Numerical results for influence the flow of MHD nanofluids on heat and mass transfer past a stretched surface

Abstract: Due to its signi cant applications in physics, chemistry, and engineering, some interest has been given in recent years to research the boundary layer ow of magnetohydrodynamicnano uids. Thenumerical resultswere analyzed for temperature pro le, concentration pro le, reduced number of Nusselt and reduced number of Sherwood. It has also been shown that the magnetic eld, the Eckert number, and the thermophoresis parameter boost the temperature eld and raise the thermal boundary layer thickness while the Prandtl number reduces the temperature eld at high values and lowers the thermal boundary layer thickness. However, if Lewis number is higher than the unit and the Eckert number increases, the concentration pro les decrease as well. Ultimately, the concentration pro les are reduced for the variance of the Brownian motion parameter and the Eckert number, where the thickness of the boundary layer for the mass friction feature is reduced.


Introduction
One of the most important emerging developments of the 21 st century is nanotechnology. It has been commonly used in manufacturing and nanometersized materials have special physical and chemical properties. With its increasingly signi cant and complex e ect on a wide range of industries, including biotechnology, oil, electron-*Corresponding Author: Nader Y. Abd Elazem, Department of Basic Science, Pyramids Higher Institute for Engineering and Technology, 6 th of October City, Giza, Egypt, E-mail: naderel-nafrawy@yahoo.com, nader_47@hotmail.com ics and consumer goods, it promises to change our lives in this decade. In many engineering processes with applications in industries such as extrusion, melt-spinning, heat rolling, wire drawing, glass-ber processing, plastic and rubber sheet manufacturing, cooling of a large metal plate in a bath that may be an electrolyte, etc., ow over a stretching surface is an important issue.The authors in [1] investigate the elastic deformation e ects on the boundary layer ow of an incompressible second grade two phase nano uid model over a stretching surface in the presence of suction and partial slip boundary condition.
In industry, polymer sheets and laments are manufactured by continuous extrusion of the polymer from a die to a windup roller, which is located at a nite distance away [2]. There are many applications in engineering and industry for boundary layer ow behavior over a stretching surface [3]. Having a low heat transfer in a uid would cause limited heat transfer and can lead to limited heat transfer e ciency. Due to the high thermal conductivity of metal particles, adding them to a uid would increase the thermal conductivity and also heat transfer of the resultant mixture uid. Choi's [4] initial analysis of the term "nano uid" identi ed a liquid suspension containing ultra-ne particles. Nanoparticles (e.g., Copper (Cu), Silver (Ag), Alumina (Al O ), Titanium (TiO )) range from 1 to 100 nm in diameter [5].The base liquid's thermal conductivity is improved by ( % − % ) if it is suspended by a low volumetric fraction (less than %) of nanoparticles [6][7][8]. In Ref. [9], the authors examined improvements in the thermal conductivity of uids (such as oil, water, and ethylene glycol mixture) which are poor heat transfer by suspending nano/micro or large particle materials in these uids. Kuznetsov and Nield [10] investigated the effect of nanoparticles on the natural convection boundarylayer ow through a vertical plate, using a model in which Brownian motion and thermophoresis are represented.
In many applications in the polymer and metallurgy industries, hydro-magnetic techniques are used [23]. Hence, the in uence of the magnetic eld has attracted signi cant interest in recent years due to its high applications in physics, chemistry, and engineering [24]. MHD free convection ow of Sodium Alginate nano uid on a solid sphere with prescribed wall temperature is investigated by [25]. The authors examine the free convection ow of Casson nano uid in the prsence of magnetic eld. The relevant partial di erential equations are rst converted into non-dimensional equations by using appropriate transformation and then computed by utilizing the Keller box method. MHD peristaltic transport of copperwater nano uid in an artery with mild stenosis for di erent shapes of nanoparticles is studied in [26]. The characteristics of MHD, heat sink/source, and convective boundary conditions in chemically reactive radia-tive Powell-Eyring nano uid ow via Darcy channel using a nonlinearly settled stretching sheet/surface are used in Rasool and Sha q [27]. In ref. [28], the authors concern with the examination of heat transfer rate, mass and motile microorganisms for convective second grade nano uid ow.
Considering these facts, and motivated by above discussed papers in the eld of nano uids. There is a physical justi cation to study the prsent article, there is enhance in the dimensionless of the stream function f , temperature θ, and volume of nanoparticles φ, respectively, (see Fig. 1 in ref. [19]). Due to the existing of Prandtl number in the momentum equation in ref. [19]. All the above previous research did not address this point. So, we studied the MHD ow of nano uid over a stretched surface on heat and mass transfer to extend to the work of Abd Elazem [19]. Because of the importance this study for engineers and researchers in nearly every branch in engineering and science. Nuclear power plants, gas turbines and the various propulsion devices for aircraft, missiles, satellites and space vehicles are examples of such engineering areas. The governing equations of the present study are solved numerically solved by using the Chebyshev pseudospectral technique [29][30][31]. The e ect of di erent parameters on temperature, concentration, local skin friction f ( ), reduced Nusselt number, and reduced Sherwood number is investigated through tables and graphs.

Analysis
A uniform magnetic eld of force Bo is imposed in the y−direction according to Ref. [8] (see Fig. ). For the current study, the basic steady conservation of mass, mo- subject to the boundary conditions: Where, u and v are the velocity components along the axes x and y, respectively, ρ f is the density of the base uid, ν is the kinematic viscosity, σ * is the electrical conductivity, p is the uid pressure, α = k f (ρc) f is the thermal di usivity, D B is the Brownian di usion coe cient, D T is the thermophoretic di usion coe cient, τ = (ρc)p (ρc) f is the ratio between the e ective heat capacity of the nanoparticle material and heat capacity of the uid with ρ being the density, c is the volumetric volume expansion coe cient and ρp is the density of the particles, cp is the uid speci c heat at constant pressure, µ f is the viscosity of the base uid and s is suction (or injection) parameter, respectively. T is the temperature of the uid, C is the fraction of the volume of nanoparticles, Tw is the temperature of the stretching surface, Cw is the fraction of the volume of nanoparticles on the stretching surface, T∞ is the ambient temperature and C∞ is the fraction of the volume of ambient nanoparticles. Under the related work [10].
Where ψ is a stream function provided by Well then, Eq. ( ) identically satis ed. For converting Eqs. ( ) − ( ) with the boundary conditions ( ) into the following nonlinear ordinary di erential equations a similarity solution in Ref. [10] was implemented.
with the boundary conditions: Where primes indicate di erentiation for η and Pr, Nb, Nt, Ec, M, and Le are Prandtl number, Brownian motion parameter, thermophoresis parameter, Eckert number, magnetic parameter, and Lewis number, respectively. The physical parameters below are described by: Here, gravitational acceleration, volumetric expansion coe cient of the uid, nanoparticle volume fraction at the surface, ambient nanoparticle volume fraction attained as y tends to be in nite, and local Rayleigh number, respectively, are also the symbols g, β, φw , φ∞, and Rax. In addition, f , θ, and φ are the dimensionless of the stream function, temperature, and volume of nanoparticles respectively. It distinguishes the local skin friction C f , the reduced Nusselt number Nur and the reduced Sherwood where, Rex is the local Reynolds number based on the stretching velocity uw(x).

. Chebyshev pseudospectral di erentiation matrix technique
A numerical solution based on Chebyshev collocation approximations can be considered as a suitable choice for many practical problems (as described in the literature review and for example Canuto et al. [29] and Peyret [30]). Accordingly, Chebyshev collocation method will be applied for the presented model. The derivatives of the function f (x) at the Gauss-Lobatto points, x k = cos kπ L * , which are the linear combination of the values of the function f (x) [31], where where, a n m,l = n l where k, j = , , , ..., L * and γ * = γ * l = , γ * j = for j = , , , ..., L − . The round o errors (see Appendix A) incurred during computing di erentiation matrices D (n) are investigated in [31].

. Description of the numerical method
The grid points (x i , x j ) in this situation are given as where xmax is the length of the dimensionless axial coordinate and the domain in the η−direction . The application of this method to di erential equation leads to system of algebric equations. The rst rows and last rows of the coe cients matrix of the algebric system are replaced by a suitable formulation of the boundary conditions. The Chebyshev collocation method is more accurate in comparison with other techniques for solving this kind of problems, espcially the nite di erence and the nite elements methods. The nite di erence methods replace the derivatives of a function at any point with nite di erence approximation formulas in terms of its values on a grid of mesh points that span the domain of interest. The numbers of these mesh points are two and three for the central nite di erence of second order of the rst and second derivatives, respectively. While in Chebyshev collocation method, the derivatives of a nonsingular function at any point from the Chebyshev points are expanded as a linear combination from the values of the function at all of these points. i.e. (the approximations of the derivatives are de ned over the whole domain). Therefore, Eqs. (9) -(11) with the boundary conditions (12) have been solved numerically [18,19]. Thus by applying the Chebyshev collocation ap-proximation to equations (9) -(11). The following Chebyshev collocation equations can be obtained: The computer program of the numerical method and the numerical compu-tations have been done by the symbolic computation software Mathematica 6 TM . Also, the solution of the above equations (15) -(17) for the unknowns f i , θ j and φ j with boundary conditions (12) where j = ( )L * (take L * = ) and ηmax corresponds to η∞ are obtained using the Newton-Raphson iteration technique.

Results and Discussion . Validation of the numerical solution
The results for the local skin friction f ( ) are compared with those obtained by Abd Elazem [19] for di erent values of Pr, s, and ηmax in Table 1. It is noticed that the comparison shows excellent agreement for each values of Pr, s, and ηmax. Also, dimensionless similarity functions f (η), θ(η) and φ(η) are matching with the previously published [19] as shown in Fig. 2.Therefore, it is con dent that the present results are very accurate.

. Results for temperature and solid volume fraction pro les
Equations ( ) − ( ) have been numerically solved with the boundary conditions (12) using the Chebyshev pseudospectral technique. It is found that as the distance increases from the solid boundaries, both the temperature and the concentration pro les start at unity near the wall and reach to vanish. In the case of Nt = Nb = . , Pr = Le = , s = . and M = , Figure  serves to highlight the present numerical results for Ec = − . , − . , . , . on temperature pro les. It is demonstrated that with an increase in Ec, the temperature pro les and thermal boundary layer thickness are enhanced. Physically, the ohmic heating e ect due to the effects on electromagnetic work is found to be produced an increase in the uid temperature, and thus a decrease in the surface temperature gradient. Further, it is found that the e ect of viscous heating leads to an increase in the temperature; this e ect is more pronounced in the presence of the magnetic led. It is acknowledged that an increase in Nt, the temperature pro le accelerates also, the temperature pro le raises the elevation of the Eckert number as shown in Fig. 4 (positive Ec values correspond to wall heating, while the opposite is true for Ec negative values).
The temperature decreases as Pr increases. This is in agreement with the physical fact that the thermal boundary layer thickness decreases with increasing Pr. Further, according to Fig. 5, the numerical results for the pro les of θ(η) for (a) Ec = − . and (b) Ec = . when Pr = . , , , at Nt = Nb = . , Le = , s = . , and M = . Physically the thermal boundary-layer thickness, as predicted, is less than the boundary-layer thickness of the momentum when Pr . Also, the temperature function decreases (see Fig. 5). At high values of the Prandtl number Pr (values of Pr , decrease conductivity, and increase pure convection). Besides the typical matching of temperature pro les [19] at values ( < Pr ≤ ). Notwithstanding this, it should also be noticed that with the rise in Ec from − . to 0.01, there is a slightly greater di erence in Ec = . if Pr = . , (see Fig. 5).
By contrast, it is clearly shown that an increase in M and Ec increases the thermal boundary layer thickness and the temperature pro les. Physically, application of a transverse magnetic eld to an electrically conducting uid creates a resistive-type force called the Lorentz force. This conclusion meets the logic that the magnetic eld exerts a retarding force on the free-convection ow in the boundary layer and increase its temperature (see Figure 6).
In the case of Le > , the thickness of the boundarylayer for mass friction function is smaller than the thermal boundary-layer thickness (see Fig. 7). Variations in the concentration function increased with Ec growing far from the boundary. Concentration function pro les generally decrease with the rise in Lewis number as in Fig. 7. It can be seen that an increment in Lewis number decreases the     solid volume fraction of nano uid pro les. Physically, This is due to the fact that mass transfer rate of nano uid increases as Lewis number increases. It also reveals that the concentration gradient at surface of the stretching sheet increases. Moreover, the concentration at the surface of stretching sheet decreases as Lewis number increases. Finally, Figure 8 shows the variation of Concentration proles with Nb and Ec when Nt = . , Pr = , Le = . , s = . , and M = . It's clear that the thickness of the boundary layer for mass friction function decreased as Nb increased. Besides, the concentration pro les decrease with the increase of the Eckert number. Physically, the Brownian motion parameter helps to measure the strength of the Brownian di usion of the nanoparticles in the ow eld. Due to the Brownian di usion, the nanoparticles tend to move away from the surface of the sheet and as result a   . , and s = . . It is reported that the reduced Nusselt number is a decreasing function while an increasing function is the reduced Sherwood number. Physically, It is also found that the impact of Joule heating on electromagnetic operation has resulted in a rise in the uid temperature and therefore a decrease in the gradient of the surface temperature. Furthermore, the actual impact of viscous heating results in a temperature increase; this e ect is more pronounced in the presence of the magnetic eld. As shown in Table 3, it is clear that the reduced Nusselt number is a monotonous function (i.e. it is an increasing function at Ec = − . , whereas a decreasing function at Ec = . , . ).

Conclusion
The e ects of various physical parameters on nano uids that ow past a stretch-ing surface were explored. This study is very important for engineers and researchers in nearly every branch in engineering and science. Nuclear Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Con ict of interest:
The authors state no con ict of interest.

A Chebyshev collocation approximation A. Rounding o error analysis
The round o errors incurred during computing di erentiation matrices D (n) investigated by [31]. Elbarbary and Elsayed [31] show that, the elements of the rst order di erentiation matrix, there would be round o error as in: The element d is the major elements concerning its values. Accordingly, it bears the major error responsibility comparing the other elements. Meanwhile, Baltensperger and Trummer [32] show that, the error in the evaluation of the element D O from the classical matrix D is of order O(N δ) where δ is the machine precision and D * − D = N π δ, whereas in Elbarbary and El-sayed [31] nd the error of order O(N δ) where, This can be taken into consideration, as itself, as modi ng the classical matrix D. Due to, with error upper bound Also, Elbarbary and El-sayed [31] show that the error bound for the second order derivatives can be given by N − and the error bound for the third order derivatives is given by Elbarbary and El-sayed [31] Finally, the error bound for the fourth order derivatives is given by Elbarbary and El-sayed [31] Table 4 lists the computed errors in the elements d and D .  Consider the following boundary value problem [33] f The exact solution is given by Table 5 represents the values of f (η) for the exact solution, Table 5: Values of f (η) for the exact solution, the shooting method and the present method. the shooting method and the present method. The error of Shootting method (E e,Shooting ) and (the error E e,ChC ) of the present method is given in Table 6.