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Heat and mass transfer of nanofluid through an impulsively vertical stretching surface using the spectral relaxation method
Boundary Value Problems volume 2015, Article number: 161 (2015)
Abstract
In this paper, we investigate heat and mass transfer in a magnetohydrodynamic nanofluid flow due to an impulsively started stretching surface. The flow is subject to a heat source, a chemical reaction, Brownian motion and thermophoretic parameters which are assumed to be significant. We have further assumed that the nanoparticle volume fraction at the wall may be actively controlled. The physical problem is modeled using systems of nonlinear differential equations which have been solved numerically using the spectral relaxation method. Comparing with previously published results by Khan and Pop (Int. J. Heat Mass Transf. 53:24772483, 2010) shows an excellent agreement. Some of the particular findings are that the skin friction coefficient decreases with an increase in the nanoparticle volume fraction, the heat transfer coefficient decreases with an increase in the nanoparticle volume fraction and that the mass transfer coefficient increases with an increase in the nanoparticle volume fraction.
Introduction
The term nanofluid denotes a liquid in which nanoscale particles are suspended in a base fluid with low thermal conductivity such as water, oils and ethylene glycol. In recent years, the concept of nanofluid has been proposed as a route for increasing the performance of heat transfer liquids. Due to the increasing importance of nanofluids, there is now a large amount of literature on convective transport of nanofluids and problems linked to a stretching surface. Choi [1] initially pointed out that addition of these nanoparticles to the base fluid appreciably enhances the effective thermal conductivity of the fluid. An excellent collection of articles on this topic can be found in [2, 3] and Das et al. [4]. A nonhomogenous equilibrium model proposed by Buongiorno [5] revealed that the massive increase in the thermal conductivity occurs due to the presence of two main effects; namely the Brownian diffusion and the thermophoretic diffusion of nanoparticles. The study of a steady boundary layer flow of a nanofluid towards a stretching sheet was reported by Khan and Pop [6]. Radiation effects on the viscous flow of a nanofluid and heat transfer over a nonlinearly stretching sheet were studied by Hady et al. [7]. Kuznetsov and Nield [8] carried out a numerical investigation of mixed convection in the nanofluid flow over a vertical flat plate. In related work, Nield and Kuznetsov [9] studied the ChengMinkowycz problem for the natural convection in nanofluid flow over a flat plate. Yacob et al. [10] studied the stagnation point flow of a nanofluid flow due to a stretching/shrinking sheet using a shooting technique together with a fourthfifth order RungeKutta method. Recently, results of MHD mixed convection in unsteady nanofluid flow due to a stretching/shrinking surface with suction/injection were reported by Haroun et al. [11]. In this study the model equations were solved using a spectral relaxation method. Stagnation point flow of a nanofluid with heat generation/absorption and suction/blowing was investigated by Hamad and Ferdows [12]. Rashidi and Erfani [13] used the modified differential transform method to investigate boundary layer flow due to stretching surfaces. Some excellent articles on the flow of nanofluids include those by Rashidi et al. [14], Anwar Bég et al. [15] and Garoosi et al. [16]. Some interesting results on discrete problems were presented by [17, 18].
Magnetohydrodynamic (MHD) flow and heat and mass transfer over a stretching surface have many important technological and industrial applications such as in micro MHD pumps, micro mixing of physiological samples, biological transportation and in drug delivery. An excellent collection of articles on this topic can be found in [19, 20]. The application of magnetic field produces a Lorentz force which assists in mixing processes as an active micromixing technology technique. Hence, transportation of conductive biological fluids in micro systems may greatly benefit from theoretical research in this area (see Yazdi et al. [21]). Studies on magnetohydrodynamics (MHD) free convective boundary layer flow of nanofluids are very limited. More recently, Chamkha and Aly [22] studied magnetohydrodynamics (MHD) free convective boundary layer flow of a nanofluid along a permeable isothermal vertical plate in the presence of heat generation or absorption effects. Matin et al. [23] studied magnetohydrodynamics (MHD) mixed convective flow of nanofluid over a stretching sheet. Magnetohydrodynamics (MHD) forced convective flow of nanofluid over a horizontal stretching flat plate with variable magnetic field including the viscous dissipation was investigated by Nourazar et al. [24]. The effect of a transverse magnetic field on the flow and heat transfer over a stretching surface was examined by Anjali Devi and Thiyagarajan [25].
Despite all the previous work, there is still a lot that is unknown about the flow and heat and mass transfer properties of different nanofluids. For instance, the composition and make of nanoparticles may have an impact on the performance of nanofluid as a heat transfer medium. The aim of the present study is to analyze the effects of Brownian motion parameter and thermophoresis parameter on unsteady boundary layer flow heat and mass transfer of a nanofluid flow past an impulsively stretching surface in the presence of a chemical reaction and an applied magnetic field. The model equations are solved using the spectral relaxation method (SRM) that was recently proposed by Motsa [26]. The spectral relaxation method promises fast convergence with good accuracy, has been successfully used in a limited number of boundary layer flow, heat and mass transfer studies (see [27, 28]). A comparative study for a special case is presented, which shows good agreement with Khan and Pop [6].
Governing equations
Consider the twodimensional unsteady boundary layer flow heat and mass transfer in a nanofluid past an impulsively stretching vertical surface situated at \(y = 0\) with stretching velocity \(u(x) = ax\), where a is a constant. The temperature and nanoparticle concentration at the stretching surface are \(T_{w}\) and \(C_{w}\), respectively, and those of the ambient nanofluid are \(T_{\infty}\) and \(C_{\infty}\), respectively. The x and y directions are taken along and perpendicular to the sheet, respectively. Here we focus mainly in the region \(x, y\ge0\). The Boussinesq approximation is applied here. The continuity, momentum, energy and concentration equations of an unsteady, incompressible nanofluid boundary layer flow are as follows (see Kuznetsov and Nield [8]):
where u and v are the fluid velocity and normal velocity components along x and ydirections, respectively; \(\mu_{nf}\), \(\rho_{nf}\), σ, \(B_{0}\), g are the effective dynamic viscosity of the nanofluid, nanofluid density, electrical conductivity, the uniform magnetic field in the ydirection and gravitational acceleration; \(\beta_{T}\), \(\beta_{C}\), T, C, \(\alpha _{nf}\), \(\tau^{*}\) (\(= (\rho c)_{p}/(\rho c)_{f}\)) are the volumetric thermal expansion coefficient, volumetric solutal expansion coefficient, temperature of fluid in the boundary layer, fluid solutal concentration, the thermal diffusivity of the nanofluid, the ratio of effective heat capacity of the nanoparticle material to heat capacity of the fluid; \(D_{B}\), \(D_{T}\), \(T_{\infty}\), K are the Brownian motion coefficient, the thermophoretic diffusion coefficient, mean fluid temperature and the chemical reaction parameter.
The boundary conditions are
and the initial conditions are
where a is the stretching/shrinking rate and stagnation flow rate parameters, with \(a< 0\) for shrinking, \(a > 0\) for stretching.
The effective dynamic viscosity of the nanofluid was given by Brinkman [29] as
where ϕ and \(\mu_{f}\) are the solid volume fraction of nanoparticles and the dynamic viscosity of the base fluid.
In equations (1) to (4), the quantities \((\rho c_{p})_{nf}\), \(\rho_{nf}\) and \(\alpha_{nf}\) are given by
where \(\nu_{nf}\), \(\rho_{nf}\), \((\rho cp)_{nf}\), \(k_{nf}\), \(k_{f}\), \(k_{s}\), \(\rho_{s}\), \((\rho c_{p})_{f}\), \((\rho c_{p})_{s}\) are the nanofluid kinematic viscosity, the electrical conductivity, the nanofluid heat capacitance, thermal conductivity of the nanofluid, thermal conductivity of the fluid, the thermal conductivity of the solid fractions, the density of the solid fractions, the heat capacity of the base fluid, the effective heat capacity of nanoparticles, respectively (see AbuNada [30]).
The continuity equation (1) is satisfied by introducing a stream function \(\psi(x,y)\) such that
Introducing the following nondimensional variables (see Liao [31]):
where η, ξ and τ are dimensionless variables and the dimensionless time, \(f(\xi,\eta)\) is the dimensionless stream function, \(\theta(\xi,\eta)\) is the dimensionless temperature and \(\phi(\xi,\eta)\) is the dimensionless solute concentration. By using (10) the governing equations (2) to (4) along with the boundary conditions (5) are reduced to the following twopoint boundary value problem:
subject to the boundary conditions
where the prime denotes differentiation with respect to η, \(\alpha_{f}=k_{f}/(\rho c_{p})_{f}\) and \(\nu_{f}=\mu_{f}/\rho_{f}\) are the thermal diffusivity and kinetic viscosity of the base fluid, respectively. Other nondimensional parameters appearing in equations (11) to (13) Ha, \(\mathit{Gr}_{t}\), \(\mathit{Gr}_{c}\), Pr, \(N_{b}\), \(N_{T}\), Sc, and γ denote the Hartman number, the local temperature Grashof number and the local concentration Grashof number (see Mahdy [32] and Hsiao [33]), the Prandtl number, Brownian motion parameter and thermophoresis parameter (see Khan and Pop [6], Nadeem and Saleem [34]), the Schmidt number and scaled chemical reaction parameter. These parameters are defined mathematically as
The nanoparticle volume fraction parameters \(\phi_{1}\) and \(\phi_{2}\) are defined as
Skin friction, heat and mass transfer coefficients
The skin friction coefficient \(C_{f}\), the local Nusselt number \(\mathit{Nu}_{x}\) and the local Sherwood number \(\mathit{Sh}_{x}\) characterize the surface drag, wall heat and mass transfer rates, respectively.
The shearing stress at the surface of the wall \(\tau_{w}\) is defined as
where \(\mu_{nf}\) is the coefficient of viscosity.
The skin friction coefficient is obtained as
and using equation (18) in (19) we obtain
The heat transfer rate at the surface flux at the wall is defined as
where \(k_{nf}\) is the thermal conductivity of the nanofluid.
The local Nusselt number (heat transfer coefficient) is defined as
Using equation (21) in equation (22), the dimensionless wall heat transfer rate is obtained as
The mass flux at the wall surface is defined as
and the local Sherwood number (mass transfer coefficient) is obtained as
The dimensionless wall mass transfer rate is obtained as
where \(\mathit{Re}_{x}\) represents the local Reynolds number and is defined as
Cases of special interest
In this section some particular cases of equations (11) to (13) where the equations are reduced to ordinary differential equations are considered.
Case (1): initial steadystate flow. For steady flow when \(\phi= 0\) (regular fluid), we have \(\xi= 0\) corresponding to \(t = 0\), thus \(f(\eta,0)= f(\eta)\), \(\theta(\eta,0) = \theta(\eta)\) and \(\Phi(\eta,0)= \Phi(\eta)\). In this case equations (11) to (13) reduce to
subject to the boundary conditions
where prime denotes differentiation with respect to η. Equation (28) subject to the boundary conditions (31) admits the exact solution (see Liao [31])
where \(\operatorname{erf}(v)\) is the error function defined as
Case (2): final steadystate flow. In this case, we have \(\xi= 1\) (\(t \rightarrow\infty\)), corresponding to \(f(\eta,1)= f(\eta)\), \(\theta(\eta,1) = \theta(\eta)\) and \(\phi(\eta,1)= \phi(\eta)\).
Equations (11) to (13) reduce to the following forms:
subject to the boundary conditions (31). Equations (11) to (13) were solved using the SRM, Motsa [26]. The SRM is an iterative procedure that employs the GaussSeidel type of relaxation approach to linearize and decouple the system of differential equations. Further details of the rules of the SRM can be found in [27, 28]. The linear terms in each equation are evaluated at the current iteration level (denoted by \(r+1\)) and nonlinear terms are assumed to be known from the previous iteration level (denoted by r). The linearized form of (11) to (13) is
where
It must be noted that equations (37)(39) are now linear and, being decoupled, can be solved sequentially to obtain approximate solutions for \(f(\eta,\xi)\), \(\theta(\eta,\xi)\) and \(\phi(\eta,\xi)\). In this study, the Chebyshev spectral collocation method was used to discretize in η and finite differences with central differencing for derivatives was used to discretize in ξ. Starting from initial guesses for f, θ and ϕ, equations (37)(39) were solved iteratively until the approximate solutions converged within a certain prescribed tolerance level. The accuracy of the results was validated against results from literature for some special cases of the governing equations.
Results and discussion
The nonlinear boundary value problem (11) to (13) subject to the boundary conditions (14) and (15) cannot be solved in closed form, so these equations are solved numerically using the spectral relaxation method (SRM) for Cuwater and Agwater nanofluids for \(0\leq\xi\leq1\). The thermophysical properties of the nanofluids used in the numerical simulations are given in Table 1. Extensive calculations have been performed to obtain the velocity, temperature, concentration profiles as well as skin friction, the local Nusselt number and the local Sherwood number for various values of physical parameters such as ϕ, Ha, \(\mathit{Gr}_{t}\), \(\mathit{Gr}_{c}\), Pr, \(N_{b}\), \(N_{T}\), Sc and γ.
To determine the accuracy of our numerical results, the heat and the mass transfer coefficients are compared with the published results of Khan and Pop [6] in Tables 2 and 3. Here, we have varied the \(N_{T}\) with Nb while keeping other physical parameters fixed. Tables 2 and 3 give a comparison of the SRM results with those obtained by Khan and Pop [6] when \(\mathit{Ha} = \mathit{Gr}_{t} = \mathit{Gr}_{c} = \gamma = \phi= 0\), \(\mathit{Pr} = 10\), \(\mathit{Sc} = 10\) and \(\xi=1\) for different values of the Brownian motion and thermophoresis parameters. It is observed that the present results are in good agreement with results in the literature. In Table 4, approximate solutions of the skin friction coefficient, surface heat transfer and surface mass transfer rates at different values of flow parameters are presented and compared with the SQLM solutions. Values of the skin friction coefficient, reduced Nusselt and Sherwood numbers at different values of ξ are presented in Table 4. The table also shows a comparison of the SHAM and SQLM results. As can be seen from the table, the results match perfectly well for the set accuracy level.
The effects of physical parameters on various fluid dynamic quantities are show in Figures 111.
Figures 14 illustrate the effect of the nanoparticle volume fraction ϕ on the velocity, temperature and concentration profiles, respectively, in the case of a Cuwater nanofluid. It is clear that as the nanoparticle volume fraction increases, the nanofluid velocity and the temperature profile increase while the opposite trend is observed for the concentration profile. Increasing the volume fraction of nanoparticles increases the thermal conductivity of the nanofluid, and we observe that thickening of the thermal boundary layer and the velocity in the case of an Agwater nanofluid are relatively less than in the case of a Cuwater. We also note that since the conductivity of silver is higher than that of copper, the temperature distribution in the Agwater nanofluid is higher than that in the Cuwater nanofluid. With increase in the nanoparticle volume fraction, the concentration boundary layer thickness increases for both types of nanofluids considered, and the opposite trend is observed when the concentration profile decreases.
Figure 3 shows that the skin friction coefficient \(f''(0,\xi)\) increases monotonically with increasing ξ. The result is true for both types of nanofluids. The minimum value of the skin friction in the case of the Cuwater nanofluid is achieved at a smaller value of ξ in comparison with the Agwater nanofluid. Furthermore, in this paper it is found that the Agwater nanofluid shows higher drag as compared to the Cuwater nanofluid. The dimensionless wall heat transfer rate and the dimensionless wall mass transfer rate are shown as functions of ξ in Figure 4(a) and (b), respectively. We observe that the wall heat transfer rate \(\theta'(0,\xi)\) decreases, while the opposite trend is observed in the case of the wall mass transfer rate \(\Phi'(0,\xi)\). The Cuwater nanofluid exhibits higher wall heat transfer rate as compared to the Agwater nanofluid, while the opposite trend is observed for the wall mass transfer rate. The presence of nanoparticle tends to increase the wall heat transfer rate and to decrease the wall mass transfer rate with increasing the values of dimensionless variable ξ.
Figures 57 show the influence of the Hartman number on the velocity, temperature, skin friction coefficient \(f''(0,\xi)\), the local Nusselt number \(\theta'(0,\xi)\) and the local Sherwood number \(\Phi'(0,\xi)\). The effect of Hartman number Ha is to decrease the nanofluid velocity and the wall heat transfer coefficient, whereas it increases the skin friction coefficient and the wall mass transfer coefficient. A similar observation was made by Haroun et al. [11]. The momentum boundary layer thickness decreases with increase in the Hartman number. In the case of the Cuwater nanofluid it is relatively higher than that of the Agwater nanofluid for nanofluid velocity. Figure 6 shows the skin friction coefficient as a function of ξ. It is clear that for the Cuwater nanofluid and the Agwater nanofluid, the skin friction coefficient increases when ξ increases. We note that the Agwater nanofluid exhibits higher drag to the flow as compared to the Cuwater nanofluid. Figure 7 shows the wall heat and mass transfer rates for different values of the Hartman number Ha, it is clear that the value of wall heat transfer rate decreases when ξ increases, in the case of the Agwater nanofluid it is less than that in the case of the Cuwater nanofluid. Further, the wall mass transfer rate increases when ξ increases, we observe that in the case of a Cuwater nanofluid it is less than that of an Agwater nanofluid.
Figures 8 to 9 show the effect of the thermophoretic parameter \(N_{T}\) on the concentration profile, wall heat and mass transfer rates, respectively. In the case of a Cuwater nanofluid and an Agwater nanofluid the concentration profile increases and the wall heat and mass transfer rates decrease with an increase in the thermophoretic parameter. It is observed that the concentration profile and the wall heat transfer rate in the case of the Agwater nanofluid are less than those of the Cuwater nanofluid, while the opposite trend is observed in the case of the wall mass transfer rate. We found that the wall heat transfer rate got higher value when \(\xi= 0\), and then the opposite trend is observed when \(\xi= 1\). The mass transfer rate got less value when \(\xi= 0\), while in the case \(\xi= 1\) it got the higher value. The fast flow from the stretching sheet carries with it nanoparticles leading to an increase in the mass volume fraction boundary layer thickness.
Figure 10(a) and (b) shows the effect of the Brownian motion parameter Nb on the wall heat and mass transfer rates. Figure 10(a) shows that the heat transfer rate decreases with increasing Nb. The mass transfer at the wall increases with the increase in Nb. The heat transfer rate for the Cuwater nanofluid is higher than that for the Agwater nanofluid, while the opposite is true for the mass transfer rate (see Figure 10).
Figure 11(a) and (b) shows the impact of the Soret number on the concentration profiles and the mass transfer coefficient, where the concentration profiles grow less while the mass transfer coefficient increases with an increase in the Soret number. Again, Figure 11(a) and (b) shows that as the Soret number increases, the boundary layer thickness for the solute concentration reduces. The mass transfer coefficient is increasing when the Soret number is positive.
Conclusions
We have investigated the heat and mass transfer in unsteady MHD boundary layer flow in nanofluid due to an impulsively stretching surface with chemical reaction and an applied magnetic field. Other parameters of interest in this study included the Brownian motion parameter and thermophoresis parameter. In this paper we considered Cuwater and Agwater nanofluids and assumed that the nanoparticle volume fraction can be actively controlled at the boundary surface. We have solved the model equations using the spectral relaxation method, and to benchmark our solutions we compared our results with some limiting cases from the literature. These results were found to be in a good agreement. From the numerical simulations, some results can be drawn as follows:

(i)
The velocity profiles increase with increase in the nanoparticle volume fraction, while the opposite trend is observed with increase in the value of the Hartman number.

(ii)
The temperature profiles increase with increasing nanoparticle volume fraction values.

(iii)
The skin friction decreases with an increase in the values of the nanoparticle volume fraction, while the opposite trend is observed for increasing values of the Hartman number.

(iv)
The heat transfer coefficient decreases with increase in the values of the nanoparticle volume fraction, the Hartman number, thermophoretic and Brownian motion parameters.

(v)
The mass transfer coefficient increases with an increase in the nanoparticle volume fraction, chemical reaction parameter, Hartman number and Brownian motion parameter, while the opposite trend is observed for increasing values of the thermophoretic parameter.
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Abbreviations
t: time; p: pressure; a: positive constant; \(q_{w}\): wall heat flux; \(q_{m}\): wall mass flux; Pr: Prandtl number; Sc: Schmidt number; Ha: Hartman number; \(N_{T}\): thermophoresis parameter; \(N_{b}\): Brownian motion parameter; \(\mathit{Sh}_{x}\): local Sherwood number; \(\mathit{Re}_{x}\): local Reynolds number; \(\mathit{Nu}_{x}\): local Nusselt number; \(\mathit{Gr}_{t}\): local temperature Grashof number; \(\mathit{Gr}_{c}\): local concentration Grashof number; \(k_{nf}\): thermal conductivity of nanofluid; \(C_{fx}\): skin friction coefficient; \(f(\xi,\eta)\): dimensionless stream function; \(T_{\infty}\): ambient temperature; \(C_{\infty}\): ambient concentration; g: acceleration due to gravity; \(B_{0}\): uniform magnetic field; u, v: velocity components along x, y directions; x: coordinate along the sheet; y: coordinate normal to the sheet; T: local fluid temperature; \(T_{w}\): temperature at the stretching surface; \(D_{B}\): Brownian motion coefficient; \(D_{T}\): thermophoretic diffusion coefficient; C: solutal concentration; \(C_{s}\): concentration susceptibility; \(C_{w}\): concentration at the stretching surface; \(v_{w}\): prescribed suction velocity; K: chemical reaction parameter; \(k_{s}\): solid volume fraction; \(k_{f}\): thermal conductivity of fluid. Greek symbols: \(\rho_{nf}\): nanofluid density; \(\nu_{nf}\): nanofluid kinematic viscosity; \(\mu_{nf}\): coefficient of viscosity; \((\rho cp)_{nf}\): nanofluid heat capacitance; \(\alpha_{nf}\): thermal diffusivity of nanofluid; \(\mu_{nf}\): effective dynamic viscosity nanofluid; \((cp)_{nf}\): specific heat of fluid at constant pressure; \(\tau_{w}\): shearing stress at the surface of the wall; γ: scaled chemical reaction parameter; σ: electrical conductivity; ϕ: fraction of nanoparticles; \(\phi_{1}\), \(\phi_{2}\): nanoparticle volume fraction; \(\psi(x,y)\): dimensionless stream function; \((\rho c_{p})_{f}\): heat capacity of base fluid; \(\rho_{f}\): density of base fluid; \(\mu_{f}\): dynamic viscosity of fluid; ϕ: fraction of nanoparticles; \(\rho_{s}\): density of solid fractions; \(\beta_{C}\): volumetric solutal expansion coefficient; \(\beta_{T}\): volumetric thermal expansion coefficient; \((\rho c_{p})_{s}\): effective heat capacity of nanoparticle. Subscripts: f: fluid; nf: nanofluid; s: solid.
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The authors have declared that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Keywords
 nanofluids
 impulsively stretching surface
 magnetohydrodynamic
 chemical reaction parameter
 spectral relaxation method