal Conductivity of Fluids with Nanopar- (2010c). ticles, edited by D. A. Siginer and H. P. Wang, Developments 32. D. A. Nield and A. V. Kuznetsov, Transport in Porous Media 87, 765 and Applications of Non-Newtonian Flows, ASME FED (1995), (2011). Vol. 231/MD–66, p. 99. IP: 192.168.39.211 On: Sun,33. 26J. Sep Kim, Y.2021 T. Kang, 22:03:21 and C. K. Choi, Phys. Fluid 16, 2395 (2011). 2. J. Philip and P. D. Shima, Adv., Coll. Interf. Sci. 183–184, 30 (2012). 34. L. J. Sheu, Transp. Porous Med. 88, 461 (2011a). 3. P. Keblinski, L. W. Hu, and J. L. Alvarado,Copyright:J. Appl. American Phys. Scientific Publishers Delivered by35. IngentaL. J. Sheu, World Sci. Eng. Technol. 58, 289 (2011b). 106, 094312 (2009). 36. R. Chand and G. C. Rana, Transp. Porous Med. 95, 269 (2012a). 4. N. Prabhat, J. Buongiorno, and H. Lin-Wen, Journal of Nanofluids 37. R. Chand and G. C. Rana, Int. J. Heat and Mass Transfer 55, 5417 1, 55 (2012). (2012b). 5. C. T. Nguyen, G. Roy, N. Galanis, and S. Suiro, Heat transfer 38. R. Chand and G. C. Rana, J. Fluids Eng. 134, 121203 (2012c). enhancement by using Al2O3-water nanofluid in a liquid cooling 39. R. Chand, G. C. Rana, and S. Kumar, Int. J. Appl. Mech. Engg. 18 system for microprocessors, Proceedings of the 4th WSEAS Inter- 631 (2013a). national Conference on Heat Transfer, Thermal Engineering and 40. R. Chand, G. C. Rana, A. Kumar, and V. Sharma, J. Sci. Technol. Environment (2006), pp. 103–108. 5, 32 (2013b). 6. C. Y. Tsai, H. T. Chien, P. P. Ding, B. Chan, T. Y. Luh, and P. H. 41. R. Chand, Int. J. Appl. Math. Mech. 9, 70 (2013a). Chen, Materials Letters 58, 1461 (2004). 42. R. Chand, Int. J. Nanoscience 12, 1350038 (2013b). 7. J. Routbort, Argonne National Lab, St. Gobain Corp., Michellin, 43. Y. Xuan and Q. Li, ASME J. Heat Transfer 125, 151 (2003). North America (2009). 44. M. A. El-Aziz, Int. J. Modern Phys. C 24, 1350044 (2013). 8. G. Donzelli, R. Cerbino, and A. Vailati, Phys. Rev. Lett. 102, 104503 45. M. Nakano, H. Matsuura, and D. Ju, Drug delivery system using (2009). nano-magnetic fluid, Proceedings of the 3rd International Confer- 9. S. J. Kim, I. C. Bang, J. Buongiorno, and L. W. Hu, Bulletin of the ence on Innovative Computing, Information and Control (ICICIC Polish Academy of Sciences-Technical Sciences 55, 211216 (2007). ’08), Dalian, Liaoning (2008). 10. S. C. Tzeng, C. W. Lin, and K. D. Huang, Acta Mechanica 179, 1123 46. J. Lai, V. M. Nelson, A. Nash, A. S. Hoffman, P. Yager, and (2005). P. Stayton, Lab Chip 9, 1997 (2009). 11. V. K. Wong and D. L. Omar, Advances in Mechanical Engineering 47. R. Singh and J. W. Lillard, Experimental and Molecular Pathology 2010, 519659 (2010). 86, 215 (2009). 12. J. Buongiorno, ASME Journal of Heat Transfer 128, 240 (2006). 48. L. Zhang, J. Xia, Q. Zhao, L. Liu, and Z. Zhang, 6, 537 (2010). 13. J. Rainbow, Trans. Am. Inst. Electr. Eng. 67, 1308 (1948). 49. V. Mahendran and J. Philip, Appl. Phys. Lett. 100, 073104 (2012).

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