T.Hayat,M.Ijaz Khan ,M.Waqas ,A.Alsaedi,T.Yasmeen *

1 Department of Mathematics,Quaid-I-Azam University,45320,Islamabad 44000,Pakistan

2 Nonlinear Analysis and Applied Mathematics(NAAM)Research Group,Department of Mathematics,Faculty of Science,King Abdulaziz University,P.O.Box 80257,Jeddah 21589,Saudi Arabia

3 Department of Mechanical Engineering,Imperial College London,London SW7 2AZ,UK

4 Department of Mechanical Engineering,University of Engineering&Technology Peshawar,Pakistan

1.Introduction

There are several materials like shampoos,mud,soaps,apple sauce,sugar solution,polymeric liquids,tomato paste,condensed milk,paints,and blood at low shear rate which show the characteristics of non-Newtonian fluids.The behavior of such materials cannot be explored by a single constitutive relationship because of their diverse properties.Hence different fluid models are developed in the past to describe the exact nature of non-Newtonian materials.Third grade fluid is a subclass of differential type non-Newtonian fluid.This fluid model exhibits shear thickening and shear thinning characteristics.Some studies on the third grade fluid can be seen in refs.[1–5].Further,the flow over a stretching surface is an important problem in many engineering processes with applications in various engineering and industrial processes like cooling of metallic sheets in a cooling bath,annealing and thinning of copper wires,aerodynamic extrusion of plastic and rubber sheets,drawing of plastic films and sheets,glass fiber and paper productionetc.It is worth mentioning to point out here that the stretching velocity is not linear necessarily in all the cases.The stretching velocity may be nonlinear or exponential.For example in annealing and thinning of copper wires,the desired quality product depends on the continuous stretching of surface with exponential dependence velocity distribution[6–10].

It is well known fact that the raw materials are constructed to undergo chemical reaction in industrial chemical processes to convert cheaper raw materials into higher standard products.Such chemical transformations are performed in a reactor.The reactor has a key role to providing an appropriate environment for suitable time and allowing the removal of finished products.The study of chemical reaction has important role in chemical technologies like polymer production.No doubt a chemical reaction can be classified either through homogeneous or heterogeneous processes.Reaction rate in first order chemical reaction is directly proportional to the concentration.A large amount of research work has been reported in this field.For instance,Soret and Dufour effects in three-dimensional flow over an exponentially stretching surface with porous medium,chemical reaction and heat source/sink is studied by Hayatet al.[11].Bhattacharyya and Layek[12,13]addressed the behavior of chemically reactive solute distribution in MHD boundary layer flow over a permeable stretching sheet and also described the slip effects on the boundary layer flow and mass transfer over a vertical stretching sheet.MHD stagnation point flow and heat transfer impinging on stretching sheet with chemical reaction and transpiration is examined by Maboodet al.[14].Sheikh and Abbas[15]explored the effects of thermophoresis and heat generation/absorption on MHD flow due to an oscillatory stretching sheet with chemically reactive species.Convective flow of micropolar liquid with chemical reaction and mixed convection is examined by Swapnaet al.[16].Hayat et al.[17]reported carbon nanotubes effects in the stagnation point flow towards a nonlinear stretching sheet with homogeneous-heterogeneous reactions.Characteristics of thermal radiation and chemical reaction in flow of nano fluid saturating porous medium is presented by Zhanget al.[18].Mythili and Sivaraj[19]reported the chemically reactive flow of Casson liquid towards a vertical cone.Narayana and Babu[20]analyzed thermal radiation and chemical reaction effects in flow of Jeffrey liquid over a stretched surface.Impact of Cattaneo–Christov heat flux model in flow of variable thermal conductivity fluid over a variable thicked surface with chemical reaction is reported by Hayatet al.[21].Hayatet al.[22]explored stagnation point flow with Cattaneo–Christov heat flux and homogeneous–heterogeneous reactions.Homogeneous–heterogeneous reactions in MHD flow due to an unsteady curved stretching surface are scrutinized by Imtiazet al.[23].Hayatet al.[24]studied the impact of Cattaneo–Christov heat flux in Jeffrey fluid flow with homogeneous–heterogeneous reactions.Influences of thermal radiation and chemical reaction in MHD flow by a cylinder are explored by Machireddy[25].Flow due to nonlinear stretching surface with chemical reaction and porous medium is addressed by Ziabakshet al.[26].Bhattacharya[27]developed dual solutions for stagnation point flow past a stretching/shrinking sheet with chemical reaction.Chen and Sun[28]addressed boundary layer interaction of chemical reacting flow in shock tube.Influence of chemical reaction and radiation in MHD free convective flow are addressed by Rajuet al.[29].

The study of magnetohydrodynamics(MHD) flow of an electrically conducting fluid over a stretching sheet has promising applications in modern metallurgical as well as in metal-working procedures.Many professional techniques regarding polymers require the cooling of continuous strips and filaments by drawing them from moving fluid.The final product depends greatly on the rate of cooling that is governed by the structure of the boundary layer close to the stretching sheet.Mukhopadhyayet al.[30]studied MHD flow of Casson fluid due to exponentially stretching sheet with thermal radiation.Impact of magnetohydrodynamics in bidirectional flow of nano fluid subject to second order slip velocity and homogeneous–heterogeneous reactions is reported by Hayatet al.[31].Linet al.[32]examined unsteady MHD pseudo-plastic nano fluid flow and heat transfer in a finite thin film over stretching surface with internal heat generation.Sheikholeslamiet al.[33]analyzed the effect of thermal radiation on magnetohydrodynamics nano fluid flow and heat transfer by means of two phase model.Application of the HAM-based Mathematica package BVPh 2.0 on MHD Falkner–Skan flow of nano fluid is provided by Farooqet al.[34].Shehzadet al.[35]presented an analytical study to investigate thermal radiation effects in three-dimensional flow of Jeffrey nano fluid with internal heat generation and magnetic field.

Here our main theme is to study the influences of magnetohydrodynamics(MHD) flow of third grade fluid by an exponentially stretching sheet.Mass transfer analysis is performed in the presence of first order chemical reaction.The governing nonlinear flow model is solved and homotopic solutions[36–53]of dimensionless velocity and concentration are presented.Physical quantities for various parameters of interest are examined.To our knowledge such analysis is not yet reported.

2.Formulation

Consider the two-dimensional hydromagnetic flow of third grade fluid over an exponentially stretching sheet.Mass transfer effects are taken into account in the presence of chemical reaction.An applied magnetic field of strengthB0is encountered normal to the flow direction.The magnetic Reynolds number is chosen small.Further the induced magnetic field is smaller in comparison to the applied magnetic field and is negligible.The two-dimensional boundary layer equations of an incompressible third grade fluid with mass transfer are[5,7]:

Introducing the similarity variables as:

Here ψ denotes the stream function,fthe dimensionless velocity and ? the dimensionless concentration.Employing above transformations,Eq.(1)is identically satisfied while Eqs.(2)–(4)are

here α1,α2and β denote the fluid parameters,Hathe Hartman number,Scthe Schmidt number and γ the chemical reaction parameter.It is noted that γ ＞ 0 is for destructive chemical reaction,γ ＜ 0 for generative chemical reaction and γ=0 corresponds to non-reactive species.These quantities are defined as

The skin friction coefficientCfand local Sherwood numberShxare

Dimensionless expressions of skin friction coefficient and local Sherwood number are

3.Series Solutions

In order to obtain analytical solution,the selected initial guesses and auxiliary linear operators are:

The above auxiliary linear operators satisfy the following properties

whereCi(i=1?5)indicates the arbitrary constants.

4.Zeroth-Order Deformation Problem

The corresponding problems at the zeroth order are:

in whichpis an embedding parameter and ?fand ??are the non-zero auxiliary parameters.The nonlinear operators are represented by Nfand N?.

5.m th-Order Deformation Problem

The resulting problems atmth order deformation are constructed as follows:

whereCi(i=1?5)indicates the arbitrary constants which are determined by employing the boundary conditions(24)and(26)

6.Convergence of the Developed Solutions

To construct the series solutions by homotopy analysis technique it is also necessary to check their convergence.Convergence region is the region parallel to ??axis.Hence we have plotted the ??curves of the velocityf″(0)and concentration ?′(0)in Figs.1 and 2.Permissible values for the derived solutions are found in the ranges?1.15≤?f≤?0.20 and?1.15≤??≤?0.35.

7.Discussion

Description of various pertinent parameters on the velocity and concentration distributions is the main motto of this section.Figs.3 and 4 illustrate the variation of fluid parameters α1and α2on the velocity pro filef′(η)respectively.It is observed that the velocity pro filef′(η)is an increasing function of both α1and α2.Physically viscosity of the material reduces for larger values of α1and α2due to which force between the adjacent layers decreases and thus velocity of the fluid increases.Characteristics of Hartman numberHaon velocity distribution is displayed in Fig.5.It is found that velocity pro file decreasesvialargerHa.Physically by increasing magnetic field the Lorentz force increases.More resistance is offered to the motion of fluid and thus the velocity of the fluid is reduced.Impact of fluid parameter β on velocity distribution is depicted in Fig.6.It is examined that the velocity profile increases near the wall for larger values of β and it vanishes away from the wall.Moreover the momentum boundary layer thickness is also increasing function of β.In fact β is inversely proportional to the viscosity.For larger values of β the viscosity of the fluid decreases and hence the velocity pro file enhances.Effect of Schmidt numberScon concentration distribution is displayed in Fig.7.Here concentration pro file and associated boundary layer thickness are decreased when Schmidt numberScincreases.Physically the Schmidt number is dependent on mass diffusionDand an increase in Schmidt number corresponds to a decrease in mass diffusion and the concentration.Fig.8 depicts the influence of destructive chemical reaction parameter(γ ＞ 0)on the concentration pro file ?(η).It is obvious that the fluid concentration decreases with an increase in the destructive chemical reaction parameter.In fact higher values of destructive chemical reaction parameter correspond to larger rate of destructive chemical reaction which dissipates or destroys the fluid specie more efficiently.Therefore concentration distribution decreases.Characteristics of generative chemical reaction parameter(γ＜0)on the concentration profile ?(η)is disclosed through Fig.9.This figure illustrates that concentration field has an opposite behavior for(γ＜0)when compared with(γ＞0).Physically larger values of generative chemical reaction parameter correspond to higher rate of generative chemical reaction which generates the fluid specie more efficiently and therefore concentration distribution increases.Characteristics of α1and α2on skin friction are shown in Fig.10.It is seen that skin friction coefficient increases when α1and α2are enhanced.Fig.11 depict the influences of β andHaon skin friction.It is analyzed that skin friction coefficient increases for larger β andHa.Analysis ofScand γ on local Sherwood number is presented in Fig.12.This figure shows that local Sherwood number increases for higher values ofScand γ.

Fig.1.??curve for f.

Fig.2.??curve for ?.

Fig.3.Impact of α1 on f′.

Fig.4.Impact of α2 on f′.

Fig.5.Impact of Ha on f′.

Fig.6.Impact of β on f′.

Fig.7.Impact of Sc on ?.

Fig.8.Impact of γ ＞ 0 on ?.

Fig.9.Impact of γ ＜ 0 on ?.

Fig.10.Impacts of α1 and α2 on Rex 1/2Cf.

Fig.11.Impacts of β and Ha on Rex 1/2Cf.

Fig.12.Impacts of Sc and γ on

Table 1 shows the convergence of the series solutions for different order of approximations.It is noted that 15th and 20th order of approximations are sufficient for the convergence of momentum and concentration equations respectively.Tables 2 and 3 give the comparison of present results in a limiting sense with the works done by Mukhopadhyayet al.[30],Elbashbeshy[54]and Chaudharyet al.[55].It is found that the obtained results in limiting sense are in good agreement.

Table 1Convergence of homotopy solutions when α1=α2=β=Ha=0.1,Sc=γ=1.0 and?f=??=?0.6.

Table 2Comparison for numerical values of f″(0)when α1=α2=β=Ha=0.

Table 3Comparison for numerical values of f″(0)for Ha when α1=α2=β=0.

8.Concluding Remarks

We have investigated the characteristics of free convection boundary layer flow of third grade fluid induced by exponentially stretching sheet.The present analysis leads to the following observations.

?Influences of α1,α2and β on velocity distribution are similar in a qualitative manner.

?Magnetic field retards the fluid velocity and momentum boundary layer thickness.

?Concentration distribution has opposite results for destructive(γ＞0)and generative(γ＜ 0)chemical reactions.

?An increase in Schmidt numberSccauses a decrease in the concentration distribution and the boundary layer thickness.

?Impacts of α1,α2,β andHaon skin friction are similar in a qualitative

sense.

?Local Sherwood number enhances for largerScand γ.

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