Analysis of MHD boundary layer flow of an Upper- Convected Maxwell fluid with homogeneous-heterogeneous chemical reactions

An analysis of MHD flow and mass transfer of an upper-convected Maxwell fluid with homogeneousheterogeneous first-order chemical reactions is presented. The flow is driven by a stretching surface and homogeneous-heterogeneous reactions. The governing nonlinear partial differential equations are first cast into ordinary nonlinear differential equations. Then analytical solutions are obtained in series form by the homotopy analysis method (HAM). The obtained results are compared with the existing results in the literature for some special cases and the obtained results are found to be in good agreement. The physical significance of different parameters on the velocity and the concentration fields are presented graphically and discussed. Also, the residual errors of the solutions for several sets of the parameters are obtained and presented.


Introduction
Homogeneous and heterogeneous reactions are the part of several chemically reacting systems such as combustion and catalysis.These reactions are usually seen in food processing, hydrometallurgical industry, manufacturing of ceramics and polymer products, fog formation and dispersion, chemical processing equipment design, crops damage via freezing.More often than not homogeneous and heterogeneous reactions occur in the bulk of fluid and on some catalytic surfaces respectively, which are generally very complex in nature.These reactions involve the production and consumption of reactant species at different rates both within the fluid and on the catalytic surfaces.Chaudhary and Merkin [1][2][3] developed a mathematical model of a homogeneous-heterogeneous reaction in stagnation-point flow and assumed reacting systems as first order.Merkin [4] studied the model for isothermal homogeneous-http://www.ispacs.com/journals/cna/2017/cna-00324/International Scientific Publications and Consulting Services heterogeneous reactions in flow over a flat plate.Khan and Pop [5] extended the model in [1] to mass transfer and Bachok et al. [6] obtained the unique solution in the case of stretching sheet and dual solution for the shrinking case.Further, Khan and Pop [7], Kameswaran et al. [8], Shaw et al. [9], Nandkeolyar et al. [10] and Hayat et al. [11][12][13] analyzed the model proposed in [1] for non-Newtonian fluid.Recently, Ryan and Harsha [14] used the previously developed lumped heterogeneous reaction models for carbon surfaces and a detailed homogeneous reaction model for carbon monoxide oxidation.Several researchers worked on different mathematical models by considering chemically reactive species (see Ref. [15][16][17][18][19][20][21]).None of the aforementioned researchers considered the upper convected Maxwell (UCM) fluid to analyze the model proposed by Chaudhary and Merkin [1].In the case of large deformations, the UCM model is a generalization of the Maxwell material model and the UCM serves as a general form of the visco-elastic fluid model.In view of this, Sadeghy et al. [22] reported that the wall skin friction coefficient decrease with an increase in the Deborah number for Sakiadis flow of a UCM fluid.Vajravelu et al. [23] extended the work of Akyildiz et al. [16] by considering the non-Newtonian UCM fluid over a permeable surface.Also, Mukhopadhyaya et al. [24] and Palani et al. [25] analyzed the flow of an UCM fluid over a stretching surface.Furthermore, several researchers reported the behavior of non-Newtonian fluids, namely, visco-elastic and UCM models through different geometries [26][27][28][29][30]. Motivated by these studies, in the present paper, we study the MHD UCM fluid flow at a stretching sheet using the model proposed by Chaudhary and Merkin [1].That is, this is an extension over the work of Chaudhary and Merkin [1] where we consider the effects of non-Newtonian Maxwell fluid with MHD and chemical reactionss.The presence of chemical reactions in the non-Newtonian UCM fluid led to a system of coupled nonlinear partial differential equations.These coupled equations for the momentum and the mass diffusion are reduced to a set of nonlinear coupled ordinary differential equations through a similarity transformation and are solved for various values of the pertinent parameters by an efficient analytical method known as the homotopy analysis method.The present results are not only useful for industrial applications; but also will provide a basic understanding of the physical model.

Mathematical Model
Let us consider a steady, two-dimensional boundary layer flow of a viscous incompressible and electrically conducting non-Newtonian UCM fluid with a homogeneous-heterogeneous chemical reaction.The chemical reaction model proposed by Chaudhary and Merkin [1] where the isothermal first order reaction on the catalyst surface is given by , rate = .
Here, a and b are the concentrations of the chemical species A and , B respectively, and c K and s K are the reaction rate constants.Both reaction processes are assumed to be isothermal.Equations (2.1) and (2.2) have been used for boundary layer flow over a stretching sheet by Chaudhary and Merkin [1] for a basic model with reactions.This model guarantees that the reaction rate will be zero in the external flow and thus zero at the outer edge of the boundary layer.Further, the flow region is exposed to a uniform transverse magnetic field ) 0 , , 0 ( 0 B  B and the imposition of such a magnetic field stabilizes the boundary layer flow (Vajravelu et al. [23]).It is assumed that the flow is generated by the stretching of an elastic sheet from a slit by imposing two equal and opposite forces in such a way that the velocity of the boundary sheet is linear.It is also assumed that the magnetic Reynolds number is very small and the electric field due to polarization of charges is negligible.Therefore, the first step would be to derive the boundary layer equations for our fluid of interest, which can be done by using the Cauchy equations of motion in which a source term due to the magnetic field should also be included (Bird et al. [31]).For a two-dimensional flow, the equations of continuity, the momentum and the mass transfer can be written as http://www.ispacs.com/journals/cna/2017/cna-00324/International Scientific Publications and Consulting Services v 0, u xy where u and v are the velocity components along the x and y axes respectively,  is the fluid density, σ is the electrical conductivity, 0 B is the uniform magnetic field, and where  is the coefficient of viscosity and  is the relaxation time of the period.The time derivative t   appearing in the above equation is the so called upper-convected time derivative devised to satisfy the requirements of the continuum (i.e., material objectivity and frame difference).This time derivative when applied to the stress tensor reads as follows (Bird et al. [31]), , where ij L is the velocity gradient tensor.For an incompressible fluid obeying the UCM model, the momentum and mass transfer equations can be reduced to (for details see Sadeghy et al. [22]) 12) The boundary conditions for the problem are 0 , v 0, , at 0, 0, , b 0 as .

Similarity equations
Using the boundary layer approximations and the above mentioned homogeneous and heterogeneous chemical reactions, the governing equations (2.10)-(2.12) in terms of stream function  can be written as, The stream function  In terms of the new variables, the velocity components and the dimensionless diffusion terms can be written as, where 0 The boundary conditions can be written as where the prime denotes differentiation with respect to  , , S , , and = .
In most applications, it can be assumed that the diffusion coefficients of chemical species A and B are of comparable sizes.This leads to the assumption that the diffusion coefficients

Homotopy analysis method
The pioneering work of Liao [32] has paved the way to solve nonlinear differential equations by homotopy analysis method (HAM).The method is a pairing of the perturbation technique and the concept of homotopy in topology.The main advantage in HAM is the freedom in selecting the auxiliary linear operators, initial approximations, and the auxiliary parameters.This yields fast convergence of the series solution.In many cases, convergence can be obtained only a few iterations.Various nonlinear problems have been solved via HAM which include Lane-Emden equation [33], Zakharov system [34], non-local Whitham equation [35], time-dependent Michalis-Menten equation Li et al. [36], etc.Here, we use the HAM to obtain solutions to Eqs. ( where [0,1] q  is an embedding parameter and f and g are non-zero convergence control parameters.The nonlinear differential operators f and g are given by The m th -order deformation equations are For 0 q  and 1 q  , we have 00 ˆ( ,0) ( ), ( ,0) ( ) and ˆ( ,1) ( ), g( ,1) ( ) respectively.
Therefore, as q varies from 0 to 1, ˆ( , ) we may expand ˆ( , ) f n q and ˆ( , ) g n q about q by way of Taylor's series expansion, obtaining .
The values of the auxiliary parameters f and g are chosen in such a way that the series in Eq. (3.51) are convergent at 1 q  , i.e., 00 .
The general solutions are

Error analysis
The HAM treatment provides great freedom in adjusting the convergence region of the series solution (3.52).We may control convergence by selecting the optimal values of the auxiliary parameters f and g .In order to select the optimal values, we evaluate the residual error and minimize over f and g .For the m th -order solution, the exact squared residual error is given by 22 00 ( , ) ( In practice, it is often difficult to evaluate the integral in (4.54).Instead, we consider an average squared residual error  [15] and Vajravelu et al. [23] (see Table 1), for some special cases and the results are found to be in good agreement.Table 2 shows residual errors of 10 -4 − 10 -7 for 3 rd -order HAM solutions.

Results and discussion
In this section, we discuss the effects of pertinent parameters on the velocity            In the present study, we have investigated MHD flow and mass transfer of an upper-convected Maxwell fluid with homogeneous-heterogeneous first-order chemical reactions.The presence of chemical reactions in the non-Newtonian UCM fluid led to a system of coupled nonlinear partial differential equations.These equations for the momentum and the mass diffusion are reduced to a set of nonlinear coupled ordinary differential equations through a similarity transformation and are solved for several sets of values of the pertinent parameters by an efficient analytical method known as the homotopy analysis method.As expected, the velocity distribution decreases with increasing magnetic parameter.Hence an increase in the velocity gradient (that is, the skin friction) is noticed.A similar trend is seen with the Maxwell parameter.A decrease in the concentration field is observed with increasing values of the chemical reaction parameter, while in the case of the Schmidt number the opposite trend is observed.

D
are diffusion species coefficients of A and B , respectively.As mentioned earlier, the fluid of interest in the present work obeys the UCM fluid model.For a Maxwell fluid the extra tensor ij  can be related to the deformation rate tensor ij d by an equation of the form 2, the stretching velocity of the boundary, with c > 0 for stretching and c < 0 for shrinking, and 0 a is a positive constant.http://www.ispacs.com/journals/cna/2017/cna-00324/International Scientific Publications and Consulting Services boundary conditions (3.44) and (3.45).Once the general solutions are available, we may analytically solve the linear equations (3.42) and (3.43) for 1, 2,3,... m 

gff
Figs. 3-5 elucidate the variation of , and sc K K

Figure 1 :
Figure 1: Variation of   f  with  for different values of

Figure 3 :
Figure 3: Variation of   g  with  for different values of K .

Figure 4 :
Figure 4: Variation of   g  with  for different values of

Figure 5 :
Figure 5: Variation of   g  with  for different values of

Figure 6 :Figure 7 :
Figure 6: Variation of ''(0) f  with  for different values of n M

Figure 8 :
Figure 8: Variation of (0) g with s K for different values of c S .

Table 2 :
HAM results of skin friction coefficient for various values of  ,