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- Open Access
On unsteady MHD mixed convection in a nanofluid due to a stretching/shrinking surface with suction/injection using the spectral relaxation method
- Nageeb A Haroun^{1},
- Precious Sibanda^{1}Email author,
- Sabyasachi Mondal^{1} and
- Sandile S Motsa^{1}
https://doi.org/10.1186/s13661-015-0289-5
© Haroun et al.; licensee Springer. 2015
- Received: 24 September 2014
- Accepted: 15 January 2015
- Published: 31 January 2015
Abstract
In this study we investigate heat and mass transfer in magnetohydrodynamic mixed convection flow of a nanofluid over an unsteady stretching/shrinking sheet. The flow is subject to a heat source, viscous dissipation and Soret and Dufour effects 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 we have solved numerically using the recent spectral relaxation method. In addition to the discussion on physical heat and mass transfer processes, we also show that the spectral relaxation technique is an accurate technique for solving nonlinear boundary value problems.
Keywords
- nanofluids
- dimensionless suction/injection
- magnetohydrodynamic
- Soret and Dufour effects
- heat generation parameter
1 Introduction
Nanofluids are suspensions of metallic, non-metallic or polymeric nano-sized powders in a base liquid which are used to increase the heat transfer rate in various applications. 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 a large amount of literature on convective heat transport in nanofluids and problems linked to a stretching surface. An excellent collection of articles on this topic can be found in [1–4]. The majority of the previous studies have been restricted to boundary layer flow and heat transfer in nanofluids. Following the early work by Crane [5], Khan and Pop [6] were among the first researchers to study nanofluid flow due to a stretching sheet. Other researchers studied various aspects of flow and heat transfer in a fluid of infinite extent; see, for instance, Chen [7] and Abo-Eldahab and Abd El-Aziz [8]. A mathematical analysis of momentum and heat transfer characteristics of the boundary layer flow of an incompressible and electrically conducting viscoelastic fluid over a linear stretching sheet was carried out by Abd El-Aziz [9]. In addition, radiation effects on viscous flow of a nanofluid and heat transfer over a nonlinearly stretching sheet were studied by Hady et al. [10]. Theoretical studies include, for example, modeling unsteady boundary layer flow of a nanofluid over a permeable stretching/shrinking sheet by Bachok et al. [11]. Rohni et al. [12] developed a numerical solution for the unsteady flow over a continuously shrinking surface with wall mass suction using the nanofluid model proposed by Buongiorno [13].
The effect of an applied magnetic field on nanofluids has substantial applications in chemistry, physics and engineering. These include cooling of continuous filaments, in the process of drawing, annealing and thinning of copper wire. Drawing such strips through an electrically conducting fluid subject to a magnetic field can control the rate of cooling and stretching, thereby furthering the desired characteristics of the final product. Such an application of a linearly stretching sheet of incompressible viscous flow of MHD was discussed by Pavlov [14]. In other work, Jafar et al. [15] studied the effects of magnetohydrodynamic (MHD) flow and heat transfer due to a stretching/shrinking sheet with an external magnetic field, viscous dissipation and Joule effects.
A model for magnetohydrodynamic flow over a uniformly stretched vertical permeable surface subject to a chemical reaction was suggested by Chamkha [16]. An analysis of the effects of a chemical reaction on heat and mass transfer on a magnetohydrodynamic boundary layer flow over a wedge with ohmic heating and viscous dissipation in a porous medium was done by Kandasamy and Palanimani [17]. Rashidi and Erfani [18] studied the steady MHD convective and slip flow due to a rotating disk with viscous dissipation and ohmic heating. Rashidi et al. [19] found approximate analytic solutions for an MHD boundary-layer viscoelastic fluid flow over a continuously moving stretching surface using the homotopy analysis method. Rashidi and Keimanesh [20] used the differential transform method and Padé approximants to solve the equations that model MHD flow in a laminar liquid film from a horizontal stretching surface. The effect of a transverse magnetic field on the flow and heat transfer over a stretching surface were examined by Anjali-Devi and Thiyagarajan [21]. The influence of a chemical reaction on heat and mass transfer due to natural convection from vertical surfaces in porous media subject to Soret and Dufour effects was also studied by Postelnicu [22].
Despite all this 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 the nanoparticles may have an impact on the performance of the nanofluid as a heat transfer medium. In this paper we investigate unsteady MHD mixed convection boundary layer with suction/injection subject to a number of source terms including Dufour and Soret effects, heat generation, an applied magnetic field and viscous dissipation. Various numerical and or semi-numerical methods can and have been used to solve the equations that model this type of boundary layer flow. These equations are non-similar and coupled. In this paper we use the spectral relaxation method (SRM) that was recently proposed by Motsa [23]. This spectral relaxation method promises fast convergence with good accuracy, has been successfully used in a limited number of boundary layer flow and heat transfer problems (see [24, 25]). In this paper we discuss the fluid flow and heat transfer as well as highlight the strengths of the solution method.
2 Governing equations
3 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.
4 Cases of special interest
In this section we highlight two particular cases where equations (2.12) to (2.14) reduce to ordinary differential equations.
4.1 Initial steady flow
4.2 Final steady state flow
5 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 the SRM result with Suali et al. [ 32 ] for the skin friction coefficient for different stretching/shrinking sheet rates
λ | Suali et al. [ 32 ] | SRM result |
---|---|---|
\(\boldsymbol{f''(0,1)}\) | \(\boldsymbol{f''(0,1)}\) | |
4 | −7.086378 | −7.086378 |
3 | −4.276545 | −4.276542 |
0.2 | 1.051130 | 1.051130 |
0.1 | 1.146561 | 1.146561 |
−0.2 | 1.373886 | 1.373886 |
−0.5 | 1.495672 | 1.495670 |
Table 4 gives the skin friction coefficient for selected stretching λ parameter values. Here we note that as the stretching rate decreases, the skin friction coefficient increases. These results are in good agreement with those obtained by Suali et al. [32].
The axial velocity in the case of an Ag-water nanofluid is comparatively higher than that in the case of a Cu-water nanofluid. The temperature distribution in an Ag-water nanofluid is higher than that in a Cu-water nanofluid and this is explained by the observation that the thermal conductivity of silver is higher than that of copper. The concentration boundary layer thickness is higher for the case of a Cu-water than that for the case of an Ag-water nanofluid.
Figure 3 shows that the skin friction coefficient decreases monotonically with increasing ξ. The result is true for both types of fluids. The maximum value of the skin friction in the case of a Cu-water nanofluid is achieved at a smaller value of ξ in comparison with an Ag-water nanofluid. Furthermore, in this paper it is found that the Ag-water nanofluid shows less drag as compared to the Cu-water 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 decreases while the opposite is true in case of the wall mass transfer rate. The Cu-water nanofluid exhibits higher wall heat transfer rate as compared to the Ag-water nanofluid, while the Cu-water nanofluid exhibits less than the Ag-water nanofluid. The presence of nanoparticle tends to increase the wall heat transfer rate and to reduce the wall mass transfer rates with increasing the values of ξ.
Figure 6 shows the skin friction coefficient as a function of ξ. It is clear that for Ag-water and Cu-water nanofluids, the skin friction reduces when ξ increases. We note that the Cu-water nanofluid exhibits higher drag to the flow as compared to the Ag-water nanofluid. Figure 7 shows the wall heat and mass transfer rates for a different Hartmann number Ha, it is clear that the value of wall heat transfer rate increases as ξ increases, in the case of an Ag-water nanofluid it is less than in the case of a Cu-water nanofluid. Further, the wall mass transfer rate increases up to the value of ξ before reducing.
The axial distributions of the wall heat and mass transfer rates are shown in Figure 11(a) and (b), respectively. The wall heat transfer rate increased with ξ, and we observe that the heat transfer rate is higher for a Cu-water nanofluid than for an Ag-water nanofluid. It is interesting to note that with suction (\(f_{w}=-1\)), the heat transfer rate is less for an Ag-water nanofluid than for a Cu-water nanofluid up to a certain value of ξ. Beyond this point, the heat transfer rate is higher for an Ag-water nanofluid as compared to a Cu-water nanofluid, while the wall mass transfer rate increases monotonically with ξ to a maximum values before reducing. It is shown that the mass transfer rate is higher for a Cu-water nanofluid than for an Ag-water nanofluid. The opposite behavior is observed in the case of suction when \(f_{w} = -1\). The mass transfer rate for an Ag-water nanofluid is higher than that for a Cu-water nanofluid up to a certain value of ξ, and beyond this critical value, the mass transfer in an Ag-water nanofluid is less than that in a Cu-water nanofluid, Figure 11(b).
Figure 13 shows that the thermal and concentration boundary layer thicknesses decrease as the stretching rate increases. For the shrinking case, when \(\lambda= -2\), the momentum boundary layer for an Ag-water nanofluid is greater than that for a Cu-water nanofluid, while the opposite is observed for the stretching case when \(\lambda = 2\). The Ag-water nanofluid thermal boundary is higher than that of a Cu-water nanofluid (see Figure 13(a)). The solutal concentration increases up to a critical η, and beyond this critical value the concentration profile decreases (see Figure 13(b)). We observe that solute concentration profiles are larger for the case of a Cu-water than those for the case of an Ag-water nanofluid for the shrinking sheet with \(\lambda= -2\), while the opposite is true for the stretching sheet with \(\lambda= 2\).
Figure 14 shows the effect of stretching/shrinking on the shear stress, while Figure 15 shows the effect of the stretching rate on the wall heat and mass transfer rates. From Figure 14 we note that the shear stress increases with the stretching/shrinking parameter. The shear stress decreases with ξ. Figure 15(a) shows that the heat transfer rate increases with increasing λ. The mass transfer at the wall decreases with the increase in λ. The heat transfer rate is larger for the case of an Ag-water nanofluid compared to that of a Cu-water nanofluid, while the opposite is true for the mass transfer rate (see Figure 15).
6 Conclusions
We have investigated heat and mass transfer in unsteady MHD mixed convection in a nanofluid due to a stretching/shrinking sheet with heat generation and viscous dissipation. Other parameters of interest in this study included the Soret and Dufour effects. In this paper we considered Cu-water and Ag-water 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 good agreement.
The numerical simulations show, inter alia, that the skin friction factor increases with both an increase in the nanoparticle volume fraction and the stretching rate and that an increase in the nanoparticle volume fraction leads to a reduction in the wall mass transfer rate.
Declarations
Acknowledgements
The authors wish to thank the University of KwaZulu-Natal for financial support.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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