There have been only a few studies on Maxwell nanofluid flow for a shrinking sheet in the recent past. However, in general, research on non-Newtonian fluid flow has gained sizeable attention because of the multiplicity of its applications in the biomedical and chemical industries . A non-Newtonian fluid is a fluid whose viscosity varies with the applied stress. The relation between the strain rate and the shear stress is nonlinear, and can be time-dependent . The constitutive equations tend to be highly nonlinear and intricate in comparison with those of a Newtonian fluid. A Maxwell fluid is a viscoelastic material having the properties of elasticity and viscosity . Unlike the Newtonian model, the upper convected Maxwell (UCM) model incorporates relaxation time.
The UCM model has been studied by many researchers, for example, Choi et al.  gave an analysis of incompressible steady two-dimensional UCM fluid flow in a porous channel. Their study included a consideration of inertia and fluid elasticity. Nandy  focused on the unsteady boundary layer flow of a Maxwell nanofluid over a permeable shrinking sheet with a Navier slip condition at the surface. The flow equations were solved using the shooting method. The homotopy analysis method was used by Rashidi et al.  to find solutions to the conservation equations for heat and mass transfer in a two-dimensional steady magnetohydrodynamic fluid flow in a porous medium . Awais  investigated heat absorption and generation in steady flow over a surface stretched linearly in its own plane using the UCM fluid model. A nanofluid is defined as a fluid with suspended solid nanoparticles that are less than 100nm in size, and solid volume fraction less than 4%, . Even at low nanoparticle volume concentration, nanofluids have been shown to have improved conductivity and thermal the performance compared to base fluids such as water and oil, .
The unsteady Maxwell fluid flow over a stretching surface subject to constructive/destructive chemical reaction was studied by Mukhopadhyay and Bhattacharyya . They showed that for a constructive chemical reaction the concentration field increased but decreased for a destructive chemical reaction . Also, in line with physical expectations, the concentration boundary-layer decreased for a destructive chemical reaction. Nandy et al.  studied forced convection in an unsteady nanofluid flow past a permeable shrinking sheet subject to heat loss due to thermal radiation. They explored the simultaneous impact of a magnetic field, thermal radiation, and unsteadiness on the heat transfer and flow properties of the fluid. Das et al.  presented simulated results for heat and mass transfer in an electrically conducting incompressible nanofluid flow near a heated stretching sheet with a convective boundary condition. The impact of an inclined magnetic field in the flow of a fluid with variable thermal conductivity was studied by Hayat et al. [13, 14]. An analysis of the significance of a heat source/ sink and temperature dependent thermal conductivity was given. Qasim and Hayat  investigated the impact of heat loss through thermal radiation in unsteady magnetohydrodynamic flow of a micropolar fluid. The influence of Joule heating and thermophoresis in a Maxwell fluid was studied in .
Som et al.  studied Ohmic dissipation and thermal radiation in flow over a stretching sheet embedded in a porous media. The results indicated that fluid injection causes a reduction in heat transfer whereas fluid suction raises the heat transfer coefficient. Hsiao  studied conjugate heat transfer with Ohmic dissipation in an incompressible Maxwell fluid close to a stagnation point. Mahapatra et al.  gave an analysis of stagnation point fluid flow over a stretching surface.
The purpose of this study is to investigate unsteady two-dimensional boundary-layer flow, heat and mass transfer in a Maxwell nanofluid flow over shrinking sheet with both slip and convective boundary conditions. The objective is to extend the study by Nandy  to include an evaluation of the impact and significance of thermophoresis and Brownian motion when the nanofluid particle volume fraction at the boundary is not actively controlled. The conservation equations are solved numerically using the spectral local linearization method, see Motsa [20, 21]. To show the accuracy of the numerical scheme, analysis of the residual errors is given for different physical parameter values. The effect of several physical parameters on the fluid properties are shown in tabular and graphical form. Comparison with published work for special cases shows an excellent agreement.
2 Problem formulations
An unsteady laminar boundary layer flow of an incompressible viscous Maxwell nanofluid over a shrinking surface in two dimensions is considered here. The shrinking sheet velocity is uw(x,t) while the mass transfer velocity is vw(x,t), t denotes time and x is measured along the sheet. Using the nanofluid model proposed by Buongiorno , the conservation equations for mass, momentum, thermal energy and nanoparticles for a Maxwell fluid are: (1) (2) (3) (4)
where u is the velocity component in the x- direction and v is the velocity component in the y-direction. The kinematic viscosity is ν, the relaxation time of the UCM fluid is k0, the thermal diffusivity is α, the variable thermal conductivity is κ(T) and the chemical reaction parameter is k1. The Brownian diffusion coefficient is DB, while the thermophoresis diffusion coefficients is DT. Here, τ is the effective heat capacity of the nanoparticle material divided by the heat capacity of the ordinary fluid, T is the fluid temperature and C is the nanoparticle volume fraction. The temperature and the nanoparticle concentration at the wall are Tw and Cw, respectively, and T∞ and C∞ denote the ambient temperature and concentration respectively. The velocity slip is proportional to the local shear stress. These equations are subject to the boundary conditions: (5)
where k0 = k(1−λ t), uw(x,t) = −ax(1−λ t)−1, uslip(x,t) = ν N1∂ u/∂ y, is the slip velocity, λ is the unsteadiness parameter and k0(> 0), a(> 0) are positive constants. This assumption is to allow for the possibility of a similarity solution. We introduce the similarity variables: (6)
where the prime denotes derivatives with respect to η. The boundary conditions are (10)
where the parameter is the non-dimensional velocity slip and vw(0,t) is the wall mass transfer velocity given by
where v0 is the constant mass flux velocity. Thus
where wall mass suction occurs when s > 0 while wall mass injection occurs when s < 0. The non-dimensional parameters in the above equations are the Maxwell parameter β( = k0a), the unsteadiness parameter A ( = λ/a), the magnetic field parameter He( = Q0/ρf cp) the heat generation parameter and Q( = Q0(1−λ t)), Nb ( = τ DB(Cw−C∞)/ν) is the Brownian motion parameter and Nt( = τ DT (Tw − T∞)/ν T∞) is the thermophoresis parameter. The Prandtl number is Pr( = ν/α), the Schmidt number is is the effective Prandtl number, is the Biot number, γ = k1(Cw−C∞) is the reaction parameter where γ < 0 denotes a destructive reaction, γ = 0 indicates that there is no reaction and γ > 0 denotes a generative reaction.
Other important physical parameters are the variable thermal conductivity κ(T), the Eckert number Ec, the local electromagnetic parameter E1, the Soret number Sr, the Dufour number Df and the radiation parameter R. These are defined as (11) (12)
where ε is a small parameter, Δ T = Tw − T∞ and the thermal conductivity parameter is K∞. The important flow attributes, the Nusselt number Nux and skin friction coefficient Cf are described by (13)
Here qw and τw are the plate heat flux and the skin friction respectively, defined as (14)
where μ is the coefficient of viscosity. Equations (13) may be written as (15) (16)
where is the local Reynolds number. Here, the Sherwood number is zero, due to the assumption of zero mass flux at the surface.
3 Method of solution
The transport equations have been solved using the iterative spectral local linearization method (SLLM), see Motsa . In principle of the SLLM algorithm is to linearize and decouple the system of equations. Motsa et al.  used the spectral local linearization method to solve the equations that model natural convection in glass-fibre production processes. Shateyi and Marewo  used the SLLM to solve equations that models an unsteady MHD flow and heat transfer. Nonetheless, this method has only been used in a limited number of studies, hence its general validation in complex systems remains to be made. For the interested reader, the SLLM algorithm is described in .
The differential equations arising from the linearization procedure are solved using a Chebyshev pseudo spectral method. The domain of the problem is transformed to the interval [−1, 1] using the transformation (b−a)(τ+1)/2. The differentiation matrix D is used to approximate the derivatives Zi(η) of the unknown variables to the matrix vector product, (17)
where the vector function at the collocation points is given by Z = [z(τ0), z(τ1), …, z(τN)]T, D = 2D/(b−a) with N+1 collocation points,  and (18)
where the superscript in D denotes higher order derivatives.
where ω1, ω2 and ω3 are non-linear operators and F,H, T are given by (20) (21) (22)
The system of equations (19) can be simplified and decoupled by linearizing the nonlinear terms. The Chebychev pseudo-spectral collocation method is utilized to integrate the decoupled system. The simple algorithm is as follows:
From the first equation, find F while treating H and T as known functions from initial guesses. This gives Fr+1
Solve for T in the second equation while treating F and H as known functions, T is known from 1 above and H is known from the initial guess. This gives Tr+1
Finally, solve for H in the last equation while treating F and T as known functions, T and H are known from 1 and 2 above. We obtain Tr+1.
Repeat steps 1-3 to find the next iterative solutions.
subject to boundary conditions: (24)
where (25) (26)
The following decoupled matrix system of equations is obtained: (27)
with corresponding boundary conditions (28)
I is an identity matrix with (N + 1) rows and columns, F, H and T are approximate values of f, ϕ and θ calculated at the collocation points. Initial approximations are required to start the iteration process, and these can be selected so as to satisfy the boundary conditions and known flow configuration. For our system, the following guesses were used as suitable initial approximations, (30)
4 Results and discussion
The solutions of the differential equations are given Tables 1 - 2 and Figures 2 - 16. In Table 1 we establish the reliability of the numerical scheme by a comparative analysis of the skin friction coefficient Cf and Nusselt number Nux with results reported by Hayat et al.  when β = 0.2, α = 0.3, Pr = 1 and s = 0.5. The other parameters have been set to zero. A good agreement is observed with the previously published work. To gain further insights as to the accuracy and convergence of the method used in this study, we have calculated residual errors as shown in Figures 2 to 4. These are calculated for different values of A, s and β. In general, the solutions have converged with an absolute residual error ∥Res∥ ≈ 10−11 after five iterations. These results sufficiently demonstrate to the accuracy and convergence of the SLLM.
The numerical computations have been done when A = −1.0, s = 2, δ = 0.25, Nb = Nt = 0.1, Sc = 0.8, Pr = 1.0, M = 0.3, ε = 0.1, Ec = 0.1, E1 = 1.0, Df = 0.1, He = 0.5, Sr = 0.1, Preff = 1.0, γ = 0.1, and Bi = 0.1. For numerical simulations, the parameter values are chosen from the previous literature on nanofluid flow such as [5, 10, 12, 16, 24] etc. Table 2 shows the Nusselt number for different Nb, M and Nt, the other parameters as stated above. The results show that Nux decreases with increasing Nb and M whereas the opposite is observed for increasing values of Nt.
Figure 5 shows that the velocity reduces with increasing magnetic field strength. This is an indication of an increase in the Lorentz force that creates a resistance to the fluid flow near the boundary slowing down the fluid motion. Figure 6 shows that the boundary layer thickness increases as the unsteadiness parameter increases. The velocity profiles decrease with increasing value of A.
Figure 7 shows that as β increases, the boundary layer thickness increases with a cross-over of profiles near the surface. The physical interpretation of this behavior is that an increase in β reduces the fluid in the flow leading to the boundary layer thickness increasing near the surface but a contrary trend is recognized away from the surface.
Figure 8 shows a reduction in the velocity with an increase in the mass suction parameter s. Figure 9 demonstrates the effect of A on the temperature profile. An increase in unsteadiness causes an increase in the solute concentration. Figure 10 shows how the temperature profiles change with respect to variations in the Maxwell parameter β. The temperature profiles increase with increasing β values. This is to be expected since higher Maxwell parameters generally suggest a more solid material able to conduct and retain heat better.
Figure 11 shows the change in temperature profiles with respect to variations in the thermophoresis parameter values. As thermophoresis increases, the temperature profiles decrease near the surface. Figure 12 shows the effect of the magnetic field parameter on θ(η) the temperature profiles. We observe that the temperature profiles increase with increasing values of the magnetic field parameter. The existence of a magnetic field in an electrically conducting fluid produces a body force that decelerates the fluid flow which in turn has the effect of retaining more heat within the boundary layer.
The impact of the unsteadiness parameter, Maxwell parameter, Brownian motion parameter, thermophoresis parameter and the Schmidt number on the concentration profiles is depicted in Figures 13 - 16. In Figures 13 and 14 an increase in the unsteadiness parameter and Maxwell parameter causes an increase in the concentration profiles near the surface but the opposite movement is observed far from the surface. Figure 15 shows the impact that the Brownian motion parameter has on the concentration profiles. An increase in the Brownian motion parameter causes a decrease in the concentration profiles. The Brownian motion tends to intensify particle displacement away from the fluid flow regime onto the surface; this phenomenon accounts for a decrease in the concentration of the nanoparticles far from the surface, thus resulting in a decrease in the nanoparticle concentration boundary layer thickness.
Figure 16 depicts the effect of the thermophoresis parameter on the concentration profiles. Thermophoresis is associated with the movement of nanoparticles from a hot to a cold wall, and since it is generated by temperature gradients, this creates a fast flow away from the moving surface. Consequently more fluid is heated away from the surface leading to an increase in the temperature within the thermal boundary layer.
In this study, we have investigated unsteady Maxwell nanofluid flow over a shrinking sheet with convective and slip boundary conditions. The conservation equations were solved using an iterative spectral local linearization method. We have given an error analysis to establish the accuracy and convergence of the method. The impact and significance of various physical parameters on the fluid properties has been demonstrated both qualitatively and quantitatively. The findings can be briefly outlined as follows;
The concentration and velocity profiles decrease whereas the temperature profile increases with increasing unsteadiness parameter.
Increasing particle Brownian motion leads to a reduction in the concentration profiles but concentration profiles increase with increasing thermophoresis.
In terms of heat transfer coefficients, increasing value of particle Brownian motion and magnetic field strength reduces the heat transfer coefficient but the opposite is observed in the case of increased thermophoresis parameter.
The authors are grateful to the University of KwaZulu-Natal and the Claude Leon Foundation, South Africa for financial support.
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About the article
Published Online: 2017-11-06
Conflict of interestConflict of interests: The authors declare that there is no conflict of interests regarding the publication of this article.