Similarity Solution of Marangoni Convection Boundary Layer Flow over a Flat Surface in a Nanofluid

Abstract

The problem of steady Marangoni boundary layer flow and heat transfer over a flat plate in a nanofluid is studied using different types of nanoparticles. The general governing partial differential equations are transformed into a set of two nonlinear ordinary differential equations using unique similarity transformation. Numerical solutions of the similarity equations are obtained using the Runge-Kutta-Fehlberg (RKF) method. Three different types of nanoparticles are considered, namely, Cu, Al₂O₃, and TiO₂, by using water as a base fluid with Prandtl number Pr=6.2. The effects of the nanoparticle volume fraction 𝝓 and the constant exponent m on the flow and heat transfer characteristics are obtained and discussed.

1. Introduction

A nanofluid is a colloidal mixture of nanosized particles (<100 nm) in a base fluid. It is known that nanofluid can tremendously enhance the heat transfer characteristics of the original (base) fluid. One such characteristic of nanofluid is the anomalous high thermal conductivity at very low concentration of nanoparticles and the considerable enhancement of convective heat transfer. Thus, nanofluids have many applications in industry such as coolants, lubricants, heat exchangers, and microchannel heat sinks. Nanoparticles are made of various materials such as oxide ceramics, and nitride ceramics. The objective of nanofluids is to achieve the best possible thermal properties with the least possible (<1%) volume fraction of nanoparticles in the base fluid [1]. There have been many studies in the literature to better understand the mechanism behind the enhanced heat transfer characteristics. An excellent collection of papers on this topic can be found in the book by Das et al. [2] and in several review papers ([3–8]). There are also several experimental studies to better understand the mechanism of heat transfer enhancement for natural convection heat transfer in nanofluids ([1, 9–12]).

Marangoni flow induced by surface tension along a liquid surface causes undesirable effects in crystal growth melts in the same manner as buoyancy-induced natural convection [13]. These undesirable effects also occur in space-based crystal growth experiments since Marangoni flow is involved in microgravity as well as in earth gravity. An excellent view of the Marangoni effect from the perspective of all three possible interfaces as motion inducing agents has been done by Tadmor [14]. It is worth mentioning that there are two existing models for Marangoni boundary layer that have been studied, namely, model for nonisobaric Marangoni boundary layer as discussed by Golia and Viviani [15] and model for Marangoni boundary layer over a flat plate studied by Christopher and Wang [13]. Marangoni boundary layer studied by Golia and Viviani [15] has been extended by Pop et al. [16] where they included the concentration equation. Chamkha et al. [17] studied the same model with Golia and Viviani [15] in which they considered the gravity effects. Hamid et al. [18] extended the problem of the thermosolutal Marangoni forced convection boundary layer flow by Pop et al. [16] when the wall is permeable. Very recently, Mat el al. [19] discussed the radiation effects on the problem of Marangoni boundary layer with permeable surface. On the other hand, nanofluid equations model as proposed by Tiwari and Das [20] has been used by Arifin et al. [21] for the Marangoni boundary layer problem by Golia and Viviani [15]. They found that the numerical results also indicate that, for both a regular fluid (𝜑=0) and a nanofluid (𝜑≠0), dual solutions exist when 𝛽<0.5. These dual solutions were not discussed by Golia and Viviani [15]. This problem has been extended by Remeli et al. [22] to the problem with suction and injection effects. Mat et al. [23] also extended the problem of Marangoni boundary layer in a nanofluid by Arifin et al. [21] to the radiation effect.

It is worth mentioning that Christopher and Wang [13] considered the Marangoni boundary layer over a flat plate where the term 𝑢_𝑒(𝑥), which is the velocity of the external flow in Golia and Viviani [15], has been neglected. The similarity solutions of the Christopher and Wang [13] problem are also different from the Golia and Viviani [15] problem. The problem of Christopher and Wang [13] has been extended by several researchers such as Al-Mudhaf and Chamkha [24] where they have presented the similarity solutions for MHD Marangoni convection in the presence of heat generation or absorption effects and Magyari and Chamkha [25] reported the exact analytical solutions of thermosolutal Marangoni flows in the presence of temperature-dependent volumetric heat source/sinks as well as of a first-order chemical reaction. Recently, Hamid et al. [26] studied the two-dimensional Marangoni convection flow past a flat plate in the presence of thermal radiation, suction, and injection effects. Further MHD thermosolutal Marangoni convection boundary layer over a flat surface considering the effects of the thermal diffusion and diffusion-thermo with fluid suction and injection has been examined by Hamid et al. [27].

It should be highlighted that the present paper presents a similarity solution for the steady Marangoni convection boundary layer flow over a static semi-infinite flat plate due to an imposed temperature gradient in a nanofluid, which extends the problem by Christopher and Wang [13] to the case of nanofluid. The nanofluid equations model proposed by Tiwari and Das [20] has been used. This model has been very successfully used in several papers [21, 28–32]. Thus, we wish to highlight that this present study is original and all the results are new. To the best of our knowledge, the present problem has not been considered before. The study of nanofluid is still at its early stage and it seems difficult to have a precise idea on the way the use of nanoparticles acts in heat transfer. A clear picture on the boundary layer flows of nanofluid is yet to emerge.

2. Problem Formulation

We consider the steady two-dimensional boundary layer flow past a semi-infinite flat plate in a water-based nanofluid containing different types of nanoparticles, namely, copper (Cu), alumina (Al₂O₃), and titania (TiO₂), with Marangoni effects. The nanofluid is assumed incompressible and the flow is assumed to be laminar. It is also assumed that the base fluid (i.e., water) and the nanoparticles are in thermal equilibrium and no slip occurs between them. The thermophysical properties of the nanofluids are given in Table 1 (see Oztop and Abu-Nada [29]). 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. It is also assumed that the temperature of the plate is 𝑇_𝑤(𝑥) and that of the ambient nanofluid is 𝑇_∞. Following [15–17, 25, 33, 34] the surface tension 𝜎 is assumed to vary linearly with temperature as follow: 𝜎=𝜎₀1−𝛾𝑇−𝑇₀, (1) where 𝜎₀ and 𝑇₀ are the characteristics surface tension and temperature, respectively, and we assume that 𝑇₀≡𝑇_∞. Equation (1) is a commonly made assumption [34]. For most liquids, the surface tension decreases with temperature; that is, 𝛾 is a positive fluid property.

Table 1 

Thermophysical properties of fluid and nanoparticles [29].

The steady boundary layer equations for a nanofluid in the coordinates 𝑥 and 𝑦 are ([13, 20]) 𝜕𝑢+𝜕𝑥𝜕𝑣𝑢𝜕𝑦=0,𝜕𝑢𝜕𝑥+𝑣𝜕𝑢=𝜇𝜕𝑦𝜌^nf_nf𝜕²𝑢𝜕𝑦²,𝑢𝜕𝑇𝜕𝑥+𝑣𝜕𝑇𝜕𝑦=𝛼_nf𝜕²𝑇𝜕𝑦² (2)(3)(4) subject to the boundary conditions 𝑣=0,𝑇=𝑇₀+𝐴𝑥^𝑚+1,𝜇_nf𝜕𝑢=𝜕𝑦𝜕𝜎𝜕𝑇𝜕𝑇𝜕𝑥at𝑦=0,𝑢=0,𝑇=𝑇_∞as𝑦⟶∞. (5) Here, 𝑢 and 𝑣 are the velocity components along the 𝑥- and 𝑦-axes, respectively, 𝑇 is the temperature of the nanofluid, m is the constant exponent of the temperature, 𝛼_nf is the thermal diffusivity of the nanofluid, 𝜌_nf is the effective density of the nanofluid, 𝑘_nf is the effective thermal conductivity of the nanofluid, and 𝜇_nf is the effective viscosity of the nanofluid, which are given by 𝛼_nf=𝑘_nf𝜌𝐶_𝑝nf,𝜌_nf=1−𝜙𝜌_𝑓+𝜙𝜌_𝑠,𝜇_nf=𝜇_𝑓1−𝜙^2.5,𝜌𝐶_𝑝nf=1−𝜙𝜌𝐶_𝑝𝑓+𝜙𝜌𝐶_𝑝𝑠,𝑘_nf𝑘_𝑓=𝑘_𝑠+2𝑘_𝑓−2𝜙𝑘_𝑓−𝑘_𝑠𝑘_𝑠+2𝑘_𝑓+𝜙𝑘_𝑓−𝑘_𝑠, (6) where 𝜙 is the nanoparticle volume fraction, 𝜌_𝑓 is the reference density of the fluid fraction, 𝜌_𝑠 is the reference velocity of the solid fraction, 𝜇_𝑓 is the viscosity of the fluid fraction, 𝑘_𝑓 is the thermal conductivity of the fluid, 𝑘_𝑠 is the thermal conductivity of the solid, and (𝜌𝐶_𝑝)_nf is the heat capacity of the nanofluid.

We look now for a similarity solution of (2)–(4) subject to the boundary conditions (5) of the following form: 𝜓=𝐶₁𝑥^(2+𝑚)/3𝑓𝜂,𝜃𝜂=𝑇−𝑇_∞𝐴𝑥^1+𝑚,𝜂=𝐶₂𝑥^(𝑚−1)/3𝑦, (7) where 𝜓 is the stream function which is defined as 𝑢=𝜕𝜓/𝜕𝑦 and 𝑣=−𝜕𝜓/𝜕𝑥. Further, 𝑚, 𝐴, 𝐶₁, and 𝐶₂ are constants with 𝐴, 𝐶₁, and 𝐶₂ given by 𝐴=Δ𝑇𝐿^𝑚+1,𝐶₁=³𝜎₀𝛾𝐴𝜇_𝑓𝜌²_𝑓,𝐶₂=³𝜎₀𝛾𝐴𝜌_𝑓𝜇²_𝑓, (8) with 𝐿 being the length of the surface and Δ𝑇 being the constant characteristic temperature. Substituting (7) into (2) and (3), we get the following ordinary differential equations: 11−𝜙^2.51−𝜙+𝜙𝜌_𝑠/𝜌_𝑓𝑓^′′′+2+𝑚3𝑓𝑓^′′−1+2𝑚3𝑓^′21=0,𝑘Pr^nf/𝑘_𝑓1−𝜙+𝜙𝜌𝐶_𝑝𝑠/𝜌𝐶_𝑝𝑓𝜃^′′+2+𝑚3−()𝑓𝑓𝜃1+𝑚^′𝜃=0, (9) and the boundary conditions (5) become 𝑓(0)1=0,1−𝜙^2.5𝑓^′′(0)(0)𝑓=−1,𝜃=1,_′(∞)=0,𝜃(∞)=0. (10) We can now determine the surface velocity 𝑢(𝑥,0)=𝑢_𝑤(𝑥) as 𝑢_𝑤(𝑥)=³𝜎₀𝛾𝐴²𝜌_𝑓𝜇_𝑓𝑥^(1+2𝑚)/3𝑓^′(0). (11) A quantity of interest is the local Nusselt number Nu_𝑥 which is defined as Nu_𝑥=𝑥𝑞_𝑤(𝑥)𝑘_𝑓[𝑇()(,𝑥,0−𝑇𝑥,∞)] (12) where 𝑞_𝑤(𝑥) is the heat flux from the surface of the plate and it is given by 𝑞_𝑤(𝑥)=−𝑘_nf𝜕𝑇𝜕𝑦_𝑦=0. (13) Using (7), (12), and (13), we get Nu_𝑥𝑘=−^nf𝑘_𝑓𝐶₂𝑥^(2+𝑚)/3𝜃^′(0). (14) The average Nusselt number Nu_𝐿 based on the average temperature difference between the temperature of the surface and the temperature far from the surface (ambient fluid) is given by Nu_𝐿=−6+3𝑚𝑘5+4𝑚𝑘^nf_𝑓Ma_𝐿^1/3Pr^−1/3𝜃^′(0), (15) where Ma_𝐿 is the Marangoni based on 𝐿 and is defined as Ma_𝐿=𝜎_𝑇𝐴𝐿^2+𝑚𝜇_𝑓𝛼_𝑓=𝜎_𝑇Δ𝑇𝐿𝜇_𝑓𝛼_𝑓. (16) Also, the total mass flow 𝑚̇ in the boundary layer per unit width is given by 𝑚̇=^∞₀𝜌_𝑓𝑢𝑑𝑦=³𝜎₀𝛾𝜌_𝑓𝜇_𝑓𝑥^(2+𝑚)/3𝑓(∞). (17)

3. Results and Discussion

The nonlinear ordinary differential equation (9) subject to the boundary conditions (10) forms a two-point boundary value problem (BVP) and is solved numerically using the Runge-Kutta-Fehlberg fourth-fifth-order (RKF45) method using Maple 12- and the algorithm RKF45 in Maple has been well tested for its accuracy and robustness [35]. In this method, it is most important to choose the appropriate finite value of the edge of boundary layer, 𝜂→∞ (say 𝜂_∞) that is between 4 and 10, which is in accordance with the standard practice in the boundary layer analysis. We begin with some initial guess value of 𝜂_∞ and solve (9) subject to the boundary conditions (10) with some particular set of parameters to obtain the surface velocity 𝑓^′(0) and the temperature gradient −𝜃^′(0). The solution process is repeated until further changes (increment) in 𝜂_∞ would not lead to any changes in the values of 𝑓^′(0) and −𝜃^′(0), or in other words, the results are independent of the value of 𝜂_∞. The initial step size employed is ℎ=Δ𝜂=0.1. Following Oztop and Abu-Nada [29], we considered the range of nanoparticles volume fraction 𝜙 as 0≤𝜙≤0.2. The Prandtl number of the base fluid (water) is kept constant at 6.2. Further, it should also be pointed out that the thermophysical properties of fluid and nanoparticles (Cu, Al₂O₃, and TiO₂) used in this study are given in Table 1. It is worth mentioning that, the present study reduces to that of a classical viscous (regular) fluid studied by Christopher and Wang [13] when 𝜙=0.

Figures 1 and 2 show the distribution of the dimensionless velocity 𝑓^′(𝜂) and temperature 𝜃(𝜂) profiles for the three types of the nanoparticles considered when the solid volume fraction of the nanofluid parameter 𝜙 = 0.1 and 𝑚=0 (which corresponds to a linear variation of the surface temperature with the distance 𝑥 measured along the flat plate), while Figures 3 and 4 display the variation with 𝜙 of the reduced surface velocity, 𝑓^′(0), and reduced temperature gradient, −𝜃^′(0), respectively. It is seen from Figure 1 that the velocity profiles in Figure 1 for Al₂O₃ and TiO₂ are almost identical while the profile for Cu is smaller. This is consistent with the variation of the reduced surface velocity 𝑓^′(0) as shown in Figure 3. In Figure 2, it is shown that the temperature profile is the highest for higher thermal diffusivity nanoparticle (Cu). On the other hand, the thermal boundary layer thickness as shown in Figure 2 decreases with a decrease in thermal diffusivity, which in turn gives rise to the −𝜃^′(0) as illustrated in Figure 4. Figures 3 and 4 display the surface velocity, 𝑓^′(0), and the surface temperature gradient, −𝜃^′(0), respectively, for different types of nanoparticles (Cu, Al₂O₃, and TiO₂) when 𝑚=0. One can see that the surface velocity, 𝑓^′(0) and the surface temperature gradient, −𝜃^′(0) decrease as 𝜙 increases for all three nanoparticles (Cu, Al₂O₃, TiO₂). It should be noticed that the entire values of −𝜃^′(0) are always positive; that is, the heat is transferred from hot surface to the cold surface. In Figure 4, we are looking at the variation of temperature gradient with the nanoparticle volume fraction 𝜙. It is observed that the reduced value of thermal diffusivity leads to higher temperature gradients and, therefore, higher enhancements in heat transfer. Nanoparticles with low thermal diffusivity, TiO₂, have better enhancement on heat transfer compared to Cu and Al₂O₃. Further, Figure 5 shows the variations of the reduced surface velocity 𝑓^′(0) with 𝑚, where 𝑚=0 refers to a linear variation of the surface temperature with the distance 𝑥 measured along the flat plate and 𝑚=1 is a quadratic variation of the surface temperature with 𝑥, while 𝑚=−0.5 refers to a temperature variation relative to the square root of 𝑥. It should also be noticed that for 𝜙=0 (regular fluid), we reproduced the variations of surface velocity obtained by Christopher and Wang [13] as illustrated by dashed lines in the figure.

Figure 1 

Dimensionless velocity profiles 𝑓′(𝜂) for different types of nanoparticles when 𝜙 = 0.1 and 𝑚=0.

Figure 2 

Dimensionless temperature profiles 𝜃(𝜂) for different types of nanoparticles when 𝜙 = 0.1 and 𝑚=0.

Figure 3 

Variation of surface velocity 𝑓′(0) with 𝜙 for different types of nanoparticles when Pr = 6.2 and 𝑚=0.

Figure 4 

Variation of −𝜃′(0) with 𝜙 for different types of nanoparticles when Pr=6.2 and 𝑚=0.

Figure 5 

Variation of surface velocity 𝑓′(0) with 𝑚 for different types of nanoparticles when Pr = 6.2, 𝜙=0 (regular fluid), and 𝜙=0.1.

Figures 6 to 11 show the dimensionless velocity 𝑓^′(𝜂) and temperature 𝜃(𝜂) profiles for different values of 𝜙 in the range 0≤𝜙≤0.2 when 𝑚=0 with different types of nanoparticles, namely, Cu, Al₂O₃, and TiO₂, respectively. It is worth mentioning that nanoparticle volume fraction is a key parameter for studying the effect of nanoparticles on flow fields and temperature distributions. More fluid is heated for higher values of nanoparticle volume fraction. Flow strength also increases with increasing of nanoparticle volume fraction. As the nanoparticle volume fraction increases, movements of particles become irregular and random due to increasing of energy exchange rates in the fluid (see [29]). It is observed from these figures that, for any type of nanoparticles, as the nanoparticle volume fraction 𝜙 increases, both the surface velocities and the temperature gradients decrease, which is again in agreement with Figures 3 and 4. It should also be noticed again, that for 𝜙=0 (regular fluid), we reproduced the velocity and temperature profiles obtained by Christopher and Wang [13].

Figure 6 

Dimensionless velocity profiles 𝑓′(𝜂) for Cu nanoparticles with 𝑚=0 and various values of 𝜙.

Figure 7 

Dimensionless temperature profiles 𝜃(𝜂) for Cu nanoparticles with 𝑚=0 and various values of 𝜙.

Figure 8 

Dimensionless velocity profiles 𝑓′(𝜂) for Al2O3 nanoparticles with 𝑚=0 and various values of 𝜙.

Figure 9 

Dimensionless temperature profiles 𝜃(𝜂) for Al2O3 nanoparticles with 𝑚=0 and various values of 𝜙.

Figure 10 

Dimensionless temperature profiles 𝜃(𝜂) for TiO2 nanoparticles with 𝑚=0 and various values of 𝜙.

Figure 11 

Dimensionless velocity profiles 𝑓′(𝜂) for TiO2 nanoparticles with 𝑚=0 and various values of 𝜙.

4. Conclusion

We have theoretically and numerically studied the problem of steady two-dimensional laminar Marangoni-driven boundary layer flow in nanofluids. It is worth mentioning that the novelty of the present paper is to study numerically the heat transfer in a liquid layer driven by Marangoni flow with various types of nanoparticles (Cu, Al₂O₃, and TiO₂) in the base fluid which has not been considered before. The nonlinear ordinary differential equation (9) subject to the boundary conditions (10) forms a two-point boundary value problem (BVP) and is solved numerically using the Runge-Kutta-Fehlberg fourth-fifth-order (RKF45) method using Maple 12, and the algorithm RKF45 in Maple has been well tested for its accuracy and robustness. Similarity solutions are obtained for the surface velocity 𝑓^′(0) and the surface temperature gradient −𝜃^′(0) as well as the velocity and temperature profiles for some values of the governing parameters, namely, the solid volume fraction of the nanofluid 𝜙 (0≤𝜙≤0.2), the constant exponent 𝑚, and the Prandtl number Pr. It was found that nanoparticles with low thermal diffusivity (TiO₂) have better enhancement on heat transfer compared to Al₂O₃ and Cu.

Acknowledgments

The authors gratefully acknowledge the financial support received in the form of a FRGS Research Grant from the Ministry of Higher Education, Malaysia, and a Research University Grant (AP-2013-009) from the Universiti Kebangsaan Malaysia. They also wish to express their sincere thanks to the reviewers for the valuable comments and suggestions.