MHD flow of Maxwell fluid with nanomaterials due to an exponentially stretching surface

In many industrial products stretching surfaces and magnetohydrodynamics are being used. The purpose of this article is to analyze magnetohydrodynamics (MHD) non-Newtonian Maxwell fluid with nanomaterials in a surface which is stretching exponentially. Thermophoretic and Brownian motion effects are incorporated using Buongiorno model. The given partial differential system is converted into nonlinear ordinary differential system by employing adequate self-similarity transformations. Locally series solutions are computed using BVPh 2.0 for wide range of governing parameters. It is observed that the flow is expedite for higher Deborah and Hartman numbers. The impact of thermophoresis parameter on the temperature profile is minimal. Mathematically, this study describes the reliability of BVPh 2.0 and physically we may conclude the study of stretching surfaces for non-Newtonian Maxwell fluid in the presence of nanoparticles can be used to obtain desired qualities.

www.nature.com/scientificreports www.nature.com/scientificreports/ and Joule heating. Jamil and Fetecau 30 considered Maxwell fluid for Helical flows between coaxial cylinders. Zheng et al. 31 constructed closed form solutions for generalized Maxwell fluid in a rotating flow. Wang and Tan 32 presented stability analysis for Maxwell fluid subject to double-diffusive convection, porous medium and soret effects. Motsa et al. 33 examined UCM flow in a porous structure. Mukhopadhyay et al. 34 elaborated thermally radiative Maxwell fluid flow over a continuously permeable expanding surface. Ramesh et al. 35 numerically investigated Maxwell fluid with nanomaterials for MHD flow in a Riga plate. Ijaz et al. 36 explored the behavior of Maxwell nanofluid flow for the motile gyrotactic microorganism in magnetic field. UCM fluid model for non-Fourier heat flux is investigated by Ijaz et al. 37 .
Nanofluids consists of ordinary liquids and nanoparticles. Nanofluids are quite useful to improve the performance of ordinary liquids. Nanofluids are developed by inserting fibers and nanometer size particles to original fluids. Nanofluids are especially significant in hybrid powered engines, pharmaceutical processes, fuel cells, microelectronics. The nanoparticles basically connect atomic structures with bulk materials. Commonly used ordinary liquids include toluene, oil, water, engine oil and ethylene glycol mixtures. The metal particles include aluminum, titanium, gold, iron or copper. The nanofluids commonly contains up to 5% fraction volume of nanoparticles to obtain significant improvement in heat transfer. Further, the magnetic nanofluids have great interest in optical switches, biomedicine, cancer therapy, cell separation, optical gratings and magnetic resonance imaging. An extensive review on nanofluids includes the attempts of 38,39 . One can also mention the previous recent studies [40][41][42][43][44][45] regarding the improvement of thermal conductivity in the nanofluids.
In view of aforementioned discussion, the MHD flow of non-viscous Maxwell fluid with nanomaterials in an exponentially stretching surface is addressed. The BVPh 2.0 which is developed on the basis of optimal homotopy analysis method (OHAM) is employed to solve nonlinear differential system [46][47][48][49][50] . The convergence of the present results is discussed by the so-called average squared residual errors. Velocity, temperature, concentration, the local Sherwood and the local Nusselt number are also examined through graphs. Figure 1 describes the MHD laminar, incompressible flow, thermal and concentration boundary layers in a surface which is stretching exponentially with velocity U w and given concentration C w and temperature T w is described using boundary layer theory as follows:

Problem Formulation
where, u is the x component and v is the y component of the Maxwell fluid velocity. Also g, D T , C, T ∞ , λ, ν, ρ f , C ∞ , T, σ, D B and α represents gravitational acceleration, thermophoretic diffusion coefficient, nanoparticle volume fraction, free stream temperature, relaxation time, the ratio between nanoparticle and original fluids www.nature.com/scientificreports www.nature.com/scientificreports/ heat capacities, kinematic viscosity, fluid density, free stream concentration, temperature, electrical conductivity, Brownian diffusion coefficient and thermal diffusivity, respectively. The subjected boundary conditions (BCs) are given by, The adequate transformations for the considered problem are taken as follows:  The non-dimensional BCs from Eqs (5)- (7) becomes The heat and mass transfer rates in terms of local Sherwood, the local Nusselt numbers, and the local skin friction coefficient are defined by where j i , q i , and τ i are the mass, heat, and momentum fluxes from the surface. These are defined as follows: In dimensionless form they are represented as: x x x f

Homotopy-Based Approach
The following methodology details should provide as a guide about OHAM aiming to solve nonlinear differential system (9)-(11) with BCs (13) and identify the variations of physical solutions of the differential system. In the framework of OHAM, we can choose auxiliary linear operators in the forms www.nature.com/scientificreports www.nature.com/scientificreports/ Obviously, the operators satisfying the below assumptions The corresponding auxiliary linear operator for f(η) is 1  and  2 corresponds to θ(η) and φ(η). In OHAM we also have flexibility to pick the initial solutions. It is mandatory that all initial solutions should satisfy the BCs (13). Therefore, we set the initial solutions as follows: where λ a = λ b = 1. We have applied BVPh 2.0 to solve nonlinear differential system (9)-(11) with BCs (13). With linear operators (17) and (18) and initial solutions (21)-(23), the Eqs (9)-(11) with BCs (13) can be solved directly by using BVPh 2.0 in quite easy and convenient way.

Results and Discussion
The OHAM approximations contains unknown convergence enhancing parameters  Table 1. It is noted that the total error is decreased by increasing the order of iteration. Optimal convergence-control parameters corresponding to 5th-order OHAM iteration are then used to check the convergence of our results at various orders of approximation. The OHAM iterations at various orders are shown in Table 2. The presented results demonstrate the high efficiency and reliability of OHAM series solutions. The graphical analysis has been accomplished for the flow pattern, concentration, temperature, the local Sherwood number and the local Nusselt number for various values of β, Le, Pr, M, N t and N b . Figure 2 illustrates the impact of β on the velocity graph for different M. It is visible that fluid flow is maximum in the ambient fluid for smaller β. However, the fluid changes its properties from Newtonian to non-Newtonian characteristics for higher β, and hence the flow shows decreasing behavior. The boundary layer thickness reduces as we increase β and M. Figure 3 displays the influence of β and M on concentration and temperature graphs. It is observed that the increase in β and M enhances concentration and temperature.     Figure 5 illustrates that due to increase in the values of N t , the concentration profile increases but temperature decreases. The overshoot in the concentration is observed that is highest concentration occurs in the ambient fluid but not at the surface.     Figure 8 shows that the local Nusselt number is a decreasing function of N b . The increase in either N t or Le decreases the local Nusselt number. The local Sherwood number is plotted as a function of N b in Fig. 9. It can be seen that local Sherwood number is an increasing function of N b and Le. However, the increase in N t decreases the local Sherwood number.