Journal of Magnetism and Magnetic Materials 416 (2016) 25–36
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Journal of Magnetism and Magnetic Materials
journal homepage: www.elsevier.com/locate/jmmm
Mixed convection in a nanofluid filled-cavity with partial slip subjected
to constant heat flux and inclined magnetic field
Muneer A. Ismael a, M.A. Mansour b, Ali J. Chamkha c,d, A.M. Rashad e,n
a
Mechanical Engineering Department, Engineering College, University of Basrah, Basrah, Iraq
Department of Mathematics, Assuit University, Faculty of Science, Assuit, Egypt
c
Mechanical Engineering Department, Prince Mohammad Bin Fahd University, Al-Khobar 31952, Saudi Arabia
d
Prince Sultan Endowment for Energy and Environment, Prince Mohammad Bin Fahd University, Al-Khobar 31952, Saudi Arabia
e
Department of Mathematics, Aswan University, Faculty of Science, Aswan 81528, Egypt
b
art ic l e i nf o
a b s t r a c t
Article history:
Received 16 February 2016
Received in revised form
2 April 2016
Accepted 4 May 2016
Available online 5 May 2016
Mixed convection in a lid-driven square cavity filled with Cu-water nanofluid and subjected to inclined
magnetic field is investigated in this paper. Partial slip effect is considered along the lid driven horizontal
walls. A constant heat flux source on the left wall is considered, meanwhile the right vertical wall is
cooled isothermally. The remainder cavity walls are thermally insulted. A control finite volume method is
used as a numerical appliance of the governing equations. Six pertinent parameters were studied these;
the orientation of the magnetic field (Φ ¼ 0–360°), Richardson number (Ri¼ 0.001–1000), Hartman
number (Ha ¼0–100), the size and position of the heat source (B ¼0.2–0.8, D¼0.3–0.7, respectively),
nanoparticles volume fraction (ϕ ¼0.0–0.1), and the lid-direction of the horizontal walls (λ ¼ 71) where
the positive sign means lid-driven to the right while the negative sign means lid-driven to the left. The
results show that the orientation and the strength of the magnetic field can play a significant role in
controlling the convection under the effect of partial slip. It is also found that the natural convection
decreases with increasing the length of the heat source for all ranges of the studied parameters, while it
is do so due to the vertical distance up to Hartman number of 50, beyond this value the natural convection decreases with lifting the heat source narrower to the top wall.
& 2016 Elsevier B.V. All rights reserved.
Keywords:
Magnetic field
Heat source
Lid-driven cavity
Partial slip
Nanofluid
1. Introduction
The developments in electronic technology, industry, environmental applications such as lubrication technologies, food processing and nuclear reactors have became more pronounced in the
recent years [1–5]. These continually developments are cumulative
accompanied with progress in heat rejection (cooling) methods.
Mixed convection has been and will continue to be pivotal in
improving the performance of the heat rejection in electronic
sources contained in enclosures. Mixed convection flow and heat
transfer in enclosures can also encountered in many other industrial applications such as, float glass manufacturing, solidification of ingots, coating or continuous reheating furnaces, and any
application under goes to a solid material motion inside a chamber. The mixed convection flow in lid-driven cavity or enclosure is
raised from two mechanisms. The first is due to shear flow caused
by the movement of one (or two) of the cavity wall(s) while the
second is due to the buoyancy flow induced by the nonn
Corresponding author.
E-mail address: [email protected] (A.M. Rashad).
http://dx.doi.org/10.1016/j.jmmm.2016.05.006
0304-8853/& 2016 Elsevier B.V. All rights reserved.
homogeneous thermal boundaries. The contribution of shear force
caused by movement of wall and the buoyancy force by temperature difference make the heat transfer mechanism complex.
Cavity flow simulation was introduced in early works of Torrance
et al. [6] and Ghia et al. [7]. Recently, lid-driven cavities are found
studied in various situations such as pure or nanofluid-filled cavities, and pure or nanofluid saturated porous medium cavities.
Selective studies of lid-driven cavities filled with pure fluids can be
referred to Koseff [1], Mohamad and Viskanta [8], Mekroussi et al.
[9], Sivasankaran et al. [10], or in nanofluid filled cavities as in
Tiwari and Das [11], Talebi et al. [12], Abu-Nada and Chamkha [13],
Chamkha and Abu-Nada [14]. The effect of uniform magnetic field
was found to have considerable effect on the rate of heat transfer
[15]. The role of magnetic field on natural convection in nanofluid
filled cavities is addressed in Sheikholeslami et al. [16] and Sheikholeslami and Rashidi [17]. Lid-driven pure fluid filled cavities as
in Oztop et al. [18], Muthtamilselvan et al. [19]. Lid-driven porous
cavity as in Khanafer and Chamkha [20], lid-driven nanofluid saturated porous cavities as in Sun and Pop [21], Chamkha and Ismael [22], and Bourantas et al. [23].
In some applications like fluoroplastic coating (e.g. Teflon)
which resists adhesion, the no-slip boundary condition imposed
26
M.A. Ismael et al. / Journal of Magnetism and Magnetic Materials 416 (2016) 25–36
Nomenclature
B
D
g
Gr
H
Ha
N
Nus
Num
p
Pr
Re
Ri
S
T
u
U
Uo
v
V
x,y
X,Y
heat source (m)
heat source position (m)
gravitational field (m s 2)
Grashof number Gr = g β ΔT H3/ν 2
cavity width (m)
Hartman number
slip constant (m2 s kg 1)
local Nusselt number
average Nusselt number
pressure (N/m2)
Prandtl number Pr = ν/α
Reynolds number Re = U0 H/ν
Richardson number Ri ¼(Gr/Re2)
dimensionless partial slip parameter St =Sb=Nμf /H
temperature (K)
velocity component along x-direction (m s 1)
dimensionless velocity component along x-direction
velocity of the moving wall (m/s)
velocity component along y-direction (m s 1)
dimensionless velocity component along y-direction
Cartesian coordinates (m)
dimensionless Cartesian coordinates
α
β
ϕ
Φ
λ
m
ν
θ
ρ
s
ψ, Ψ
thermal diffusivity (m2 s 1)
thermal expansion coefficient (K 1)
nanoparticles volume fraction
inclination angle of the magnetic field (degree)
constant moving parameter
dynamic viscosity (Pa s)
kinematic viscosity (m2 s 1)
dimensionless temperature
density (kg m 3)
electrical conductivity (S m 1)
stream function (m2 s 1), dimensionless stream
function
Subscripts
b
c
f
h
m
nf
p
t
bottom
cold
fluid
hot
average
nanofluid
porous
top
Greek symbols
on the tangential velocity cannot be held. Moreover, some surfaces
are rough or porous such that equivalent slip occurs (Wang [24]).
Also, there existsa hydrodynamic boundary slip regime for rarefied
gases when the Knudsen number is small (Sharipov and Seleznev
[25]). Dealing with such problems is strictly embargo to consider
Navier's slip-boundary condition [26] along these surfaces. Generally, the physical interpretation of the velocity slip on solid
boundary arises from the unequal wall and fluid densities, the
weak wall-fluid interaction, and the high temperature [27]. However, the studies dealing with slip boundary conditions may be
achieved in order to simulate engineering problems Fang et al. [28]
and Yoshimura and Prudhomme [29], or to solve the problem of
non-physical singularity resulting from meeting stationary and
moving walls, Navier [26], Koplik & Banavar [30], Qian and Wang
[31], Nie et al. [32], and Ismael et al. [33] have considered the
partial slip condition along two horizontal isothermal moving
walls under steady laminar mixed convection inside lid-driven
square cavity. Their results have showed that there were critical
values of the partial slip parameter at which the convection is
declined. These non-zero critical slips where found to be sensitive
to both Richardson number and the lid direction. Alternatively,
Soltani and Yilmazer [34] have reported that the wall slip can
occur in the working fluid contains concentrated suspensions.
Convective heat transfer of nanofluids in circular concentric pipes
under the influence of partial velocity slips on the surfaces and the
resulting anomalous heat transfer enhancement were investigated
by Turkyilmazoglu [35]. Recently, there are some studies consider
slip boundary condition in nanofluid fill cavity, as in Malvandi and
Ganji [36], and Mabood and Mastroberardin [37].
The present literature survey has led us to confirm that there is,
relatively, a very little published works regarding the slip boundary condition in the lid-driven cavities. Moreover, the topics of
nanofluids and magnetic field with partial slip have not clearly
arisen yet. Accordingly, the present work is prepared as an attempt
to continue in developing the mixed convection aspects. The
present geometry is a square cavity filled with Cu-Water nanofluid
subjected to an inclined magnetic field. The horizontal walls are
thermally insulted (adiabatic) and lid-driven with partial slip, the
vertical left wall is also adiabatic but contains a segment of heat
source. The right wall is isothermally cooled. It is sought that this
work will contribute in finding new parameters arrangements
those govern the performance of the lid driven cavities especially
those hold very high temperature difference where the partial slip
is inevitably exist.
2. Mathematical modeling
Consider a steady two-dimensional mixed convection flow inside a square cavity of side length H filled with Cu-water nanofluid, as depicted in Fig. 1. The coordinates x and y are chosen such
that x measures the distance along the bottom horizontal wall,
Fig. 1. Schematic diagram and boundary conditions.
M.A. Ismael et al. / Journal of Magnetism and Magnetic Materials 416 (2016) 25–36
Table 1
Thermo-physical properties of water and nanoparticles [22].
u
Property
Water
Copper (Cu)
ρ (kg m 3)
Cp (J Kg 1 K 1)
k (W m 1 K 1)
β (K 1)
s (S m 1)
997.1
4179
0.613
21 10 5
0.05
8933
385
401
1.67 10 5
5.96 10 7
⎛ ∂ 2u
∂u
∂u
1 ∂p
∂ 2u ⎞
+v
= −
+ νnf ⎜ 2 +
⎟
∂x
∂y
ρnf ∂x
∂y2 ⎠
⎝ ∂x
+
u
Iwatsu et al. [4]
Khanafer and Chamkha [20]
Present study
100
400
1000
1.94
3.84
6.33
2.01
3.91
6.33
1.93
3.91
6.31
while y measures the distance along the left vertical wall, respectively. Heat source (q″ ¼constant) is located on a part of the
left wall with length B* at x ¼D*. The remainder of the left wall is
adiabatic, while the right wall is kept isothermally at cold temperature Tc. The upper and bottom walls are adiabatic and moves
to the right or left with a speed U0. The nanofluids used in the
analysis are assumed to be incompressible and laminar, and the
base fluid (water) and the solid spherical nanoparticles (Cu) are in
thermal equilibrium. The thermo-physical properties of the base
fluid and the nanoparticles are given in Table 1 [21]. The thermophysical properties of the nanofluid are assumed constant except
for the density variation, which is calculated based on the Boussinesq approximation. Under the above assumptions, the conservation of mass, linear momentum, and also conservation of
energy equations are as follow [38]:
∂u
∂v
+
= 0,
∂y
∂x
+
u
(1)
σn B02
(v sin Φ cos Φ − u sin2 Φ),
ρnf
(2)
⎛ ∂ 2v
∂v
∂v
∂ 2v ⎞
1 ∂p
+v
= −
+ νnf ⎜ 2 + 2 ⎟
∂x
∂y
ρnf ∂y
⎝ ∂x
∂y ⎠
Table 2
Comparisons of the mean Nusselt number at the top wall, for different values of Re
at Pr¼ 0.71, Gr¼ 102.
Re
27
σnf B02
ρnf
(u sin Φ cos Φ − v cos2 Φ ) +
(ρβ ) nf
ρnf
g (T − Tc )
⎛ ∂ 2T
∂T
∂T
∂ 2T ⎞
+v
= αnf ⎜ 2 +
⎟,
∂x
∂y
∂y2 ⎠
⎝ ∂x
(3)
(4)
where u and v are the velocity components along the x and y axes,
respectively, T is the fluid temperature, p is the fluid pressure, g is
the gravity acceleration, ρnf is the density, αnf is the kinematic
viscosity.
The boundary conditions are
On the left wall, x ¼0,
u = v = 0,
∂T
q″
= −
,
∂x
k nf
( D* − 0. 5B*) ≤ y ≤ ( D* + 0. 5B*),
∂T
= 0 otherwise
∂x
On the right wall, x ¼H,
u=v=0, T =Tc , 0 ≤ y≤H
On the top wall, y¼ H (partial slip)
v=0, u=λt Uo+Nμ nf
∂u ∂T
,
=0
∂y ∂y
Fig. 2. (a) Streamlines and (b) Isotherms for Ri¼ 1, Ha¼10, ϕ ¼0.05, D ¼0.5, B ¼0.5, λt ¼ 1, λb ¼ 1.
(5)
28
M.A. Ismael et al. / Journal of Magnetism and Magnetic Materials 416 (2016) 25–36
Fig. 3. Horizontal (a) and vertical (b) velocities along mid sections of the enclosure for Ri ¼1, Ha¼ 10, ϕ ¼ 0.05, D ¼ 0.5, B ¼0.5, λt ¼1, λb ¼ 1.
previous studies (Aminossadati and Ghasemi [39], Khanafer et al.
[40], and Tukyilmazoglu [41]). This assumption is even coinciding
with Buongiorno's nanofluid model (multi-phase) when the concentration of nanoparticles is assumed constant [42] as follows:
The effective density of the nanofluid is given as:
ρnf = ( 1 − ϕ) ρf + ϕρp ,
(6)
where ϕ is the solid volume fraction of the nanofluid, ρf and ρp are
the densities of the fluid and of the solid particles respectively, and
the heat capacitance of the nanofluid is given by Khanafer et al.
[40] as,
( ρCp)nf
= ( 1 − ϕ)( ρCp) f + ϕ ( ρCp) p ,
(7)
The thermal expansion coefficient of the nanofluid can be determined by:
Fig. 4. Variation of the mean Nusselt number for Ri¼ 1, Ha¼ 10, D ¼0.5, B¼ 0.5,
λt ¼1, λb ¼ 1.
( ρβ )nf
= ( 1 − ϕ)( ρβ ) f + ϕ ( ρβ ) p ,
(8)
where βf and βp are the coefficients of thermal expansion of the
fluid and of the solid particles respectively.
Thermal diffusivity, αnf of the nanofluid is defined by Abu-Nada
and Chamkha [43] as:
αnf =
k nf
,
(ρcp )nf
(9)
In Eq. (9), k nf is the thermal conductivity of the nanofluid which
for spherical nanoparticles, according to the Maxwell-Garnett
model [44], is:
k nf
( k p + 2k f ) − 2ϕ ( k f − k p ) ,
=
kf
( k p + 2k f ) + ϕ ( k f − k p )
(10)
The effective dynamic viscosity of the nanofluid based on the
Brinkman model [45] is given by
Fig. 5. Variation of the average Nusselt number with Φ for different combinations
of λt and λb for Ri¼1, Ha¼ 10, ϕ ¼ 0.05, D¼ 0.5, B¼ 0.5.
On the bottom wall, y¼0 (partial slip)
v=0, u=λ b Uo+Nμ nf
∂u ∂T
,
=0
∂y ∂y
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 nanoparticles
volume fraction only which were proven and used in many
μ nf =
μf
( 1 − φ)2.5
,
(11)
where μf is the viscosity of the base fluid and the effective
electrical conductivity of nanofluid was presented by Maxwell [44]
as;
3 ( γ − 1) ϕ
σnf
=1+
,
σf
( γ + 2) − ( γ − 1) ϕ
where
(12)
M.A. Ismael et al. / Journal of Magnetism and Magnetic Materials 416 (2016) 25–36
29
Fig. 6. (a) Streamlines and (b) Isotherms for Φ ¼0, Ha ¼10, ϕ ¼0.05, D ¼0.5, B ¼0.5 λt ¼ 1 and λb ¼ 1.
0.8
0.9
0.7
0.8
0.7
0.5
0.4
U (X=0.5)
0.3
0.2
0.5
0.4
0.3
0.2
V (Y=0.5)
0.6
Ri=0.001
Ri=0.01
Ri=1
Ri=100
Ri=1000
0.6
Ri=0.001
Ri=0.01
Ri=1
Ri=100
Ri=1000
0.1
0.0
-0.1
-0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.3
-0.4
-0.4
-0.5
-0.5
-0.6
-0.6
-0.7
-0.7
-0.8
0.0
0.2
0.4
0.6
0.8
0.0
1.0
0.2
0.4
0.6
0.8
1.0
X
Y
Fig. 7. Horizontal (a) and vertical (b) velocities along mid sections of the enclosure for Φ ¼ 0, Ha¼ 10, ϕ¼ 0.05, D¼ 0.5, B¼ 0.5, λt ¼ λb ¼ 1.
γ=
σp
.
σf
(13)
U
∂U
∂U
∂P
1
+V
= −
+
.
∂X
∂Y
∂X
Re
⎛ νnf ⎞ ⎛ ∂ 2U
∂ 2U ⎞
⎜ ⎟ ⎜ 2 +
⎟
∂Y 2 ⎠
⎝ νf ⎠ ⎝ ∂X
⎛ ρ ⎞ ⎛ σ ⎞ Ha2
f
⎟⎟ ⎜ nf ⎟.
+ ⎜⎜
. (V sin Φ cos Φ − U cos2 Φ),
⎝ ρnf ⎠ ⎝ σf ⎠ Re
(16)
Introducing the following dimensionless set:
⎛ u, v ⎞
⎛ x, y , D*, B* ⎞
p
⎟, ( U , V )=⎜
,
⎟, P =
⎠
⎝ Uo ⎠
H
ρnf Uo2
( X , Y , D, B)=⎜⎝
θ=
( T −Tc )
q″H
k nf , Ri=
Nμf
Gr
, St =Sb=
,
Re2
H
U
(14)
into Eqs. (1)–(4) yields the following dimensionless equations:
∂U
∂V
+
= 0,
∂X
∂Y
(15)
∂V
∂V
∂P
1
+V
= −
+
.
∂Y
∂Y
∂X
Re
⎛ νnf ⎞ ⎛ ∂ 2V
(ρβ ) nf
∂ 2V ⎞
⎜ ⎟ ⎜ 2 +
. θ
⎟ + Ri.
2
∂Y ⎠
ρnf . βf
⎝ νf ⎠ ⎝ ∂X
⎛ ρ ⎞ ⎛ σ ⎞ Ha2
f
⎟⎟ ⎜ nf ⎟.
+ ⎜⎜
. (U sin Φ cos Φ − V cos2 Φ)
⎝ ρnf ⎠ ⎝ σf ⎠ Re
(17)
30
M.A. Ismael et al. / Journal of Magnetism and Magnetic Materials 416 (2016) 25–36
U = V = 0, θ = 0, 0 ≤ Y ≤ 1
On the top wall, Y = 1
V =0, U =λt +St
( partial
slip)
(19)
μ nf ∂U ∂θ
,
=0
μf ∂Y ∂Y
On the bottom wall, Y ¼0 (partial slip)
V =0, U =λ b+Sb
μ nf ∂U ∂θ
,
=0
μf ∂Y ∂Y
N
where St =Sb= H μf
The local Nusselt number is defined as:
Nus =
Fig. 8. Variation of the mean Nusselt number for Φ ¼ 0, ϕ¼ 0.05, D ¼ 0.5, B¼ 0.5,
λt ¼1, λb ¼ 1.
U
⎞
⎛ 1 ⎞ αnf ⎛
∂θ
∂θ
⎟
+V
=⎜
+
⎜
⎟,
⎝ Pr. Re ⎠ αf ⎝ ∂X 2
∂X
∂Y
∂Y 2 ⎠
∂ 2θ
∂ 2θ
(18)
gβf H 3ΔT
νf
U H
, Re = 0 , Gr =
, Ha = B0 H σf /μf
αf
νf
ν2 f
are the Prandtl number, the Reynolds number, the Grashof number
and the Hartman number, respectively.
The dimensionless boundary conditions for Eqs. (15)–(18) are
as follows:
On the left wall, X ¼0
U =V =0,
kf
∂θ
=−
,
∂X
k nf
( D−0. 5B)≤Y ≤( D+0. 5B),
On the right wall, X ¼1
∂θ
=0 otherwise
∂X
(20)
and the average Nusselt number over the heat source is defined as:
⎛
⎜1
Num=⎜
⎜B
⎝
⎞
Nus dY ⎟
⎟
⎠X= 0
D −0.5B
D +0.5B
∫
It is useful to recall the definition of the stream function
describe the fluid motion. It can be expressed as follow:
where
Pr =
1
,
θ along y = B
u=
∂ψ
∂ψ
, v= −
∂y
∂x
(21)
ψ to
(22)
3. Numerical method and validation
Eqs. (15)–(18) with the boundary conditions (19) 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
Fig. 9. (a) Streamlines and (b) Isotherms for Φ ¼0, Ri ¼1, ϕ¼ 0.05, D¼ 0.5, B¼ 0.5, λt ¼ λb ¼ 1.
M.A. Ismael et al. / Journal of Magnetism and Magnetic Materials 416 (2016) 25–36
31
0.04
0.04
0.03
0.02
0.00
-0.02
0.01
Ha=0
Ha=10
Ha=20
Ha=50
Ha=100
-0.04
-0.06
-0.08
V (Y=0.5)
U (X=0.5)
Ha=0
Ha=10
Ha=20
Ha=50
Ha=100
0.02
0.00
-0.01
-0.02
-0.10
-0.03
-0.12
0.0
0.2
0.4
0.6
0.8
0.0
1.0
0.2
0.4
0.6
0.8
1.0
X
Y
Fig. 10. Horizontal (a) and vertical (b) velocities along mid sections of the enclosure for Φ ¼0, Ri ¼ 1, ϕ ¼0.05, D ¼0.5, B¼ 0.5, λt ¼ λb ¼ 1.
5.2
5.0
9
4.8
4.6
8
6
4.2
4.0
3.8
3.6
Nu
7
Nu
4.4
Ha=0
Ha=10
Ha=20
Ha=50
Ha=100
5
3.4
3.2
3.0
4
2.8
2.6
3
Ha=0
Ha=10
Ha=20
Ha=50
Ha=100
2.4
2.2
2.0
2
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Y
1.8
0.00
0.02
0.04
0.06
0.08
0.10
φ
Fig. 11. (a) Distribution of the local Nusselt number and at ϕ¼ 0.05 and (b) variation of the mean Nusselt number for Φ ¼0, Ri¼ 1, D ¼ 0.5, B¼0.5, λt ¼ 1, λb ¼ 1.
algorithm [46]. 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. The
convergence criterion was set as 10 5. 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 Iwatsu et al. [4] and Khanafer
and Chamkha [20] in terms of the mean Nusselt number at the top
wall, for different values of Re. As we can see form Table 2, the
results are found in a good agreement with these results. These
favorable comparisons lend confidence in the numerical results to
be reported subsequently.
4. Results and discussion
Streamlines, isotherms, local velocity components, and local
and average Nusselt numbers are adopted in inspecting the results
of mixed convection aspects of the present problem. The studied
parameters which pertinently affect the flow and thermal fields
inside the considered cavity are; the orientation of the magnetic
field Φ ¼ 0–360°, the constant moving parameter λ ¼ 71, the size
and position of the constant heat source B ¼0.2–0.8, D ¼0.3–0.7,
respectively, Hartman number Ha¼ 0–100, Richardson number Ri
¼0.001–1000, and nanoparticles volume fraction ϕ ¼ 0.0–0.1.
Prandtl number and slip parameter are fixed at Pr ¼6.26, and Sb
¼St ¼ 1, respectively. For the purpose of viewing the results easily,
five parameters are fixed (unless where stated) while the remainder single one is altered and as illustrated in the following
categories.
4.1. Effect of magnetic field orientation (Φ)
To investigate the effect of the orientation angle (Φ) of the
applied magnetic field, the other parameters are fixed at Ri ¼1,
ϕ ¼0.05, Ha¼ 10, D ¼B ¼1, λt ¼1, λb ¼ 1. Fig. 2 shows the
streamlines and isotherms for various values of Φ. For Φ ¼45°, the
streamlines are formed in a main clockwise (CW) circulation with
a secondary counterclockwise (CCW) vortex formed close to the
upper wall which is being lid to the right. When Φ is increased to
90°, the expected magnetic force will act horizontally, hence, this
will skew the buoyancy effect, as a result, decreases the streamlines strength as can be seen in Fig. 2a. When Φ is set at 180° the
magnetic force will act downward, i.e. in opposite with the
buoyancy force, the case which is identical with that of Φ ¼0.
When Φ is further increased to 270°, the magnetic force will again
act horizontally, hence the main circulation is mostly similar to
that of Φ ¼ 90°, except the absence of the secondary vortex. Fig. 2b
presents the isotherms contours which manifest undisturbed
patterns with varying Φ, this due to the equivalence convections
mode (Ri ¼1) and low Hartman number as well.
The horizontal U and vertical V velocity components are plotted
32
M.A. Ismael et al. / Journal of Magnetism and Magnetic Materials 416 (2016) 25–36
Fig. 12. (a) Streamlines and (b) Isotherms for Ri ¼1, Ha¼ 10, Φ ¼ 0, ϕ¼ 0.05, D ¼0.5 λt ¼ λb ¼1.
Fig. 13. (a) Distribution of the local Nusselt number at Ha¼ 10 and (b) variation of the mean Nusselt number for Φ ¼0, Ri ¼ 1, ϕ ¼0.05, D ¼0.5, λt ¼ 1, λb ¼ 1.
in Fig. 3 along Y ¼0.5 and X ¼0.5, respectively. The U component
reveals negative values close to both horizontal walls (Fig. 3a). This
is an expected result for the bottom wall, but due to the top one,
this means that the buoyancy effect overcomes the shear effect
exerted by the top wall (λt ¼1 and λb ¼ 1). Increasing Φ reduces
the negative speed close to the moving walls. Two peaks of vertical
components V (Fig. 3b) are recorded close to the vertical walls
with higher velocity values associated with increasing Φ.
The mean Nusselt number is plotted with the nanoparticles
volume fraction for different values of Φ. As depicted in Fig. 4, it is
clear that for pure fluid (ϕ ¼ 0), a maximum convection suppression can be obtained when Φ ¼45°, while for nanofluid (ϕ 4 0),
convection suppression vanishes with increasing Φ value. However, the effect nanoparticles volume fraction will be discussed
fairly in Subsection 4.5.
It is worth mentioning here that the lid-directions of the horizontal walls was inspected for different combinations of λt and λb
with the orientation of the magnetic field Φ. The set of λt ¼1 (top
wall moves to right) and λb ¼ 1 (bottom wall moves to left)
manifests more stable performance with Φ and gives higher
Nusselt number when Φ r 25° (see Fig. 5). All others combinations
of λt,b manifests sinusoidally variation of Num with Φ. However,
the stable set λt ¼ λb ¼1 will be adopted in the next results.
4.2. Effect of Richardson number (ri)
The ratio of natural to forced convection modes is measured by
Richardson number. Its effect is studied by fixing the other independent parameters at Ha ¼10, ϕ ¼0.05, B¼ 0.5, D ¼0.5, λt ¼ λb
¼1. For low Ri range (0.001–0.01) the dominance forced convection can be characterized by the dominance shear action where
two counter rotated vortices each one is guided by a moving wall
(Fig. 6a). The corresponding isotherms (Fig. 6b) tend to be plumed
from heat source towards the cold vertical wall with isothermal
M.A. Ismael et al. / Journal of Magnetism and Magnetic Materials 416 (2016) 25–36
33
Fig. 14. (a) Streamlines and (b) Isotherms for Ri¼ 1, Ha ¼10, Φ ¼ 0, ϕ¼ 0.05, B¼ 0.5, λt ¼ λb ¼1.
8.0
5.5
7.5
7.0
6.0
5.5
4.5
D=0.3
D=0.4
D=0.5
D=0.6
D=0.7
4.0
5.0
Nu
Nu
5.0
D=0.3
D=0.4
D=0.5
D=0.6
D=0.7
6.5
4.5
3.5
3.0
4.0
3.5
2.5
3.0
2.0
2.5
1.5
0.0
(a)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Y
1.0
0
(b)
20
40
60
80
100
Ha
Fig. 15. (a) Distribution of the local Nusselt number at Ha¼ 10 and (b) variation of the mean Nusselt number for Φ ¼0, Ri¼ 1, ϕ¼ 0.05, B¼ 0.5, λt ¼ 1, λb ¼ 1.
zones localized close the moving walls and dense isotherms close
to the heat source. For the dominance natural convection
(Ra¼100–1000), the streamlines are strengthened and governed
by a single main CW circulation while the mostly horizontal isotherms reveal the dominance convection mode.
The local velocity components U and V shown in Fig. 7 demonstrate the mostly nanofluid stagnant at low Ra number.
Whereas for high Ra number these velocity components are significantly increased, nevertheless, the cavity center (core center)
still appearing as a stagnant point. However, U does not reach the
cavity walls speed (Uo) because the slip effect.
The effect of Richardson number on the mean Nusselt number
Num is shown in Fig. 8 with Hartman number. It is clear that the
Num is significantly decrease with increasing Ri up to unity where
beyond this Ri value there is no noticeable variation of Num with
Ri. This trend reveals that the convection is mainly due to the
forced convection mode which comes from the shear effect resulting from the horizontal walls movement. The variation of Num
with Hartman number will be postponed to a next category.
4.3. Effect of Hartman number
The effect of the applied magnetic field is studied by varying
the Hartman number Ha from 0 to 100 in the case of horizontal
magnetic field (Φ ¼0) while the other parameters are fixed at
Ri¼1, ϕ ¼ 0.05, B ¼0.5, D ¼0.5, λt ¼ λb ¼1. It is well known that
applying an external magnetic field, Lorentz force will be generated perpendicularly to the direction of the applied magnetic field
i.e. opposite to the buoyancy force (in this case). Therefore, Lorentz
force will act normally in the negative Y-direction. Accordingly, the
streamlines presented in Fig. 9a weaken and the main vortex
breaks up to double-eye pattern at Ha ¼50 while secondary vortex
34
M.A. Ismael et al. / Journal of Magnetism and Magnetic Materials 416 (2016) 25–36
Fig. 16. (a) Streamlines and (b) Isotherms forRi¼ 1, Ha¼ 10, Φ ¼0, D ¼ 0.5, B¼ 0.5 λt ¼ λb ¼ 1.
Fig. 17. Distribution of the local Nusselt number for Φ¼ 0, Ri¼ 1, Ha¼ 10, D ¼ 0.5,
B¼ 0.5, λt ¼ λb ¼ 1.
Fig. 18. Variation of the mean Nusselt number for Φ ¼0, Ri¼ 1, Ha¼ 10, B ¼0.5,
λt ¼ λb ¼ 1.
is squeezed to be limited close to the moving top wall. The corresponding isotherms (Fig. 9b) are transmitted from convection
pattern at Ha¼ 0 to mostly vertical pattern with increasing Ha
which indicates to the suppressed convection due to the magnetic
force effect. The U component is flattened with Ha while the peaks
of V component are significantly vanished as shown in Fig. 10. The
magnetic field suppression effect can be clearly observed by the
reduction of local and mean Nusselt numbers with increasing Ha
as shown in Fig. 11.
with small heat source, B ¼0.2, is changed to a main CW circulation occupying most of the cavity with small secondary vortex
close to the top wall. This fashion is almost seen with isothermally
heated wall cavity. Respecting with isotherms (Fig. 12b), the
plume-like and the dense isotherms fashion seen adjacent to the
heat source is not seen when B is increased more than 0.4. However, the effect of B is more evidenced by the local and mean
Nusselt numbers as shown in Fig. 13. This figure shows that the
maximum local Nusselt number occurs at the lower edge of the
heat source (Fig. 13a), and generally it is greater for smaller heat
source size. This can be understood by re-examining the isotherms
of Fig. 12b, where their pattern manifests that when increasing the
heat source size, a large portion of heat is transferred by conduction, hence, a decrease of convection heat transfer occurs accordingly as shown in Fig. 13b.
The effect of heat source position (D) on the streamlines and
isotherms is depicted in Fig. 14 for fixed source size (B¼ 0.5). It can
4.4. Effect of position (D) and dimension (B) of heating element
The effects of the size (B) and the position (D) of the heat
source will be discussed in this category for Ri¼1, Φ ¼0, ϕ ¼0.05,
Ha¼10, λt ¼ λb ¼1. Fig. 12 shows the effect of B on the streamlines and isotherms while D is fixed at 0.5. When B is increased,
the mostly equivalent counter rotating vortices pattern associated
M.A. Ismael et al. / Journal of Magnetism and Magnetic Materials 416 (2016) 25–36
Fig. 19. Variation of the mean Nusselt number for Φ¼ 0, Ha ¼10, B ¼0.5, D ¼ 0.5,
λt ¼ λb ¼ 1.
35
convective heat transfer. To understand the physical aspects of
adding the Cu nanoparticles to the pure water we would worthily
clarify that adding very high conductive solid nanoparticles will
generate a nanofluid with higher viscosity, higher density, and
higher thermal conductivity. The first two properties increase the
viscous and inertia forces, respectively, while the enhanced thermal conductivity increases the transferred thermal energy. Hence,
the distribution of the local Nusselt number presented in Fig. 17
elucidates that the enhanced viscous and inertia forces dominant
over the enhanced thermal conductivity and both buoyancy and
shear effects in addition. This is a reasonable reason for the deterioration of Nusselt number with addition of nanoparticles. Fig. 4
shows the deterioration of Num with ϕ. Nevertheless, other examinations of Nusselt number were conducted and the results are
presented in Figs. 18–20. Fig. 18 also tells the deterioration scenario
of Num with ϕ for different values of heat source position. However, inspecting ϕ for different values of Ri (Fig. 19) gave us an
encouragement issue, where a slight increase of Nusselt number
up to ϕ ¼0.04 is seen when Rir0.01 which means the dominance
of the thermal energy enhancement over the viscous and inertia
increase due to the very weak buoyancy force. This gave us an
ambitious to inspect another parameters. Already, Fig. 20 presents
the effect of ϕ for different B values. An enhancement in Num is
seen at ϕ ¼0.04 for lowest heat size, B ¼0.2. Eventually, reexamining Fig. 11b, it can be elucidated that when HaZ50, the
suppressed natural convection can serve in a continuous increase
in Num with increasing ϕ.
5. Conclusions
Fig. 20. Variation of the mean Nusselt number for Φ ¼ 0, Ha¼ 10, Ri ¼1, D ¼ 0.5,
λt ¼ λb ¼ 1.
be seen that the double-eye circulation is significantly affected by
increasing D, where the CW vortex grows up while the CCW one is
completely diminished leaving a stagnant zone as shown in
Fig. 14a. The isotherms behaviors with increasing D are similar to
those behaviors associated with B increments, where the plumelike and the dense isotherms fashion disappear when D is increased more than 0.4. The distribution of the local Nusselt
number for Φ ¼ 0, Ri ¼1, Ha¼10, ϕ ¼0.05, B ¼0.5, λt ¼1, λb ¼ 1 is
shown in Fig. 15a which presents similar patterns for all D values,
but greater under-curve area with lower heat source position. This
means that the mean Nusselt number is greater for lower D position which can be demonstrated by Fig. 15b. This can be attributed to that when the heat source becomes narrower to the top
wall, there will no enough vertical distance for the buoyancy effect
to be taken place. Hence, a substantial amount of nanofluid will
not corporate in convective heat transfer. However, when the
Hartman number is increased more than 40, this effect will diminishes resulting in increasing Num with higher D values.
4.5. Effect of nanoparticles volume fraction
The nanoparticles volume fraction is increased from ϕ ¼0 (pure
water) to ϕ ¼ 0.1 for Ri¼1, Ha ¼10, B ¼0.5, D ¼0.5, λt ¼ λb ¼ 1, and
Φ ¼ 0. The streamlines (Fig. 16a) keep its general pattern with
increasing ϕ, but their strength and intensity are appreciably decreased. The isotherms (Fig. 16b) desolate its plume like and
thermal boundary layers close the heat source and the cold wall
with increasing ϕ as an indication to the suppression of the
The studied problem is a steady laminar two dimensional
mixed convection inside a square cavity filled with Cu-water nanofluid and subjected to an externally-applied inclined magnetic
field. A constant heat flux is set on a portion of the left wall while
the right wall is kept isothermally cold. The remainder boundaries
of the cavity are adiabatic. Horizontal walls are lid-driven by
considering the partial slip effect. The following conclusions are
drawn from the present results;
1. The shortest length of the constant heat flux element gives the
maximum convective heat transfer.
2. For Hao50, the lowest heat source position gives maximum
Nusselt number, while for Ha 450, Nusselt number increases
with increasing the position of the heat source.
3. The orientation of the magnetic field can be considered as a key
control of the convective heat transfer where the suppression
exerted by the magnetic field on the Nusselt number decreases
with increasing the orientation of the applied magnetic field.
4. The most stable lid-driven direction can be attained when the
top wall moves to the right while the bottom one moves to the
left.
5. The natural convection is a decreasing function of the heat
source length for all ranges of the studied parameters, while it is
so due to the vertical distance up to Hartman number of 50
where beyond this value the Natural convection decreases with
lifting the heat source narrower to the top wall.
6. The effect of the nanoparticles volume fraction on the Nusselt
number manifests wide variety fashions, where little enhancements of Nusselt number can be obtained with very low values
of Richardson number (dominance forced convection) at volume fraction not more than 0.04. For equivalent convection
modes (Ri¼ 1), Nusselt number can be enhanced slightly with
shortest heat source length and noticeably with stronger magnetic field (Ha 450).
36
M.A. Ismael et al. / Journal of Magnetism and Magnetic Materials 416 (2016) 25–36
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