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Exact analysis for the effect of heat transfer on MHD and radiation Marangoni boundary layer nanofluid flow past a surface embedded in a porous medium
Journal of Molecular Liquids 215 (2016) 625–639
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Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq
Exact analysis for the effect of heat transfer on MHD and radiation Marangoni boundary layer nanofluid flow past a surface embedded in a porous medium Emad H. Aly a,b,⁎, Abdelhalim Ebaid c,⁎ a b c
Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah 21589, Saudi Arabia Department of Mathematics, Faculty of Education, Ain Shams University, Roxy 11757, Cairo, Egypt Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 5 September 2015 Accepted 30 December 2015 Available online 27 January 2016 Keywords: Marangoni Radiation Magnetic Heat transfer Laplace
a b s t r a c t In the presence of a magnetic field, thermal radiation and with suction/injection, Marangoni boundary layer flow past a surface embedded in a porous medium saturated by a water based nanofluid containing two different types of nanoparticles, namely Copper and Titanium dioxide, has been studied. Exact solutions of the resulted equations were solved using a new approach via Laplace transform. Influence of the velocity profile and temperature distribution for the present nanofluids was investigated under effect of the involved parameters. It was found that the velocities of the two investigated nanoparticles decrease significantly with an increase in the solid volume fraction. In addition, the effective electrical conductivity parameter is mandatory and should be taken into account on applying the magnetic field; otherwise a spurious physical sight is to be gained. As expected, the magnetic parameter decelerates the fluid velocity and increases its temperature, as well, for all nanoparticles considered and for all investigated cases of the suction/injection parameter, the temperature profiles increase as radiation parameter increases. Further, the fluid suction decreases the fluid velocity and, therefore, thickness of the hydrodynamic boundary layer, regarding, the fluid temperature and thermal boundary layer decreases as well. However, fluid injection produces the opposite effect. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The Marangoni boundary layers are basically defined as dissipative layers which may occur along the liquid–liquid or liquid–gas interfaces. The original work in this field was first addressed by Napolitano [1,2]. Such type of Marangoni flows are induced by surface tension gradients at the interface of immiscible fluids which become relevant in a microgravity environment due to the increased importance of surface forces and greater extensions and interfaces. Applications of Marangoni convection can be found in the fields of welding, crystal growth, and the convection is also necessary to stabilize the soap films and drying silicon wafers. Chamkha et al. [3] showed that the steady boundary layers can be formed along the interface of two immiscible fluids in surface driven flows that may be generated not only with Marangoni effects, but also with the existence of the buoyancy effects due to gravity and external pressure gradient. Napolitano [4] pointed out that the field equations in the bulk fluids for non-Marangoni boundary layers do not depend explicitly on the geometry of the interface when using the arc length as coordinates, and that the distance normal to the interface involves ⁎ Corresponding author. E-mail addresses: [email protected], [email protected] (E.H. Aly), [email protected], [email protected] (A. Ebaid).
http://dx.doi.org/10.1016/j.molliq.2015.12.108 0167-7322/© 2015 Elsevier B.V. All rights reserved.
the mean curvature of its hydrostatic and dynamic shapes. Together with the other surface balance equations, this introduces kinematic, thermal and pressure couplings for the flow fields in the two fluids. In addition, it has been shown in Ref. [5] that the fields are uncoupled when the momentum and energy resistivity ratios of the two layers and the viscosity ratio of the two fluids are much less than one. Several useful numerical studies on Marangoni boundary layers of the classical fluids in various geometries were introduced in Refs. [3,6–8,10,11]. Recently, the field of nanofluid flows has become one of the popular areas of research. Scientists have declared that a more effective way for cooling is to use nanofluids in which nanometer-sized particles are added to a base fluid. Such base fluids are, usually, water, oil, acetone and ethylene glycol, while the common nanoscale particles are Copper (Cu), Silver (Ag), Gold (Au), Aluminum oxide (Al2O3), Titanium dioxide (TiO2) and Copper Oxide (CuO), see for example Refs. [12–19]. It is now well known that the nanoparticles have high thermal conductivity, hence, Lee et al. [20], Xuan and Li [21], and Das et al. [22] have showed that the mixed fluids have better thermal properties. Commonly, nanofluids contain up to a 5% volume fraction of nanoparticles to ensure effective heat transfer enhancements. Also, nanofluids have novel properties that make them potentially useful in many applications in heat transfer and they exhibit enhanced the thermal conductivity and the convective heat transfer coefficient compared to the base fluid.
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Typical thermal conductivity enhancements are in the range 15% − 40% over the base fluid and heat transfer coefficient enhancements have been found up to 40% [23]. Recently, Jalilpour et al. [24] studied the MHD boundary layer flow of a nanofluid towards a stretching surface with suction or blowing and surface heat flux. They found that the magnitude of the reduced Nusselt number decreases with an increase in magnetic number, thermophoresis parameter and Lewis number. Further, the reduced Sherwood number decreases with increasing magnetic number and thermophoresis parameter and increases with increasing Lewis number. Further studies on some physical concepts over the stretching sheet are to be found in Refs. [25–30]. The thermal radiation has been also received considerable attention because of its effect in high operating temperature process. In Ref. [31], Pathak and Maheshwari have analyzed the influence of radiation on an unsteady free convection flow bounded by an oscillating plate with variable wall temperature. Shateyi [32] studied the effects of thermal radiation, buoyancy and suction/blowing on natural convection heat and mass transfer over a semi-infinite stretching surface. Besides, the authors in Ref. [33] have investigated the radiation effects on the MHD free convection flow past an impulsively started vertical plate with variable surface temperature and concentration. Ishak [34] analyzed the radiation effects on the flow and heat transfer over a moving plate in a parallel stream. Arifin et al. [35] have investigated the influence of nanoparticles over the Marangoni boundary layer flow using a model introduced by Tiwari and Das [36]. Many extended works were done later, see for example Buongiorno [37], Trisaksri and Wongwises [38], Wang and Mujumdar [39], and Kakaç and Pramuanjaroenkij [40]. In addition, Hamid et al. [41] studied the radiation effects on the Marangoni boundary layer flow past a flat plate in nanofluid. Their work was then updated by Sastry et al. [42] who investigated effect of heat transfer on MHD Marangoni boundary layer with Cu, Al2O3, and TiO2 nanofluids, further, Mat et al. [43] studied radiation effect on Marangoni convection boundary layer flow of Cu, Al2O3, and TiO2 nanofluids as well. Recently, Poursamad et al. [44] studied the Marangoni effect in nematic liquid crystals due to a Gaussian light beam absorption. The objective of the current research is to extend the just mentioned studies, where both effect of the radiation and the Marangoni convection boundary condition are to be affected on the flow and heat transfer of two different nanofluids past a plat embedded in a porous medium saturated by water in the presence of a magnetic field and with suction/injection. The exact analytical solutions for the stream function and temperature distribution are to be constructed via a new approach using Laplace transform. Hence, the effect of nanoparticle volume fraction, magnetic field, permeability, thermal radiation and wall mass transfer on the nanofluid velocity and its temperature is discussed with the help of several plots. Regarding the advantages of Laplace transform over the other methods, there are well known solutions for many standardized differential equations of second-order with variable coefficients, such as Legendre's equation, Kummer's equation, Bessel's equation, Airy's equation,…, etc. However, if the differential equation under consideration does not follow any of these types, we have to transform it to one of them; otherwise, another technique of solution has to be searched. Since most of the differential equations that we usually investigate are of forms that differ than those mentioned above, then Laplace transform comes to be one of the best tools to achieve this task. In addition, Laplace transform gives the researchers an opportunity to deduce analytical solution in a closed-integral form and/ or an exact form when the involved integrals can be theoretically evaluated, as seen in the rest of the present work. 2. Basic equations In this research, we consider a steady two-dimensional Marangoni boundary layer flow past a porous surface in a water based nanofluid
containing two different types of nanoparticles, namely Copper (Cu), where its electrical conductivity is higher, and Titanium dioxide (TiO2), with a lower electrical conductivity. It is assumed that the fluid is incompressible, the flow is laminar and the base fluid and particles are in thermal equilibrium, i.e. no slip occurs between them. The thermophysical properties of nanoparticles are given in Table 1. Further, we consider a Cartesian coordinate system (x,y), where x and y are the coordinates measured along the plate and normal to it, respectively, and the flow takes place at y ≥0. In addition, Tw and T∞ are the temperature of the plate and ambient fluid, respectively. Furthermore, it is assumed that the surface tension γ is to vary linearly with temperature as [45] γ ¼ γ0 ½1 − γðT − T ∞ Þ;
ð1Þ
where γ0 is the surface tension at the interface and γ is the rate of change of surface tension with temperature (a positive fluid property). It is also assumed that a uniform magnetic field, H0 is imposed in the direction normal to the surface (see Fig. 1). Regarding the above physical model, the steady state boundary layer equations for a nanofluid in the Cartesian coordinates are given by ∂u ∂v þ ¼ 0; ∂x ∂y
ð2Þ
u
2 μ nf σ nf 2 ∂u ∂u μ nf ∂ u u; − H u− þv ¼ ρnf 0 σ nf k ∂x ∂y ρnf ∂y2
u
knf ∂ T ∂T ∂T 1 ∂qr − ; þv ¼ ∂x ∂y ρC p nf ∂y ρC p nf ∂y2
ð3Þ
2
ð4Þ
subject to the following boundary conditions v ¼ 0; T ¼ T ∞ þ ax2 ; μ nf u ¼ 0; T→T ∞ ; as y→∞;
∂u ∂σ ∂T ¼ at y ¼ 0; ∂y ∂T ∂x
ð5Þ
where u and v are the velocity components of the nanofluid in the x and y directions, respectively, T is the temperature of the nanofluid, μnf is the effective dynamic viscosity, ρnf is the effective density, σnf is the effective electrical conductivity, k is the permeability of the porous medium, knf is the thermal conductivity, (ρCp)nf is the heat capacitance, Cp is the specific heat at constant pressure, qr is the radiative heat flux and a ¼ ΔT L , where ΔT is the constant characteristic temperature, and L is the length of the surface. Further, ()nf denotes the nanofluid quantities which are defined as follows [46,47] ρnf ¼ ð1 − ϕÞρ f þ ϕρs ; μ nf ¼
μf ð1 − ϕÞ2:5
ð6aÞ
;
ð6bÞ
2
σ nf
3 σs 3 − 1 ϕ 6 7 σ 6 f 7 ¼ 61 þ 7σ f ; σs σs 4 5 þ2 − −1 ϕ σf σf
ð6cÞ
Table 1 Thermophysical properties of the water and nanoparticles [16,50]. Physical properties
Pure water
Cu
TiO2
Cp (J/kg K) ρ (kg/m3) k (W/m K) σ (Ω/m)−1
4179 997.1 0.613 0.05
385 8933 401 5.96 × 107
686.2 4250 8.9538 1 × 10−12
E.H. Aly, A. Ebaid / Journal of Molecular Liquids 215 (2016) 625–639
"
1
ρC p s 1−ϕþ ϕ ρC p f
627
#
1 ks þ 2k f þ 2ϕ ks − k f þ R θ″ðηÞ Pr ks þ 2k f − ϕ ks − k f
ð12Þ
þf ðηÞθ0ðηÞ − 2f 0ðηÞθðηÞ ¼ 0;
and the boundary Condition (5) becomes f ð0Þ ¼ f w ; f 0ðηÞ → 0;
Fig. 1. Schematic diagram of the problem.
ρC p
knf
nf
¼ ð1 − ϕÞ ρC p f þ ϕ ρC p s ;
ð6dÞ
f
ð6eÞ
4σ ∂T 4 ; 3k ∂y
ð7Þ
where σ⁎ and k⁎ are the Stefan–Boltzmann constant and mean absorption coefficient, respectively. It is assumed that the temperature differences within the flow such as the term T4 may be expressed as a linear function of the temperature. Hence, on expanding T4 in a Taylor series about T∞ and neglecting the higher order terms, one obtains T 4 ≅ 4T 3∞ T − 3T 4∞ :
ð8Þ
Therefore, Eq. (4) reduces to 2
μ
H 20 σ f 1 ρ f ξ1 ξ2 Þ
is the magnetic field parameter, K p ð¼ ρ fk ξ 1ξ Þ is the permeability parameter, Prð¼
where ϕ is the solid volume fraction, ()f and ()s refer the basic fluid and nanoparticles, respectively. Now regarding the approximation of Rosseland for radiation [48,49], the radiative heat flux qr is simplified as
u
ð13Þ
where the prime denotes differentiation with respect to η, Mð¼
" # ks þ 2k f þ 2ϕ ks − k f kf ; ¼ ks þ 2k f − ϕ ks − k f
qr ¼ −
f ″ð0Þ ¼ − 2ð1 − ϕÞ2:5 ; θð0Þ ¼ 1; θðηÞ → 0 as η → ∞;
ν f ðρC p Þ f Þ kf
1 2
16σ T 3∞ Þ fνfk
is the Prandtl number, Rð¼ 3ðρC p Þ
is the
thermal radiation parameter, fw is the constant mass transfer parameter with fw N 0 for suction and fw b 0 for injection. The surface velocity can then determine by " #1=3 ðσ 0 γaÞ2 uw ¼ xf 0ð0Þ: ρf μ f
ð14Þ
Another quantity of interest is also the local Nusselt number Nux which is defined as Nux ¼
xqw ðxÞ ∂T ; qw ¼ − knf : þ qr k f ðT − T ∞ Þ ∂y y¼0
ð15Þ
where qw(x) is the heat flux from the surface of the plate. Now on using Eq. (10), we get Nux ¼ −
knf 0 þ R Pr ξ1 x θ ð0Þ: kf
ð16Þ
3. Exact solutions 2
knf ∂ T ∂T ∂T 16σ T 3∞ ∂ T þ : þv ¼ 2 ∂x ∂y ρC p nf ∂y 3 ρC p nf k ∂y2
ð9Þ
3.1. Exact solution of f(η) Regarding the conditions of f(η) in Eq. (13), it can be deduced as
Now, the dimensionless variables may be introduced as follows [35] T − T∞ ; η ¼ ξ1 y; where ψðηÞ ¼ ξ2 xf ðηÞ; θðηÞ ¼ ax2 !1=3 !1=3 σ 0 γaμ f σ 0 γaρ f ξ1 ¼ ; ξ ¼ ; 2 μ 2f ρ2f
f ðηÞ ¼ c1 þ c2 e−βη ; c1 ¼ f w − c2 ; c2 ¼
ð10Þ
−2 β2
ð1 − ϕÞ2:5 ;
ð17Þ
where β can be obtained by substituting the last relation into Eq. (11) to get
where η is the similarity variable, f(η) is the dimensionless stream function and θ(η) is the dimensionless temperature. Further, ψ is the stream function defined in the usual way to identically satisfy Eq. (2) and v ¼ − ∂ψ . as u ¼ ∂ψ ∂y ∂x On substituting Eqs. (6a)–(6e) and (10) into (3) and (9), we therefore obtain the following nonlinear ordinary differential equations " # ρs 2 f 000 ð η Þ þ 1 − ϕ þ ϕ f ðηÞf ″ðηÞ − ½ f 0ðηÞ 2:5 ρ ð1−ϕ2Þ0 1 3 f σs 3 −1 ϕ C 6B σ 1 17 6B 7 f C −6B1 þ CM þ 7 f 0ðηÞ ¼ 0; 2:5 A σs σs 4@ ð1 − ϕÞ K p 5 þ2 − −1 ϕ σf σf 1
ð11Þ
20
1 3 σs 3 −1 ϕ B 6 C σf 1 ρ 1 17 6B 7 C β 3 −f w 1−ϕ þ ϕ s β 2 −6B1 þ β CM þ 2:5 K 7 σs σs 4@ A ρf p5 ð1−ϕÞ2:5 ð 1−ϕ Þ þ2 − −1 ϕ σf σf " # ρ ð18Þ −2 1−ϕ þ ϕ s ð1−ϕÞ2:5 ¼ 0; ρf "
#
where the positive resulted value is only to be considered to obtain the physical solution. For more details about deducing the exact solutions of f(η) in this section, the reader is advised to see the papers by Wang [51], Aly and Ebaid [14] and Aly and Vajravelu [52]. 3.2. Exact solution of θ(η) using Laplace transform Regarding the advantages of Laplace transform mentioned in the current introduction, we apply this method here to solve the
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temperature Eq. (12) with its appropriate Condition (13). Inserting Eq. (17) into Eq. (12) with supposing that t = − e−βη, yields tθ″ðt Þ þ ðn − mt Þθ0ðt Þ þ 2mθðt Þ ¼ 0;
ð19Þ
subject to the following set of boundary conditions θð0− Þ ¼ 0; θð − 1Þ ¼ 1:
ð20Þ
where n ¼ 1−
" # τ 2 2τ f w þ 2 ð1 − ϕÞ2:5 ; m ¼ 3 ð1 − ϕÞ2:5 ; β β β
1−1 " # B C 1 1 ks þ 2k f þ 2ϕ ks − k f þR C τ¼B @ A : ðρC p Þs Pr ks þ 2k f − ϕ ks − k f 1 − ϕ þ ϕ ρC ð pÞ f
ð21aÞ
0
ð21bÞ
On applying Laplace transform to Eq. (19), we get sðm − sÞΘ0ðsÞ þ ½ðn − 2Þs þ 3mΘðsÞ ¼ 0;
ð22Þ
where Θ(s) is the Laplace transform of θ(t). Now on integrating Eq. (22), we obtain ΘðsÞ ¼
c s3 ðs − mÞ−n−1
;
ð23Þ
where c is constant of the integration to be determined. On applying the inverse Laplace transform to Eq. (23), we have θðt Þ ¼
c 2Γð−n−1Þ
Z
t 0
ðt−μ Þ2 μ −n−2 emμ dμ;
n b − 1:
ð24Þ
It is clear from this equation that the boundary condition θ(0) = 0 is automatically satisfied. Besides, the other boundary condition θ(− 1) = 1 gives c by c ¼ −Z
2Γð − n − 1Þ 0 −1
:
ð25Þ
ð1 þ μ Þ2 μ −n−2 emμ dμ
Therefore, θ(t) is given in a closed form as Z
t
θðt Þ ¼ − Z
0 0
ðt − μ Þ2 μ −n−2 emμ dμ
−1
:
ð26Þ
ð1 þ μ Þ2 μ −n−2 emμ dμ
Performing the integrations in Eq. (26), we get the following exact solution for θ(t) in terms of the generalized incomplete gamma function: θðt Þ ¼
m2 t 2 Γð−n−1; 0; −mt Þ þ 2mtΓð−n; 0; −mtÞ þ Γð−n þ 1; 0; −mt Þ ; ð27Þ m2 Γð−n−1; 0; mÞ−2mΓð−n; 0; mÞ þ Γð−n þ 1; 0; mÞ
which can be finally expressed in terms of η as follows θðηÞ ¼
m2 e−2βη Γ −n−1; 0; me−βη −2me−βη Γ −n; 0; me−βη þ Γ −n þ 1; 0; me−βη : 2 m Γð−n−1; 0; mÞ−2mΓð−n; 0; mÞ þ Γð−n þ 1; 0; mÞ
ð28Þ
It should be noted that the combinations of the considered parameters have to be achieved the following condition " # τ 2 2:5 f þ ð1 − ϕÞ − 2 N 0: β w β2
ð29Þ
In this section, it was deduced that simpler special functions arise on applying Laplace transform as a tool of solution while other complex special functions arise on using other techniques, see for example Hamad [53]. 4. Results and discussion In the presence of the magnetic field and radiation, a steady two-dimensional Marangoni boundary layer flow past a porous surface in a water based nanofluid containing two different types of nanoparticles, namely Copper (Cu) and Titanium dioxide (TiO2), has been investigated in the case of suction, impermeable and injection. The governing equations of the stream flow and temperature have been solved using Laplace transform, thus, producing several plots to represent the effect of various parameters on the physical phenomena. In particular, influence of the velocity and temperature for both Cu–water, represented by solid curves, and TiO2–water, delineated by dotted curves, nanofluids is to be studied under effect of the following parameters; nanoparticle volume fraction (ϕ), magnetic (M), permeability (Kp ), thermal radiation and wall mass transfer (fw). The thermophysical properties of the water and present nanoparticles are given in Table 1, where Prandtl number of the base fluid (water) is kept in the present analysis at 6.2. The influence of nanoparticle volume fraction parameter (ϕ) on the velocity profiles f ′ (η) and temperature distribution θ(η) for both Cu–water and TiO2 –water nanofluids is depicted in Figs. 2(a–c) and 3(a–c) when the surface wall is in the case of injection (fw N 0), impermeable (fw = 0) and suction (fw b 0). For all of these cases and both types of nanofluids, it is observed that the velocities decrease significantly with an increase in ϕ. Therefore, the momentum boundary layer thickness decreases and tends asymptotically to zero as the distance increases from the boundary. In addition, as higher values of the thermal conductivity are accompanied by higher values of thermal diffusivity, this makes that the boundary layer thickness is sensitive to the change in ϕ. Moreover, it is noticed that TiO 2–water nanofluid is of higher velocity than Cu–water nanofluid. This is because that the density of TiO 2 is lower than of Cu, which means that the solid Titanium dioxide is lighter in motion compared to Copper. Further, it is observed from these figures that the increase in ϕ is to increase the temperature of the current two types of nanofluids. However, the thermal boundary layer for nanoparticles, namely Cu–water is greater than TiO 2–water. This is due to the higher conductivity of the solid particles Cu compared with TiO2. Furthermore, the thickness of the boundary layer increases as fw decreases with a remarkable effect of fw in the suction case with higher value of ϕ. Figs. 4(a–c) and 5(a–c) show effect of the magnetic parameter M on the velocity and temperature profile, respectively, of Cu–water and TiO2–water nanofluids at three different cases of fw parameter (N 0, = 0, b0). Form these figures and for the both types of nanofluids, it is seen that application of a magnetic field results a resistive-type force called Lorentz force. This force opposes the flow and, therefore, decelerates the fluid velocity and increases its temperature. In addition, as the strength of the magnetic field increases, the hydrodynamic and thermal boundary layers decrease and increase, respectively. This agrees with the fact that the nanoparticles dissipate energy in the form of heat which causes the thermal boundary layer thickness to increase and ultimately a localized rise in
E.H. Aly, A. Ebaid / Journal of Molecular Liquids 215 (2016) 625–639
Fig. 2. Velocity profiles f′(η) for variation of ϕ when M = 1 and Kp = 1 at (a) fw = 1, (b) fw = 0 and (c) fw = −1.
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Fig. 3. Temperature distribution θ(η) for variation of ϕ when M = 1, Kp = 0.5, R = 1 at (a) fw = 0.1, (b) fw = 0 and (c) fw = −0.1.
E.H. Aly, A. Ebaid / Journal of Molecular Liquids 215 (2016) 625–639
Fig. 4. Velocity profiles f′(η) for variation of M when ϕ = 0.1 and Kp = 1 at (a) fw = 1, (b) fw = 0 and (c) fw = −1.
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Fig. 5. Temperature distribution θ(η) for variation of M when ϕ = 0.1, Kp = 0.5, R = 1 at (a) fw = 0.1, (b) fw = 0 and (c) fw = −0.1.
E.H. Aly, A. Ebaid / Journal of Molecular Liquids 215 (2016) 625–639
temperature of the fluid occurs. Further, as the density of TiO 2 is lower than of Cu, the velocity of TiO2 –water is higher than of Cu–water. Moreover, the thickness of the boundary layer increases as fw decreases from 1 to − 1.
633
The impact of the porous parameter Kp on the velocities and the temperature distribution is displayed in Figs. 6(a–c) and 7(a–c), respectively. It can be noted from these figures that, as Kp increases, which means a rising in the holes of the porous structure, the velocities and temperature
Fig. 6. Velocity profiles f′(η) for variation of Kp when ϕ = 0.1 and M = 1 at (a) fw = 1, (b) fw = 0 and (c) fw = −1.
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E.H. Aly, A. Ebaid / Journal of Molecular Liquids 215 (2016) 625–639
Fig. 7. Temperature distribution θ(η) for variation of Kp when ϕ = 0.1, M = 1, R = 1 at (a) fw = 0.1, (b) fw = 0 and (c) fw = −0.1.
increase in the both types of Cu–water and TiO2–water nanofluids at the investigated three cases of fw. This is, of course, agreed with the physical view. It should be noted here that, for K≤0.1, velocities of Cu–water and
TiO2–water nanofluids are nearly identical, especially at fw = 0 and fw = −1 (injection). While, as (K→∞), the current two types of nanofluids have nearly the same temperature distribution when fw = 1 (suction).
E.H. Aly, A. Ebaid / Journal of Molecular Liquids 215 (2016) 625–639
As shown in Fig. 8(a–c), the temperature profiles increase as radiation parameter R increases for all nanoparticles considered and for all investigated cases of the f w parameter. It is known that the
635
radiation parameter, being the reciprocal of Stephan number, is the measure of relative importance of the thermal radiation transfer to the conduction heat transfer. Thus, larger values of R are indicative
Fig. 8. Temperature distribution θ(η) for variation of R when ϕ = 0.1, M = 1, Kp = 0.5 at (a) fw = 0.1, (b) fw = 0 and (c) fw = −0.1.
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of larger amount of radiative heat energy being poured into the system, causing a rise in θ(η). Therefore, the radiation shows a dominance of the thermal radiation over conduction, thus, it can be used to control the thermal boundary layers quite effectively. Figs. 9(a–b) and 10(a–b) depict the variation of velocity and temperature, respectively, at different selected values for the suction/injection parameter fw. From these figures, it is observed that imposition of fluid suction (fw N 0) has the tendency to decrease the fluid velocity and, therefore, thickness of the hydrodynamic boundary layer, regarding, the fluid temperature and thermal boundary layer decrease as well. However, fluid injection (fw b 0) produces the opposite effect, namely an increase in the fluid velocity and temperature. It should be noted that these profiles satisfy the far field boundary conditions asymptotically. In addition, for the injection case, Fig. 9(a) and (b) refers to remarkable and slightly higher velocity, respectively, for TiO2–water nanofluid, as a result, a remarkable increase in the boundary layer thickness has been observed in the suction case than in the injection case. Further, from Fig. 10, it is shown that the temperature profile is the more for higher thermal conductivity nanoparticle (Cu). Figs. 11, 12 and 13 display variation of the local Nusselt number with the parameters of nanoparticle volume fraction (ϕ), radiation
R and location x for different values of fw in both cases of Cu–water and TiO2–water nanofluids. Here, it should be noticed that the entire values of − θ ′ (0) are always positive, i.e. the heat is transferred from hot surface to the cold region. From Fig. 11, it is seen that local Nusselt number, increases as R increases, however as shown in Fig. 12, it decreases as ϕ increases in the presence of radiation. It can be also noticed that the heat transfer rate at the surface increases in the presence of radiation. This result agrees with the physical view, since the effect of radiation is to increase the rate of energy transport to the fluid, thereby increasing the temperature of the fluid, as shown in Fig. 8. It is observed from Fig. 13 that the fluid suction and injection have the tendency to increase and decrease, respectively, the heat transfer rate at the wall. From these figures and in the injection case it should be noticed that nanoparticles with low thermal conductivity, TiO2 , have better enhancement on heat transfer compared to Cu. However, for the suction case, the same result is to be obtained until a specific value of ϕ where this behavior becomes opposite, as shown in Fig. 13. Further, this critical value of ϕ, where Nux equals for both Cu–water and TiO2–water, decreases as fw increases, as shown by the solid circles in this figure.
Fig. 9. Velocity profiles f′(η) when M = 1 and Kp = 1 for variation of fw (a) injection and (b) suction.
E.H. Aly, A. Ebaid / Journal of Molecular Liquids 215 (2016) 625–639
637
Fig. 10. Temperature distribution θ(η) when ϕ = 0.1, M = 1, Kp = 0.5 and R = 1 for variation of fw (a) injection and (b) suction.
Marangoni boundary layer flow past a porous surface in a water based nanofluid containing two different types of nanoparticles, namely
Copper (Cu) and Titanium dioxide (TiO2), has been studied. Exact solutions of the stream flow and temperature have been obtained using Laplace transform. Influence of the velocity and temperature for both nanofluids has been investigated under effect of the following
Fig. 11. Variation of the local Nusselt number Nux as a function of R for fw = −0.1, 0, 0.1 when ϕ = 0.1, M = 1 and Kp = 0.5.
Fig. 12. Variation of the local Nusselt number Nux as a function of ϕ for R = 1, 2, 3 in suction fw = 0.1 when M = 1 and Kp = 0.5.
5. Conclusion
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E.H. Aly, A. Ebaid / Journal of Molecular Liquids 215 (2016) 625–639
Fig. 13. Variation of the local Nusselt number Nux as a function of ϕ at various values of fw when M = 1, Kp = 0.5 and R = 1.
parameters; nanoparticle volume fraction (ϕ), magnetic (M), permeability (Kp), thermal radiation (R), and wall mass transfer (fw) in the case of suction, impermeable and injection. The main results of the present study can be epitomized as follows: • the velocities of the two investigated nanoparticles decrease significantly with an increase in ϕ, however, TiO2–H2O nanofluid is of higher velocity than Cu–H2O nanofluid, • increase in ϕ is to increase the temperature of the current two types of nanofluids. However, the thermal boundary layer of Cu–H2O is greater than TiO2–H2O, • the effective electrical conductivity makes a very considerable and remarkable effect on the nanofluids' flow, therefore, this parameter is mandatory and should be taken into account on applying the magnetic field, otherwise a spurious physical sight is to be gained. The parameter M decelerates the fluid velocity and increases its temperature, in addition, as M increases, the hydrodynamic and thermal boundary layers decrease and increase, respectively, • as Kp increases, the velocities and temperature increase in the both types of Cu–H2O and TiO2–H2O nanofluids at the investigated three cases of fw, further, when (K → ∞), the current two types of nanofluids have nearly the same temperature distribution in the suction case, • for all nanoparticles considered and for all investigated cases of the fw parameter, the temperature profiles increase as radiation parameter R increases, in addition, the radiation can be used to control the thermal boundary layers quite effectively, • the fluid suction (fw N 0) decreases the fluid velocity and, therefore, thickness of the hydrodynamic boundary layer, regarding, the fluid temperature and thermal boundary layer decreases as well, however, fluid injection (fw b 0) produces the opposite effect, namely an increase in the fluid velocity and temperature, • for TiO2 –H2O nanofluid, a remarkable increase in the boundary layer thickness has been obtained in the suction case than in the injection case, further, the temperature profile is the more for higher thermal conductivity nanoparticle (Cu), • the local Nusselt number, increases as R increases, however, it decreases as ϕ increases, • the fluid suction and injection have the tendency to increase and decrease, respectively, the heat transfer rate at the wall which increases in the presence of radiation, • nanoparticles with low thermal conductivity, TiO 2 , have better enhancement on heat transfer compared to Cu. References [1] L.G. Napolitano, Microgravity fluid dynamics, 2nd Levitch Conference, Washington, 1978.
[2] L.G. Napolitano, Marangoni boundary layers, Proceedings of the 3rd European Symposium on Material Science in Space, Grenoble, ESA SP-142, June 1979. [3] A.J. Chamkha, I. Pop, H.S. Takhar, Marangoni mixed convection boundary layer flow, Meccanica 41 (2006) 219–232. [4] L.G. Napolitano, Surface and buoyancy driven free convection, Acta Astronaut. 9 (1982) 199–215. [5] L.G. Napolitano, C. Golia, Coupled Marangoni boundary layers, Acta Astronaut. 8 (1981) 417–434. [6] K. Arafune, A. Hirarta, Interactive solutal and thermal Marangoni convection in a rectangular open boat, Numer. Heat Transfer, Part A 34 (1998) 421–429. [7] A. Croll, W. Muller-Sebert, R. Nitsche, The critical Marangoni number for the onest of time-dependent convection in silicon, Mater. Res. Bull. 24 (1989) 995–1004. [8] I. Pop, A. Postelnicu, T. Grosan, Thermosolutal Marangoni forced convection boundary layers, Meccanica 36 (2001) 555–571. [10] E. Magyari, A.J. Chamkha, Exact analytical results for the thermosolutal MHD Marangoni boundary layers, Int. J. Therm. Sci. 47 (2008) 848–8572. [11] C.E. Nanjundappa, I.S. Shivakumara, R. Arunkumar, Bénard–Marangoni ferroconvection with magnetic field dependent viscosity, J. Magn. Magn. Mater. 322 (2010) 2256–2263. [12] N.S. Akbar, A.W. Butt, Magnetic field effects for copper suspended nanofluid venture through a composite stenosed arteries with permeable wall, J. Magn. Magn. Mater. 381 (2015) 285–291. [13] M. Sheikholeslami, M.G. Bandpy, R. Ellahi, A. Zeeshan, Simulation of MHD CuO–water nanofluid flow and convective heat transfer considering Lorentz forces, J. Magn. Magn. Mater. 369 (2014) 69–80. [14] E.H. Aly, A. Ebaid, Exact analytical solution for suction and injection flow with thermal enhancement of five nanofluids over an isothermal stretching sheet with effect of the slip model: a comparative study, Abstr. Appl. Anal. (2013), Article ID 721578 (in the special issue: Mathematical and Computational Analysis of Flow and Transport Phenomena; 12 pages). [15] K. Vajravelu, K.V. Prasad, C.-O. Ng, The effect of variable viscosity on the flow and heat transfer of a viscous Ag–water and Cu–water nanofluids, J. Hydrodyn. 25 (2013) 1–9. [16] H.F. Oztop, E. Abu-Nada, Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow 29 (2008) 1326–1336. [17] E.H. Aly, H.M. Sayed, Thermal radiation effects on magnetohydrodynamic boundarylayer flow due to a moving extensible surface in nanofluids, J. Comput. Theor. Nanosci. 11 (2014) 1756–1765. [18] M. Sheikholeslami, M.G. Bandpy, R. Ellahi, M. Hassan, S. Soleimani, Effects of MHD on Cu–water nanofluid flow and heat transfer by means of CVFEM, J. Magn. Magn. Mater. 349 (2014) 188–200. [19] N.S. Akbar, Ferromagnetic CNT suspended H2O + Cu nanofluid analysis through composite stenosed arteries with permeable wall, Physica E 72 (2015) 70–76. [20] S. Lee, S.U.S. Choi, J.A. Eastman, Measuring thermal conductivity of fluids containing oxide nanoparticles, ASME J. Heat Transf. 9 (1999) 280. [21] Y. Xuan, Q. Li, Heat transfer enhancement of nanofluids, Int. J. Heat Fluid Flow 64 (2000) 58. [22] S.K. Das, N. Putra, W. Roetzel, Pool boiling characteristics of nano-fluids, Int. J. Heat Mass Transf. 46 (2003) 851–862. [23] D.M. Yu, J.L. Routbort, S.U.S. Choi, Review and comparison of nanofluid thermal conductivity and heat transfer enhancements, Heat Transfer Eng. 29 (2008) 432–460. [24] B. Jalilpour, S. Jafarmadar, D.D. Ganji, A.B. Shotorban, H. Taghavifar, Heat generation/ absorption on MHD stagnation flow of nanofluid towards a porous stretching sheet with prescribed surface heat flux, J. Mol. Liq. 195 (2014) 194–204. [25] M.M. Rashidi, N. Vishnu Ganesh, A.K. Abdul Hakeem, B. Ganga, Buoyancy effect on MHD flow of nanofluid over a stretching sheet in the presence of thermal radiation, J. Mol. Liq. 198 (2014) 234–238. [26] E.H. Aly, A. Ebaid, New exact solutions for boundary-layer flow of a nanofluid past a stretching sheet, J. Comput. Theor. Nanosci. 10 (2013) 2591–2595. [27] A. Ebaid, M.D. Aljoufi, A.M. Wazwaz, An advanced study on the solution of nanofluid flow problems via Adomian's method, Appl. Math. Lett. 46 (2015) 117–122. [28] N.S. Akbar, A. Ebaid, Z.H. Khan, Numerical analysis of magnetic field on Eyring– Powell fluid flow towards a stretching sheet, J. Magn. Magn. Mater. 382 (2015) 355–358. [29] E.H. Aly, Radiation and MHD boundary layer stagnation-point of nanofluid flow towards a stretching sheet embedded in a porous medium: analysis of suction/injection and heat generation/absorption with effect of the slip model, Math. Probl. Eng. Vol. 2015 (2015), Article ID 563547 (in the special issue: Macroscopic/Mesoscopic Computational Materials Science Modeling and Engineering; 20 pages). [30] N.S. Akbar, Z.H. Khan, S. Nadeem, The combined effects of slip and convective boundary conditions on stagnation-point flow of CNT suspended nanofluid over a stretching sheet, J. Mol. Liq. 196 (2014) 21–25. [31] G. Pathak, C.H. Maheshwari, Effect of radiation on unsteady free convection flow bounded by an oscillating plate with variable wall temperature, Int. J. Appl. Mech. Eng. 11 (2006) 371–382. [32] S. Shateyi, Thermal radiation and buoyancy effects on heat and mass transfer over a semi-infinite stretching surface with suction and blowing, J. Appl. Math. (2008), Article ID 414830 (12 pages). [33] S. Suneetha, N.B. Redd, V.R. Prasad, Thermal radiation effects on MHD free convection flow past an impulsively started vertical plate with variable surface temperature and concentration, J. Nav. Archit. Mar. Eng. 2 (2008) 57–70. [34] A. Ishak, Radiation effects on the flow and heat transfer over a moving plate in a parallel stream, Chin. Phys. Lett. 26 (2009) 1–4 (034701).
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[45] E. Magyari, A.J. Chamkha, Exact analytical solutions for thermosolutal Marangoni convection in the presence of heat and mass generation or consumption, Heat Mass Transf. 43 (2007) 965–974. [46] K. Khanafer, K. Vafai, M. Lightstone, Buoyancy-driven heat transfer enhancement utilizing nanofluids, Int. J. Heat Mass Transf. 46 (2003) 3639–3653. [47] J.C. Maxwell, A Treatise on Electricity and Magnetism, second ed. Oxford University Press, Cambridge, 1904 435–441. [48] M.Q. Brewster, Thermal Radiative Transfer Properties, John Wiley and Sons, New York, 1992. [49] R. Cortell, Radiation effects for the Blasius and Sakiadis flows with a convective surface boundary condition, Appl. Math. Comput. 206 (2008) 832–840. [50] M. Sheikholeslami, M. Gorji-Bandpy, D.D. Ganji, MHD free convection in an eccentric semi-annulus filled with nanofluid, J. Taiwan Inst. Chem. Eng. 45 (2014) 1204–1216. [51] C.Y. Wang, Analysis of viscous flow due to a stretching sheet with surface slip and suction, Nonlinear Anal. Real World Appl. 10 (2009) 375–380. [52] E.H. Aly, K. Vajravelu, Exact and numerical solutions of MHD nano boundary-layer flows over stretching surfaces in a porous medium, Appl. Math. Comput. 232 (2014) 191–204. [53] M.A.A. Hamad, Analytical solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field, Int. Commun. Heat Mass Transfer 38 (2011) 487–492.
Contents lists available at ScienceDirect
Journal of Molecular Liquids journal homepage: www.elsevier.com/locate/molliq
Exact analysis for the effect of heat transfer on MHD and radiation Marangoni boundary layer nanofluid flow past a surface embedded in a porous medium Emad H. Aly a,b,⁎, Abdelhalim Ebaid c,⁎ a b c
Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah 21589, Saudi Arabia Department of Mathematics, Faculty of Education, Ain Shams University, Roxy 11757, Cairo, Egypt Department of Mathematics, Faculty of Science, University of Tabuk, Tabuk 71491, Saudi Arabia
a r t i c l e
i n f o
Article history: Received 5 September 2015 Accepted 30 December 2015 Available online 27 January 2016 Keywords: Marangoni Radiation Magnetic Heat transfer Laplace
a b s t r a c t In the presence of a magnetic field, thermal radiation and with suction/injection, Marangoni boundary layer flow past a surface embedded in a porous medium saturated by a water based nanofluid containing two different types of nanoparticles, namely Copper and Titanium dioxide, has been studied. Exact solutions of the resulted equations were solved using a new approach via Laplace transform. Influence of the velocity profile and temperature distribution for the present nanofluids was investigated under effect of the involved parameters. It was found that the velocities of the two investigated nanoparticles decrease significantly with an increase in the solid volume fraction. In addition, the effective electrical conductivity parameter is mandatory and should be taken into account on applying the magnetic field; otherwise a spurious physical sight is to be gained. As expected, the magnetic parameter decelerates the fluid velocity and increases its temperature, as well, for all nanoparticles considered and for all investigated cases of the suction/injection parameter, the temperature profiles increase as radiation parameter increases. Further, the fluid suction decreases the fluid velocity and, therefore, thickness of the hydrodynamic boundary layer, regarding, the fluid temperature and thermal boundary layer decreases as well. However, fluid injection produces the opposite effect. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The Marangoni boundary layers are basically defined as dissipative layers which may occur along the liquid–liquid or liquid–gas interfaces. The original work in this field was first addressed by Napolitano [1,2]. Such type of Marangoni flows are induced by surface tension gradients at the interface of immiscible fluids which become relevant in a microgravity environment due to the increased importance of surface forces and greater extensions and interfaces. Applications of Marangoni convection can be found in the fields of welding, crystal growth, and the convection is also necessary to stabilize the soap films and drying silicon wafers. Chamkha et al. [3] showed that the steady boundary layers can be formed along the interface of two immiscible fluids in surface driven flows that may be generated not only with Marangoni effects, but also with the existence of the buoyancy effects due to gravity and external pressure gradient. Napolitano [4] pointed out that the field equations in the bulk fluids for non-Marangoni boundary layers do not depend explicitly on the geometry of the interface when using the arc length as coordinates, and that the distance normal to the interface involves ⁎ Corresponding author. E-mail addresses: [email protected], [email protected] (E.H. Aly), [email protected], [email protected] (A. Ebaid).
http://dx.doi.org/10.1016/j.molliq.2015.12.108 0167-7322/© 2015 Elsevier B.V. All rights reserved.
the mean curvature of its hydrostatic and dynamic shapes. Together with the other surface balance equations, this introduces kinematic, thermal and pressure couplings for the flow fields in the two fluids. In addition, it has been shown in Ref. [5] that the fields are uncoupled when the momentum and energy resistivity ratios of the two layers and the viscosity ratio of the two fluids are much less than one. Several useful numerical studies on Marangoni boundary layers of the classical fluids in various geometries were introduced in Refs. [3,6–8,10,11]. Recently, the field of nanofluid flows has become one of the popular areas of research. Scientists have declared that a more effective way for cooling is to use nanofluids in which nanometer-sized particles are added to a base fluid. Such base fluids are, usually, water, oil, acetone and ethylene glycol, while the common nanoscale particles are Copper (Cu), Silver (Ag), Gold (Au), Aluminum oxide (Al2O3), Titanium dioxide (TiO2) and Copper Oxide (CuO), see for example Refs. [12–19]. It is now well known that the nanoparticles have high thermal conductivity, hence, Lee et al. [20], Xuan and Li [21], and Das et al. [22] have showed that the mixed fluids have better thermal properties. Commonly, nanofluids contain up to a 5% volume fraction of nanoparticles to ensure effective heat transfer enhancements. Also, nanofluids have novel properties that make them potentially useful in many applications in heat transfer and they exhibit enhanced the thermal conductivity and the convective heat transfer coefficient compared to the base fluid.
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Typical thermal conductivity enhancements are in the range 15% − 40% over the base fluid and heat transfer coefficient enhancements have been found up to 40% [23]. Recently, Jalilpour et al. [24] studied the MHD boundary layer flow of a nanofluid towards a stretching surface with suction or blowing and surface heat flux. They found that the magnitude of the reduced Nusselt number decreases with an increase in magnetic number, thermophoresis parameter and Lewis number. Further, the reduced Sherwood number decreases with increasing magnetic number and thermophoresis parameter and increases with increasing Lewis number. Further studies on some physical concepts over the stretching sheet are to be found in Refs. [25–30]. The thermal radiation has been also received considerable attention because of its effect in high operating temperature process. In Ref. [31], Pathak and Maheshwari have analyzed the influence of radiation on an unsteady free convection flow bounded by an oscillating plate with variable wall temperature. Shateyi [32] studied the effects of thermal radiation, buoyancy and suction/blowing on natural convection heat and mass transfer over a semi-infinite stretching surface. Besides, the authors in Ref. [33] have investigated the radiation effects on the MHD free convection flow past an impulsively started vertical plate with variable surface temperature and concentration. Ishak [34] analyzed the radiation effects on the flow and heat transfer over a moving plate in a parallel stream. Arifin et al. [35] have investigated the influence of nanoparticles over the Marangoni boundary layer flow using a model introduced by Tiwari and Das [36]. Many extended works were done later, see for example Buongiorno [37], Trisaksri and Wongwises [38], Wang and Mujumdar [39], and Kakaç and Pramuanjaroenkij [40]. In addition, Hamid et al. [41] studied the radiation effects on the Marangoni boundary layer flow past a flat plate in nanofluid. Their work was then updated by Sastry et al. [42] who investigated effect of heat transfer on MHD Marangoni boundary layer with Cu, Al2O3, and TiO2 nanofluids, further, Mat et al. [43] studied radiation effect on Marangoni convection boundary layer flow of Cu, Al2O3, and TiO2 nanofluids as well. Recently, Poursamad et al. [44] studied the Marangoni effect in nematic liquid crystals due to a Gaussian light beam absorption. The objective of the current research is to extend the just mentioned studies, where both effect of the radiation and the Marangoni convection boundary condition are to be affected on the flow and heat transfer of two different nanofluids past a plat embedded in a porous medium saturated by water in the presence of a magnetic field and with suction/injection. The exact analytical solutions for the stream function and temperature distribution are to be constructed via a new approach using Laplace transform. Hence, the effect of nanoparticle volume fraction, magnetic field, permeability, thermal radiation and wall mass transfer on the nanofluid velocity and its temperature is discussed with the help of several plots. Regarding the advantages of Laplace transform over the other methods, there are well known solutions for many standardized differential equations of second-order with variable coefficients, such as Legendre's equation, Kummer's equation, Bessel's equation, Airy's equation,…, etc. However, if the differential equation under consideration does not follow any of these types, we have to transform it to one of them; otherwise, another technique of solution has to be searched. Since most of the differential equations that we usually investigate are of forms that differ than those mentioned above, then Laplace transform comes to be one of the best tools to achieve this task. In addition, Laplace transform gives the researchers an opportunity to deduce analytical solution in a closed-integral form and/ or an exact form when the involved integrals can be theoretically evaluated, as seen in the rest of the present work. 2. Basic equations In this research, we consider a steady two-dimensional Marangoni boundary layer flow past a porous surface in a water based nanofluid
containing two different types of nanoparticles, namely Copper (Cu), where its electrical conductivity is higher, and Titanium dioxide (TiO2), with a lower electrical conductivity. It is assumed that the fluid is incompressible, the flow is laminar and the base fluid and particles are in thermal equilibrium, i.e. no slip occurs between them. The thermophysical properties of nanoparticles are given in Table 1. Further, we consider a Cartesian coordinate system (x,y), where x and y are the coordinates measured along the plate and normal to it, respectively, and the flow takes place at y ≥0. In addition, Tw and T∞ are the temperature of the plate and ambient fluid, respectively. Furthermore, it is assumed that the surface tension γ is to vary linearly with temperature as [45] γ ¼ γ0 ½1 − γðT − T ∞ Þ;
ð1Þ
where γ0 is the surface tension at the interface and γ is the rate of change of surface tension with temperature (a positive fluid property). It is also assumed that a uniform magnetic field, H0 is imposed in the direction normal to the surface (see Fig. 1). Regarding the above physical model, the steady state boundary layer equations for a nanofluid in the Cartesian coordinates are given by ∂u ∂v þ ¼ 0; ∂x ∂y
ð2Þ
u
2 μ nf σ nf 2 ∂u ∂u μ nf ∂ u u; − H u− þv ¼ ρnf 0 σ nf k ∂x ∂y ρnf ∂y2
u
knf ∂ T ∂T ∂T 1 ∂qr − ; þv ¼ ∂x ∂y ρC p nf ∂y ρC p nf ∂y2
ð3Þ
2
ð4Þ
subject to the following boundary conditions v ¼ 0; T ¼ T ∞ þ ax2 ; μ nf u ¼ 0; T→T ∞ ; as y→∞;
∂u ∂σ ∂T ¼ at y ¼ 0; ∂y ∂T ∂x
ð5Þ
where u and v are the velocity components of the nanofluid in the x and y directions, respectively, T is the temperature of the nanofluid, μnf is the effective dynamic viscosity, ρnf is the effective density, σnf is the effective electrical conductivity, k is the permeability of the porous medium, knf is the thermal conductivity, (ρCp)nf is the heat capacitance, Cp is the specific heat at constant pressure, qr is the radiative heat flux and a ¼ ΔT L , where ΔT is the constant characteristic temperature, and L is the length of the surface. Further, ()nf denotes the nanofluid quantities which are defined as follows [46,47] ρnf ¼ ð1 − ϕÞρ f þ ϕρs ; μ nf ¼
μf ð1 − ϕÞ2:5
ð6aÞ
;
ð6bÞ
2
σ nf
3 σs 3 − 1 ϕ 6 7 σ 6 f 7 ¼ 61 þ 7σ f ; σs σs 4 5 þ2 − −1 ϕ σf σf
ð6cÞ
Table 1 Thermophysical properties of the water and nanoparticles [16,50]. Physical properties
Pure water
Cu
TiO2
Cp (J/kg K) ρ (kg/m3) k (W/m K) σ (Ω/m)−1
4179 997.1 0.613 0.05
385 8933 401 5.96 × 107
686.2 4250 8.9538 1 × 10−12
E.H. Aly, A. Ebaid / Journal of Molecular Liquids 215 (2016) 625–639
"
1
ρC p s 1−ϕþ ϕ ρC p f
627
#
1 ks þ 2k f þ 2ϕ ks − k f þ R θ″ðηÞ Pr ks þ 2k f − ϕ ks − k f
ð12Þ
þf ðηÞθ0ðηÞ − 2f 0ðηÞθðηÞ ¼ 0;
and the boundary Condition (5) becomes f ð0Þ ¼ f w ; f 0ðηÞ → 0;
Fig. 1. Schematic diagram of the problem.
ρC p
knf
nf
¼ ð1 − ϕÞ ρC p f þ ϕ ρC p s ;
ð6dÞ
f
ð6eÞ
4σ ∂T 4 ; 3k ∂y
ð7Þ
where σ⁎ and k⁎ are the Stefan–Boltzmann constant and mean absorption coefficient, respectively. It is assumed that the temperature differences within the flow such as the term T4 may be expressed as a linear function of the temperature. Hence, on expanding T4 in a Taylor series about T∞ and neglecting the higher order terms, one obtains T 4 ≅ 4T 3∞ T − 3T 4∞ :
ð8Þ
Therefore, Eq. (4) reduces to 2
μ
H 20 σ f 1 ρ f ξ1 ξ2 Þ
is the magnetic field parameter, K p ð¼ ρ fk ξ 1ξ Þ is the permeability parameter, Prð¼
where ϕ is the solid volume fraction, ()f and ()s refer the basic fluid and nanoparticles, respectively. Now regarding the approximation of Rosseland for radiation [48,49], the radiative heat flux qr is simplified as
u
ð13Þ
where the prime denotes differentiation with respect to η, Mð¼
" # ks þ 2k f þ 2ϕ ks − k f kf ; ¼ ks þ 2k f − ϕ ks − k f
qr ¼ −
f ″ð0Þ ¼ − 2ð1 − ϕÞ2:5 ; θð0Þ ¼ 1; θðηÞ → 0 as η → ∞;
ν f ðρC p Þ f Þ kf
1 2
16σ T 3∞ Þ fνfk
is the Prandtl number, Rð¼ 3ðρC p Þ
is the
thermal radiation parameter, fw is the constant mass transfer parameter with fw N 0 for suction and fw b 0 for injection. The surface velocity can then determine by " #1=3 ðσ 0 γaÞ2 uw ¼ xf 0ð0Þ: ρf μ f
ð14Þ
Another quantity of interest is also the local Nusselt number Nux which is defined as Nux ¼
xqw ðxÞ ∂T ; qw ¼ − knf : þ qr k f ðT − T ∞ Þ ∂y y¼0
ð15Þ
where qw(x) is the heat flux from the surface of the plate. Now on using Eq. (10), we get Nux ¼ −
knf 0 þ R Pr ξ1 x θ ð0Þ: kf
ð16Þ
3. Exact solutions 2
knf ∂ T ∂T ∂T 16σ T 3∞ ∂ T þ : þv ¼ 2 ∂x ∂y ρC p nf ∂y 3 ρC p nf k ∂y2
ð9Þ
3.1. Exact solution of f(η) Regarding the conditions of f(η) in Eq. (13), it can be deduced as
Now, the dimensionless variables may be introduced as follows [35] T − T∞ ; η ¼ ξ1 y; where ψðηÞ ¼ ξ2 xf ðηÞ; θðηÞ ¼ ax2 !1=3 !1=3 σ 0 γaμ f σ 0 γaρ f ξ1 ¼ ; ξ ¼ ; 2 μ 2f ρ2f
f ðηÞ ¼ c1 þ c2 e−βη ; c1 ¼ f w − c2 ; c2 ¼
ð10Þ
−2 β2
ð1 − ϕÞ2:5 ;
ð17Þ
where β can be obtained by substituting the last relation into Eq. (11) to get
where η is the similarity variable, f(η) is the dimensionless stream function and θ(η) is the dimensionless temperature. Further, ψ is the stream function defined in the usual way to identically satisfy Eq. (2) and v ¼ − ∂ψ . as u ¼ ∂ψ ∂y ∂x On substituting Eqs. (6a)–(6e) and (10) into (3) and (9), we therefore obtain the following nonlinear ordinary differential equations " # ρs 2 f 000 ð η Þ þ 1 − ϕ þ ϕ f ðηÞf ″ðηÞ − ½ f 0ðηÞ 2:5 ρ ð1−ϕ2Þ0 1 3 f σs 3 −1 ϕ C 6B σ 1 17 6B 7 f C −6B1 þ CM þ 7 f 0ðηÞ ¼ 0; 2:5 A σs σs 4@ ð1 − ϕÞ K p 5 þ2 − −1 ϕ σf σf 1
ð11Þ
20
1 3 σs 3 −1 ϕ B 6 C σf 1 ρ 1 17 6B 7 C β 3 −f w 1−ϕ þ ϕ s β 2 −6B1 þ β CM þ 2:5 K 7 σs σs 4@ A ρf p5 ð1−ϕÞ2:5 ð 1−ϕ Þ þ2 − −1 ϕ σf σf " # ρ ð18Þ −2 1−ϕ þ ϕ s ð1−ϕÞ2:5 ¼ 0; ρf "
#
where the positive resulted value is only to be considered to obtain the physical solution. For more details about deducing the exact solutions of f(η) in this section, the reader is advised to see the papers by Wang [51], Aly and Ebaid [14] and Aly and Vajravelu [52]. 3.2. Exact solution of θ(η) using Laplace transform Regarding the advantages of Laplace transform mentioned in the current introduction, we apply this method here to solve the
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temperature Eq. (12) with its appropriate Condition (13). Inserting Eq. (17) into Eq. (12) with supposing that t = − e−βη, yields tθ″ðt Þ þ ðn − mt Þθ0ðt Þ þ 2mθðt Þ ¼ 0;
ð19Þ
subject to the following set of boundary conditions θð0− Þ ¼ 0; θð − 1Þ ¼ 1:
ð20Þ
where n ¼ 1−
" # τ 2 2τ f w þ 2 ð1 − ϕÞ2:5 ; m ¼ 3 ð1 − ϕÞ2:5 ; β β β
1−1 " # B C 1 1 ks þ 2k f þ 2ϕ ks − k f þR C τ¼B @ A : ðρC p Þs Pr ks þ 2k f − ϕ ks − k f 1 − ϕ þ ϕ ρC ð pÞ f
ð21aÞ
0
ð21bÞ
On applying Laplace transform to Eq. (19), we get sðm − sÞΘ0ðsÞ þ ½ðn − 2Þs þ 3mΘðsÞ ¼ 0;
ð22Þ
where Θ(s) is the Laplace transform of θ(t). Now on integrating Eq. (22), we obtain ΘðsÞ ¼
c s3 ðs − mÞ−n−1
;
ð23Þ
where c is constant of the integration to be determined. On applying the inverse Laplace transform to Eq. (23), we have θðt Þ ¼
c 2Γð−n−1Þ
Z
t 0
ðt−μ Þ2 μ −n−2 emμ dμ;
n b − 1:
ð24Þ
It is clear from this equation that the boundary condition θ(0) = 0 is automatically satisfied. Besides, the other boundary condition θ(− 1) = 1 gives c by c ¼ −Z
2Γð − n − 1Þ 0 −1
:
ð25Þ
ð1 þ μ Þ2 μ −n−2 emμ dμ
Therefore, θ(t) is given in a closed form as Z
t
θðt Þ ¼ − Z
0 0
ðt − μ Þ2 μ −n−2 emμ dμ
−1
:
ð26Þ
ð1 þ μ Þ2 μ −n−2 emμ dμ
Performing the integrations in Eq. (26), we get the following exact solution for θ(t) in terms of the generalized incomplete gamma function: θðt Þ ¼
m2 t 2 Γð−n−1; 0; −mt Þ þ 2mtΓð−n; 0; −mtÞ þ Γð−n þ 1; 0; −mt Þ ; ð27Þ m2 Γð−n−1; 0; mÞ−2mΓð−n; 0; mÞ þ Γð−n þ 1; 0; mÞ
which can be finally expressed in terms of η as follows θðηÞ ¼
m2 e−2βη Γ −n−1; 0; me−βη −2me−βη Γ −n; 0; me−βη þ Γ −n þ 1; 0; me−βη : 2 m Γð−n−1; 0; mÞ−2mΓð−n; 0; mÞ þ Γð−n þ 1; 0; mÞ
ð28Þ
It should be noted that the combinations of the considered parameters have to be achieved the following condition " # τ 2 2:5 f þ ð1 − ϕÞ − 2 N 0: β w β2
ð29Þ
In this section, it was deduced that simpler special functions arise on applying Laplace transform as a tool of solution while other complex special functions arise on using other techniques, see for example Hamad [53]. 4. Results and discussion In the presence of the magnetic field and radiation, a steady two-dimensional Marangoni boundary layer flow past a porous surface in a water based nanofluid containing two different types of nanoparticles, namely Copper (Cu) and Titanium dioxide (TiO2), has been investigated in the case of suction, impermeable and injection. The governing equations of the stream flow and temperature have been solved using Laplace transform, thus, producing several plots to represent the effect of various parameters on the physical phenomena. In particular, influence of the velocity and temperature for both Cu–water, represented by solid curves, and TiO2–water, delineated by dotted curves, nanofluids is to be studied under effect of the following parameters; nanoparticle volume fraction (ϕ), magnetic (M), permeability (Kp ), thermal radiation and wall mass transfer (fw). The thermophysical properties of the water and present nanoparticles are given in Table 1, where Prandtl number of the base fluid (water) is kept in the present analysis at 6.2. The influence of nanoparticle volume fraction parameter (ϕ) on the velocity profiles f ′ (η) and temperature distribution θ(η) for both Cu–water and TiO2 –water nanofluids is depicted in Figs. 2(a–c) and 3(a–c) when the surface wall is in the case of injection (fw N 0), impermeable (fw = 0) and suction (fw b 0). For all of these cases and both types of nanofluids, it is observed that the velocities decrease significantly with an increase in ϕ. Therefore, the momentum boundary layer thickness decreases and tends asymptotically to zero as the distance increases from the boundary. In addition, as higher values of the thermal conductivity are accompanied by higher values of thermal diffusivity, this makes that the boundary layer thickness is sensitive to the change in ϕ. Moreover, it is noticed that TiO 2–water nanofluid is of higher velocity than Cu–water nanofluid. This is because that the density of TiO 2 is lower than of Cu, which means that the solid Titanium dioxide is lighter in motion compared to Copper. Further, it is observed from these figures that the increase in ϕ is to increase the temperature of the current two types of nanofluids. However, the thermal boundary layer for nanoparticles, namely Cu–water is greater than TiO 2–water. This is due to the higher conductivity of the solid particles Cu compared with TiO2. Furthermore, the thickness of the boundary layer increases as fw decreases with a remarkable effect of fw in the suction case with higher value of ϕ. Figs. 4(a–c) and 5(a–c) show effect of the magnetic parameter M on the velocity and temperature profile, respectively, of Cu–water and TiO2–water nanofluids at three different cases of fw parameter (N 0, = 0, b0). Form these figures and for the both types of nanofluids, it is seen that application of a magnetic field results a resistive-type force called Lorentz force. This force opposes the flow and, therefore, decelerates the fluid velocity and increases its temperature. In addition, as the strength of the magnetic field increases, the hydrodynamic and thermal boundary layers decrease and increase, respectively. This agrees with the fact that the nanoparticles dissipate energy in the form of heat which causes the thermal boundary layer thickness to increase and ultimately a localized rise in
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Fig. 2. Velocity profiles f′(η) for variation of ϕ when M = 1 and Kp = 1 at (a) fw = 1, (b) fw = 0 and (c) fw = −1.
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Fig. 3. Temperature distribution θ(η) for variation of ϕ when M = 1, Kp = 0.5, R = 1 at (a) fw = 0.1, (b) fw = 0 and (c) fw = −0.1.
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Fig. 4. Velocity profiles f′(η) for variation of M when ϕ = 0.1 and Kp = 1 at (a) fw = 1, (b) fw = 0 and (c) fw = −1.
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Fig. 5. Temperature distribution θ(η) for variation of M when ϕ = 0.1, Kp = 0.5, R = 1 at (a) fw = 0.1, (b) fw = 0 and (c) fw = −0.1.
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temperature of the fluid occurs. Further, as the density of TiO 2 is lower than of Cu, the velocity of TiO2 –water is higher than of Cu–water. Moreover, the thickness of the boundary layer increases as fw decreases from 1 to − 1.
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The impact of the porous parameter Kp on the velocities and the temperature distribution is displayed in Figs. 6(a–c) and 7(a–c), respectively. It can be noted from these figures that, as Kp increases, which means a rising in the holes of the porous structure, the velocities and temperature
Fig. 6. Velocity profiles f′(η) for variation of Kp when ϕ = 0.1 and M = 1 at (a) fw = 1, (b) fw = 0 and (c) fw = −1.
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Fig. 7. Temperature distribution θ(η) for variation of Kp when ϕ = 0.1, M = 1, R = 1 at (a) fw = 0.1, (b) fw = 0 and (c) fw = −0.1.
increase in the both types of Cu–water and TiO2–water nanofluids at the investigated three cases of fw. This is, of course, agreed with the physical view. It should be noted here that, for K≤0.1, velocities of Cu–water and
TiO2–water nanofluids are nearly identical, especially at fw = 0 and fw = −1 (injection). While, as (K→∞), the current two types of nanofluids have nearly the same temperature distribution when fw = 1 (suction).
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As shown in Fig. 8(a–c), the temperature profiles increase as radiation parameter R increases for all nanoparticles considered and for all investigated cases of the f w parameter. It is known that the
635
radiation parameter, being the reciprocal of Stephan number, is the measure of relative importance of the thermal radiation transfer to the conduction heat transfer. Thus, larger values of R are indicative
Fig. 8. Temperature distribution θ(η) for variation of R when ϕ = 0.1, M = 1, Kp = 0.5 at (a) fw = 0.1, (b) fw = 0 and (c) fw = −0.1.
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of larger amount of radiative heat energy being poured into the system, causing a rise in θ(η). Therefore, the radiation shows a dominance of the thermal radiation over conduction, thus, it can be used to control the thermal boundary layers quite effectively. Figs. 9(a–b) and 10(a–b) depict the variation of velocity and temperature, respectively, at different selected values for the suction/injection parameter fw. From these figures, it is observed that imposition of fluid suction (fw N 0) has the tendency to decrease the fluid velocity and, therefore, thickness of the hydrodynamic boundary layer, regarding, the fluid temperature and thermal boundary layer decrease as well. However, fluid injection (fw b 0) produces the opposite effect, namely an increase in the fluid velocity and temperature. It should be noted that these profiles satisfy the far field boundary conditions asymptotically. In addition, for the injection case, Fig. 9(a) and (b) refers to remarkable and slightly higher velocity, respectively, for TiO2–water nanofluid, as a result, a remarkable increase in the boundary layer thickness has been observed in the suction case than in the injection case. Further, from Fig. 10, it is shown that the temperature profile is the more for higher thermal conductivity nanoparticle (Cu). Figs. 11, 12 and 13 display variation of the local Nusselt number with the parameters of nanoparticle volume fraction (ϕ), radiation
R and location x for different values of fw in both cases of Cu–water and TiO2–water nanofluids. Here, it should be noticed that the entire values of − θ ′ (0) are always positive, i.e. the heat is transferred from hot surface to the cold region. From Fig. 11, it is seen that local Nusselt number, increases as R increases, however as shown in Fig. 12, it decreases as ϕ increases in the presence of radiation. It can be also noticed that the heat transfer rate at the surface increases in the presence of radiation. This result agrees with the physical view, since the effect of radiation is to increase the rate of energy transport to the fluid, thereby increasing the temperature of the fluid, as shown in Fig. 8. It is observed from Fig. 13 that the fluid suction and injection have the tendency to increase and decrease, respectively, the heat transfer rate at the wall. From these figures and in the injection case it should be noticed that nanoparticles with low thermal conductivity, TiO2 , have better enhancement on heat transfer compared to Cu. However, for the suction case, the same result is to be obtained until a specific value of ϕ where this behavior becomes opposite, as shown in Fig. 13. Further, this critical value of ϕ, where Nux equals for both Cu–water and TiO2–water, decreases as fw increases, as shown by the solid circles in this figure.
Fig. 9. Velocity profiles f′(η) when M = 1 and Kp = 1 for variation of fw (a) injection and (b) suction.
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Fig. 10. Temperature distribution θ(η) when ϕ = 0.1, M = 1, Kp = 0.5 and R = 1 for variation of fw (a) injection and (b) suction.
Marangoni boundary layer flow past a porous surface in a water based nanofluid containing two different types of nanoparticles, namely
Copper (Cu) and Titanium dioxide (TiO2), has been studied. Exact solutions of the stream flow and temperature have been obtained using Laplace transform. Influence of the velocity and temperature for both nanofluids has been investigated under effect of the following
Fig. 11. Variation of the local Nusselt number Nux as a function of R for fw = −0.1, 0, 0.1 when ϕ = 0.1, M = 1 and Kp = 0.5.
Fig. 12. Variation of the local Nusselt number Nux as a function of ϕ for R = 1, 2, 3 in suction fw = 0.1 when M = 1 and Kp = 0.5.
5. Conclusion
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Fig. 13. Variation of the local Nusselt number Nux as a function of ϕ at various values of fw when M = 1, Kp = 0.5 and R = 1.
parameters; nanoparticle volume fraction (ϕ), magnetic (M), permeability (Kp), thermal radiation (R), and wall mass transfer (fw) in the case of suction, impermeable and injection. The main results of the present study can be epitomized as follows: • the velocities of the two investigated nanoparticles decrease significantly with an increase in ϕ, however, TiO2–H2O nanofluid is of higher velocity than Cu–H2O nanofluid, • increase in ϕ is to increase the temperature of the current two types of nanofluids. However, the thermal boundary layer of Cu–H2O is greater than TiO2–H2O, • the effective electrical conductivity makes a very considerable and remarkable effect on the nanofluids' flow, therefore, this parameter is mandatory and should be taken into account on applying the magnetic field, otherwise a spurious physical sight is to be gained. The parameter M decelerates the fluid velocity and increases its temperature, in addition, as M increases, the hydrodynamic and thermal boundary layers decrease and increase, respectively, • as Kp increases, the velocities and temperature increase in the both types of Cu–H2O and TiO2–H2O nanofluids at the investigated three cases of fw, further, when (K → ∞), the current two types of nanofluids have nearly the same temperature distribution in the suction case, • for all nanoparticles considered and for all investigated cases of the fw parameter, the temperature profiles increase as radiation parameter R increases, in addition, the radiation can be used to control the thermal boundary layers quite effectively, • the fluid suction (fw N 0) decreases the fluid velocity and, therefore, thickness of the hydrodynamic boundary layer, regarding, the fluid temperature and thermal boundary layer decreases as well, however, fluid injection (fw b 0) produces the opposite effect, namely an increase in the fluid velocity and temperature, • for TiO2 –H2O nanofluid, a remarkable increase in the boundary layer thickness has been obtained in the suction case than in the injection case, further, the temperature profile is the more for higher thermal conductivity nanoparticle (Cu), • the local Nusselt number, increases as R increases, however, it decreases as ϕ increases, • the fluid suction and injection have the tendency to increase and decrease, respectively, the heat transfer rate at the wall which increases in the presence of radiation, • nanoparticles with low thermal conductivity, TiO 2 , have better enhancement on heat transfer compared to Cu. References [1] L.G. Napolitano, Microgravity fluid dynamics, 2nd Levitch Conference, Washington, 1978.
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