Eur. Phys. J. Plus (2015) 130: 86 DOI 10.1140/epjp/i2015150864
THE EUROPEAN PHYSICAL JOURNAL PLUS
Regular Article
Flow and heat transfer of ferroﬂuids over a ﬂat plate with uniform heat ﬂux W.A. Khan1 , Z.H. Khan2,3 , and R.U. Haq4,a 1
2 3 4
Department of Mechanical and Mechatronics Engineering, University of Waterloo 200, University of West Waterloo, Ontario, N2L 3G1, Canada School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China Department of Mathematics, University of Malakand, Dir (Lower), Khyber Pakhtunkhwa, Pakistan Department of Mathematics, QuaidiAzam University 45320, Islamabad, 44000, Pakistan Received: 2 March 2015 c Societ` Published online: 30 April 2015 – a Italiana di Fisica / SpringerVerlag 2015 Abstract. The present work is dedicated to analyze the ﬂow and heat transport of ferroﬂuids along a ﬂat plate subjected to uniform heat ﬂux and slip velocity. A magnetic ﬁeld is applied in the transverse direction to the plate. Moreover, three diﬀerent kinds of magnetic nanoparticles (Fe3 O4 , CoFe2 O4 , MnZnFe2 O4 ) are incorporated within the base ﬂuid. We have considered two diﬀerent kinds of base ﬂuids (kerosene and water) having poor thermal conductivity as compared to solid magnetic nanoparticles. Selfsimilar solutions are obtained and are compared with the available data for special cases. A simulation is performed for each ferroﬂuid mixture by considering the dominant eﬀects of slip and uniform heat ﬂux. It is found that the present results are in an excellent agreement with the existing literature. The variation of skin friction and heat transfer is also performed at the surface of the plate and then the better heat transfer and of each mixture is analyzed. Kerosenebased magnetite Fe3 O4 provides the higher heat transfer rate at the wall as compared to the kerosenebased cobalt ferrite and MnZn ferrite. It is also concluded that the primary eﬀect of the magnetic ﬁeld is to accelerate the dimensionless velocity and to reduce the dimensionless surface temperature as compared to the hydrodynamic case, thereby increasing the skin friction and the heat transfer rate of ferroﬂuids.
a
Nomenclature
Greek symbols
B0 σ cp cf f k M N ux Pr qw Rex T T∞ u U∞ v x y
α β ϕ η μ ν σ ρ ρcp θ Ψ
Magnetic ﬁeld intensity Electric conductivity Speciﬁc heat [J/kg·K] Friction coeﬃcient Dimensionless stream function Thermal conductivity Magnetic parameter Local Nusselt number Prandtl number of base ﬂuid Wall heat ﬂux [W/m2 ] Local Reynolds number Local ﬂuid temperature [K] Free stream temperature [K] xcomponent of velocity [m/s] Free stream velocity [m/s] ycomponent of velocity [m/s] Distance along the plate [m] Distance normal to the plate [m]
email: ideal
[email protected]
Thermal diﬀusivity [m2 /s] Slip parameter Volume fraction of ferroﬂuid Similarity variable Absolute viscosity [N·s/m2 ] Kinematic viscosity [m2 /s] Electric conductivity Density [kg/m3 ] Heat capacity [kg/m3 ·K] Dimensionless temperature Stream function
Subscripts nf f s
Nanoﬂuid Base ﬂuid Solid ferroparticles
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Eur. Phys. J. Plus (2015) 130: 86
1 Introduction Usually magnetite nanoparticles perform a random motion within the base ﬂuid, so through applying a magnetic ﬁeld the motion of those particles becomes uniform. So this phenomenon is applicable at a global level especially in electronic devices, mechanical engineering, spacecraft population, materials science, medical applications, optics and heat transfer process. Because of the temperature gradient the external magnetic ﬁeld is imposed on the ferroﬂuid with varying vulnerability output in a nonuniform magnetic body force, which is indicative of a form of heat transfer called thermomagnetic convection. This kind of heat transfer process can be helpful when the conventionalconvection heat transfer is insuﬃcient. The ﬂow and heat transfer from Newtonian and nonNewtonian ﬂuids along ﬂat horizontal plates under diﬀerent hydrodynamic and thermal boundary conditions have been well documented in several books including [1–4] and open literature including [5–11]. Recently, the use of nanoﬂuids along ﬂat plates to enhance the heat transfer rate is gaining interest among researchers and several studies can be found in the literature as [12–18], among others. MHD ﬂow and heat transfer from ﬂuids along plates have many engineering applications in diﬀerent industries like petroleum, geothermal and aerodynamics ones. Most of the earlier studies including [19–24] investigate MHD convection heat transfer from surfaces under no slip condition. There are, however, many situations, where ﬂuids exhibit boundary slip, e.g. microscale ﬂuid dynamics in MEMS. The studies related to slip boundary can be found in [25–27]. Like nanoﬂuids, ferroﬂuids are suspensions of ferromagnetic particles of diameter approximately 10 nm stabilized by surfactants in carrier liquids, such as water or oil. Ferroﬂuids have the capability to reduce the friction at the surface. If strong magnetic particles are incorporated in the base ﬂuid and applied to the surface, this one made of neodymium can cause the magnet to glide across smooth surfaces with minimal resistance. The heat transfer rate and ﬂow properties of these ﬂuids can be controlled by applying an external magnetic ﬁeld [28–30]. Due to various novel interesting properties, ferroﬂuids have been used in many engineering applications including heat transfer, the magnetically controlled thermal ﬂow, sealing technology, biomedicine, printer inks, magnetorheological ﬂuids and shock absorbers [31–34]. In the presence of magnetic ﬁeld, ferroﬂuids become strongly magnetized and they can exhibit extremely large enhancement in thermal conductivity due to the eﬃcient transport of heat through percolating ferroparticle paths. Special magnetic nanoﬂuids with tunable thermalconductivity–to–viscosity ratio can be used as multifunctional smart materials that can remove heat and also reduce vibrations. Such ﬂuids may ﬁnd applications in microﬂuidic devices and microelectromechanical systems. These ferroﬂuid properties are derived using the volume fraction of solid nanoparticles in combination with the properties of base ﬂuid and nanoparticles. This model, originally developed by Choi [12], has been used by [13,15,35] among others. The studies related to the use of ferroﬂuids have been conducted by [36–38] to enhance heat transfer in the boundary layer, and the convective stability of ferroﬂuids in magnetic ﬁeld by [39–47] could be helpful in understanding the physics behind the ﬂow of ferroﬂuids along plates. The above literature review reveals that no study exists to investigate the MHD boundary layer ﬂow and heat transfer of ferroﬂuids along a ﬂat plate with slip velocity. This is the main objective of the present study. For this purpose, we have considered a homogeneous model to study the forced convective ﬂow and heat transfer of ferroﬂuids along a ﬂat plate subjected to a uniform wall heat ﬂux. The eﬀects of magnetic ﬁeld and slip velocity on the dimensionless velocity, temperature, skin friction and heat transfer rate are investigated for diﬀerent magnetic nanoparticles in water and kerosene oil.
2 Problem formulation Consider the forced convection heat transfer of selected ferroﬂuids along a stationary ﬂat plate in a constant magnetic ﬁeld with uniform surface heat ﬂux. The plate is embedded in a medium saturated with water or kerosenebased ferroﬂuids. The ﬂow is assumed to be laminar, steady, incompressible, and twodimensional. The base ﬂuids and the selected nanoparticles are assumed to be in thermal equilibrium. In the presence of magnetic ﬁeld, the ferroparticles moments almost instantly orient along the magnetic ﬁeld lines and when the magnetic ﬁeld is removed, the particles moments are quickly randomized. This orientation along the magneticﬁeld lines shows a certain precise positioning of the ferroparticles depending upon the position of the magnetic ﬁeld (see ﬁg. 1). The hydrodynamic slip is assumed at the ﬂuidsolid interface. The viscous dissipation and radiation are neglected in the analysis. The ambient temperature is assumed to be constant. Using an orderofmagnitude analysis, the standard boundary layer equations for this problem can be written as follows: ∂u ∂v + = 0, ∂x ∂y ∂u ∂u ∂ 2 u σB 2 (x) +v = υnf 2 − (u − U∞ ), u ∂x ∂y ∂y ρnf ∂T ∂T ∂2T +v = αnf 2 , u ∂x ∂y ∂y
(1) (2) (3)
Eur. Phys. J. Plus (2015) 130: 86
Page 3 of 10
Fig. 1. Schematic of the boundary layer ferroﬂuid ﬂow over a ﬂat plate.
where u and v are the velocity components along the x and yaxes, T is the temperature, υnf and αnf are the eﬀective kinematic viscosity and thermal diﬀusivity of the ferroﬂuid, respectively. The transverse magnetic ﬁeld assumed to be a function of the distance from the origin is deﬁned as B(x) = B0 x−1/2 with B0 = 0, where x is the coordinate along the plate and B0 is the magneticﬁeld strength. The eﬀective properties of ferroﬂuids may be expressed in terms of the properties of base ﬂuid and ferroparticles and the volume fraction of solid ferroparticles as follows: ⎧ μnf μf , μnf = , υnf = ⎪ ⎪ ⎪ ρ (1 − φ)2.5 nf ⎪ ⎪ ⎪ knf ⎪ ⎪α = ⎨ , ρnf = (1 − φ)ρf + φρs , nf ρnf (Cp )nf (4) ⎪ ⎪ (ρCp )nf = (1 − φ)(ρCp )f + φ(ρCp )s , ⎪ ⎪ ⎪ ⎪ ⎪ k + 2kf − 2φ(kf − ks ) k ⎪ ⎩ nf = s , kf ks + 2kf + φ(kf − ks ) where knf is the thermal conductivity of the ferroﬂuid, (ρCp )nf is the heat capacity of ferroﬂuid and φ is the volume fraction of solid ferroparticles. The hydrodynamic and thermal boundary conditions for the problem are given by ⎧ ⎨u = γ ∂u , v = 0, −k ∂T = q , at y = 0, nf w ∂y ∂y (5) ⎩ as y → ∞, u → U∞ , v → 0, T → T∞ . where γ is the slip parameter and U∞ is the free stream velocity. We look for a similarity solution of eqs. (1)–(3) of the following form: y T − T∞ η= Rex , θ(η) = Rex , (6) ψ = νf Rex f (η), x qw x/kf where η is the similarity variable and Rex = U∞ x/νf is the local Reynolds number based on the free stream velocity and νf is the kinematic viscosity of the base ﬂuid. The stream function Ψ is deﬁned as u = ∂Ψ/∂y and v = −∂Ψ/∂x. Employing the similarity variables (6), eqs. (1)–(3) reduce to the following nonlinear system of ordinary diﬀerential equations: 1 2.5 (1 − φ + φρs /ρf ) f f + M (1 − f ) = 0, (7) f + (1 − φ) 2 (knf /kf ) 1 1 θ + (f θ − f θ) = 0, (8) [1 − φ + φ ((ρCp )s /(ρCp )f )] Pr 2 subjected to the boundary conditions (6) which become ⎧ β ⎨f (0) = 0, f (0) = f (0), (1 − φ)2.5 ⎩ f (η) → 1, θ(η) → 0 as η → ∞,
θ (0) = −
knf , kf
(9)
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Eur. Phys. J. Plus (2015) 130: 86
Table 1. Comparison of skin friction coeﬃcients for speciﬁc values of velocity slip and magnetic parameters in the absence of ferromagnetic particles. β
M
Blasius
Cortell [49]
Rahman
Yazdi
Present
[50]
[51]
work
[48] 0
0.5
0.5
0.3321
0.33206
–
–
0.33206
1
–
–
1.0440
1.0440
1.04400
–
–
–
0.6987
0.6987
0.69872
Table 2. Thermophysical properties of base ﬂuids and magnetic nanoparticles [46, 47, 52]. Physical properties
Base ﬂuids
Magnetic nanoparticles
Water
Kerosene
Fe3 O4
CoFe2 O4
MnZnFe2 O4
ρ (kg/m )
997
783
5180
4907
4900
cp (j/Kg·k)
4179
2090
670
700
800
k (W/m·k)
0.613
0.15
9.7
3.7
5
3
Here, primes denote diﬀerentiation with respect to η, P r = (μcp )f /kf is the Prandtl number, β = U∞ γ/νf is the dimensionless slip parameter and M = σB02 /ρU∞ is the magnetic parameter. The physical quantities of interest are the skin friction coeﬃcient Cf and the local Nusselt number N ux , which are deﬁned as cf =
τwx , 2 ρf U∞
N ux =
xqw , kf (Tw − T∞ )
where τwx is the surface shear stresses along the xdirections and qw is the heat ﬂux, and are given by
∂u ∂T , qw = −knf . τwx = μnf ∂y y=0 ∂y y=0 Reduced dimensionless forms of eq. (10) take the form, after ﬁxing Re = Cf Re1/2 = x
f (0)
2.5 , (1 − φ) knf 1 . Nu = Re−1/2 x kf θ(0)
(10)
(11)
Uw x νf ,
(12) (13)
3 Results and discussion The coupled nonlinear twopoint boundary value problem, eqs. (7) and (8), combined with the boundary condition (9) are solved numerically using the RungeKuttaFehlberg method after converting the boundary value problem in to the initialvalue problem with the help of the shooting technique. The proposed numerical method is programmed in MATLAB with a step size η = 0.01 and used to solve the coupled system in the interval 0 ≤ η ≤ ηmax , where ηmax is the ﬁnite value of the similarity variable η for the farﬁeld boundary conditions. For the asymptotic boundary conditions, the result converges for high values of the similarity variable ηmax = 12, for all values of the parameters involved. The eﬀects of volume fraction of solid ferroparticles, magnetic and velocity slip parameters on the dimensionless velocity, temperature, skin friction and Nusselt numbers are investigated for the three selected ferroparticles; magnetite, cobalt ferrite and MnZn ferrite with two diﬀerent base ﬂuids, water and kerosene. The values of Prandtl number for the base ﬂuids, kerosene and water are taken to be 21 and 6.2, respectively. The eﬀect of the volume fraction of solid ferroparticles φ is studied in the range 0 ≤ φ ≤ 0.2, where φ = 0 represents the pure ﬂuid water or kerosene. The thermophysical properties of the base ﬂuids (water and kerosene) and the ferroparticles (magnetite, cobalt ferrite and MnZn ferrite) are listed in table 1. Table 2, shows the variation of the thermophysical properties of water and kerosenebased ferroﬂuids with the volume fraction of solid ferroparticles. It is clear that the density and thermal conductivity of each ferroﬂuid increase with the volume fraction of solid ferroparticles. It is observed that waterbased ferroﬂuids have higher densities and
Eur. Phys. J. Plus (2015) 130: 86
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Table 3. Variation of the skin friction with the volume fraction of solid nanoparticles for diﬀerent values of velocity slip and magnetic parameters. β=0
φ M =0
M =1
β=2 M =2
M =0
M =1
M =2
Waterbased ferroﬂuids Fe3 O4
CoFe2 O4
MnZnFe2 O4
0.01
0.34324
1.05914
1.46424
0.24437
0.34578
0.37573
0.1
0.45131
1.21292
1.66309
0.28287
0.36201
0.38832
0.2
0.59517
1.43092
1.94460
0.31974
0.38007
0.40242
0.01
0.34278
1.05902
1.46416
0.24418
0.34576
0.38180
0.1
0.44694
1.21148
1.66210
0.28153
0.36176
0.38820
0.2
0.58624
1.42757
1.94228
0.31781
0.37962
0.40219
0.01
0.34277
1.05902
1.46416
0.24418
0.34576
0.37572
0.1
0.44683
1.21144
1.66208
0.28149
0.36175
0.38820
0.2
0.58601
1.42749
1.94222
0.31776
0.37961
0.40219
0.01
0.34557
1.05980
1.46470
0.24533
0.34593
0.37580
0.1
0.47336
1.22039
1.66825
0.28942
0.36332
0.38896
0.2
0.63950
1.44827
1.95670
0.32879
0.38235
0.40356
0.01
0.34500
1.05964
1.46458
0.24509
0.34589
0.37579
0.1
0.46804
1.21856
1.66698
0.28788
0.36300
0.38881
0.2
0.62891
1.44402
1.95372
0.32671
0.38180
0.40328
0.01
0.34498
1.05964
1.46458
0.24509
0.34589
0.37578
0.1
0.46791
1.21851
1.66695
0.28784
0.36299
0.38880
0.2
0.62863
1.44391
1.95365
0.32665
0.38179
0.40328
Kerosenebased ferroﬂuids Fe3 O4
CoFe2 O4
MnZnFe2 O4
thermal conductivities than kerosenebased ferroﬂuids. This is due to the higher density and thermal conductivity of water as shown in table 1. The present results are found in good agreement with the published data. For each mixture of water and kerosenebased ferroparticles, the variation of skin friction and Nusselt number with the volume fraction of solid ferroparticles for diﬀerent values of velocity slip and magnetic parameters is presented in tables 3 and 4, respectively. Table 3 shows that the skin friction increases/decreases with increasing magnetic/velocity slip parameters, respectively. Table 4 shows the increase in the Nusselt number with both magnetic and velocity slip parameters. It is interesting to note that kerosenebased ferroﬂuids have higher skin friction and Nusselt numbers than waterbased ferroﬂuids. The variation of the dimensionless velocity with volume fraction of solid ferroparticles and magnetic parameters is shown in ﬁgs. 2 for the three selected ferroparticles, respectively. In each case, the dimensionless velocity increases at the surface and within the boundary layer with the volume fraction of solid ferroparticles and the magnetic parameter. In the absence of magnetic ﬁeld, the dimensionless velocity is found to be smaller. As the magnetic ﬁeld is applied, it arranges the ferroparticles in order and enhances the dimensionless velocity. Consequently, the hydrodynamic boundary layer thickness decreases. It is also noticed that the dimensionless velocity increases with the volume fraction of solid ferroparticles. The eﬀects of nanoparticle volume fraction and magnetic parameter on dimensionless temperature are shown in ﬁgs. 3(a) and (b) for three diﬀerent ferroﬂuids. Due to the higher Prandtl number of kerosene, the thermal boundary layer thickness as well as the dimensionless surface temperatures is smaller for each kerosenebased ferroﬂuid. Due to increase in thermal conductivity with nanoparticle volume fraction (table 2), in each case, the dimensionless surface temperature decreases with an increase in the ferroparticle volume fraction and magnetic parameter. The eﬀects of magnetic and slip parameters on the skin friction are investigated in ﬁgs. 4 and 5, respectively, for water and kerosenebased ferroﬂuids. It can be seen that the skin friction is minimum for pure ﬂuids in each case. As the volume fraction of solid nanoparticles increases, the skin friction also increases. It is due to an increase in density with the ferroparticle volume fraction. It is important to note that kerosenebased ferroﬂuids have higher skin friction in each case. This is due to lower densities and higher Prandtl numbers of kerosenebased ferroﬂuids. Figures 4(a)–(c) illustrate that the skin friction increases with magnetic ﬁeld. This increase in skin friction increases pressure drop, which is not favorable to hydraulic engineers. Figures 5(a)–(c) illustrate the eﬀects of slip parameter on the skin friction coeﬃcient in the presence of magnetic ﬁeld. When there is no slip, ﬁg. 5(a) shows that there is high resistance to ﬂuid ﬂow.
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Eur. Phys. J. Plus (2015) 130: 86
Table 4. Variation of the Nusselt numbers with the volume fraction of solid nanoparticles for diﬀerent values of velocity slip and magnetic parameters. β=0
φ M =0
β=2
M =1
M =2
M =0
M =1
M =2
Waterbased ferroﬂuids
Fe3 O4
CoFe2 O4
MnZnFe2 O4
0.01
0.88810
1.20061
1.30706
1.70454
1.99423
2.06952
0.1
1.28045
1.65723
1.79827
2.49038
2.78388
2.87232
0.2
1.85096
2.31637
2.50562
3.66044
3.95812
4.06042
0.01
0.87871
1.18854
1.29401
1.68805
1.97568
2.05034
0.1
1.15876
1.50571
1.63495
2.27419
2.54806
2.62984
0.2
1.52897
1.92546
2.08546
3.07881
3.33868
3.42672
0.01
0.88245
1.19358
1.29948
1.69497
1.98377
2.05874
0.1
1.20686
1.56795
1.70242
2.36516
2.64993
2.73494
0.2
1.65431
2.08270
2.25547
3.32224
3.60255
3.69745
Kerosenebased ferroﬂuids
Fe3 O4
CoFe2 O4
MnZnFe2 O4
0.01
1.35242
1.86980
2.05115
3.09818
3.64864
3.79469
0.1
2.15964
2.82685
3.08952
5.07056
5.65194
5.83813
0.2
3.44803
4.33828
4.72385
8.32249
8.94696
9.18055
0.01
1.34707
1.86336
2.04417
3.08701
3.63701
3.78272
0.1
2.08374
2.73678
2.99218
4.91058
5.48577
5.66791
0.2
3.22404
4.07609
4.44143
7.84059
8.44961
8.67363
0.01
1.35021
1.86780
2.04906
3.09520
3.64673
3.79284
0.1
2.12913
2.79741
3.05878
5.03200
5.62205
5.80887
0.2
3.35711
4.24670
4.62816
8.20623
8.84478
9.07959
Cobalt ferrite
Magnetite
1
M=1
f (η)
f (η)

0.7
0.6
β=2 0
1
2
3
η
φ = 0.2, 0.1, 0
0.8
M=0
0.5
0.9
0.6
Kerosene (Pr = 21)
0.5
4
Water (Pr = 6.2)
(b)
1
2
3
η
4
M=0
0.7 Water (Pr = 6.2)
0.6
Kerosene (Pr = 21)
β=2 0
5

M=0
0.7
Water (Pr = 6.2)
φ = 0.2, 0.1, 0
0.8
f (η)
φ = 0.2, 0.1, 0
0.8
(a)
M=1
M=1
0.9
0.9

Manganesezinc ferrite
1
1
5
0
(c)
Kerosene (Pr = 21)
β=2
0.5
1
2
3
4
5
η
Fig. 2. Eﬀects of the volume fraction of ferroparticles and magnetic parameters on the dimensionless velocity for diﬀerent ferroﬂuids.
Eur. Phys. J. Plus (2015) 130: 86
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Magnetite 0.8
Cobalt ferrite 0.8
Water (Pr = 6.2) Kerosene (Pr = 21)
0.6
Kerosene (Pr = 21)
Water (Pr = 6.2) Kerosene (Pr = 21)
0.6
0.4
φ = 0.1, 0
θ(η)
M=0
θ(η)
θ(η)
0.8
Water (Pr = 6.2)
0.6
M=0
0.4
Manganesezinc ferrite
M=0
0.4
φ = 0.1, 0
φ = 0.1, 0
0.2
0.2
M=2
0.2 M=2
β = 0.5
M=2
β = 0.5
β = 0.5
0
0 0
1
2
3
η
(a)
0 0
1
2
η
(b)
3
0
1
2
3
η
(c)
Fig. 3. Eﬀects of the volume fraction of ferroparticles and magnetic parameters on the dimensionless temperature for diﬀerent ferroﬂuids.
0.26
β=2
0.355
Fe3O4
0.38
Fe3O4
Fe3O4
0.35
MnZnFe2O4
0.25
0.385 1/2
W
0.27
β=2
Rex Cf
Re1/2 x Cf
at er
0.36
M=2
Re1/2 x Cf
β=2
0.28
M=1 Ke ro se ne W at er
Ke ro se ne
M=0
MnZnFe2O4
MnZnFe2O4
CoFe2O4
Ke ro se ne W at er
0.39
0.365
0.29
CoFe2O4
0.345
CoFe2O4
0.375
0.24 0
(a)
0.025
0.05
φ
0.075
0.1
0
(b)
0.025
0.05
φ
0.075
0
0.1
(c)
0.025
0.05
0.075
0.1
φ
Fig. 4. Variation of the skin friction coeﬃcient with φ and with varying magnetic parameter M for diﬀerent ferroﬂuids.
As the slip parameter increases, the ﬂow resistance decreases and as a result skin friction also decreases for each ferroﬂuid. It is noticed that the magnetite nanoparticles with kerosene or water as the base ﬂuids show the highest resistance among other two ferroparticles for the same volume fraction. The variation of Nusselt number with volume fraction of solid ferroparticles are shown in ﬁgs. 6 and 7 for diﬀerent values of magnetic and slip parameters, respectively. Both water and kerosene are used as base ﬂuids. Due to higher Prandtl number of kerosene, the thermal boundary layer thickness for kerosene is found to be smaller (ﬁgs. 3(a)–(c)). This is why the Nusselt numbers for kerosene are found to be higher than for water, as shown in ﬁgs. 6(a)–(c). It is also observed that the Nusselt numbers increase with increasing magnetic ﬁeld. The same behavior was observed by Jafari et al. [30] using CFD simulations. Figures 7(a)–(c) show the eﬀects of the slip parameter on the Nusselt numbers for water and kerosenebased ferroﬂuids. In case of no slip, the Nusselt numbers are found to be lower and they increase with increasing slip parameter. Again, the Nusselt numbers for kerosenebased ferroﬂuids are higher than for waterbased ferroﬂuids in each case. Figures 8 (a) and (b) provide the stream lines for the base ﬂuid and the magnetite nanoﬂuid. We can further observe that due to its higher density, the magnetite nanoﬂuid gives the dominant contribution near the surface of the sheet (η = 0).
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Eur. Phys. J. Plus (2015) 130: 86
0.9
0.5
Fe3 O4 0.48
CoFe2O4 0
0.05
0.1
0.15
0.2
ne
MnZnFe2 O4 0.32
CoFe2O4 0.05
0.1
0.15
CoFe2O4 0
0.2
φ
(b)
at er
Fe3 O4
MnZnFe2O4
0.46
φ
(a)
0.34
Fe3 O4
MnZnFe2 O4 0.8
W
1/2
0.52
β=2
Ke ro se
Rex C f
1/2
Rex C f
1
β=1
M = 0.5
0.36
1/2
0.54
Ke ro se ne W at er
β=0
M = 0.5
Rex C f
0.56
M = 0.5
Ke ro se ne W at er
1.1
0.05
0.1
0.15
0.2
φ
(c)
Fig. 5. Variation of the skin friction coeﬃcient with the volume fraction of solid ferroparticles and with varying velocity slip parameters for diﬀerent ferroﬂuids.
W
3
ate
5
β=2
4
W
ate
r
0
0.05
(a)
0.1
φ
0.15
0.2
M=2
5
β=2
W
3
3
2
2 0
(b)
6
4
2 1
0.05
0.1
φ
CoFe2O4
0.15
Ke ro se ne
M=1
Nux
6
MnZnFe2O4
7
Re
r
CoFe2O4
Fe3O4
8
1/2 x
β=2
Nux
4
1/2
M=0
MnZnFe2O4
7
Rex
5
Ke ro se ne
CoFe2O4
6
9
Fe3O4
8
MnZnFe2O4
7
Re1/2 Nux x
9
Fe3O4
Ke ro se ne
8
0.2
0
(c)
0.05
0.1
ate
r
0.15
0.2
φ
Fig. 6. Variation of the Nusselt number with the volume fraction of solid ferroparticles and with varying magnetic parameter for diﬀerent ferroﬂuids.
4 Conclusions The present study investigates the magnetohydrodynamics ﬂow and heat transfer of ferroﬂuids along a ﬂat plate with slip velocity. For each ferroﬂuid, it is concluded that: – – – – –
The eﬀect of the slip parameter is to reduce friction and increase heat transfer rate. The magnetic ﬁeld tends to increase both skin friction and heat transfer rate. The skin friction and heat transfer rates increase with nanoparticle volume fraction. Kerosenebased ferroﬂuids have higher skin friction and heat transfer rates than waterbased ferroﬂuids. Kerosenebased magnetite Fe3 O4 provides the higher heat transfer rate at the wall as compared to the kerosenebased cobalt ferrite and MnZn ferrite. – The dominant eﬀects of the ferroﬂuid are visualized in the stream lines as compared to the simple base ﬂuid.
Eur. Phys. J. Plus (2015) 130: 86
Page 9 of 10
CoFe2O4
Re1/2 Nux x
Nux 1/2
Rex
Ke ro se ne
CoFe2O4
2.5
M=0 2
β=0 W
1.5
MnZnFe2O4
4
r ate
CoFe2O4
M=2
3
β=0 W
2
MnZnFe2O4
7
Re1/2 Nux x
MnZnFe2O4
Fe3O4
8
Ke ro se ne
3
Fe3O4
r ate
6
Ke ro se ne
Fe3O4
M=2 5
β=1
4
W
3
ate
r
1 2 0
0.05
0.1
0.15
φ
(a)
1
0.2
0
0.05
0.1
0.15
0.2
φ
(b)
0
0.05
0.1
0.15
0.2
φ
(c)
Fig. 7. Variation of the Nusselt number with the volume fraction of solid ferroparticles and with varying velocity slip parameters for diﬀerent ferroﬂuids.
5
M = β = 0.5, φ = 0.2
M=β=φ=0
5
4
3
3
η
η
4
(a)
2
2
1
1
0 0
2
4
x
0 0
(b)
1
2
3
x
Fig. 8. Variation of stream lines for (a) base ﬂuid and for (b) ferroﬂuids.
4
5
Page 10 of 10
Eur. Phys. J. Plus (2015) 130: 86
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