Similarity Analysis for Effects of Variable Diffusivity and Heat Generation / Absorption on Heat and Mass Transfer for a MHD Stagnation-Point Flow of a Convective Viscoelastic Fluid over a Stretching Sheet with a Slip Velocity

A mathematical analysis has been carried out for stagnation-point heat and mass transfer of a viscoelastic fluid over a stretching sheet with surface slip velocity, concentration dependent diffusivity, thermal convective boundary conditions, and heat source/sink. The governing partial differential equations are reduced to a system of nonlinear ordinary differential equations using Lie group analysis. Numerical solutions of the resulting ordinary differential equations are obtained using shooting method. The influences of various parameters on velocity, temperature, and mass profiles have been studied. Also, the effects of various parameters on the local skin-friction coefficient, the local Nusselt number, and the local Sherwood number are given in graphics form and discussed.


Introduction
Practical applications have demand to find the solution of differential equations governing the motion of non-Newtonian fluids.Several types of non-Newtonian fluids exist, but one of the public ones is the viscoelastic fluid; they can exhibit a response that resembles that of an elastic solid under some circumstances or the response of a viscous liquid under other circumstances.In nature, fluids that exhibit this behavior are macromolecular such as polymeric fluids used to make plastic articles, food systems such as dough used to make bread and pasta, and biological fluids such as synovial fluids found in joints.Flow of a viscoelastic fluid and heat transfer phenomena over a stretching surface are encountered in a wide range of chemical and engineering applications, particularly in manufacturing process of artificial film, artificial fibers, polymer extrusion, drawing of plastic films and wires, glass fiber and paper production, manufacture of foods, and crystal growing.Rajagopal et al. [1] analyzed the flow of secondorder fluid over a stretched surface.Dandapat and Gupta [2] extended this problem to present the exact analytical solutions of the nonlinear equations governing this problem.Vajravelu and Rollins [3] studied heat transfer characteristics in a viscoelastic fluid over a stretching sheet with frictional heating and internal heat generation/absorption.
Flow and mass transfer of a viscoelastic fluid over a stretching sheet were studied by many authors.Kelly et al. [4] investigated the analytical solution for the problem of heat and mass transfer of an MHD viscoelastic fluid over a porous stretching sheet.Sanjayanand and Khan [5] carried out the problem of heat and mass phenomena in a viscoelastic fluid flow over an exponentially stretching impermeable sheet.Cortell [6] presented the numerical analysis of flow and mass transfer for two MHD viscoelastic fluids, namely, secondgrade and second-order non-Newtonian fluids, over a porous stretching sheet.Mahmoud [7] studied the effects of variable viscosity, surface slip velocity, and chemical reaction on the flow and mass transfer of a viscoelastic fluid immersed on porous medium over a stretching surface.Heat transfer for an electrically conducting viscoelastic fluid flow over a stretching sheet under the influence of a magnetic field in presence of heat source is considered to be of significant importance due to its application in many engineering problems such as magnetohydrodynamic (MHD) generator, plasma studies, nuclear reactors, oil exploration, geothermal energy extractions, and the boundary-layer control in the field of aerodynamics.Recently, researches in these fields have been studied by several authors.Abel et al. [8] studied the problem of heat transfer for an incompressible electrically conduction viscoelastic fluid flow over a linear stretching sheet in the presence of viscous and ohmic dissipation.Singh [9] investigated the analytical solution of the effects of uniform magnetic field, thermal radiation, and heat source on the flow and heat transfer of a viscoelastic fluid over a stretching flat plat; two types of heating processes are studied: (i) prescribed power law surface temperature (PST) and (ii) prescribed power law surface heat flux (PHT).The influence of the variable thermal conductivity on the flow of an MHD viscoelastic fluid over a stretching surface in the presence of thermal radiation, viscous dissipation, and a nonuniform heat source was carried out by Abel and Mahesha [10], they studied the problem under the two different types of boundary conditions PST and PHF boundary conditions.Hsiao [11] examined the effects of magnetic and electric dissipation on the MHD flow of a second grade fluid in the presence of heat source/sink.The analytical solution for the flow of two types of viscoelastic fluids in the presence of Joule heating and viscous dissipation and internal heat generation/absorption over a stretched surface has been found by Chen [12].Nandeppanavar et al. [13] considered the problem of heat transfer of a MHD viscoelastic fluid over an impermeable stretching sheet with nonuniform heat source/sink.
In all the previous researches, the diffusion coefficient is taken to be constant.However, there are some problems where the diffusion coefficient varies with the concentration over a certain range.The diffusion coefficient can often be approximated by a linear or exponential dependence equation [14][15][16].
In recent years, the stagnation-point flow of viscoelastic fluid past a stretching surface has attracted the interest of many investigators.Mahapatra and Gupta [17] studied the laminar steady stagnation-point flow of a viscoelastic fluid over a stretching surface; they studied the flow when the stretching velocity of the surface is more (less) than the freestream velocity.Ayub et al. [18] extended the results of Mahapatra and Gupta [17]; they solved the problem analytically using HAM method.Li et al. [19] investigated the problem of steady mixed convection flow of viscoelastic fluid over a heated or cooled vertical flat plate near a stagnation point.
Lie group analysis, also called symmetry analysis, was developed by Sophius Lie to find point transformations which map a given partial differential equation to ordinary differential equation [20][21][22][23][24][25].An extensive literature exist on this subject, and Bhuvaneswari et al. [26] examine the effect of chemical reaction on the natural convection flow and mass transfer of a steady viscous fluid past a semi-infinite inclined surface using Lie group analysis.Abd-el-Malek and Hassan [27] studied the symmetry analysis for the steady twodimensional boundary-layer stagnation-point flow of Rivlin-Ericksen fluid of second-grade with a uniform suction.Jalil et al. [28] calculated Lie symmetries of the equations of mixed convection flow and mass transfer over a stretching surface with suction/injection.The similarity solution for the effect of generation/absorption, suction/blowing on the flow, and mass transfer of a nanofluid over a stretching sheet in a porous medium near a stagnation point is investigated by Hamad and Ferdows [29].
The aim of the present study is to find new similarity transformations with their corresponding similarity solutions and to investigate heat and mass transfer of an MHD viscoelastic fluid moving over a stretching surface near a stagnation point in the presence of slip velocity, heat source/sink, and the concentration dependent thermal diffusivity.

Mathematical Formulation of the Problem
Consider the two-dimensional steady flow of an incompressible electrically conducting viscoelastic fluid past a stretching sheet which coincides with the plane  > 0 in the presence of heat source and a uniform magnetic field applied normal to the plate.The magnetic Reynolds number of the flow is taken to be small enough so that the induced magnetic field is assumed to be negligible.The down surface of the plate is heated by convection from a hot fluid of temperature   which provides a heat transfer coefficient ℎ  , as shown in Figure 1.Under the boundary-layer approximations the governing equations of the continuity, momentum, energy, and concentration for the steady flow can be written as [7] subject to the boundary conditions Introducing the following dimensionless variables where Re =  0 /] is the Reynolds number, then the dimensionless forms of ( 1)-(2b) are The dimensionless boundary conditions are where  =  0  0 /] is the viscoelastic parameter,  =  2 0 / 0 is the magnetic parameter, Pr = ]  / is the Prandtl number,  = / 0   is the heat generation ( > 0)/absorption ( < 0) parameter, Sc = ]/  is the Schmidt number,  =  1 ] √ Re/ is the slip parameter, and  = (ℎ  /)√]/ 0 is the convective parameter.Here we assumed that the variable concentration diffusivity given by the linear relation [16] where  is a constant and  is the concentration diffusivity parameter.
Introducing stream function  as then ( 4)-(5b) will have the form The related boundary conditions are

Symmetry Analysis
A symmetry admitted by ( 8)-( 10) is a transformation mapping any solution of ( 8)-( 10) into another solution of ( 8)- (10).Consider the one-parameter () Lie group of infinitesimal transformations leaving ( 8)- (10) where  is the group parameter and   ,   ,   ,   ,   , and    are infinitesimal elements of the group.The associated Lie algebra of infinitesimal symmetry is the set of the vector field of the form where   = , , , and   for  = 1, . . ., 4, respectively,   = ,  for  = 1, 2, respectively.The action of  is extended to all derivatives appearing in ( 8)- (10) through the fourth prolongation pr (4) where where , , , , and  = 1, 2 ,  = 1, 2, . . ., 4, and   is the total derivative operator defined as The operator  is a point symmetry of ( 8)- (10) if where   = 0,  = 1, 2, 3, for ( 8)-( 10).Expanding the system (17) with the aid of mathematica and setting the coefficients of like derivatives of , , , and   in system equal zero, one obtains an overdetermined linear system of PDEs for infinitesimal   ,   ,   ,   ,   , and    .Solving these sets of equations, we obtain the components of symmetries as follows: where  1 ,  2 ,  3 ,  4 , and  5 are constants.The characteristic equation for obtaining the similarity transformation would then be The resulting similarity variable is and the similarity functions are where  is an integration constant.Moreover, in our problem to satisfy the boundary conditions (11),  3 ,  4 , and  5 must equal zero.This implies that the similarity variable (20) where  =  2 / 1 , which may be taken to equal zero, and   is the velocity of the stretching surface.Substituting ( 22) into ( 8)- (11) we get the following system of ordinary differential equations: with the corresponding boundary conditions where The parameter  measures the ratio between the free-stream velocity to the plate velocity; that is, where  > 1, the freestream velocity is greater than the plate velocity, and while  < 1, the free-stream velocity is smaller than the plate velocity.The exact solution for (23), subjected to boundary conditions (26a) and (26b) can be obtained for ( =  = 0) [4] Also, we can solve (25) analytically with  = 0 by using the transformation Then (25) will be transformed to subject to boundary conditions The exact solution to (31) satisfying the boundary conditions (32) in terms of  is where Γ() is Gamma function and Γ(, ) is the incomplete Gamma function.The rate of heat transfer is given as The physical quantities of interest are the local skinfriction coefficient   , the local Nusselt number Nu  , and the local Sherwood number Sh  , which are defined, respectively, as Re 1/2  = − (1 − 3  (0))   (0) , where Re  = √   /] indicates the local Reynolds number.

Numerical Results and Discussion
For this present system of nonlinear differential equations ( 23)-( 25), subject to boundary conditions (26a) and (26b), numerical computations have been carried out by fourthorder Runge-Kutta integration with the shooting scheme.
In order to assess the accuracy of our results, we have compared our results for ( 23) and ( 25), subjected to boundary conditions (26a) and (26b) with the analytical solutions ( 29) and (35) as shown in Table 1.
The comparison shows a good agreement.Numerical results are plotted in Figures 2-16 to exhibit the influences of the various parameters on the velocity, heat, and mass transfer with the boundary layer.The values of Sc increase from 0.1 (low weight diffusing gas species) through 0.5 (oxygen diffusing) to 1 (denser hydrocarbon derivatives as the diffusing species), 3, and 5 for more denser species [30].
Figures 2 and 3 depict the variations in the velocity for different numerical values of viscoelastic parameter  for   = 1.5 and  = 0.5, respectively.From Figure 2 it can be notice that the velocity   decreases with an increase in  near the surface, and the reverse is true away from the surface.It was observed from Figure 3 that velocity   increases with an increase in  near the surface, and the reverse is true away from the surface.Figures 4 and 5 show the velocity distribution for various values of the magnetic field .From Figure 4 we can notice that for  = 1.5 an increase in the value of magnetic parameter  leads to an increase in the velocity profile within the boundary layer.But Figure 5   decreases the velocity ; this is because the application of a transverse magnetic field to an electrically conducting fluid gives rise to a resistive-type force called the Lorentz force.This force has the tendency to slow down the motion of the fluid in the boundary layer.Figures 6, 7, and 8 illustrate the influence of slip velocity parameter on the velocity and temperature profiles with  = 0.5 and 1.5, respectively.It shows that slip parameter increases the velocity   for  = 1.5 and decreases for  = 0.5 and vice versa with respect to the temperature . Figure 9 shows the temperature profile for different values of the convective parameter ; one can find that the temperature profiles and their corresponding thicknesses of the boundary layer increase as  increases.
Figure 10 illustrates the effect of  on the concentration gradient; one can observe that the concentration gradient decreases when  increases for both  = 0.5 and  = 1.5.Figure 11 presents typical profiles for the temperature for various values of a heat source ( > 0) or a heat sink ( < 0), respectively.As shown, the temperature is increasing with increasing heat source, but it decreases with increasing heat sink.In all cases, the magnitude of  effect causes an increasing in the thermal boundary-layer thickness.
Figure 12 illustrates the variation of the temperature profiles  over a range of Prandtl number Pr.It is noticed that the temperature is decreasing as Pr increases.The influence Prandtl number Pr on the thickness of the thermal boundary layer is according to the well-known relation   / ≈ Pr −1/2 , where   is the thickness of the thermal boundary layer and  is the thickness of the velocity boundary layer.So, the thickness of the thermal boundary layer increases as Pr decreases, and hence the temperature  increases with the decrease of Pr.
Figure 13 presents the behavior of the concentration profile for the variation of Schmidt number Sc; it is seen that the decrease in the values of Sc increases the concentration.Figures 14, 15, and 16 display the effect the parameter  on the velocity, temperature, and concentration, respectively.It is clear that the velocity increases as the parameter  increases, but the temperature and the concentration decrease by increasing .

Mathematical Problems in Engineering
Figure 17 describes the effect of the viscoelastic parameter  and the magnetic parameter  on local skin-friction coefficient for two cases  = 1.5 and  = 0.5 keeping all other parameters fixed.It is noticed that as  increases, the local skin-friction coefficient decreases for  = 1.5 and  = 0.5.Also, it is observed that for a fixed value of  the local skinfriction coefficient increases with increasing  for  = 1.5 and  = 0.5.
The variations of the local Nusselt number with  for various values of  for two cases  = 0.5, and 1.5 when all other parameters fixed are shown in Figure 18.It is found that the local Nusselt number increases with increasing  for  = 1.5 while it decreases for  = 0.5 with fixed value of .Also, the local Nusselt number increases with increasing  for  = 1.5 and  = 0.5 at a fixed value of .number decreases with the parameter  increasing for  = 0.5 and  = 1.5 for a fixed value of Pr.However, it increases by increasing Pr at a fixed value of  and for  = 0.5 and  = 1.5.
The influence of the parameters , Sc on the local Sherwood number has been described in Figure 20.It is shown that dimensionless concentration rates at the wall has a higher value when the parameters , Sc has high values for  = 0.5 and  = 1.5.

Conclusion
In this work, we have used Lie group method to obtain similarity reductions of nonlinear boundary-layer equations for the two-dimensional MHD steady, laminar flow, and mass transfer of a viscoelastic fluid near a stagnation point; this flow is over a stretching surface with slip velocity; the surface is heated from down by a hot fluid.By determining Lie transformation group under which the given partial differential equations are invariant, we obtained the invariants and the symmetries of these equations.The resulting system of nonlinear ordinary differential equations is solved numerically using the shooting method coupled with Runge-Kutta scheme.The effects of the viscoelastic parameter , magnetic parameter , slip velocity parameter , convective parameter , concentration diffusivity parameter , Prandtl number Pr, Schmidt number Sc, and free-stream velocity  parameter  have been studied.It was found that with the increasing in viscoelastic parameter the velocity decreases near the surface, and the reverse is true away from the surface for  = 1.5, whereas the velocity increases with an increase in  near the surface, and the reverse is true away from the surface for  = 0.5.The convective parameter  increases the temperature and thicknesses of the boundary layer.A concentration diffusivity parameter  decreases the concentration.

Figure 1 :
Figure 1: Physical model and the coordinate system.

Figure 2 :
Figure 2: Velocity profiles for various values of  with  = 1.5.

Figure 6 :
Figure 6: Velocity profiles for various values of .

Figure 19 Figure 14 : 1 𝑆Figure 15 :
Figures 14,15,and 16 display the effect the parameter  on the velocity, temperature, and concentration, respectively.It is clear that the velocity increases as the parameter  increases, but the temperature and the concentration decrease by increasing .Figure17describes the effect of the viscoelastic parameter  and the magnetic parameter  on local skin-friction coefficient for two cases  = 1.5 and  = 0.5 keeping all other parameters fixed.It is noticed that as  increases, the local skin-friction coefficient decreases for  = 1.5 and  = 0.5.Also, it is observed that for a fixed value of  the local skinfriction coefficient increases with increasing  for  = 1.5 and  = 0.5.The variations of the local Nusselt number with  for various values of  for two cases  = 0.5, and 1.5 when all other parameters fixed are shown in Figure18.It is found that the local Nusselt number increases with increasing  for  = 1.5 while it decreases for  = 0.5 with fixed value of .Also, the local Nusselt number increases with increasing  for  = 1.5 and  = 0.5 at a fixed value of .Figure19shows the parameters  and Pr effects on the local Nusselt number.It is observed that the local Nusselt