Heat and mass transfer of nanofluid through an impulsively vertical stretching surface using the spectral relaxation method
- Nageeb AH Haroun^{1},
- Precious Sibanda^{1},
- Sabyasachi Mondal^{1}Email author,
- Sandile S Motsa^{1} and
- Mohammad M Rashidi^{2}
Received: 29 May 2015
Accepted: 25 August 2015
Published: 15 September 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. 53:2477-2483, 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.
Keywords
1 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 non-homogenous 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 Cheng-Minkowycz 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 fourth-fifth order Runge-Kutta 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 magneto-hydrodynamics (MHD) free convective boundary layer flow of nanofluids are very limited. More recently, Chamkha and Aly [22] studied magneto-hydrodynamics (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 magneto-hydrodynamics (MHD) mixed convective flow of nanofluid over a stretching sheet. Magneto-hydrodynamics (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].
2 Governing equations
2.1 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.
3 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 (2): final steady-state flow. In this case, we have \(\xi= 1\) (\(t \rightarrow\infty\)), corresponding to \(f(\eta,1)= f(\eta)\), \(\theta(\eta,1) = \theta(\eta)\) and \(\phi(\eta,1)= \phi(\eta)\).
4 Results and discussion
Physical properties | Base fluid (water) | Copper (Cu) | Silver (Ag) |
---|---|---|---|
\(C_{p}\) (J/kgK) | 4,179 | 385 | 235 |
ρ (Kg/m^{3}) | 997.1 | 8,933 | 10,500 |
k (W/mK) | 0.613 | 401 | 429 |
α × 10^{7} (m^{2}/s) | 1.47 | 1,163.1 | 1,738.6 |
β × 10^{5} (K^{−1}) | 21 | 1.67 | 1.89 |
Comparison of values of \(\pmb{-\theta'(0,\xi)}\) for various values of \(\pmb{N_{T}}\) and Nb with \(\pmb{\phi=0}\) (regular fluid), \(\pmb{\mathit{Ha} = \mathit{Gr}_{t} = \mathit{Gr}_{c} = \gamma= 0}\) , \(\pmb{\xi= 1}\) , \(\pmb{\mathit{Pr} = 10}\) , \(\pmb{\mathit{Sc} = 10}\)
\(\boldsymbol{N_{T}}\) | \(\boldsymbol{N_{b}= 0.1}\) | \(\boldsymbol{N_{b}= 0.2}\) | \(\boldsymbol{N_{b}= 0.3}\) | |||
---|---|---|---|---|---|---|
[ 6 ] | Present results | [ 6 ] | Present results | [ 6 ] | Present results | |
0.1 | 0.9524 | 0.9519 | 0.5056 | 0.5052 | 0.2522 | 0.2522 |
0.2 | 0.6932 | 0.6930 | 0.3654 | 0.3662 | 0.1816 | 0.1841 |
0.3 | 0.5201 | 0.5219 | 0.2731 | 0.2760 | 0.1355 | 0.1394 |
0.4 | 0.4026 | 0.4040 | 0.2110 | 0.2117 | 0.1046 | 0.1044 |
0.5 | 0.3211 | 0.3185 | 0.1681 | 0.1639 | 0.0833 | 0.0779 |
Comparison of values of \(\pmb{-\phi'(0,\xi)}\) for various values of \(\pmb{N_{T}}\) and Nb with \(\pmb{\phi=0}\) (regular fluid), \(\pmb{\mathit{Ha} = \mathit{Gr}_{t} = \mathit{Gr}_{c} = \gamma= 0}\) , \(\pmb{\xi= 1}\) , \(\pmb{\mathit{Pr} = 10}\) , \(\pmb{\mathit{Sc} = 10}\)
\(\boldsymbol{N_{T}}\) | \(\boldsymbol{N_{b}= 0.1}\) | \(\boldsymbol{N_{b}= 0.2}\) | \(\boldsymbol{N_{b}= 0.3}\) | |||
---|---|---|---|---|---|---|
[ 6 ] | Present results | [ 6 ] | Present results | [ 6 ] | Present results | |
0.1 | 2.1294 | 2.1294 | 2.3819 | 2.3817 | 2.4100 | 2.4097 |
0.2 | 2.2740 | 2.2745 | 2.5152 | 2.5145 | 2.5150 | 2.5134 |
0.3 | 2.5286 | 2.5242 | 2.6555 | 2.6513 | 2.6088 | 2.6047 |
0.4 | 2.7952 | 2.7883 | 2.7818 | 2.7787 | 2.6876 | 2.6862 |
0.5 | 3.0351 | 3.0413 | 2.8883 | 2.8944 | 2.7519 | 2.7574 |
Comparison of the SRM solutions for \(\pmb{f''(\xi,0)}\) , \(\pmb{-\theta '(\xi,0)}\) , and \(\pmb{-\phi'(\xi,0)}\) against those of the SQLM at different values of ξ , \(\pmb{N_{T} = 0.1}\) , \(\pmb{\mathit{N_{b}} = 0.1}\) , \(\pmb{\mathit{Pr} = 7}\) , \(\pmb{\mathit{Gr}_{t} = 0.1}\) , \(\pmb{\mathit{Gr}_{c} = 0.1}\) , \(\pmb{\mathit{Sc} = 1}\) , \(\pmb{\phi=0.2}\) , \(\pmb{\gamma=2}\) , \(\pmb{\mathit{Ha}= 3}\)
ξ | \(\boldsymbol{f''(\xi,0)}\) | \(\boldsymbol{-\theta'(\xi ,0)}\) | \(\boldsymbol{-\phi'(\xi,0)}\) | |||
---|---|---|---|---|---|---|
SRM | SQLM | SRM | SQLM | SRM | SQLM | |
0.1 | −1.024404 | −1.024404 | 0.861024 | 0.861024 | 0.372386 | 0.372386 |
0.2 | −1.062742 | −1.062742 | 0.864900 | 0.864900 | 0.389380 | 0.389380 |
0.3 | −1.088333 | −1.088333 | 0.872739 | 0.872739 | 0.386305 | 0.386305 |
0.4 | −1.108200 | −1.108200 | 0.882753 | 0.882753 | 0.372517 | 0.372517 |
0.5 | −1.124261 | −1.124261 | 0.894651 | 0.894651 | 0.350129 | 0.350129 |
0.6 | −1.136890 | −1.136890 | 0.908632 | 0.908632 | 0.318838 | 0.318838 |
0.7 | −1.145390 | −1.145390 | 0.925350 | 0.925350 | 0.276272 | 0.276272 |
0.8 | −1.146964 | −1.146964 | 0.946411 | 0.946411 | 0.216141 | 0.216141 |
0.9 | −1.127531 | −1.127531 | 0.977047 | 0.977047 | 0.118365 | 0.118365 |
1.0 | −4.252384 | −4.252384 | 1.495226 | 1.495226 | 0.463421 | 0.463421 |
Figures 1-4 illustrate the effect of the nanoparticle volume fraction ϕ on the velocity, temperature and concentration profiles, respectively, in the case of a Cu-water 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 Ag-water nanofluid are relatively less than in the case of a Cu-water. We also note that since the conductivity of silver is higher than that of copper, the temperature distribution in the Ag-water nanofluid is higher than that in the Cu-water 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.
5 Conclusions
- (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.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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