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Mixed convection of MHD flow in nanofluid filled and partially heated wavy walled lid-driven enclosure
International Communications in Heat and Mass Transfer 86 (2017) 42–51
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Mixed convection of MHD flow in nanofluid filled and partially heated wavy walled lid-driven enclosure
MARK
Hakan F. Öztopa, Ahmad Sakhriehb,c, Eiyad Abu-Nadad,⁎, Khaled Al-Saleme a
Department of Mechanical Engineering, Technology Faculty, Firat University, Elazig, Turkey Department of Mechanical and Industrial Engineering, American University of Ras Al Khaimah, 10021, United Arab Emirates c Mechanical Engineering Department, The University of Jordan, Amman 11942, Jordan d Department of Mechanical Engineering, Khalifa University of Science and Technology, P.O. Box 127788, Abu Dhabi, United Arab Emirates e Department of Mechanical Engineering, College of Engineering, King Saud University, Riyadh, Saudi Arabia b
A R T I C L E I N F O
A B S T R A C T
Keywords: Wavy walled cavity Lid-driven flow Nanofluids Magnetic field Partial heating
A computational work has been done to investigate the effects of mixed convection of MHD flow in nanofluid filled and partially heated wavy walled lid-driven enclosure. Finite difference method is used to solve governing equations of mixed convection for different parameters as Hartmann number, Richardson number, nanoparticle volume rate in partially heated and wavy walled enclosure. It is found that the rate of heat transfer decreases with increasing the Hartmann number. The rate of heat transfer can be enhanced or reduced by increasing the volume fraction of nanoparticles based on Hartmann and Richardson numbers.
1. Introduction Mixed convection heat transfer and fluid flow in complex shaped geometry are important in engineering due to its wide applications, such as cooling of electronical devices, heat exchangers, solar collectors, cooling or heating of buildings. Besides, partial heating of confined spaces attracted many researchers recently. Partial heater was applied in some papers in open literature as seen in review of Oztop et al. [1]. There are many example on partial cooling or heating applications as Guimaraes and Menon [2]. Pioneer of computational studies of natural convection in a nanofluid filled enclosures is Khanafer et al. [3]. Oztop and Abu-Nada [4] conducted a computational solution on natural convection in enclosures with Cu-water nanofluid. They found that heat transfer increases with increasing of length of partial heater and nanoparticle volume fraction. Also, effects of combined convection (natural + forced convection) are studied in nanofluid filled systems. In this context, Mehrez et al. [5] focused on entropy generation analysis in the assisting flow of Cu-water nanofluid in an inclined open cavity and they observed that the main important parameter is the inclination angle. Hussain et al. [6] investigated the effects of magnetic field on entropy generation due to mixed convection of water-alumina nanofluid flow in a double lid driven cavity with discrete heating.
⁎
In some problems in engineering, the geometry can be curvilinear. Sheremet et al. [7] solved a problem computationally to make analysis of MHD free convection in a wavy open porous tall cavity filled with nanofluids. The cavity has a corner heater. They obtained that heat transfer enhancement with Rayleigh number (Ra) and heat transfer reduction with Hartmann (Ha) number, while magnetic field inclination angle leads to non-monotonic changes of the heat transfer. Heated wavy walled cavity was applied to a melting problem by Kousksou et al. [8]. In their case, the bottom wavy wall is heated isothermally and the problem solved via finite volume method. Waviness of the wall plays important role on the temperature distribution and flow field and they observed that the rate of the melting increases with the elevation in the magnitude of the amplitude value of the wavy surface. As an original work on insulated wavy walled cavity with natural convection of Al2O3/water nanofluids is studied by Abu-Nada and Oztop [9]. Influence of a magnetic field on the natural convection and entropy generation was studied by Cho [10] for Cu–water nanofluid in a closed space with complex-wavy surfaces. Their obtained results showed that the mean Nusselt number decreases and entropy generation increases with an increasing wave amplitude. Abu-Nada and Chamkha [11] solved a problem of mixed convection in a water-CuO filled lid-driven cavity with wholly heated wavy wall. They observed that heat transfer
Corresponding author. E-mail addresses: [email protected] (A. Sakhrieh), [email protected] (E. Abu-Nada).
http://dx.doi.org/10.1016/j.icheatmasstransfer.2017.05.011
0735-1933/ © 2017 Elsevier Ltd. All rights reserved.
International Communications in Heat and Mass Transfer 86 (2017) 42–51
H.F. Öztop et al.
B Bo cP g H Ha h k Nu Pr qw Ra Re Ri T u, v U, V x′ , y′ x, y
geometry parameter magnetic field strength specific heat at constant pressure (kJ kg− 1 K− 1) gravitational acceleration (m s− 2) height and width of the cavity (m) Hartmann number local heat transfer coefficient (W m− 2 K− 1) thermal conductivity (W m− 1 K− 1) Nusselt number Prandtl number heat flux (W m− 2) Rayleigh number Reynolds number Richardson number temperature (K) dimensional x and y components of velocity (m s− 1) dimensionless x and y components of velocity dimensional coordinates (m) dimensionless coordinates
Subscripts avg C cp f H nf p w
Greek symbols α
thermal expansion coefficient (K− 1) numerical tolerance Nanoparticle volume fraction kinematic viscosity (m2 s− 1) dimensionless temperature dimensional stream function (m2 s− 1) dimensionless stream function ∞∞φσ dimensional vorticity (s− 1) dimensionless vorticity density (kg m− 3) electrical conductivity (Ω− 1 m− 1) volume fraction of nanoparticles, (dimensionless) dynamic viscosity (N s m− 2)
β ε φ ν θ ψ Ψ ω Ω ρ σ φ μ
Nomenclature
average cold centi poise fluid hot nanofluid particle wall
fluid thermal diffusivity (m2 s− 1)
3. Governing equations and problem formulation
enhancement with Rayleigh number and heat transfer reduction with Hartmann number, while magnetic field inclination angle leads to nonmonotonic changes of the heat transfer rate. Bondereva et al. [12] solved a problem numerically to simulate the heat transfer in a wavy walled and nanofluid filled cavity with a corner heater. They found that an increase in the Ha number leads to an attenuation of the convective flow and heat transfer reduction. Study on laminar mixed convection was performed by Akbarinia et al. [13] to see the effects of nanofluid in horizontal curved tubes. Other related works can be found in refs. Sheikholeslami et al. [14], Sathiyamoorthy and. Chamkha [15] and Kefayati et al. [16]. The main aim of this work is to study the effects of magnetic field on mixed convection heat transfer and fluid flow in a wavy walled cavity with lid-driven wall filled with nanofluid. The originality of this work is the heated wavy bottom wall.
Fig. 1 shows a schematic diagram of the wavy walled differentially heated enclosure in a presence of magnetic field. The nanofluid is assumed incompressible and the flow is assumed to be laminar. It is assumed that the base fluid (i.e. water) and the nanoparticles are in thermal equilibrium and no slip occurs between them. Thermo-physical properties of the nanofluid are assumed to be constant except for the density variation, which is approximated by the Boussinesq model. The governing equations for the laminar, two-dimensional, steady state natural convection in the presence of magnetic field in terms of the stream function-vorticity formulation are written as Vorticity
∂ ⎛ ∂ψ ⎞ ∂ ⎛ ∂ψ ⎞ μnf ⎛ ∂ω ∂ω ⎞ ∂T ⎞ ω ω − = + + βnf g ⎛ ρnf ⎝ ∂x ′2 ∂x ′ ⎝ ∂y′ ⎠ ∂y′ ⎝ ∂x ′ ⎠ ∂y′2 ⎠ ⎝ ∂x ′ ⎠ σnf Bo2 ∂2ψ + ρnf ∂x 2 ⎜
2. Definition of considered physical model The considered physical model is depicted in Fig. 1. In this model, the wavy wall is heated partially. The gravity acts in −y direction and magnetic field acts parallel to x-axis. The height of the hill is given by B and length of it is L which are governing parameters on heat and fluid flow. The ceiling of the cavity moves from left to right with constant velocity. Geometrically, the length and width of the cavity, which are given by W and H, are equal. The vertical walls are considered adiabatic.
⎟
⎜
⎟
(1)
Energy
∂ ⎛ ∂ψ ⎞ ∂ ⎛ ∂ψ ⎞ ∂ 2T ∂ 2T ⎞ T T − = αnf ⎛ 2 + x y′2 ⎠ ∂x ′ ⎝ ∂y′ ⎠ ∂y′ ⎝ ∂x ′ ⎠ ∂ ∂ ′ ⎝ ⎜
⎟
⎜
⎟
(2)
Kinematics
∂ 2ψ ∂ 2ψ + = −ω ∂x ′2 ∂y′2
43
(3)
International Communications in Heat and Mass Transfer 86 (2017) 42–51
H.F. Öztop et al.
the Chon et al. model [17]:
knf
df = 1 + 64.7φ0.7640 ⎛⎜, ⎞⎟ d ⎝ p⎠
kbf
0.3690
⎛, kf ⎞ ⎜ ⎟ ⎝ kp ⎠
0.7476
Pr 0.9955 Re1.2321 (8)
This model is found to be appropriate for studying heat transfer enhancement using nanofluids (Akbarinia and Behzadmehr [13], AbuNada et al. [18]). The correlation for dynamic viscosity of CuO-water nanofluid is derived using the available experimental data of Nguyen et al. [19]. The correlation that gives the viscosity of nanofluid as a function of temperature and volume fraction of nanoparticles is given in [19,20] as:
μCuO (cp) = −0.6967 + −
19652.74 T3
15.937 T
1356.14 T2
+ 1.238φ +
φ
− 0.259φ2 − 30.88 T
φ2
+ 0.01593φ3 + 4.38206 T + 147.573
φ T2
(9) The horizontal and vertical velocities are given by the following relations,
∂ψ , ∂y′
u=
v=−
(10)
∂ψ . ∂x ′
(11)
The following dimensionless variables are introduced:
ψ y′ u v T − TC x′ ω ;y= ;Ω= ;Ψ= ;U= ;V= ;θ= Up H Up Up TH − TC H H Up H
x=
Using the previous dimensionless quantities, the governing equations are re-written in dimensionless form as:
∂ ∂x
( ) ∂Ψ
Ω ∂y
−
∂ ∂y
(Ω ) = ∂Ψ ∂x
⎡
⎤
1 ⎢ 1 ⎥ , Re ⎢ ⎛ ρp ⎞ ⎥ ⎜ (1 − φ) + φ ⎟ ⎢ ⎝ ρf ⎥ ⎠⎦ ⎣ β
p + Ri ⎡φ β + (1 − φ) ⎤ ⎣ f ⎦
(
( )+ ∂θ ∂x
1 ∂ ⎛ ∂Ψ ⎞ ∂ ⎛ ∂Ψ ⎞ θ θ − = (ρcp )p ∂x ⎝ ∂y ⎠ ∂y ⎝ ∂x ⎠ (1 − φ) + φ (ρc ) ⎜
Fig. 1. a) Problem geometry, b) Grid distribution.
∂2Ω ∂x 2
⎟
p f
+
∂2Ω ∂y 2
)
Ha2 ∂2Ψ Re ∂x 2
(12)
2 2 ⎛∂ θ + ∂ θ⎞ 2 x y2 ⎠ ∂ ∂ ⎝
⎜
⎟
∂ 2Ψ ∂ 2Ψ + = −Ω 2 ∂x ∂y 2 αnf =
(14)
where the dimensionless numbers are given as.
keff (ρcp )nf
(13)
(4)
Ra =
The effective properties of nanofluid are expressed as (Khanafer et al. [3]; Abu-Nada and Chamkha [11]):
νf σnf gβH 3 (TH − TL ) Ra ; Pr = ; Ha = Bo H , Ri = νf αf αf ρnf νf Pr Re2
(15)
The dimensionless horizontal and vertical velocities are converted
ρnf = (1 − φ) ρf + φρp
(5)
to:
βnf = (1 − φ) βf + φβp
(6)
U=
(ρcp )nf = (1 − φ)(ρcp )f + φ (ρcp )p
(7)
∂Ψ , ∂y
V=−
σnf = (1 − φ) σf + σβp Effective thermal conductivity of the nanofluid is approximated by
∂Ψ . ∂x
The dimensionless boundary conditions can be written as:
44
(16)
International Communications in Heat and Mass Transfer 86 (2017) 42–51
H.F. Öztop et al.
Fig. 2. Validation of Isotherms and streamlines for natural convection with magnetic field, Ha = 50, Pr = 0.025, and Ra = 105 (a) present results isotherms (b) present results streamlines (c) Chamkha and Sathiyamoorthy [8] isotherms (d) Chamkha and Sathiyamoorthy [8]streamlines.
I.
On the left wall i. e. , x = 0, Ψ = 0, Ω = −
II.
On the right wall i. e. , x = 1, Ψ = 0, Ω = −
III.
On the top walls: Ψ = 0, Ω = −
IV.
On the bottom wall:
1. 2.
∂2Ψ dθ , =0 ∂x 2 dx
h=
∂2Ψ , ∂n2
∂2Ψ − 2, ∂y
knf = −
hH kf
qw ∂T ∂x
knf ∂θ Nu = −⎜⎛ ⎟⎞ ⎝ kf ⎠ ∂x
=0
(21)
(22)
Where (knf/kf) is calculated using Eq. (8). The average Nusselt number is defined as:
The physical quantities of interest are the local Nusselt number and the mean Nusselt number Nu . The local Nusselt numbers is defined as
Nu =
(20)
Substituting Eqs. (20), (21) into Eq. (19), and using the dimensionless quantities, the local Nusselt number along the left wall can be written as:
θ=1 dθ dy
qw TH − TL
The thermal conductivity is expressed as: (18)
∂ 2Ψ , θ=0 ∂y 2
On the heater surface Ψ = 0, Ω = − On the flat surface Ψ = 0, Ω =
∂2Ψ dθ , =0 ∂x 2 dx
1
Nu = (19)
∫ Nu (y) dy 0
(23)
To evaluate Eq. (23), a 1/3 Simpson's rule of integration is used.
The heat transfer coefficient is expressed as:
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International Communications in Heat and Mass Transfer 86 (2017) 42–51
H.F. Öztop et al.
Fig. 3. Validation of Isotherms and streamlines for mixed convection with magnetic field, Ri = 100, Pr = 0.71, Re = 100, and Ha = 25 (a) present results isotherms (b) present results (c) Kefayati et al. [9] isotherms (d) Kefayati et al. [9] streamlines.
4. Validation
for different governing parameters such as Hartmann numbers, Richardson numbers and nanoparticle volume fraction. Fig. 4 presents the streamlines (on the left) and isotherms (on the right) for different Richardson numbers for Re = 100, ϕ = 0.05, Ha = 10, B/H = 0.2 and 0.5. It is noticed that Reynolds number is fixed for whole work. As seen from the figure, the shape of isotherms fits with the geometry of the considered model up to middle of the cavity then it behaves as a temperature distribution inside a rectangular cavity. An egg shaped main circulation cell is formed near the ceiling of the cavity but its length is related with Richardson number while other parameters are constant. The flow strength is increased with increasing of Richardson number. Fig. 5 illustrates the isotherms (on the left) and streamlines (on the right) to see the effects of Hartmann numbers with different parameters (Table 1) as Ri = 1, Re = 100, ϕ = 0.05, B/H = 0.2 and 0.5. As clearly seen from the Fig. 5(a) and (b), Hartmann number makes important effect on both temperature distribution and flow field. In the absence of Hartmann number, only one cell is formed inside the cavity. For
The present numerical solution was validated with other published work in literature for natural convection with magnetic field in a square cavity against the work of Sathiyamoorthy and Chamkha [15] for the case of Ha = 50, Pr = 0.025, and Ra = 105. Further validation of the present numerical solution is validated for mixed convection with magnetic field in a square cavity against the work of Kefayati et al. [16] for the case of Ri = 100, Pr = 0.71, Re = 100, and Ha = 25. As shown from the Figs. 1 and 2, the present work agrees very well with other results reported in literature. Further validation of the present work against nanofluid filled and wavy enclosure wall surfaces is found elsewhere [9,11]. 5. Results and discussion A computational study (Fig. 3) is performed to study of effects of magnetic field in a partially heated wavy walled lid-driven enclosure
46
International Communications in Heat and Mass Transfer 86 (2017) 42–51
H.F. Öztop et al.
Fig. 4. Streamlines (on the left) and isotherms (on the right) for different Richardson number for Re = 100, ϕ = 0.05, Ha = 10, B/ H = 0.2, 0.5, a) Ri = 0.1, b) Ri = 1, c) Ri = 10.
(a)
(c)
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International Communications in Heat and Mass Transfer 86 (2017) 42–51
H.F. Öztop et al.
-0.045
0.05 0.15
5 0.2 0.35
0.00178115
5 0.4 5 5 . 0 5 6 0. 5 0.7 5 8 . 0
(a)
-0.1
0.1 5 0.2 0.3 4 0. 0. 5
0.6 0.75
(b) Fig. 5. Streamlines (on the left) and isotherms (on the right) for different Hartmann numbers for Ri = 1, Re = 100, ϕ = 0.05, B/H = 0.2, 0.5, a) Ha = 50, b) Ha = 0.
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International Communications in Heat and Mass Transfer 86 (2017) 42–51
H.F. Öztop et al.
Table 1 Variation of average Nusselt number for different parameters. Ri
ϕ
Ha
g
W
Nu
1 1 1 1 1 1 1 1 1 1 1 1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 10 10 10 10 10 10 10 10
0 0 0,05 0, 0.05 0.02 0.03 0 0.05 0.0 0.05 0.05 0.0 0.05 0.0 0.0 0.05 0.05 0.05 0.05 0.05 0.0 0.05 0.03 0.05 0.0 0.03 0.05 0.05
0 0 0 0 0 50 50 50 10 10 10 50 0.0 0.0 0.0 10 10 10 10 30 50 0.0 0.0 10 10 50 50 50 100
0.1 0.2 0.2 0,3 0.3 0.2 0.2 0.2 0.3 0.3 0.1 0.1 0.3 0.2 0.2 0.2 0.2 0.2 0.3 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.2 0.5 0.05 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
4.895 4.60 4.41 4.184 4.019 1.11 1.015 0.986 3.45 3.413 3.89 1.29 3.03 3.857 3.813 2.56 3.017 2,02 2917 1,29 1,06 7.054 6.458 6.473 5.992 5.497 5.562 5.043 3.529
Fig. 7 illustrates the variation of local Nusselt number for Reynolds number (Re = 100), B/H = 0.5 and two values of nanoparticles volume fractions ϕ = 0 and ϕ = 0.05 for the case corresponding to different geometry heights (B = 0.1, 0.2, 0.3). As shown from the figure, the local Nusselt number is strongly dependent on the top height of the geometry. The local Nusselt number decreases with increasing the height of the geometry. Moreover, the local Nusselt number decreases with increasing of the Hartmann number. The meaning of almost horizontal profile of the local Nusselt number is convection turns to conduction due to decreasing of kinetic energy when magnetic field applied to the fluid. Differences among the values increase with increasing of geometry heights.
Ha = 50, isotherms are almost parallel to each other due to retarding of flow motion. Also, two circulation cells are formed and they turn in different directions. Fig. 6 presents the isotherms (on the left) and streamlines (on the right) to see the effects of geometry and magnetic field on temperature distribution and flow field for Ri = 1 and ϕ = 0.05. Fig. 6(a) gives the results for g = 0.3 and Ha = 10. As seen from the figure, egg shaped single cell is formed inside the cavity and the flow turns in clockwise direction with ψmin = −0.09. For the strongest considered magnetic field (Ha = 50) a secondary eddy is formed in the lower portion of the cavity. Isotherms are clustered near the hill and conduction mode of heat transfer becomes dominant at right and left side of the hill due to motionless fluid. As an expected, the shape of the cavity is the main effective parameter on heat and fluid flow.
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International Communications in Heat and Mass Transfer 86 (2017) 42–51
H.F. Öztop et al.
Fig. 6. Isotherms (on the left) and Streamlines (on the right) for Ri = 1 and ϕ = 0.05, a) g = 0.3, Ha = 10, b) g = 0.1, Ha = 10, c) g = 0.1, Ha = 50.
(a)
(b)
(c)
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International Communications in Heat and Mass Transfer 86 (2017) 42–51
H.F. Öztop et al.
6. Conclusions In this numerical work, finite volume method is used to solve governing equations of sinusoidal shaped partial heater in a nanofluid filled cavity under magnetic field. Results for the streamlines and isotherms contours for various parameters were presented and discussed. The heat transfer and flow characteristics depends strongly Richardson and Hartmann numbers. It was found that heat transfer can be enhanced or reduced by increasing the volume fraction of nanoparticles based on Hartmann and Richardson numbers. Acknowledgement First and last authors extend their appreciation to the International Scientific Partnership Program ISPP at King Saud University for funding this research work through ISPP# 0030. References
(a)
[1] H.F. Öztop, P. Estellé, W.M. Yan, K. Al-Salem, J. Orfi, O. Mahian, A brief review of natural convection in enclosures under localized heating with and without nanofluids, Int. Commun. Heat Mass Transfer 60 (2015) 37–44. [2] P.M. Guimaraes, G.J. Menon, Natural nanofluid-based cooling of a protuberant heat source in a partially-cooled enclosure, Int. Commun. Heat Mass Transfer 45 (2013) 23–31. [3] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transf. 46 (2003) 3639–3653. [4] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosure filled with nanofluids, Int. J. Heat Fluid Flow 29 (2008) 1326–1336. [5] Z. Mehrez, A. ElCafsi, A. Belghith, P. Le Quéré, The entopy generation analysis in the mixed convective assisting flow of Cu-water nanofluid in an inclined open cavity, Adv. Powder Technol. 26 (2015) 1442–1453. [6] S. Hussain, K. Mehmood, M. Sagheer, MHD mixed convection and entropy generation of water-alumina nanofluid flow in a double lid driven cavity with discrete heating, J. Magn. Magn. Mater. 419 (2016) 140–155. [7] M.A. Sheremet, H.F. Oztop, I. Pop, K. Al-Salem, MHD free convection in a wavy open porous tall cavity filled with nanofluids under an effect of corner heater, Int. J. Heat Mass Transf. 103 (2016) 955–964. [8] T. Kousksou, M. Mahdaoui, A. Ahmed, A. Ait Msaad, Melting over a wavy surface in a rectangular cavity heated from below, Energy 64 (2014) 212–219. [9] E. Abu-Nada, H. Oztop, Numerical anaysis of Al2O3/water nanofluids natural convection in a wavy walled cavity, Numer. Heat Transfer 59 (2011) 403–419. [10] C.C. Cho, Heat transfer and entropy generation of natural convection in nanofluidfilled square cavity with partilally-heated wavy surface, Int. J. Heat Mass Transf. 77 (2014) 818–827. [11] E. Abu-Nada, A.J. Chamkha, Mixed convection flow of a nanofluid in a lid-driven cavity with a wavy wall, Int. Commun. Heat Mass Transfer 57 (2014) 36–47. [12] N.S. Bondareva, M.A. Sheremet, H.F. Oztop, N. Abu-Hamdeh, Heatline visualization of MHD natural convection in an inclined wavy open porous cavity filled with a nanofluid with a local heater, Int. J. Heat Mass Transf. 99 (2016) 872–881. [13] A. Akbarinia, A. Behzadmehr, Numerical study of laminar mixed convection of a nanofluid in horizontal curved tubes, Appl. Therm. Eng. 27 (2007) 1327–1337. [14] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, S. Soleimani, MHD natural convection in a nanofluid filled inclined enclosure with sinusoidal wall using CVFEM, Neural Comput. & Applic. 24 (2014) 873–882. [15] M. Sathiyamoorthy, A. Chamkha, Effect of magnetic filed on natural convection flow in a liquid gallium filled square enclosure for linearly heated side wall(s), Int. J. Therm. Sci. 49 (2010) 1856–1865. [16] G.H.R. Kefayati, M. Gorji-Bandby, H. Sjjadi, D.D. Ganji, Simulation of MHD mixed convection in al id-driven cavity with linearly heated wall, Sci. Iran. B 19 (4) (2012) 1053–1065. [17] C.H. Chon, K.D. Kihm, S.P. Lee, S.U.S. Choi, Empirical correlation finding the role of temperature and particle size for nanofluid (Al2O3) thermal conductivity enhancement, Appl. Phys. Lett. 87 (2005) 153107. [18] E. Abu-Nada, Z. Masoud, H.F. Oztop, A. Campo, Effect of nanofluid variable properties on natural convection in enclosures, Int. J. Therm. Sci. 49 (2010) 479–491. [19] C.T. Nguyen, F. Desgranges, G. Roy, N. Galanis, T. Mare, S. Boucher, H. Angue Minsta, Temperature and particle-size dependent viscosity data for water-based nanofluids - hysteresis phenomenon, Int. J. Heat Fluid Flow 28 (2007) 1492–1506. [20] E. Abu-Nada, Effects of variable viscosity and thermal conductivity of CuO-water nanofluid on heat transfer enhancement in natural convection: mathematical model and simulation, ASME J. Heat Transf. 132 (2010) 052401.
(b)
(c) Fig. 7. Variation of local Nusselt number along the heating part for different parameters for Re = 100,, B/H = 0.5 (a) g = 0.1, (b) g = 0.2, (c) g = 0.3.
51
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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt
Mixed convection of MHD flow in nanofluid filled and partially heated wavy walled lid-driven enclosure
MARK
Hakan F. Öztopa, Ahmad Sakhriehb,c, Eiyad Abu-Nadad,⁎, Khaled Al-Saleme a
Department of Mechanical Engineering, Technology Faculty, Firat University, Elazig, Turkey Department of Mechanical and Industrial Engineering, American University of Ras Al Khaimah, 10021, United Arab Emirates c Mechanical Engineering Department, The University of Jordan, Amman 11942, Jordan d Department of Mechanical Engineering, Khalifa University of Science and Technology, P.O. Box 127788, Abu Dhabi, United Arab Emirates e Department of Mechanical Engineering, College of Engineering, King Saud University, Riyadh, Saudi Arabia b
A R T I C L E I N F O
A B S T R A C T
Keywords: Wavy walled cavity Lid-driven flow Nanofluids Magnetic field Partial heating
A computational work has been done to investigate the effects of mixed convection of MHD flow in nanofluid filled and partially heated wavy walled lid-driven enclosure. Finite difference method is used to solve governing equations of mixed convection for different parameters as Hartmann number, Richardson number, nanoparticle volume rate in partially heated and wavy walled enclosure. It is found that the rate of heat transfer decreases with increasing the Hartmann number. The rate of heat transfer can be enhanced or reduced by increasing the volume fraction of nanoparticles based on Hartmann and Richardson numbers.
1. Introduction Mixed convection heat transfer and fluid flow in complex shaped geometry are important in engineering due to its wide applications, such as cooling of electronical devices, heat exchangers, solar collectors, cooling or heating of buildings. Besides, partial heating of confined spaces attracted many researchers recently. Partial heater was applied in some papers in open literature as seen in review of Oztop et al. [1]. There are many example on partial cooling or heating applications as Guimaraes and Menon [2]. Pioneer of computational studies of natural convection in a nanofluid filled enclosures is Khanafer et al. [3]. Oztop and Abu-Nada [4] conducted a computational solution on natural convection in enclosures with Cu-water nanofluid. They found that heat transfer increases with increasing of length of partial heater and nanoparticle volume fraction. Also, effects of combined convection (natural + forced convection) are studied in nanofluid filled systems. In this context, Mehrez et al. [5] focused on entropy generation analysis in the assisting flow of Cu-water nanofluid in an inclined open cavity and they observed that the main important parameter is the inclination angle. Hussain et al. [6] investigated the effects of magnetic field on entropy generation due to mixed convection of water-alumina nanofluid flow in a double lid driven cavity with discrete heating.
⁎
In some problems in engineering, the geometry can be curvilinear. Sheremet et al. [7] solved a problem computationally to make analysis of MHD free convection in a wavy open porous tall cavity filled with nanofluids. The cavity has a corner heater. They obtained that heat transfer enhancement with Rayleigh number (Ra) and heat transfer reduction with Hartmann (Ha) number, while magnetic field inclination angle leads to non-monotonic changes of the heat transfer. Heated wavy walled cavity was applied to a melting problem by Kousksou et al. [8]. In their case, the bottom wavy wall is heated isothermally and the problem solved via finite volume method. Waviness of the wall plays important role on the temperature distribution and flow field and they observed that the rate of the melting increases with the elevation in the magnitude of the amplitude value of the wavy surface. As an original work on insulated wavy walled cavity with natural convection of Al2O3/water nanofluids is studied by Abu-Nada and Oztop [9]. Influence of a magnetic field on the natural convection and entropy generation was studied by Cho [10] for Cu–water nanofluid in a closed space with complex-wavy surfaces. Their obtained results showed that the mean Nusselt number decreases and entropy generation increases with an increasing wave amplitude. Abu-Nada and Chamkha [11] solved a problem of mixed convection in a water-CuO filled lid-driven cavity with wholly heated wavy wall. They observed that heat transfer
Corresponding author. E-mail addresses: [email protected] (A. Sakhrieh), [email protected] (E. Abu-Nada).
http://dx.doi.org/10.1016/j.icheatmasstransfer.2017.05.011
0735-1933/ © 2017 Elsevier Ltd. All rights reserved.
International Communications in Heat and Mass Transfer 86 (2017) 42–51
H.F. Öztop et al.
B Bo cP g H Ha h k Nu Pr qw Ra Re Ri T u, v U, V x′ , y′ x, y
geometry parameter magnetic field strength specific heat at constant pressure (kJ kg− 1 K− 1) gravitational acceleration (m s− 2) height and width of the cavity (m) Hartmann number local heat transfer coefficient (W m− 2 K− 1) thermal conductivity (W m− 1 K− 1) Nusselt number Prandtl number heat flux (W m− 2) Rayleigh number Reynolds number Richardson number temperature (K) dimensional x and y components of velocity (m s− 1) dimensionless x and y components of velocity dimensional coordinates (m) dimensionless coordinates
Subscripts avg C cp f H nf p w
Greek symbols α
thermal expansion coefficient (K− 1) numerical tolerance Nanoparticle volume fraction kinematic viscosity (m2 s− 1) dimensionless temperature dimensional stream function (m2 s− 1) dimensionless stream function ∞∞φσ dimensional vorticity (s− 1) dimensionless vorticity density (kg m− 3) electrical conductivity (Ω− 1 m− 1) volume fraction of nanoparticles, (dimensionless) dynamic viscosity (N s m− 2)
β ε φ ν θ ψ Ψ ω Ω ρ σ φ μ
Nomenclature
average cold centi poise fluid hot nanofluid particle wall
fluid thermal diffusivity (m2 s− 1)
3. Governing equations and problem formulation
enhancement with Rayleigh number and heat transfer reduction with Hartmann number, while magnetic field inclination angle leads to nonmonotonic changes of the heat transfer rate. Bondereva et al. [12] solved a problem numerically to simulate the heat transfer in a wavy walled and nanofluid filled cavity with a corner heater. They found that an increase in the Ha number leads to an attenuation of the convective flow and heat transfer reduction. Study on laminar mixed convection was performed by Akbarinia et al. [13] to see the effects of nanofluid in horizontal curved tubes. Other related works can be found in refs. Sheikholeslami et al. [14], Sathiyamoorthy and. Chamkha [15] and Kefayati et al. [16]. The main aim of this work is to study the effects of magnetic field on mixed convection heat transfer and fluid flow in a wavy walled cavity with lid-driven wall filled with nanofluid. The originality of this work is the heated wavy bottom wall.
Fig. 1 shows a schematic diagram of the wavy walled differentially heated enclosure in a presence of magnetic field. The nanofluid is assumed incompressible and the flow is assumed to be laminar. It is assumed that the base fluid (i.e. water) and the nanoparticles are in thermal equilibrium and no slip occurs between them. Thermo-physical properties of the nanofluid are assumed to be constant except for the density variation, which is approximated by the Boussinesq model. The governing equations for the laminar, two-dimensional, steady state natural convection in the presence of magnetic field in terms of the stream function-vorticity formulation are written as Vorticity
∂ ⎛ ∂ψ ⎞ ∂ ⎛ ∂ψ ⎞ μnf ⎛ ∂ω ∂ω ⎞ ∂T ⎞ ω ω − = + + βnf g ⎛ ρnf ⎝ ∂x ′2 ∂x ′ ⎝ ∂y′ ⎠ ∂y′ ⎝ ∂x ′ ⎠ ∂y′2 ⎠ ⎝ ∂x ′ ⎠ σnf Bo2 ∂2ψ + ρnf ∂x 2 ⎜
2. Definition of considered physical model The considered physical model is depicted in Fig. 1. In this model, the wavy wall is heated partially. The gravity acts in −y direction and magnetic field acts parallel to x-axis. The height of the hill is given by B and length of it is L which are governing parameters on heat and fluid flow. The ceiling of the cavity moves from left to right with constant velocity. Geometrically, the length and width of the cavity, which are given by W and H, are equal. The vertical walls are considered adiabatic.
⎟
⎜
⎟
(1)
Energy
∂ ⎛ ∂ψ ⎞ ∂ ⎛ ∂ψ ⎞ ∂ 2T ∂ 2T ⎞ T T − = αnf ⎛ 2 + x y′2 ⎠ ∂x ′ ⎝ ∂y′ ⎠ ∂y′ ⎝ ∂x ′ ⎠ ∂ ∂ ′ ⎝ ⎜
⎟
⎜
⎟
(2)
Kinematics
∂ 2ψ ∂ 2ψ + = −ω ∂x ′2 ∂y′2
43
(3)
International Communications in Heat and Mass Transfer 86 (2017) 42–51
H.F. Öztop et al.
the Chon et al. model [17]:
knf
df = 1 + 64.7φ0.7640 ⎛⎜, ⎞⎟ d ⎝ p⎠
kbf
0.3690
⎛, kf ⎞ ⎜ ⎟ ⎝ kp ⎠
0.7476
Pr 0.9955 Re1.2321 (8)
This model is found to be appropriate for studying heat transfer enhancement using nanofluids (Akbarinia and Behzadmehr [13], AbuNada et al. [18]). The correlation for dynamic viscosity of CuO-water nanofluid is derived using the available experimental data of Nguyen et al. [19]. The correlation that gives the viscosity of nanofluid as a function of temperature and volume fraction of nanoparticles is given in [19,20] as:
μCuO (cp) = −0.6967 + −
19652.74 T3
15.937 T
1356.14 T2
+ 1.238φ +
φ
− 0.259φ2 − 30.88 T
φ2
+ 0.01593φ3 + 4.38206 T + 147.573
φ T2
(9) The horizontal and vertical velocities are given by the following relations,
∂ψ , ∂y′
u=
v=−
(10)
∂ψ . ∂x ′
(11)
The following dimensionless variables are introduced:
ψ y′ u v T − TC x′ ω ;y= ;Ω= ;Ψ= ;U= ;V= ;θ= Up H Up Up TH − TC H H Up H
x=
Using the previous dimensionless quantities, the governing equations are re-written in dimensionless form as:
∂ ∂x
( ) ∂Ψ
Ω ∂y
−
∂ ∂y
(Ω ) = ∂Ψ ∂x
⎡
⎤
1 ⎢ 1 ⎥ , Re ⎢ ⎛ ρp ⎞ ⎥ ⎜ (1 − φ) + φ ⎟ ⎢ ⎝ ρf ⎥ ⎠⎦ ⎣ β
p + Ri ⎡φ β + (1 − φ) ⎤ ⎣ f ⎦
(
( )+ ∂θ ∂x
1 ∂ ⎛ ∂Ψ ⎞ ∂ ⎛ ∂Ψ ⎞ θ θ − = (ρcp )p ∂x ⎝ ∂y ⎠ ∂y ⎝ ∂x ⎠ (1 − φ) + φ (ρc ) ⎜
Fig. 1. a) Problem geometry, b) Grid distribution.
∂2Ω ∂x 2
⎟
p f
+
∂2Ω ∂y 2
)
Ha2 ∂2Ψ Re ∂x 2
(12)
2 2 ⎛∂ θ + ∂ θ⎞ 2 x y2 ⎠ ∂ ∂ ⎝
⎜
⎟
∂ 2Ψ ∂ 2Ψ + = −Ω 2 ∂x ∂y 2 αnf =
(14)
where the dimensionless numbers are given as.
keff (ρcp )nf
(13)
(4)
Ra =
The effective properties of nanofluid are expressed as (Khanafer et al. [3]; Abu-Nada and Chamkha [11]):
νf σnf gβH 3 (TH − TL ) Ra ; Pr = ; Ha = Bo H , Ri = νf αf αf ρnf νf Pr Re2
(15)
The dimensionless horizontal and vertical velocities are converted
ρnf = (1 − φ) ρf + φρp
(5)
to:
βnf = (1 − φ) βf + φβp
(6)
U=
(ρcp )nf = (1 − φ)(ρcp )f + φ (ρcp )p
(7)
∂Ψ , ∂y
V=−
σnf = (1 − φ) σf + σβp Effective thermal conductivity of the nanofluid is approximated by
∂Ψ . ∂x
The dimensionless boundary conditions can be written as:
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(16)
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H.F. Öztop et al.
Fig. 2. Validation of Isotherms and streamlines for natural convection with magnetic field, Ha = 50, Pr = 0.025, and Ra = 105 (a) present results isotherms (b) present results streamlines (c) Chamkha and Sathiyamoorthy [8] isotherms (d) Chamkha and Sathiyamoorthy [8]streamlines.
I.
On the left wall i. e. , x = 0, Ψ = 0, Ω = −
II.
On the right wall i. e. , x = 1, Ψ = 0, Ω = −
III.
On the top walls: Ψ = 0, Ω = −
IV.
On the bottom wall:
1. 2.
∂2Ψ dθ , =0 ∂x 2 dx
h=
∂2Ψ , ∂n2
∂2Ψ − 2, ∂y
knf = −
hH kf
qw ∂T ∂x
knf ∂θ Nu = −⎜⎛ ⎟⎞ ⎝ kf ⎠ ∂x
=0
(21)
(22)
Where (knf/kf) is calculated using Eq. (8). The average Nusselt number is defined as:
The physical quantities of interest are the local Nusselt number and the mean Nusselt number Nu . The local Nusselt numbers is defined as
Nu =
(20)
Substituting Eqs. (20), (21) into Eq. (19), and using the dimensionless quantities, the local Nusselt number along the left wall can be written as:
θ=1 dθ dy
qw TH − TL
The thermal conductivity is expressed as: (18)
∂ 2Ψ , θ=0 ∂y 2
On the heater surface Ψ = 0, Ω = − On the flat surface Ψ = 0, Ω =
∂2Ψ dθ , =0 ∂x 2 dx
1
Nu = (19)
∫ Nu (y) dy 0
(23)
To evaluate Eq. (23), a 1/3 Simpson's rule of integration is used.
The heat transfer coefficient is expressed as:
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H.F. Öztop et al.
Fig. 3. Validation of Isotherms and streamlines for mixed convection with magnetic field, Ri = 100, Pr = 0.71, Re = 100, and Ha = 25 (a) present results isotherms (b) present results (c) Kefayati et al. [9] isotherms (d) Kefayati et al. [9] streamlines.
4. Validation
for different governing parameters such as Hartmann numbers, Richardson numbers and nanoparticle volume fraction. Fig. 4 presents the streamlines (on the left) and isotherms (on the right) for different Richardson numbers for Re = 100, ϕ = 0.05, Ha = 10, B/H = 0.2 and 0.5. It is noticed that Reynolds number is fixed for whole work. As seen from the figure, the shape of isotherms fits with the geometry of the considered model up to middle of the cavity then it behaves as a temperature distribution inside a rectangular cavity. An egg shaped main circulation cell is formed near the ceiling of the cavity but its length is related with Richardson number while other parameters are constant. The flow strength is increased with increasing of Richardson number. Fig. 5 illustrates the isotherms (on the left) and streamlines (on the right) to see the effects of Hartmann numbers with different parameters (Table 1) as Ri = 1, Re = 100, ϕ = 0.05, B/H = 0.2 and 0.5. As clearly seen from the Fig. 5(a) and (b), Hartmann number makes important effect on both temperature distribution and flow field. In the absence of Hartmann number, only one cell is formed inside the cavity. For
The present numerical solution was validated with other published work in literature for natural convection with magnetic field in a square cavity against the work of Sathiyamoorthy and Chamkha [15] for the case of Ha = 50, Pr = 0.025, and Ra = 105. Further validation of the present numerical solution is validated for mixed convection with magnetic field in a square cavity against the work of Kefayati et al. [16] for the case of Ri = 100, Pr = 0.71, Re = 100, and Ha = 25. As shown from the Figs. 1 and 2, the present work agrees very well with other results reported in literature. Further validation of the present work against nanofluid filled and wavy enclosure wall surfaces is found elsewhere [9,11]. 5. Results and discussion A computational study (Fig. 3) is performed to study of effects of magnetic field in a partially heated wavy walled lid-driven enclosure
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Fig. 4. Streamlines (on the left) and isotherms (on the right) for different Richardson number for Re = 100, ϕ = 0.05, Ha = 10, B/ H = 0.2, 0.5, a) Ri = 0.1, b) Ri = 1, c) Ri = 10.
(a)
(c)
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-0.045
0.05 0.15
5 0.2 0.35
0.00178115
5 0.4 5 5 . 0 5 6 0. 5 0.7 5 8 . 0
(a)
-0.1
0.1 5 0.2 0.3 4 0. 0. 5
0.6 0.75
(b) Fig. 5. Streamlines (on the left) and isotherms (on the right) for different Hartmann numbers for Ri = 1, Re = 100, ϕ = 0.05, B/H = 0.2, 0.5, a) Ha = 50, b) Ha = 0.
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Table 1 Variation of average Nusselt number for different parameters. Ri
ϕ
Ha
g
W
Nu
1 1 1 1 1 1 1 1 1 1 1 1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 10 10 10 10 10 10 10 10
0 0 0,05 0, 0.05 0.02 0.03 0 0.05 0.0 0.05 0.05 0.0 0.05 0.0 0.0 0.05 0.05 0.05 0.05 0.05 0.0 0.05 0.03 0.05 0.0 0.03 0.05 0.05
0 0 0 0 0 50 50 50 10 10 10 50 0.0 0.0 0.0 10 10 10 10 30 50 0.0 0.0 10 10 50 50 50 100
0.1 0.2 0.2 0,3 0.3 0.2 0.2 0.2 0.3 0.3 0.1 0.1 0.3 0.2 0.2 0.2 0.2 0.2 0.3 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.2 0.5 0.05 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
4.895 4.60 4.41 4.184 4.019 1.11 1.015 0.986 3.45 3.413 3.89 1.29 3.03 3.857 3.813 2.56 3.017 2,02 2917 1,29 1,06 7.054 6.458 6.473 5.992 5.497 5.562 5.043 3.529
Fig. 7 illustrates the variation of local Nusselt number for Reynolds number (Re = 100), B/H = 0.5 and two values of nanoparticles volume fractions ϕ = 0 and ϕ = 0.05 for the case corresponding to different geometry heights (B = 0.1, 0.2, 0.3). As shown from the figure, the local Nusselt number is strongly dependent on the top height of the geometry. The local Nusselt number decreases with increasing the height of the geometry. Moreover, the local Nusselt number decreases with increasing of the Hartmann number. The meaning of almost horizontal profile of the local Nusselt number is convection turns to conduction due to decreasing of kinetic energy when magnetic field applied to the fluid. Differences among the values increase with increasing of geometry heights.
Ha = 50, isotherms are almost parallel to each other due to retarding of flow motion. Also, two circulation cells are formed and they turn in different directions. Fig. 6 presents the isotherms (on the left) and streamlines (on the right) to see the effects of geometry and magnetic field on temperature distribution and flow field for Ri = 1 and ϕ = 0.05. Fig. 6(a) gives the results for g = 0.3 and Ha = 10. As seen from the figure, egg shaped single cell is formed inside the cavity and the flow turns in clockwise direction with ψmin = −0.09. For the strongest considered magnetic field (Ha = 50) a secondary eddy is formed in the lower portion of the cavity. Isotherms are clustered near the hill and conduction mode of heat transfer becomes dominant at right and left side of the hill due to motionless fluid. As an expected, the shape of the cavity is the main effective parameter on heat and fluid flow.
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Fig. 6. Isotherms (on the left) and Streamlines (on the right) for Ri = 1 and ϕ = 0.05, a) g = 0.3, Ha = 10, b) g = 0.1, Ha = 10, c) g = 0.1, Ha = 50.
(a)
(b)
(c)
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6. Conclusions In this numerical work, finite volume method is used to solve governing equations of sinusoidal shaped partial heater in a nanofluid filled cavity under magnetic field. Results for the streamlines and isotherms contours for various parameters were presented and discussed. The heat transfer and flow characteristics depends strongly Richardson and Hartmann numbers. It was found that heat transfer can be enhanced or reduced by increasing the volume fraction of nanoparticles based on Hartmann and Richardson numbers. Acknowledgement First and last authors extend their appreciation to the International Scientific Partnership Program ISPP at King Saud University for funding this research work through ISPP# 0030. References
(a)
[1] H.F. Öztop, P. Estellé, W.M. Yan, K. Al-Salem, J. Orfi, O. Mahian, A brief review of natural convection in enclosures under localized heating with and without nanofluids, Int. Commun. Heat Mass Transfer 60 (2015) 37–44. [2] P.M. Guimaraes, G.J. Menon, Natural nanofluid-based cooling of a protuberant heat source in a partially-cooled enclosure, Int. Commun. Heat Mass Transfer 45 (2013) 23–31. [3] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement in a two-dimensional enclosure utilizing nanofluids, Int. J. Heat Mass Transf. 46 (2003) 3639–3653. [4] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosure filled with nanofluids, Int. J. Heat Fluid Flow 29 (2008) 1326–1336. [5] Z. Mehrez, A. ElCafsi, A. Belghith, P. Le Quéré, The entopy generation analysis in the mixed convective assisting flow of Cu-water nanofluid in an inclined open cavity, Adv. Powder Technol. 26 (2015) 1442–1453. [6] S. Hussain, K. Mehmood, M. Sagheer, MHD mixed convection and entropy generation of water-alumina nanofluid flow in a double lid driven cavity with discrete heating, J. Magn. Magn. Mater. 419 (2016) 140–155. [7] M.A. Sheremet, H.F. Oztop, I. Pop, K. Al-Salem, MHD free convection in a wavy open porous tall cavity filled with nanofluids under an effect of corner heater, Int. J. Heat Mass Transf. 103 (2016) 955–964. [8] T. Kousksou, M. Mahdaoui, A. Ahmed, A. Ait Msaad, Melting over a wavy surface in a rectangular cavity heated from below, Energy 64 (2014) 212–219. [9] E. Abu-Nada, H. Oztop, Numerical anaysis of Al2O3/water nanofluids natural convection in a wavy walled cavity, Numer. Heat Transfer 59 (2011) 403–419. [10] C.C. Cho, Heat transfer and entropy generation of natural convection in nanofluidfilled square cavity with partilally-heated wavy surface, Int. J. Heat Mass Transf. 77 (2014) 818–827. [11] E. Abu-Nada, A.J. Chamkha, Mixed convection flow of a nanofluid in a lid-driven cavity with a wavy wall, Int. Commun. Heat Mass Transfer 57 (2014) 36–47. [12] N.S. Bondareva, M.A. Sheremet, H.F. Oztop, N. Abu-Hamdeh, Heatline visualization of MHD natural convection in an inclined wavy open porous cavity filled with a nanofluid with a local heater, Int. J. Heat Mass Transf. 99 (2016) 872–881. [13] A. Akbarinia, A. Behzadmehr, Numerical study of laminar mixed convection of a nanofluid in horizontal curved tubes, Appl. Therm. Eng. 27 (2007) 1327–1337. [14] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, S. Soleimani, MHD natural convection in a nanofluid filled inclined enclosure with sinusoidal wall using CVFEM, Neural Comput. & Applic. 24 (2014) 873–882. [15] M. Sathiyamoorthy, A. Chamkha, Effect of magnetic filed on natural convection flow in a liquid gallium filled square enclosure for linearly heated side wall(s), Int. J. Therm. Sci. 49 (2010) 1856–1865. [16] G.H.R. Kefayati, M. Gorji-Bandby, H. Sjjadi, D.D. Ganji, Simulation of MHD mixed convection in al id-driven cavity with linearly heated wall, Sci. Iran. B 19 (4) (2012) 1053–1065. [17] C.H. Chon, K.D. Kihm, S.P. Lee, S.U.S. Choi, Empirical correlation finding the role of temperature and particle size for nanofluid (Al2O3) thermal conductivity enhancement, Appl. Phys. Lett. 87 (2005) 153107. [18] E. Abu-Nada, Z. Masoud, H.F. Oztop, A. Campo, Effect of nanofluid variable properties on natural convection in enclosures, Int. J. Therm. Sci. 49 (2010) 479–491. [19] C.T. Nguyen, F. Desgranges, G. Roy, N. Galanis, T. Mare, S. Boucher, H. Angue Minsta, Temperature and particle-size dependent viscosity data for water-based nanofluids - hysteresis phenomenon, Int. J. Heat Fluid Flow 28 (2007) 1492–1506. [20] E. Abu-Nada, Effects of variable viscosity and thermal conductivity of CuO-water nanofluid on heat transfer enhancement in natural convection: mathematical model and simulation, ASME J. Heat Transf. 132 (2010) 052401.
(b)
(c) Fig. 7. Variation of local Nusselt number along the heating part for different parameters for Re = 100,, B/H = 0.5 (a) g = 0.1, (b) g = 0.2, (c) g = 0.3.
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