Eur. Phys. J. Plus (2015) 130: 86 DOI 10.1140/epjp/i2015-15086-4
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, Quaid-i-Azam University 45320, Islamabad, 44000, Pakistan Received: 2 March 2015 c Societ` Published online: 30 April 2015 – a Italiana di Fisica / Springer-Verlag 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 , Mn-ZnFe2 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. Self-similar 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. Kerosene-based magnetite Fe3 O4 provides the higher heat transfer rate at the wall as compared to the kerosene-based cobalt ferrite and Mn-Zn 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] x-component of velocity [m/s] Free stream velocity [m/s] y-component of velocity [m/s] Distance along the plate [m] Distance normal to the plate [m]
e-mail: 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 non-uniform 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 conventional-convection heat transfer is insufficient. The flow and heat transfer from Newtonian and non-Newtonian 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. micro-scale 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, magneto-rheological 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 thermal-conductivity–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 kerosene-based ferrofluids. The flow is assumed to be laminar, steady, incompressible, and two-dimensional. 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 magnetic-field 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 fluid-solid interface. The viscous dissipation and radiation are neglected in the analysis. The ambient temperature is assumed to be constant. Using an order-of-magnitude 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
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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 y-axes, 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 magnetic-field 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
Mn-ZnFe2 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 x-directions 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 non-linear two-point boundary value problem, eqs. (7) and (8), combined with the boundary condition (9) are solved numerically using the Runge-Kutta-Fehlberg method after converting the boundary value problem in to the initial-value 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 far-field 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 Mn-Zn 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 Mn-Zn ferrite) are listed in table 1. Table 2, shows the variation of the thermophysical properties of water- and kerosene-based 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 water-based 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
Water-based ferrofluids Fe3 O4
CoFe2 O4
Mn-ZnFe2 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
Kerosene-based ferrofluids Fe3 O4
CoFe2 O4
Mn-ZnFe2 O4
thermal conductivities than kerosene-based 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 kerosene-based 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 kerosene-based ferrofluids have higher skin friction and Nusselt numbers than water-based 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 kerosene-based 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 kerosene-based ferrofluids have higher skin friction in each case. This is due to lower densities and higher Prandtl numbers of kerosene-based 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
Water-based ferrofluids
Fe3 O4
CoFe2 O4
Mn-ZnFe2 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
Kerosene-based ferrofluids
Fe3 O4
CoFe2 O4
Mn-ZnFe2 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
-
Manganese-zinc 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
Manganese-zinc 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
Mn-ZnFe2O4
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
Mn-ZnFe2O4
Mn-ZnFe2O4
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 kerosene-based 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 kerosene-based ferrofluids are higher than for water-based 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
Mn-ZnFe2 O4 0.32
CoFe2O4 0.05
0.1
0.15
CoFe2O4 0
0.2
φ
(b)
at er
Fe3 O4
Mn-ZnFe2O4
0.46
φ
(a)
0.34
Fe3 O4
Mn-ZnFe2 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
Mn-ZnFe2O4
7
Re
r
CoFe2O4
Fe3O4
8
-1/2 x
β=2
Nux
4
-1/2
M=0
Mn-ZnFe2O4
7
Rex
5
Ke ro se ne
CoFe2O4
6
9
Fe3O4
8
Mn-ZnFe2O4
7
Re-1/2 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. Kerosene-based ferrofluids have higher skin friction and heat transfer rates than water-based ferrofluids. Kerosene-based magnetite Fe3 O4 provides the higher heat transfer rate at the wall as compared to the kerosenebased cobalt ferrite and Mn-Zn 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
Re-1/2 Nux x
Nux -1/2
Rex
Ke ro se ne
CoFe2O4
2.5
M=0 2
β=0 W
1.5
Mn-ZnFe2O4
4
r ate
CoFe2O4
M=2
3
β=0 W
2
Mn-ZnFe2O4
7
Re-1/2 Nux x
Mn-ZnFe2O4
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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