ar ticle

Copyright © 2017 by American Scientific Publishers

Journal of Nanofluids Vol. 6, pp. 1–11, 2017 (www.aspbs.com/jon)

All rights reserved. Printed in the United States of America

Inclined MHD Mixed Convection and Partial Slip of Nanofluid in a Porous Lid-Driven Cavity with Heat Source-Sink: Effect of Uniform and Non-Uniform Bottom Heating M. Rashad1 , Sameh E. Ahmed2 , Waqar A. Khan3, ∗ , and M. A. Mansour4 1

Department of Mathematics, Aswan University, Faculty of Science, Aswan, 81528, Egypt Department of Mathematics, Faculty of Sciences, South Valley University, 83523 Qena, Egypt 3 Department of Mechanical and Industrial Engineering, College of Engineering, Majmaah University Majmaah 11952, Kingdom of Saudi Arabia 4 Department of Mathematics, Assuit University, Faculty of Science, Assuit, Egypt 2

KEYWORDS: Magnetohydrodynamics, Mixed Convection, Square Enclosure, Partial Slip, Nanofluid.

1. INTRODUCTION The lid-driven rectangular cavity is one of the most widely studied problems in thermal sciences. These cavities have many practical engineering and industrial applications. Several researchers including1–15 have considered various combinations of these lid-driven cavities and temperature gradients. Sharif1 studied mixed convective heat transfer in two-dimensional shallow rectangular driven cavities. They observed interesting behaviors of the flow and thermal fields with increasing inclination. Jeng and Tzeng2 investigated numerically the heat transfer in a lid-driven cavity filled with water-saturated aluminum foam. It was shown that the local heat transfer rate increases with the Reynolds number for a constant Grashof number. The effect of heating location and size on mixed convection in a lid-driven cavity was investigated by Sivakumar et al.3 They found that the heat transfer rate is enhanced on reducing the heating section in the hot wall. Nath et al.4 ∗

Author to whom correspondence should be addressed. Email: [email protected] Received: 11 October 2016 Accepted: 1 November 2016

J. Nanofluids 2017, Vol. 6, No. 2

2169-432X/2017/6/001/011

investigated a combined e?ect of convective heat and mass transfer on a hydromagnetic electrically conducting viscous incompressible fluid through a porous medium in a non uniformly heated vertical channel. Bhuvaneswari et al.5 conducted a numerical study on magneto-convection in a square cavity with a sinusoidal temperature distribution on both walls. They concluded that the heat transfer rate increases with increasing amplitude ratio. In a series of papers, Sivasankaran et al.6–8 conducted several numerical studies on mixed convection in square cavities in different situations under different boundary conditions with/without magnetic field. They performed simulations on different Richardson numbers, sizes and locations of the heater and the cavity inclination angles and found that the flow behavior and heat transfer rate inside the cavity are strongly affected by the presence of the magnetic field. Teamah et al.9 studied double-diffusive laminar mixed convection in an inclined cavity. They investigated the effects of inclination of the cavity on the flow, thermal and mass fields. Ahmed et al.10 conducted a numerical study of laminar MHD mixed convection in an inclined lid-driven square cavity with opposing temperature gradients. They showed that the rate of heat transfer along the doi:10.1166/jon.2017.1324

1

ARTICLE

A numerical simulation of inclined magneto-hydrodynamic mixed convection and partial slip in a square liddriven cavity filled with a Darcian nanofluid saturated porous medium is performed. The left and right walls of the lid driven cavity are moving up with constant speed in vertical direction and partial slip effect is considered along the lid driven horizontal walls. The bottom wall of the cavity is considered to be having uniform or sinusoidal temperature distributions. The remainder cavity walls are assumed to be adiabatic. The developed equations of the mathematical model are non-dimensionalized and then solved numerically subject to appropriate boundary conditions by the control finite volume method. Comparisons with previously published works are presented and found to be in excellent agreement. A parametric study is also performed and a set of graphical results is presented and discussed to demonstrate interesting features of the solution.

ARTICLE

Inclined MHD Mixed Convection and Partial Slip of Nanofluid in a Porous Lid-Driven Cavity with Heat Source-Sink

heated walls is enhanced on increasing either Hartmann number or inclination angle. Mondal and Sibanda11 studied the effect of buoyancy ratio on an unsteady double diffusive natural convection flow in an inclined lid-driven enclosure with different magnetic field angles. They found that different angles of the magnetic field may suppress the convection flow and that the direction of the magnetic field influences the flow pattern. Guo and Sharif12 studied mixed convection heat transfer in a two-dimensional rectangular cavity with constant heat flux from partially heated bottom wall while the isothermal sidewalls are moving in the vertical direction. They investigated the influence of the Richardson number, heat source length, placement of the heat source, and aspect ratio of the cavity, on the maximum temperature and the Nusselt number. Oztop and Dagtekin13 solved the mixed convection problem in a vertical two-sided lid-driven differentially heated square cavity. They considered three cases depending on the direction of moving walls. Ghasemi and Aminossadati14 explored unsteady laminar mixed convection heat transfer in a two-dimensional square cavity. They considered different configurations of sliding wall movement and a series of Richardson numbers and Strouhal numbers. Chamkha15 formulated the problem of unsteady, laminar, mixed convection flow in a square cavity in the presence of internal heat generation or absorption and a magnetic field. Both the top and bottom horizontal walls of the cavity are insulated while the left and right vertical walls are kept at constant and different temperatures. He considered two cases of thermal boundary conditions corresponding to aiding and opposing flows. Iwatsu et al.16 performed several parametric studies for the flow and heat transfer of a viscous fluid contained in a square cavity. They investigated the nature of flow for the different values of the mixed convection parameter. Nanofluids are obtained by dispersing nanometer-sized particles of metals like copper, aluminum, silver and silicon or their oxides in a base fluid such as water, ethylene glycol or oil. These fluids provide excellent thermal performance and can be used as coolants in several applications like electronics cooling, transportation, defense, nuclear, space and biomedical. Several numerical studies17–38 have been conducted for a Steady state two-dimensional laminar mixed convection in different lid-driven cavities filled with water-based nanofluids. Alinia et al.17 investigated mixed convection of a nanofluid consisting of water and SiO2 in an inclined enclosure cavity. Their results reveal that addition of nanoparticles enhances heat transfer in the cavity remarkably and causes significant changes in the flow pattern. Karimipour et al.18 studied the laminar mixed convection of water-Cu nanofluid in an inclined shallow driven cavity using the lattice Boltzmann method. They investigated the effects of the cavity inclination angle and nanoparticles volume fraction at three states of free, force 2

Rashad et al.

and mixed convection domination. Esfe et al.19 considered mixed-convection fluid flow and heat transfer in a square cavity filled with Al2 O3 –water nanofluid. They investigated the effects of different key parameters on the fluid flow and heat transfer inside the cavity. Elshehabey and Ahmed20 employed Buongiorno’s nanofluid model to study MHD mixed convection of a lid-driven cavity filled with nanofluid. They considered sinusoidal temperature and nanoparticle volume fraction distributions on both vertical sides. Muthtamilselvan and Kandaswamy21 conducted a numerical study to investigate the transport mechanism of mixed convection in a lid-driven enclosure filled with nanofluids. They obtained numerical solutions for a wide range of parameters and copper-water nanofluids. Chamkha and Abu-Nada22 focused on the numerical modeling of mixed convection flow in single and double-lid square cavities filled with a water–Al2 O3 nanofluid. They conducted a parametric study and illustrated the effects of the presence of nanoparticles and the Richardson number on the flow and heat transfer characteristics in both cavities. In another study, Abu-Nada and Chamkha23 focused on the numerical modeling of mixed convection flow in a lid-driven inclined square enclosure filled with water– Al2 O3 nanofluid. In this study, they also investigated the effects of the presence of nanoparticles and enclosure inclination angle on the flow and heat transfer characteristics. Later on, Abu-Nada and Chamkha24 extended their work to mixed convection flow in a lid-driven cavity with a wavy wall filled with a water-CuO nanofluid. They conducted a parametric study and found that the presence of nanoparticles causes significant heat transfer augmentation for all values of Richardson numbers and bottom wall geometry ratios. Kalteh et al.25 dealt with the numerical solution of mixed convection flow in a lid-driven square cavity with a triangular heat source filled with water-based nanofluid. They found that suspending the nanoparticles in pure fluid leads to a significant heat transfer increase. Gümgüm and Sezgin26 performed a numerical study on unsteady mixed convection flow of nanofluids in lid-driven enclosures filled with aluminum oxide and copper–water based nanofluids. They concluded that the average Nusselt number increases with an increase in volume fraction, and decreases with an increase in both the Richardson number and heat source length. Talebi et al.27 executed a numerical investigation of mixed convection flows through a copper–water nanofluid in a square lid-driven cavity. They investigated the effects of solid volume fraction of nanofluids on hydrodynamic and thermal characteristics and found that the solid concentration affects on the flow pattern and thermal behavior. Nemati et al.28 employed Lattice Boltzmann Method to investigate the mixed convection flows utilizing nanofluids in a lid-driven cavity and investigated the effects of Reynolds number and solid volume fraction for different nanofluids on hydrodynamic and thermal characteristics. Aminossadati et al.29 presented a J. Nanofluids, 6, 1–11, 2017

Rashad et al.


J. Nanofluids, 6, 1–11, 2017

Motivated by the investigations mentioned above, the main objective of the present study is to investigate the effect of slip on the MHD mixed convection flow of a Cuwater nanofluid in a square enclosure filled with a porous medium with heat source/sink.

2. MATHEMATICAL MODELING Consider a steady two-dimensional mixed convection flow inside a porous square lid-driven cavity of length H filled with nanofluid, as shown in Figure 1. The coordinates x and y are chosen such that x measures the distance along the bottom horizontal wall, while y measures the distance along the left vertical wall, respectively. The bottom wall is considered to be having uniform or non-uniform temperature distributions. The left and right walls are moving up with constant speed in vertical direction and partial slip effect is considered along the lid driven horizontal walls. The remainder cavity walls are assumed to be adiabatic. The nanofluids used in the analysis are assumed to be incompressible and laminar, and the base fluid (water) and the solid spherical nanoparticles (Cu, Ag, Al2 O3 and TiO2 ) are assumed to be in thermal equilibrium. The thermophysical properties of the base fluid and the nanoparticles are given in Table I. The thermo-physical properties of the nanofluid are assumed constant except for the density variation, which is based on the Boussinesq approximation. Under the above assumptions, the conservation of mass, linear momentum, angular momentum and in the case of mixed convection and also conservation of energy equations are given in Refs. [36–38]. u v + =0 x y

Fig. 1.

(1)

Physical model and coordinates system.

3

ARTICLE

numerical study of mixed convection in a two-sided liddriven cavity filled with a water–Al2 O3 nanofluid. Their results show that an Adaptive Network-based Fuzzy Inference System (ANFIS) can successfully be used to predict the fluid velocity and temperature as well as the heat transfer rate of the cavity, with reduced computation time and without compromising the accuracy. Mansour et al.30 dealt with mixed convection in a square lid-driven cavity partially heated from below and filled with water-base nanofluid containing various volume fractions of Cu, Ag, Al2 O3 and TiO2 . They discovered that an increase in solid volume fraction leads to decrease both fluid velocity and temperature, however, it leads to increase the corresponding average Nusselt number. Muthtamilselvan and Doh31 investigated mixed convection in a lid-driven square cavity filled with Cu–water nanofluid in the presence of internal heat generation. They found that Richardson number strongly affect the fluid flow and heat transfer in the cavity. Cho et al.32 performed a numerical investigation to determine the mixed convection heat transfer characteristics of water-based nanofluids confined within a lid-driven cavity with wavy surfaces. They showed that for all considered values of the Richardson number, the mean Nusselt number increases with an increasing volume fraction of nanoparticles. Tiwari and Das33 investigated the behavior of nanofluids inside a two-sided lid-driven square cavity. They considered three case depending on the direction of the moving walls. Arani et al.34 investigated mixed convection flow of Cu–water nanofluid inside a lid-driven square cavity with adiabatic horizontal walls and sinusoidal heating on sidewalls. They studied the effects of variations of Richardson number, phase deviation of sinusoidal heating, and volume fraction of nanoparticles on flow and temperature fields. Sheikhzadeh et al.35 focused on the study of mixed convection in a lid-driven enclosure filled with nanofluids using variable thermal conductivity and variable viscosity. They found significant differences between the magnitudes of heat transfer enhancement in the enclosure for two employed models. Moumni et al.36 investigated mixed convection fluid flow and heat transfer of different water-based nanofluids in a two-sided facing lid-driven cavity. They analyzed the flow and temperature patterns and found the significant heat transfer enhancement. Sheremet and Pop37 studied mixed convection inside a lid-driven square cavity filled with water based nanofluid. They showed that the governing parameters have substantial effects on the flow and heat transfer characteristics. Elshehabey and Ahmed38 studied Buongiorno’s nanofluid model for MHD mixed convection in a lid-driven cavity filled with nanofluid. They demonstrated that, the presence of an inclined magnetic field in the flow region leads to lose the fluid movement and the fluid flow is dominated by the movement of the upper wall in the case of the highest values of the buoyancy ratio.


Table I. Thermophysical properties of water and nanoparticles. Property Cp k

Water

Copper (Cu)

997.1 4179 0.613 21 × 10−5 0.05

8933 385 401 167 × 10−5 596 × 107

2 nf u 1 p u u 2 u +nf u u +v = − + − x y nf x x 2 y 2 K n B02 vsincos −usin2

nf 2 nf v 1 p v v 2 v +nf + u +v = − − 2 2 x y nf y x y K +

ARTICLE

nf gT −T0

nf 2 T T T 2 T Q0 +v = nf + T −T0

u + 2 2 x y x y Cp nf

(2)

where is the solid volume fraction of the nanofluid, f and p are the densities of the fluid and of the solid fractions respectively, and the heat capacitance of the nanofluid given is by Khanafer et al.39 as; (7)

The thermal expansion coefficient of the nanofluid can be determined by:

(3) nf =

(8)

knf cp nf

(9)

(4)

T = Th or x T = Th + Th − Tc sin H

On the top wall y = H

In Eq. (9), knf is the thermal conductivity of the nanofluid which estimated in the present study according to Yu and Choi model47 that can be considered an updated model of Maxwell-Garnetts model (Maxwell42 ): kp + 2kf − 2 1 + 3 kf − kp

knf = kf kp + 2kf + 1 + 3 kf − kp

(10)

where in Eq. (10) represents the ratio of nanolayer thickness to nanoparticle radius and thereby accounts for the size dependent nature of thermal conductivity of nanofluids. This model attributes the enhanced thermal conductivity of nanofluid to the solidlike layering of liquid at nanoparticle/base fluid interface. Here, is set equal to 0.1, which implies a nanolayer thickness of 1 nm. The effective dynamic viscosity of the nanofluid based on the Brinkman model42 is given by nf =

f 1 − 25

(11)

where f is the viscosity of the fluid fraction and the effective electrical conductivity of nanofluid was presented by Maxwell43 as (5)

Numerous formulations for the thermo-physical properties of nanofluids are proposed in the literature. In the present study, we are adopting the relations which depend on the 4

(6)

where f and p are the coefficients of thermal expansion of the fluid and of the solid fractions respectively. Thermal diffusivity, nf of the nanofluid isdefined by Oztop and Abu-Nada41 as:

u = v = 0

u = v = 0 T = Tc ⎧ T ⎪ ⎪ ⎨x = H u = x = 0 On the right wall ⎪ ⎪ ⎩v = V + N v r 0 nf x ⎧ T ⎪ ⎪ ⎨x = 0 u = x = 0 On the left wall ⎪ ⎪ ⎩v = V + N v l 0 nf x

nf = 1 − f + p

nf = 1 − f + p

Where u and v are the velocity components along the xand y-axes respectively, T is the fluid temperature, p is the fluid pressure, g is the gravity acceleration, nf is the density, nf is the dynamic viscosity, nf is the kinematic viscosity. The boundary conditions are On the bottom wall y = 0

nanoparticles volume fraction only and which were proven and used in many previous studies39 40 as follows: The effective density of the nanofluid is given as:

cp nf = 1 − cp f + cp p

nf B02 + usincos −vcos2

nf +

Rashad et al.

nf 3 − 1 = 1+ f + 2 − − 1

(12)

where = p /f . The electrical conductivity nf of the nanofluid is defined by nf 3p /f − 1 = 1+ f p /f + 2 − p /f − 1

(13)

J. Nanofluids, 6, 1–11, 2017

Rashad et al.


Introducing the following dimensionless set: x y u v X= Y = U= V = H H V0 V0 T − Th Gr p

=

Ri =

P= Th − Tc nf V02 Re2 N f Q0 H 2 Sr = Sl =

Q= H Cp f f

Table II. Comparisons of the mean Nusselt number at the top wall, for different values of Re at Pr = 071, Gr = 102 . Re

(14)

into Eqs. (1)–(5) yields the following dimensionless equations: U V + =0 (15) X Y 2 U U P 1 nf U 2 U U U +V =− + + − X Y X Re f X 2 Y 2 Da f nf Ha2 + nf f Re

× U sin cos − V cos2

(17) 2 nf 2 1 U + 2 +V = 2 X Y Pr Re f X Y cp f Q + (18) Re Pr cp nf 3 2 Where Pr = f /f , Re = V0 H /f , Gr = gf H T /f , Ha = B0 H f / f are respectively the Prandtl number, the Reynolds number the Grashof number, the magnetic number and the Eckret number, respectively. The dimensionless boundary condition for Eqs.(15)– (18) are as follows: On the bottom wall

Y = 0

U = V = 0

=1

Khanafer and Chamkha45

Present study

194 384 633

201 391 633

193 391 631

The local Nusselt number is defined as: −knf Nu = kf Y Y =0

(20)

and the average Nusselt number is defined as:

1 Num = Nu dX

(21)

0

Equations (16)–(19) with the boundary conditions (20) have been solved numerically using the collocated finite volume method. The first upwind and central difference approaches have been used to approximate the convective and diffusive terms, respectively. The resulting discretized equations have been solved iteratively, through alternate direction implicit ADI, using the SIMPLE algorithm.44 The velocity correction has been made using the Rhie and Chow interpolation. For convergence, under-relaxation technique has been employed. To check the convergence, the mass residue of each control volume has been calculated and the maximum value has been used to check the convergence. 10−5 was set as the convergence criterion. A uniform grid resolution of 81 × 81 is found to be suitable. In order to verify the accuracy of present method, the obtained results in special cases are compared with the results obtained by Khanafer and Chamkha45 and Iwatsu et al.46 in terms of the mean Nusselt number at the top wall, for different values of Re. As we can see form Table II, the results are found in a good agreement with these results. These favorable comparisons lend confidence in the numerical results to be reported subsequently.

or = sin X

0.00022

On the top wall Y = 1

U = V = 0

0.00020

=0

0.00018

On the right wall ψmax

=0 X nf V V = r + S r f X

X=1

U=

0.00016 0.00014 0.00012 0.00010

On the left wall

0.00008

=0 X = 0 U = X nf V V = l + S l f X J. Nanofluids, 6, 1–11, 2017

20

(19)

40

60

80

100

120

Grid size Fig. 2. Grid independence study for uniform heating case at Ha = 10, = /4, Ri = 1, = 5%, Q = 1, l = 1, r = −1, Sr = Sl = 1.

5

ARTICLE

× V sin cos − U sin

(16) 2 2 V P 1 nf V V V V +V =− + U + − X Y Y Re f X 2 Y 2 Da nf Ha2 f nf + + Ri nf f nf f Re 2

100 400 1000

Iwatsu et al.46


1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

Rashad et al.

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 3. Streamlines (left) and isotherms (right) for non-uniform bottom heating (solid = 0% and dash = 5% at Ha = 10, = /4, Ri = 1, Q = 1, l = 1, r = 1, Sr = Sl = 0

0% ≤ ≤ 5%, the partial slip parameter; 0 ≤ Sr , Sl ≤ 20, the Hartmann number; 0 ≤ Ha ≤ 100 and the generation/absorption parameter; −2 ≤ Q ≤ 2. The referenced case for all the figures is Ha = 10, = /4, Ri = 1, = 4%, Q = 1, l = 1, r = 1, Sr = Sl = 1. All the obtained results are presented in terms of streamlines and isotherms contours, horizontal and vertical velocity components at the enclosure mid-section, temperature distributions at the enclosure mid-section and the rate of heat transfer at the heated wall. Figure 3 displays the contours of streamlines and isotherms for different values of nanoparticles volume fraction for the case of non-uniform bottom heating. It is found that the streamlines shows two symmetrical

ARTICLE

3. RESULTS AND DISCUSSION The numerical results obtained from the investigation of the effects of an inclined magnetic field and partial slip conditions on the mixed convection flow of nanofluids in a square cavity filled with an isotropic porous medium are discussed here. The bottom wall of the cavity is considered to be having uniform or non-uniform temperature distributions. The non-uniform condition is represented, here, by the sinusoidal distributions. Water is considered as a base fluid with Pr = 62 and Copper is considered as nanoparticles. During the calculations, the Grashof number and Darcy number are fixed at 104 and 10−4 , respectively, while the ranges of the other key-parameters are considered to be as follow: the nanoparticles volume fraction;

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 4. Streamlines (left) and isotherms (right) for uniform bottom heating (solid = 0% and dash = 5% at Ha = 10, = /4, Ri = 1, Q = 1, l = 1, r = 1, Sr = Sl = 0

6

J. Nanofluids, 6, 1–11, 2017

Rashad et al.


0.015

1.0

Sr=Sl=1 Sr=Sl=3 Sr=Sl=5 Sr=Sl=20

0.010


0.8

0.005

0.6

Y

V

Uniform bottom heating

0.000 Uniform bottom heating

–0.005

0.4

–0.010 0.2 –0.015 0.0

2.0×10–4

1.5×10–4

1.0×10–4

5.0×10–4

0.0

–5.0×10–4

–1.0×10–4

–1.5×10–4

–2.0×10–4

0.0

0.2

0.4

0.6

0.8

1.0

0.8

1.0

X 0.015

U


0.010

1.0

0.005 0.8 0.000

V Non-uniform bottom heating

Non-uniform bottom heating –0.005

Y


0.4

–0.010 –0.015

0.2

0.0

0.2

0.4

0.6

X 2.0×10–4

1.5×10–4

1.0×10–4

5.0×10–4

0.0

–5.0×10–4

–1.0×10–4

–1.5×10–4

0.0

U Fig. 5. Profiles of the horizontal velocity component at the enclosure mid-section for different values of Sr and Sl.

vortices formed near the vertical walls. The main reason for this behavior is due to the sheer force resulting from the movement of the vertical walls. The isotherms gather near the heated wall indicating small temperature differences in this region. However, at the remaining area of the enclosure, a weak distribution for the isotherms is observed indicating large temperature gradient in this area. For uniform bottom heating, the streamlines and isotherms contours are presented in Figure 4 for different values of nanoparticles volume fraction. Like the non-uniform case, the fluid motion is represented by two clockwise and anticlockwise eddies near the vertical walls, whereas the isotherms lines distribute in the whole cavity with the same rate. In addition, there is a gathering of the isotherms can be seen near the top corner indicating small temperature gradients in these areas. For the two cases, as the nanoparticles volume fraction increases, the mixed convection decreases and the rates of fluid motion and temperature distributions at = 0% are greater than those J. Nanofluids, 6, 1–11, 2017

Fig. 6. Profiles of the vertical velocity component at the enclosure midsection for different values of Sr and Sl.

of = 5%. The physical explanation of this behavior is due to the viscosity of the nanofluid that increases as the addition of the nanoparticles to the base fluid increases ( = 5% . Figures 5 and 6 show the profiles of the horizontal and vertical velocity components at the enclosure midsection for different values of partial slip parameter Sr for both uniform and non-uniform bottom heating cases. It is observed that a clear reduction in the profiles of the velocity components is obtained by increasing the partial slip parameter. The reason for this behavior is due to the tangential slip velocity which proportional to the shear stress that increases as Sr increases which in turn slowdown the fluid motion. The effect of partial slip represented by the variations of Sr on the temperature distributions at the enclosure midsection for uniform and non-uniform bottom heating and the profiles of the local Nusselt number at the heated wall is depicted in Figures 7 and 8, respectively. It observed that the effect of Sr on the temperature profiles is clearer in the case of uniform bottom heating than the case of nonuniform heating and this is due to that the dominance of the sinusoidal effect than the partial slip effect. In addition, for both cases, the increase in Sr causes a decrease in the 7

ARTICLE

0.6


Rashad et al.

1.0

0.548 Sr=Sl=1 Sr=Sl=3 Sr=Sl=5 Sr=Sl=20

0.546 0.544

0.8

0.542


0.6

θ 0.540

Y

Ha=0 Ha=10 Ha=25 Ha=50 Ha=100


0.538

0.4

0.536 0.534

0.2

0.532 0.0

0.2

0.4

0.6

0.8

0.0

1.0

U

X 1.0 Non-uniform bottom heating

0.37

Non-uniform bottom heating

0.8 0.36 0.6

0.35

θ


0.34

Ha=0 Ha=10 Ha=25 Ha=50 Ha=100

0.4

0.33

0.2

0.32

Fig. 7. Profiles of the temperature at the enclosure mid-section for different values of Sr and Sl.

temperature profiles in the left half of the enclosure and an increase in the temperature distributions in the right half. Based on that, the temperature gradients in the right half of the enclosure is smaller than those at the left half, which in turn, leads to opposite behavior for the local Nusselt number to the temperature behavior. 1.66


1.64 1.62

Nu

1.60 1.58


1.56 1.54 1.52 0.0

0.2

0.4

0.6

0.8

1.0

X Fig. 8. Profiles of the local Nusselt number at heated wall for different values of Sr and Sl.

8

2.0×10–4

1.5×10–4

1.0

1.0×10–4

0.8

5.0×10–4

0.6

X

0.0

0.4

–5.0×10–4

0.2

–1.0×10–4

0.0

–1.5×10–4

0.0 –2.0×10–4

ARTICLE

Y

U Fig. 9. Profiles of the horizontal velocity component at the enclosure mid-section for different values of the Hartmann number Ha.

Figure 9 shows the profiles of the horizontal velocity component at the enclosure mid-section for different values of the Hartmann number Ha for uniform and nonuniform bottom heating. It is evident that the increase in the Hartmann number causes a slightly decrease in the velocity profiles and this behavior is observed for the both cases considered in the current paper. The physical explanation for this profile is due to the Lorentz force resulting from the presence of magnetic field which causes a reduction in the mixed convection and this force increases as Ha increases. However, the profiles of the nanofluid temperature at the enclosure mid-section have an enhancement by increasing Ha as it can be seen form Figure 10. Figure 11 displays the profiles of the local Nusselt number at the heated wall for different values of the Hartmann number Ha. It is clear that the increase in the Hartmann number decreases the thermal boundary layer which decreases the local Nusselt number. Figures 12–14 show the effect of heat generation/absorption parameter Q on the profiles of the horizontal velocity component, temperature distributions and the local Nusselt number for uniform and non-uniform J. Nanofluids, 6, 1–11, 2017

Rashad et al.


1.0


Ha=0 Ha=10 Ha=25 Ha=50 Ha=100

0.546 0.544 0.542

Uniform bottom heating 0.8

0.6

θ 0.540

Y 0.4

0.538

Q=–2 Q=–1 Q=0 Q=1 Q=2

0.536 0.2 0.534 0.532

2.0×10–4

1.5×10–4

U

Non-uniform bottom heating

0.37

1.0×10–4

X

5.0×10–4

1.0

0.0

0.8

–5.0×10–4

0.6

–1.0×10–4

0.4

–1.5×10–4

0.2

–2.0×10–4

0.0 0.0

Fig. 12. Profiles of the horizontal velocity component at the enclosure mid-section for different values of the heat generation/absorption parameter Q.

0.36 0.35 0.34 0.33 0.32

Q=2

0.58

Uniform bottom heating 0.0

0.2

0.4

0.6

0.8

0.56

1.0

x

Q=1

0.54

Fig. 10. Profiles of the temperature at the enclosure mid-section for different values of the Hartmann number Ha.

θ

0.52

Q=0

0.50

bottom heating. As it expected, the increase in the heat generation/absorption parameter Q enhances the mixed convection which in turn increases the nanofluid flow and the nanofluid temperature, whereas the local Nusselt number decreases as Q increases. The main reason

Q=-1

0.48 0.46

Q=-2

0.44 0.0

0.2

0.4

0.6

0.8

1.0

X 0.40 0.39 0.38 0.37 0.36 0.35 0.34 θ 0.33 0.32 0.31 0.30 0.29 0.28 0.27 0.26


Ha=0 Ha=10 Ha=25 Ha=50 Ha=100

1.64 1.62

Nu

1.60 1.58 1.56 1.54 1.52 0.0

0.2

0.4

0.6

0.8

1.0

X Fig. 11. Profiles of the local Nusselt number at the heated wall for different values of the Hartmann number Ha. J. Nanofluids, 6, 1–11, 2017

Non-uniform bottom heating Q=2 Q=1 Q=0 Q=-1 Q=-2

0.0

0.2

0.4

0.6

0.8

1.0

X Fig. 13. Profiles of the temperature at the enclosure mid-section for different values of the heat generation/absorption parameter Q.

9

ARTICLE

for this behaviour is due to the total heat generation rate which increases as Q increases which in turn increases the nanofluid temperature and decreases the thermal boundary layer thickness.

Ha=0 Ha=10 Ha=25 Ha=50 Ha=100

θ


2.7 2.6 2.5 2.4 2.3 2.2 2.1 2.0 Nu 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2

3. The effect of partial slip condition on the temperature distributions using uniform bottom heating is clearer than the non-uniform heating. 4. The increase in the Hartmann number causes a reduction in the velocity component profiles and heat transfer rate, whereas it enhances the nanofluid temperature. 5. An enhancement in the velocity profiles and temperature distributions is obtained by increasing the heat generation/absorption parameter, whereas the local Nusselt number takes the inverse behavior.

Q=-2 Uniform bottom heating Q=-1

Q=0

Q=1

Q=2 0.0

0.2

0.4

0.6

0.8

1.0

0.8

1.0

X

5 0

Q=–2 Q=–1 Q=0 Q=1 Q=2

–5

ARTICLE

Nu –10

Non-uniform bottom heating –15 –20 0.0

0.2

0.4

0.6

X Fig. 14. Profiles of the local Nusselt number at the heated wall for different values of the heat generation/absorption parameter Q.

4. CONCLUSIONS The problem of MHD mixed convection flow of a nanofluid in a square cavity filled with uniform and isotropic porous media in the presences of effect of heat generation/absorption was studied in the current paper. Two cases for the temperature boundary conditions were considered, namely, uniform boundary condition and non-uniform boundary conditions. The governing equations were presented in dimensional form then converted to dimensionless form using suitable non-dimensional quantities. The collocated finite volume method was used to solve the resulting system and the pressure distributions were obtained using SIMPLE algorithm. Comparisons with previously published results were performed and found to be in a very good agreement. From the present study, it can be concluded that: 1. An increase in the nanoparticles volume fraction causes a reduction in the fluid flow and temperature distributions. 2. A clear reduction in the profiles of horizontal and vertical velocity component can be obtained by increasing the partial slip parameter. 10

Rashad et al.

References and Notes

1. M. A. R. Sharif, Applied Thermal Engineering 27, 1036 (2007). 2. T. M. Jeng and S. C. Tzeng, Numer. Heat Transfer A 54, 178 (2008). 3. V. Sivakumar, S. Sivasankaran, P. Prakash, and J. Lee, Comput. Math. Appl. 59, 3053 (2010). 4. P. R. Nath, P. M. V. Prasad, and D. R. V. PrasadaRao, Comput. Math. Appl. 59, 803 (2010). 5. M. Bhuvaneswari, S. Sivasankaran, and Y. J. Kim, Numer. Heat Transfer A 59, 167 (2011). 6. S. Sivasankaran, V. Sivakumar, and P. Prakash, Int. J. Heat Mass Transfer 53, 4304 (2010). 7. S. Sivasankaran, A. Malleswaran, J. Lee, and P. Sundar, Int. J. Heat Mass Transfer 54, 512 (2011). 8. S. Sivasankaran, V. Sivakumar, and A. K. Hussein, International Communications in Heat and Mass Transfer 46, 112 (2013). 9. M. A. Teamah, M. M. Sorour, W. M. El-Maghlany, and A. Afifi, Alexandria Engineering Journal 52, 227 (2013). 10. S. E. Ahmed, M. A. Mansour, and A. Mahdy, Nuclear Engineering and Design 265, 938 (2013). 11. S. Mondal and P. Sibanda, International Journal of Heat and Mass Transfer 90, 900 (2015). 12. G. Guo and M. A. R. Sharif, Int. J. Therm. Sci. 43, 465 (2004). 13. H. F. Oztop and I. Dagtekin, Int. J. Heat Mass Transfer 47, 1761 (2004). 14. B. Ghasemi and S. M. Aminossadati, Numer. Heat Transfer A 54, 726 (2008). 15. A. J. Chamkha, Numer. Heat Transfer A 41, 529 (2002). 16. R. Iwatsu, J. M. Hyun, and K. Kuwahara, Int. J. Heat Mass Transfer 36 1601 (1993). 17. M. Alinia, D. D. Ganji, and M. Gorji-Bandpy, International Communications in Heat and Mass Transfer 38, 1428 (2011). 18. A. Karimipour, M. H. Esfe, M. R. Safaei, D. T. Semiromi, S. Jafari, and S. N. Kazi, Physica A: Statistical Mechanics and its Applications 402, 150 (2014). 19. M. H. Esfe, M. Akbari, A. Karimipour, M. Afrand, O. Mahian, and S. Wongwises, International Journal of Heat and Mass Transfer 85, 656 (2015). 20. H. M. Elshehabey and S. E. Ahmed, International Journal of Heat and Mass Transfer 88, 181 (2015). 21. M. Muthtamilselvan, P. Kandaswamy, and J. Lee, Communications in Nonlinear Science and Numerical Simulation 15, 1501 (2010). 22. A. J. Chamkha and E. Abu-Nada, European Journal of MechanicsB/Fluids 36, 82 (2012). 23. E. Abu-Nada and A. J. Chamkha, European Journal of MechanicsB/Fluids 29, 472 (2010). 24. E. Abu-Nada and A. J. Chamkha, International Communications in Heat and Mass Transfer 57, 36 (2014). 25. M. Kalteh, K. Javaherdeh, and T. Azarbarzin, Powder Technology 253, 780 (2014). 26. S. Gümgüm and M. T. Sezgin, Journal of Computational and Applied Mathematics Part B 259, 730 (2014). 27. F. Talebi, A. H. Mahmoudi, and M. Shahi, International Communications in Heat and Mass Transfer 37, 79 (2010). J. Nanofluids, 6, 1–11, 2017

Rashad et al.


28. H. Nemati, M. Farhadi, K. Sedighi, E. Fattahi, and A. A. R. Darzi, International Communications in Heat and Mass Transfer 37, 1528 (2010). 29. S. M. Aminossadati, A. Kargar, and B. Ghasemi, International Journal of Thermal Sciences 52, 102 (2012). 30. M. A. Mansour, R. A. Mohamed, M. M. Abd-Elaziz, and Sameh E. Ahmed, International Communications in Heat and Mass Transfer 37, 1504 (2010). 31. M. Muthtamilselvan and D. H. Doh, Applied Mathematical Modelling 38, 3164 (2014). 32. C. C. Cho, C. L. Chen, and C. K. Chen, International Journal of Thermal Sciences 68, 181 (2013). 33. R. K. Tiwari and M. K. Das, International Journal of Heat and Mass Transfer 50, 2002 (2007). 34. A. A. A. Arani, S. M. Sebdani, M. Mahmoodi, A. Ardeshiri, and M. Aliakbari, Superlattices and Microstructures 51, 893 (2012). 35. G. A. Sheikhzadeh, M. E. Qomi, N. Hajialigol, and A. Fattahi, Results in Physics 2, 5 (2012). 36. H. Moumni, H. Welhezi, R. Djebali, and E. Sediki, Applied Mathematical Modelling 39, 4164 (2015).

37. M. A. Sheremet and I. Pop, Applied Mathematics and Computation 266, 792 (2015). 38. H. M. Elshehabey and S. E. Ahmed, International Journal of Heat and Mass Transfer 88, 181 (2015). 39. K. M. Khanafer, A. M. Al-Amiri, and I. Pop, European Journal of Mechanics B Fluids 26, 669 (2007). 40. A. M. Al-Amiri, K. M. Khanafer, and I. Pop, Int. J. Heat Mass Transfer 50, 1771 (2007). 41. H. F. Oztop and E. Abu-Nada, Int. J. Heat Fluid Flow 29, 1326 (2008). 42. H. C. Brinkman, J. Chem. Phys. 20, 571 (1952). 43. J. C. Maxwell, A Treatise on Electricity and Magnetism, Clarendon, UK (1973). 44. S. V. Patnkar, Numerical Heat Transfer and Fluid Flow, Hemisphere, New York (1980). 45. K. M. Khanafer and A. J. Chamkha, International Journal of Heat and Mass Transfer 31, 1354 (1999). 46. R. Iwatsu, J. M. Hyun, and K. Kuwahara, International Journal of Heat and Mass Transfer 36, 1601 (1993). 47. W. Yu and S. Choi, Journal of Nanoparticle Research 5, 167 (2003).

ARTICLE

J. Nanofluids, 6, 1–11, 2017

11