Cooling and heating of liquids are significant in many engineering applications and transportation industries. In general, many common heat transfer fluids such as ethylene glycol, oil and water are not efficient heat transfer fluids for their low thermal conductivities and poor physical and chemical characteristics . In contrast, fluids with added nanometer sized metals such as titanium, aluminum, gold, silver, copper and iron or their oxides and composite materials have significantly higher thermal conductivities than these base fluids. The nanoparticles are commonly made by a high-energy-pulsed process from a conductive material. For these reasons, it is prudent to combine base fluids with nanoparticles to achieve both the characteristics of a base fluid and the physical properties of the nanoparticles [2,3]. Suspending nanoparticles affects the base fluid’s homogeneity and the randomness of molecular motion leading to higher conductive rates and better convective heat transfer performances compared to base fluids [4–7]. There are however mechanisms, which include particle agglomeration, particle shape/surface area, nanoparticle size, temperature and liquid layering on the nanoparticle liquid interface that still need to be fully understood.
Nanofluids have large application in industries, like micro-electro-mechanical or microprocessors electronic systems as well as in the field of biotechnology . Several authors, for instance, [9,10] conducted theoretical and experimental investigations to demonstrate that nanofluids increased the heat transfer properties of base fluid. The numerical solution of the equations that describe the combined effects of thermophoresis and Brownian motion on the nanofluid flow through a flat surface in a saturated porous medium was examined by Nield and Kuznetsov . Kuznetsov and Nield  examined a problem on natural convection boundary layer nanofluid flow with showing the effects of thermophoresis and Brownian motion. Buongiorno  studied that particle diffusion and thermophoresis play an important role in the flow of a nanofluid. The classical concept of a boundary layer corresponds to a thin region next to a solid surface where viscous forces are important, Blasius . Viscous forces play an essential role in processes such as glass fiber drawing, crystal growing and plastic extrusion. For the case of the Blasius problem, Sakiadis  focused on the concept of a boundary layer flow induced by a moving plate in a quiescent ambient fluid. Further studies in this area were made by Schlichting and Gersten , Bejan , White  and Mukhopadhyay [19,20].
The magnetohydrodynamic (MHD) boundary layer flow of a viscous incompressible fluid which is electrically conducted has huge applications in industry and technological sectors, like cooling of nuclear reactors, the extrusion of plastics, textile industry, purification of crude oil, polymer technology, geothermal energy extractions, metallurgy, drug delivery and biological transportation [21, 22]. The application of the magnetic field produces a Lorentz force that increase the mixing processes as an active micro mixing mechanism. The transport system of conductive biological fluids in the micro systems will be benefitted from research . The simultaneous occurrence of buoyancy and magnetic field forces on fluid flow were investigated by many researchers [24–28]. In these investigations, many authors have used a no-slip boundary condition. Recently, some researchers have investigated boundary layer flow assuming a slip condition at the boundary [29–35].
Stagnation point flow, illustrating the fluid motion near a stagnation region, exists near solid bodies in that fluid flow. Chiam  studied fluid flow on a steady stagnation-point through an elastic surface with equal free stream and stretching velocities.
The mail focus of this study is to expand the work of Nandy and Mahapatra  to unsteady flow over a permeable stretching flat plate with an internal heat source or sink. For the nanoparticles we have used the boundary condition suggested recently by Kuznetsov and Nield  where active control of the nanoparticle volume fraction at the surface is not possible. Appropriate similarity transformations reduce the conservation equations to a set of nonlinear ordinary differential equations. These equations are solved by using a spectral relaxation method, numerically (see Motsa et al. ).
2 Problem formulations
Consider an unsteady, viscous, incompressible, laminar nanofluid flow past a stretching flat plate. The flow is restricted to the area y ≥ 0, y and x are the coordinate system normal to the stretching flat plate and along the stretching surface, respectively. The concentration and temperature on the surface are ϕw and Tw, respectively, and away from the surface the values are ϕ∞ and T∞, respectively. We assume that at a time t ≥ 0, the plate stretching velocity is uw = bx. The free stream velocity is u∞ = ax. Here, b is constant with b > 0 corresponding to a stretching plate and b < 0 to a shrinking plate. The fluid flow is subjected to a uniform magnetic field of strength B0 which is applied in a normal direction to the plane y = 0 and to the heat source/sink. The fluid is electrically conducting with constant properties, except for the density in the buoyancy term in the momentum equation. With these assumptions, the Oberbeck-Boussinesq and the boundary layer approximations, the unsteady momentum, energy and concentration equations can be described as (see Nandy and Mahapatra ); (1) (2) (3) (4) where t, u and v represent the time, the fluid velocity components along the x and y axis, respectively, νf is the fluid kinematic viscosity, σ is the fluid electrical conductivity, ρf∞ is the density far from the plate surface, g is the acceleration due to gravity, ρf is the fluid density, T is the fluid temperature, β is the coefficient of thermal expansion, ρp is the density of the particles, (ρ cp)f is fluid heat capacity, ϕ is nanoparticle volume fraction, αf is thermal diffusivity, q‴ is the rate of internal heat sink (< 0) or heat source (>0) coefficient, The ratio of the effective heat capacity of nanoparticle material with the heat capacity of the fluid is τ, DT denotes the thermophoresis diffusion coefficient and DB is the Brownian diffusion coefficient.
The boundary conditions are (see Kuznetsov and Nield , Nandy and Mahapatra  and Kuznetsov and Nield ) (5) with the initial conditions where L is a proportionality constant. Vw denotes the surface mass flux. Negative values of Vw imply fluid suction and positive values of Vw imply injection. Also, Vw = 0 corresponds to an impermeable plate.
q‴ is defined according as follows, (see [40–42]) (6) where B* and A* are temperature-dependent internal heat source/sink and coefficients of space-dependent, respectively and η is a dimensionless variable defined in Eq. (7). The first term in the Eq. (6) represents the dependence of the internal heat sink or source on the space coordinates while the second term represents its dependence on the temperature. Now, for the case when the case A* < 0 and B* < 0, indicates to internal heat sink while both A* > 0 and B* > 0, indicates to internal heat source.
We introduce non-dimensional variables given by (7) where θ(η,ξ) and Φ(η,ξ) are the non-dimensional temperature and concentration, respectively and the stream function ψ is defined as
Substituting (7) into Eqs. (1) - (4) gives (8) (9) (10) with boundary conditions (11) where the differentiation with respect to η is denoted as prime, S is the dimensionless suction/blowing parameter, γ is the dimensionless slip factor, and the stretching parameter is ε where
In Eqs. (8)-(11) the parameters are the magnetic parameter M, the buoyancy parameter λ, Grx is the local Grashof number, Pr is the Prandtl number, Rex is the local Reynolds number, Nt is the thermophoresis parameter, Nr is the buoyancy ratio parameter, Nb is the Brownian motion parameter, and Le is the Lewis number. These parameters are defined as: (12)
We note that in the case of assisting flow λ > 0, and for opposing flow λ < 0, λ = 0 corresponds to free convection. The skin friction coefficient Cf and the local Nusselt number Nux are defined as (13) where τw is the shear stress along the stretching surface, qw is the wall heat flux, respectively defined as (14)
Hence using Eq.(14) we get (15)
3 Method of solution
The spectral relaxation method (see ) was used to solve the system of non-similar equations (8)–(10) with the boundary conditions (11). In the SRM framework, we obtain the iterative scheme (16) (17) (18) (19) with boundary conditions (20) where (21) (22) (23)
In Equations (16)–(23) the indices r and r + 1 denote the previous and current iteration levels, respectively. Starting from initial approximations denoted by f0, u0, θ0, and Φ0, equations (9)–(11) are solved iteratively for fr + 1(η,ξ), ur + 1(η,ξ), θr + 1(η,ξ), and Φr + 1(η,ξ) (r = 0,1,2,…). Equations (8)–(10) were discretized using the Chebyshev spectral collocation method in the η direction while the discretization in the ξ direction uses the implicit finite difference method. This leads to a system of linear equations of the form (24) (25) (26) (27) where (28) (29) (30) (31)
Here I is an (N + 1) × (N + 1) identity matrix, and [ . ]d are diagonal matrices of order (N + 1) × (N + 1).
In applying the SRM a computational domain of extent L = 20 was chosen in the η-direction. Through numerical experimentation, this value was found to give accurate results for all the selected physical parameters. Increasing the value of η do not change the results to a significant extent. The number of collocation points is used in the spectral method discretization is Nx = 100 in all cases. The calculations are carried until some desired, tolerance level ε is attained. The tolerance level was defined to be the maximum infinity norm of the difference between the values of the calculated quantities, that is
To ensure the accuracy of the results, a sufficiently small step size Δξ was used. The step size was chosen to be small enough such that further reduction in step size did not change the results of the flow properties.
4 Results and discussion
The transformed system of coupled nonlinear ordinary differential equations (8)-(10) including boundary conditions (11) are solved numerically using the spectral relaxation method. The numerical results are presented here to show the velocity, concentration and temperature profiles, the rate of heat transfer and the skin friction coefficient for different physical parameter values. These results are presented both graphically and in tabular form. In the numerical simulations the default parameter values used are, unless otherwise specified; Pr = 6.8, ε = 0.5, λ = 0.5, M = 0.1, Nr = 0.5, A* = 2.0, Nb = 0.5, B* = 1.0, Nt = 0.5, S = 0.1, Le = 10, γ = 0.1 and ξ = 0.5.
To verify the accuracy of our numerical scheme, a comparison of the computed skin friction coefficient is made with earlier results of Anwar et al.  in Table 1. Here we observe an excellent agreement validating the accuracy of the current numerical results. The residual error in the numerical simulations against the number of iterations for different values of γ, ε, S and M is shown in Figure 1. These results again confirm that the numerical method used in this study converges.
Figure 2(a) displays the variation of f′ with respect to η for several values of γ when ε < 1, ε = 1 and ε > 1. Here we note that f′ increases with γ when ε > 1 while the opposite is true when ε < 1. An increase in the slip parameter has the effect of reducing the velocity at the wall. The velocity decreases asymptotically to zero at the edge of the boundary layer. The boundary layer thickness decreases as γ increases. We note that when ε = 1, increasing γ gives no further changes in the velocity profiles because at this stage the external stream velocity becomes equal to the stretching velocity. This causes a frictionless Hiemenz flow .
Figure 2(b) explores the effect of λ and Nr on the velocity profiles. It is seen that the velocity increases as the buoyancy force parameter increases. This is due to the fact that the assisting flow (λ > 0) induces a favorable pressure gradient which enhances the fluid flow in the boundary layer and as a result the momentum boundary layer increases whereas the reverse occurs for the opposing flow when λ < 0.
Figure 3 shows the effects of S and M on f′ with respect to η. Here we note that f′(η, ξ) decreases with increasing values of S and M. This behaviour is due to the fact that M increases the resistive forces on the flat plate which in turn reduces the fluid velocity and hence the motion of the fluid is slowed down. It is known that the wall suction (S > 0) has the tendency to decrease the momentum boundary layer thickness which is the cause of reduction in the velocity. But the the opposite behaviour can be seen for fluid injection (S < 0).
The internal heat source or sink in the boundary layer has an influence on the temperature fields as illustrated in Figure 4(a). It is clear that increasing A* and B* increases the temperature distribution within the fluid and the thermal boundary layer thickness increases. Figure 4(b) shows the variation of the temperature profiles for various values of the slip and stretching parameters. The temperature profiles increase with increasing γ when ε < 1 but the opposite trend is observed when ε > 1 as the thermal boundary layer thickness decreases.
Figure 5(a) depicts the effect of λ and Nr on the temperature profiles. We note that with the increasing value of buoyancy force parameter decreases the nanofluid temperature. Moreover, positive λ values induce a favorable pressure gradient that enhances the nanofluid temperature in the boundary layer when Nr increases. Consequently, the thermal boundary layer increases while the opposite is observed for negative λ values. Figure 5(b) demonstrates the influence of Le on the temperature distribution within the boundary layer in the presence of magnetic influence, buoyancy force, heat sink/source, thermophoresis/Brownian motion. It is observed that the temperature profiles including the thickness of the temperature boundary layer increases with increases in the values of Le.
Figure 6(a) suggests that, in so far as the boundary layer temperature is concerned, thermophoresis plays the same role as the fluid Brownian motion. Thus, the fluid temperature increases with both Nt and Nb in the boundary layer region. The physical reason for this behaviour is that increased random motion of the nanoparticles increases the fluid temperature which enhances the thickness of the thermal boundary layer profiles. Figure 6(b) depicts the response of the temperature profiles to changing suction and magnetic field parameter values. Moreover, as would be expected, we note that the fluid temperature decreases with increase in suction but is higher in the case of injection.
The traditional model is revised such that the nanofluid particle volume fraction on the boundary is passively controlled. The effect of this change in boundary conditions can be seen in Figures 7 to 9 for different parameters. The impact of heat sink/source parameters B* and A* on the concentration profiles is presented in Figure 7(a). It can be seen that, away from the boundary, the concentration profiles increase with an increase in the values of A* and B*. The effect of γ and ε on nanoparticle concentration is shown in Figure 7(b). An increase in the value of partial slip parameter value leads to an increase in the nanoparticle concentration when ε < 1 while the opposite trend is observed for ε > 1. In addition, the concentration increases and attains its highest value in the vicinity of the stretching plate near η = 0.9 and then decreases to the zero. Figure 8(a) displays the effect of λ and Nr on the nanoparticle concentration profile distributions. The nanoparticle concentration decreases asymptotically with increasing values of λ. Furthermore, in the case of opposing flow, the nanoparticle concentration decreases with increasing Nr. Figure 8(b) shows the nanoparticle concentration profiles for several values of Le. It is seen that the nanoparticle volume fraction decreases with increases in Le and this manifested through the reduction in the thickness of the concentration boundary layer. Additionally, the nanoparticle concentration profiles decrease asymptotically to zero at the edge of the boundary layer.
The nanoparticle concentration profiles are presented in Figure 9(a) for various values of the thermophoresis parameter. The results show that the nanoparticle concentration profiles increase with increasing Nt away the boundary layer region. It is interesting to note that the distinctive peaks in the profiles occur in regions adjacent to the surface for higher values of thermophoresis parameter. This means that the nanoparticle concentration profile takes higher value near the plate. Figure 9(b) shows that the nanoparticle concentration profiles increase with increasing M up to a certain value of η, beyond which the opposite trend is observed. It is clear that as S increases the concentration profiles increase to the highest value in the vicinity of the plate and then decreases to zero in the quiescent fluid. As a result, the concentration boundary layer thickness increases close to the plate surface and decreases far from the surface with increasing S. Figure 9(c) shows that the concentration profile for various values of Nb. It is seen that the concentration profiles decrease with increasing Nb. The concentration profiles attain their maximum value near the η = 1.0.
Figure 10(a) shows the variation of the skin friction coefficient with ξ in response to changes in A* and B*. The skin friction coefficient decreases with increases in both A* and B* when ξ increases. This suggests that A* and B* can be useful parameters for reducing the drag coefficient. Variations of the skin friction coefficient as a function of ξ for different values of the slip γ and stretching ε parameters are shown in Figure 10(b). We note that when ε < 1 the skin friction coefficient increases with increasing ξ, and decreases with increasing γ. The highest surface shear stress occurs with the no-slip velocity condition.
The variation of skin friction coefficient with λ and Nr is shown in Figure 11(a). We observe that the skin friction coefficient decreases with increasing λ. In the case of opposing flow, the skin friction coefficient increases with ξ and decreases with an increase in Nr. Figure 11(b) shows that the skin friction coefficient decreases with increasing Lewis numbers but increases with increasing ξ.
Figure 12(a) shows the variation of the skin friction coefficient with different values of Nt and Nb. The skin friction coefficient decreases as Nt and Nb increase but increases with increasing ξ. Figure 12(b) shows the skin friction coefficient with respect to ξ for different values of M and S. The skin friction coefficient increases with increasing M and S. Figures 13 - 14 show the effect of Pr on temperature and concentration profiles with same values of other parameters where dashed line indicates Nb = 0.1 and solid line for Nb = 0.5. The nature of those figures are similar to the Figures 4(a) and 9(b).
In this work, the unsteady stagnation boundary layer flow of a magnetohydrodynamic nanofluid over a stretching flat plate with velocity slip was investigated. The equations that model the boundary layer equations were solved numerically using the spectral relaxation method. A parametric study was performed to explore the impact of various physical parameters on the flow, and heat and mass transfer characteristics. The internal heat source or sink is shown to enhance the nanofluid motion while reducing the skin friction coefficient and the rate of heat transfer at the stretching surface.
In this study we have shown that increasing the Brownian motion parameter and the Lewis number reduces nanoparticle concentration profiles near the boundary layer region due to increasing mass transfer. The velocity, temperature and concentration profiles and the heat and mass transfer rate on the stretching plate are strongly influenced by the slip parameter. Heat transfer rates decrease with increasing velocity slip parameter values.
The authors are grateful to the Claude Leon Foundation and the University of KwaZulu-Natal, South Africa for the financial support.
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About the article
Published Online: 2017-05-20
Citation Information: Open Physics, Volume 15, Issue 1, Pages 323–334, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2017-0036.
© 2017 Sami M. Ahamed et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0