S.R.Mishra ,M.M.Bhatti*

1 Department of Mathematics,Siksha ‘O’Anusandhan University,Khandagiri,Bhubaneswar 751030,Odisha,India

2 Shanghai Institute of Applied Mathematics and Mechanics,Shanghai University,Shanghai 200072,China

1.Introduction

The study of heat transfer over a stretching surface is important in many industrial applications such as glass fiber production,hot rolling and wire drawing,the aerodynamic extrusion of plastic sheets,the continuous casting,paper production,glass blowing,metal spinning and drawing plastic films.The quality of the final product depends on the rate of heat transfer at the stretching surface.McCormack and Crane[1]have provided the comprehensive discussion on boundary layer flow caused by stretching of an elastic flat sheet moving in its own plane with a velocity varying linearly with distance.The problem of heat and mass transfer on a stretching sheet with suction or blowing was investigated by Gupta and Gupta[2].Many authors presented some mathematical results on flow and heat transfer and a good amount of work can be found in the papers by Liao and Pop[3]and Nazaret al.[4].Nadeemet al.[5]considered the three-dimensional flow through a porous stretching sheet under the effects of a magnetic field.Akbaretal.[6]investigated numerically the flow ofnon-Newtonian hyperbolic tangent fluid through a stretching sheet under the impact of a magnetic field.Akbaret al.[7]presented a dualsolution for Magnetohydrodynanic(MHD)stagnation point flow of Prandtl fluid through a shrinking sheet.Recently,Akbaret al.[8]analyzed numerically the effects of magnetic on non-Newtonian Ree-Eyring fluid through a stretching sheet.

The heat,mass and momentum transport on a continuously moving or stretching sheet has several applications in electrochemistry and polymer processing.The application of boundary layer techniques to mass transfer has been of considerable assistance in developing the theory of separation processes and chemical kinetics.The heat transfer problem associated with the boundary layer micropolar fluid flow under different physical conditions has been studied by several authors[9–11].Takharet al.[12]for example,considered diffusion of a chemically reactive species from a stretching sheet.The free convective flow of viscoelastic fluid in a vertical channel with Dufour effect was studied by Mishraet al.[13].Also,effects of chemical reaction,heat and mass transfer on non-linear Magnetohydrodynamic(MHD)laminar boundary layer flow over a wedge with suction or injection was studied by AnjaliDeviand Kandasamy[14].Baagetal.[15]studied Magnetohydrodynamic(MHD)micropolar fluid flow toward a stagnation point on a vertical surface with heat source and chemical reaction.Mohantyet al.[16]presented chemical reaction effect on Magnetohydrodynamic(MHD)Jeffery fluid over a stretching sheet with heat generation/absorption.Recently,Tripathyet al.[17]studied the effects of chemical reaction effect on Magnetohydrodynamic(MHD)free convective surface over a moving vertical plane through porous medium.Further,they have[18]presented numerical analysis of hydromagnetic micropolar fluid along a stretching sheet with non-uniform heat source and chemical reaction.

The porous media heat transfer problems have several practical engineering applications such as geothermal systems,crude oil extraction,and ground-water pollution.Also,the study of chemical reaction with heat transfer in the porous medium has important engineering applicationse.g.tubular reactors,oxidation of solid materials and synthesis of ceramic materials.In all the above studies,the effects of both the viscous and Joule heating were neglected since they were of the same order as well as negligibly small[19].The effect which bears a greatimportance on heattransferis viscous dissipation.When the viscosity of the fluid is high,the dissipation term becomes important.For many cases,such as polymer processing which is operated at a very high temperature,viscous dissipation cannot be neglected.El-Amin[20]studied the combined effectofviscous dissipation and Joule heating on Magnetohydrodynamic(MHD)forced convection flow over a nonisothermal horizontal cylinder embedded in a fluid-saturated porous medium.Abo-Eldahab and El-Aziz[21]studied the effect of Ohmic heating on a mixed convection boundary-layer flow of a micropolar fluid from a rotating cone with power-law variation in surface temperature.Dashet al.[22]presented the numerical approach to boundary layer stagnation-point flow past a stretching/shrinking sheet.Pal and Mondal[23]examined the effect of thermal radiation on Magnetohydrodynamic(MHD)non-Darcy flow and heat transfer over a stretching sheet in the presence of Ohmic dissipation.Mohantyet al.[24]studied the Numerical investigation on heat and mass transfer effect of the micropolar fluid over a stretching sheet.Mishraet al.[25]studied the flow of heat and mass transfer on Magnetohydrodynamic(MHD)free convection in a micropolar fluid with a heatsource.Few more investigation on the said topic can be found from references[26–30].

Hence our aim is to study an unsteady Magnetohydrodynamic(MHD)viscous fluid past through a stretching surface embedded in a porous plate having uniform heat source/sink with constant heat and mass fluxes.The viscous dissipation and Ohmic heating terms are considered for high-speed fluid.The governing equations of the problem contain the partialdifferentialequations which are transformed by similarity technique into dimensionless ordinary coupled non-linear differential equations.The obtained dimensionless equations are solved numerically by means of Successive Linearization method(SLM)along with Chebyshev Spectral Collocation method.It is necessary to investigate in detail the distributions of velocity,temperature and concentration across the boundary layer in addition to the surface skin friction coefficient,Nusselt number,and Sherwood number.The layout of this paper is as follows:after the introduction in Section 1,Section 2 deals with the mathematical formulation of the problem.Section 3 describes the transformation model.Section 4 characterizes the solution methodology and finally Section 5 devoted to numericaland graphicalresults of all the emerging parameters.

2.Mathematical Formulation

Consider an unsteady two dimensional laminar boundary layer flow of a viscous incompressible fluid over an elastic,vertical and impermeable stretching sheet which emerges vertically in the upward direction from a narrow slot with velocity

which varies hydrodynamic boundary layer stretching sheet both along spatial and time coordinate.It is apt to note that,the expressions forUw(x,t)andTw(x,t)in Eqs.(1)and(2)are valid only for timet＜α?1unless α=0.The Expression(2)for the temperatureTw(x,t)of the sheet represents a situation in which the sheet temperature increases(reduces)ifTrefis positive(negative)fromT∞at the leading edge in proportion tox2and such that the amount of temperature increase(reduction)along the sheet increases with time.The strength is applied transversely to the direction of the flow.The magnetic Reynolds number of the flow is taken to be sufficiently small enough,so that the induced magnetic field can be neglected in comparison with applied magnetic field so thatB(0,B0,0),whereB0is the uniform magnetic field acting normal to the plate.The model of first order chemical reaction has been considered following Bhattacharyya[26].Under the usual boundary layer approximation,the governing equations can be written in the following form:

The continuity equation

The corresponding boundary conditions for the regime are

whereQ0represents the heat source whenQ0＞0 and the heat sinkQ0＜0.

3.Transformation Model

We now introduce dimensionless variablesfand θ and similarity variable η as

The velocity components are readily obtained as

The mathematical problem de fined in Eqs.(4)–(6)are then transformed into a set of ordinary differential equations and their associated boundary conditions

where prime denotes ordinary differentiation with respect to η andfwdenotes suction parameter

Fromthe engineering pointof view,the mostimportantcharacteristics of the flow are the local skin-friction coefficientCfx,the local Nusselt numberNuxand the Sherwood numberShxwhich are,respectively,de fined by

4.Method of Solution

We apply the Successive linearization method to Eq.(11)with their boundary conditions in Eq.(14),by setting[35,36]

wherefIare the unknown functions which are obtained by iteratively solving the linearized version of the governing equation and assuming thatfI(0≤N≤I?1)are known from previous iterations.Our algorithm starts with an initial approximationf0which satis fies the given boundary conditions in Eq.(14)according to SLM.The suitable initialguess for the governing flow problem is

where L and N are the linear and non-linear part of Eq.(11).By substituting Eq.(20)in Eq.(11)and taking the linear terms only,we get

We solve Eq.(23)numerically by a well-known method namely Chebyshev spectral collocation method[30–34].For numerical implementation,the physical region[0,∞]is truncated to[0,Γ]where Γ is taken to be sufficient large.The truncated region is transformed in to[?1,1],using the transformation

We de fine the following discretization between the interval[?1,1].Now,we can apply Gause–Lobatto collocation points to de fine the nodes between[?1,1]by

with(N+1)number of collocation points.Chebyshev spectral collocation method is based on the concept of differentiation matrix D.This differentiation matrix maps a vector of the function values G=[f(Ω0),…,f(ΩN)]Tthe collocation points to a vector G′is de fined as

The entries ofmatrix D can be computed by the method proposed by Bhattiet al.[35,36].Now,applying the spectral method,with derivative matrices on linearized equation Eqs.(23)and(24),we getthe following linearized matrix system

In the above equation Bs，I?1(s=0，1，…3)are(N+1)×(N+1)diagonal matrices with Bs，I?1(ΩJ)on the main diagonal and

After employing the boundary conditions Eq.(30),the solutions forfIare obtained by iteratively solving the system Eq.(29)starting from the initial approximation given in Eq.(21).We obtained the solution forf(η)from solving Eq.(31)and now Eqs.(12)and(13)are now linear,therefore,we will apply the Chebyshev pseudo-spectral method directly;we get

Table 1Numerical values of?f′′(0),?θ′(0)and??′(0)for different values of Sc，γ，S，Ec，A，K P ，F(xiàn) and M

with their corresponding boundary conditions boundary conditions

where H=(θ(ΩJ) ,?(ΩJ) ),R is the set of linear coupled equation of temperature and concentration,S is a vector of zeros,and all vectors in Eq.(33)are converted into diagonal matrix.We imposed the boundary conditions in Eqs.(34)and(35)on the first and last rows of R and S,respectively.

5.Numerical Results and Discussion

This section deals with graphical and numerical results of all the emerging parameters arising in the governing flow problem(Fig.1).All the numerical computations are performed with the help of computational software“Matlab”in order to analyze effects of Hartmann numberM,local inertia parameter F,porous parameterKP, unsteady parameterA,PrandtlnumberPr,Heatsource parameterS,chemicalreaction parameterγ,EckertnumberEcand SchmidtnumberSc.Particularly,we discussed their impact graphically on velocity pro file,temperature pro file and concentration pro file.For numerical computations,we have considered the following parametric values:Sc=1，Ec=0.2，A=0.9，M=1.2，KP=0.1，fw=0.5，γ=1，Pr=6，S=0.5，F(xiàn)=0.2.

Fig.1.Geometry of the flow problem.

The expression for skin friction coefficient,localNusselt number and Sherwood number are computed against all the parameters and shown in Table 1.In order to examine the accuracy of the present technique(SLM),we have presented a comparison in Tables 2 and 3.The numeric value ofΓusually depends upon the physicalparametersinvolved in the problem.For our current numerical computations,we have considered Γ=20 and the number of collocation pointsN=80.These parametric values are excellent combination for all the physical parameters and provide the best results.The numerical comparison for?f′(0)is presented with existing published results[37]by taking F=0 as a special case of our study as shown in Table 2.Table 3 described the numerical comparison of?θ′(0)with previously published data[2,39].From Tables 2 and 3,we observed that the present results are in excellent agreement with previously published results which con firms the validity of the present methodology.

Figs.2 and 3 show the variation of velocity pro file against multiple values ofHartmann numberM,localinertia parameter F，porous parameterKPand unsteady parameterA.It is depicted from Fig.2 that when local inertia parameter F increases then the velocity of the fluid decreases but very slowly.Physically,when the inertial force is applied to any moving fluid then it opposes the fluid and hence the velocity of the fluid tends to decrease.However,the behavior ofthe velocity pro file remains similar against the porosity parameterKP. It can be observed from Fig.3 that when the unsteady parameter increases then it causes a reduction in the velocity pro file.We also found that when the Hartmann numberMincreases then it tends to diminish the velocity pro file.Physically,when the magnetic field is applied to any conducting lf uid,an opposite force originated which is known as the Lorentz force which tends to reduce the velocity ofthe fluid.Figs.4 and 5 are preparedfor temperature pro file against different values of porous parameterKP,Heat source parameterS,Prandtl numberPrand Eckert numberEc.It is depicted from Fig.4 that when the porosity parameterKPincreases then the temperature pro file reduces because an increment in porosity parameterKPopposes the flow and hence more force is required to move the fluid.As a result,the temperature pro file increases.Moreover,we also observed that with the increment in heat source parameterS,the temperature pro file also increases but very slightly.From Fig.5,we can see that for higher values of Prandtl numberPrthe temperature pro file diminishes.An incrementin Prandtlnumber is related to weaker thermal diffusivity,and those fluids which have weaker thermal diffusivity contains lower temperature.From this figure,we can also notice that the Eckert numberEcshows opposite behavior on temperature pro file as compared to the Prandtl numberPr.It can be noticed from Fig.6 that large values of Schmidt numberScand chemical reaction parameter γ cause a significant reduction in the concentration pro file.

Table 2Numerical comparison of?f′(0)with existing published results for different values of F

Table 3Numerical comparison of?θ′(0)with existing published results for different values of Pr by taking f w =S=Ec=M=F=γ=S=0

Fig.2.Velocity pro file for different values of K P and F.

Fig.3.Velocity pro file for different values of M and A.

Fig.4.Temperature pro file for different values of K P and S.

Fig.5.Temperature pro file for different values of Ec and Pr.

Fig.6.Temperature pro file for different values of Sc and γ.

6.Conclusions

In this article,we have considered the simultaneous effects of chemical reaction and ohmic heating with heat and mass transfer flow through a stretching sheet.The effect of porosity and magnetic field are also taken into consideration.The governing flow problem is simplified with the help of similarity transformation model.The resulting highly nonlinear coupled differential equation is solved numerically with the help of successive linearization method(SLM)and the Chebyshev spectral collocation method.The impact of all physical parameters is discussed with the help of graphs and tables.The major outcomes of the present study are described below:

?Velocity of the fluid tends to decrease due to inertia parameter and porosity parameter.

?With the increment in Hartmann number and unsteady parameter,velocity of the fluid rises.

?Porosity parameter and heat source parameter enhance the temperature pro file.

?Prandtl number and Eckert number show differentbehaviors on temperature pro file.

?Schmidt number and chemical reaction parameter cause a reduction in the concentration pro file.

?Numerical comparison is also presented by taking F=0 as a special case ofour study and found that the presentresults are in good agreement which also con firms the validity of the present methodology.

Nomenclature

Aunsteady parameter

apositive constant

B0applied Magnetic field

Cconcentration of solute

Cbdrag coefficient

Cfxlocal skin friction coefficient

CPspecific heat at constant pressure

Cwconcentration of solute at sheet

C∞concentration of solute at in finity

D′ molecular diffusivity

EcEckert number

F local inertia coefficient

KC*chemical reaction rate constant

KP*permeability of porous medium

kthermal conductivity

MHartmann number

NuxNusselt number

PrPrandtl number

Q0heat source coefficient

Rexlocal Reynolds number

Sheat source parameter

ScSchmidt number

Shxlocal Sherwood number

Tnon-dimensional temperature

TwWall temperature

T∞Temperature at in finity

ttime

Uwsheet velocity

u,vvelocity components

x,yCartesian coordinate

α stretching rate

β thermal expansion coefficient

γ chemical reaction parameter

η similarity variable

θ dimensionless temperature

μ fluid viscosity

μedynamic viscosity of fluid

ν kinematic viscosity

ρ density of the fluid

σ electrical conductivity

τwwall shear stress

? dimensionless concentration

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