MHD Convective Flow of Jeffrey Fluid Due to a Curved Stretching Surface with Homogeneous-Heterogeneous Reactions

This paper looks at the flow of Jeffrey fluid due to a curved stretching sheet. Effect of homogeneous-heterogeneous reactions is considered. An electrically conducting fluid in the presence of applied magnetic field is considered. Convective boundary conditions model the heat transfer analysis. Transformation method reduces the governing nonlinear partial differential equations into the ordinary differential equations. Convergence of the obtained series solutions is explicitly discussed. Characteristics of sundry parameters on the velocity, temperature and concentration profiles are analyzed by plotting graphs. Computations for pressure, skin friction coefficient and surface heat transfer rate are presented and examined. It is noted that fluid velocity and temperature through curvature parameter are enhanced. Increasing values of Biot number correspond to the enhancement in temperature and Nusselt number.


Introduction
The study of non-Newtonian fluids has gained special focus of the recent researchers and engineers. Such motivation of the researchers is due to various applications of non-Newtonian fluids in technology and industrial areas. Unlike the viscous materials, the non-Newtonian fluids cannot be explained using well known Navier-Stokes theory. A single constitutive relationship cannot describe the characteristics of non-Newtonian liquids. The facts of non-Newtonian fluids are distinct than the viscous materials. The order of differential system in non-Newtonian fluid situation is higher than the viscous material. There are many proposed models of non-Newtonian fluids with diverse properties. These fluids in general have been classified into three categories known as the rate, the differential and the integral types. The most common and simplest model of non-Newtonian fluids is Jeffrey fluid. Such fluid has time derivative instead of convected derivative. Aspects of retardation and relaxation times are described by this fluid model. MHD flow of Jeffrey fluid in a cylindrical tube has been studied by Tripathi et al. [1]. Influences of slip and heat transfer on MHD peristaltic flow of Jeffrey fluid have been examined by Das [2]. Variable thermal conductivity of Jeffrey fluid in presence of thermal jump has been analyzed by Hamad et al. [3]. Shehzad et al. [4] presented the nonlinear thermal radiation effect in three dimensional flow of Jeffrey nanofluid. Ellahi and Hussain [5] examined slip feature in flow of Jeffrey fluid. Hayat et al. [6] studied stagnation point flow of Jeffrey nanofluid in presence of Newtonian heating. Flow of Jeffrey fluid due to oscillation of disks has been studied by Reddy et al. [7]. Farooq et al. [8] analyzed MHD flow of Jeffrey fluid in presence of Newtonian heating.
Many chemically reacting systems involve homogeneous-heterogeneous reactions for example in biochemical systems, combustion and catalysis. The correlation between homogeneous and heterogeneous reactions is very complex. Some of the reactions have the ability to proceed very slowly or not at all except in the presence of a catalyst. Fog formation and dispersion, food processing, ceramics and polymer production, hydrometallurgical industry etc. show obvious involvement of chemical reaction. Merkin [9] studied homogeneous-heterogeneous reactions in flow of viscous fluid over a flat plate. He considered homogeneous reaction for cubic autocatalysis and heterogeneous reaction on the catalyst surface. It is shown that surface reaction dominants near the plate. Homogenous-heterogeneous reactions with equal diffusivities have been examined by Chaudhary and Merkin [10]. Homogeneous-heterogeneous reactions in stretched flow of viscous fluid have been investigated by Bachok et al. [11]. Khan and Pop [12] presented stretched flow of viscoelastic fluid in presence of homogeneous-heterogeneous reactions. Shaw et al. [13] studied homogeneous-heterogeneous reactions in flow of micropolar fluid. Homogeneous-heterogeneous reactions in nanofluid flow over a permeable stretching surface have been analyzed by Kameswaran et al. [14]. Hayat et al. [15] investigated three dimensional flow of nanofluid in presence of second order slip velocity and homogeneousheterogeneous reactions. Hayat et al. [16] also examined Cattaneo-Christov heat flux in MHD flow of Oldroyd-B fluid. Here homogeneous-heterogeneous reactions are considered.
Two-phase flow has wide applications in many industrial processes such as natural gas networks, spray processes, lubrication and nuclear reactor cooling. Main difference between single phase flow and multiphase flow is the existence of flow pattern which indicates a flow situation uniquely defined by the temporal and spatial distribution of the two immiscible phases. Three typical gas-liquid flow patterns are bubble flow, slug flow and churn flow. Research interests in the characterization of flow patterns lie on the fact that different flow patterns have distinct nonlinear dynamical properties. The recent advances in the study of multiphase flow are presented by Gao et al. [17][18][19][20].
Fluid flow by a stretching surface has promising applications in engineering and industrial processes such as in paper production, manufacture of foods, glass fiber, drawing of wires and plastic films, liquid films in condensation process, crystal growing, manufacturing and extraction of polymer and rubber sheets etc. Flow caused by stretching of a sheet has been examined by Crane [21]. After that stretched flow problems under different configurations have been examined by several researchers. Cortell [22] studied radiative nonlinear heat transfer in flow over a stretching sheet. Hsiao [23,24] presented mixed convection effect in MHD flow of viscoelastic fluid past a stretching sheet with ohmic dissipation. Hsiao [25] examined MHD mixed convection for viscoelastic fluid past a porous wedge. Slip effect in stretched flow of nanofluid have been studied by Malvandi et al. [26]. Hsiao [27] investigated MHD stagnation point flow of nanofluid over a stretching sheet with mixed convection and partial slip. Sheikholeslami et al. [28] discussed effect of thermal radiation in magnetohydrodynamic nanofluid flow over a stretching sheet. Lin et al. [29] analyzed flow of pseudo-plastic nanoliquid over a stretching surface. In all these articles flat sheet is stretched and Cartesian coordinate system is used for mathematical modeling. Sajid et al. [30] provided fluid flow due to curved stretching sheet. They used curvilinear coordinate system in order to obtain the governing equations. They found that pressure is not negligible inside the boundary layer as in the case of a flat stretching sheet. Naveed et al. [31] studied MHD flow by a curved stretching surface. Time-dependent fluid flow due to curved stretching/shrinking surface has been presented by Rosca and Pop [32]. Radiative flow of nanofluid by a curved stretching surface with partial slip has been examined by Abbas et al. [33].
Main objective of present study is to extend the flow analysis of Sajid et al. [30] into following directions. Firstly, to model flow analysis for Jeffrey fluid. Secondly to predict the influence of homogeneous-heterogeneous reactions. Thirdly to examine heat transfer analysis in the presence of convective boundary conditions. Series solutions of present problem are computed by homotopy analysis method (HAM) [34][35][36][37][38][39][40][41][42]. The behaviors of different parameters on the physical quantities have been examined. Pressure, surface drag force and heat transfer rate are also studied. We hope that this study will lead to further investigations in future for various flow geometries and different flow models.

Model development
Consider two-dimensional flow of Jeffrey fluid induced by a curved stretching sheet at r = R. Stretching of sheet is taken in the xdirection with velocity u = u w . A magnetic field of strength B 0 is applied in the rdirection. Also the bottom surface of sheet is heated by convection from a hot fluid at temperature T f while ambient fluid temperature is T 1 . Homogeneousheterogeneous reactions of two chemical species A and B are considered. For cubic autocatalysis, the homogeneous reaction is while heterogeneous reaction on the catalyst surface is where rate constants are defined by k c and k s and the chemical species A and B have concentrations a and b. Governing equations of present boundary layer flow problem are with boundary conditions where the velocity components in (r,x) direction are (v,u) respectively, p denotes the pressure, ρ the density, σ the electrical conductivity, ν the kinematic viscosity, c > 0 the stretching constant, λ 1 the ratio of relaxation to retardation times, λ 2 the retardation time, h the convective heat transfer coefficient, T the temperature, k f the thermal conductivity and α Ã the thermal diffusivity. Using the following transformations Eq (3) is satisfied automatically and Eqs 4-8 can be reduced as follows: is the heterogeneous reaction strength and Now eliminating pressure P between Eqs 11 and 12, we obtain with boundary conditions Pressure can now be determined from Eq 12 as Here it is assumed that both chemical species have equal diffusion coefficients D A and D B , i.e. δ = 1 and thus FðxÞ þ gðxÞ ¼ 1: Now Eqs 14 and 15 yield with the boundary conditions Skin friction coefficient C f and Nusselt number Nu are where τ rx represents surface shear stress and q w the wall heat flux which are given by Finally, we have where local Reynolds number is defined as Re x = cx 2 / ν.

m th -order deformation problems
The mth order deformation problems are The general solutions (f m , θ m , F m ) comprising the special solutions ðf Ã where the constants c i (i = 1-8) through the boundary conditions (44) have the values ;

Results and Discussion
In this section the effects of different parameters on the velocity, temperature and concentration fields are investigated through plots.  An increase in λ 1 corresponds to increase in relaxation time. It means particle needs much more time to come back from perturbed system to equilibrium system and consequently the fluid velocity decreases. Fig 5 shows impact of curvature parameter K on the velocity profile f 0 (ξ). Here increment in the magnitude of velocity profile is subjected to the enhanced values of K. Table 1. HAM solutions convergence when K = 0.01, λ 1 = 0.5, λ = 0.9, γ 1 = 0.1, M = 0.3, Pr = 1, Sc = k 1 = 0.9 and k 2 = 0.7.

Dimensionless temperature profile
Temperature profile θ(ξ) is plotted in Fig 6 to show the effect of Prandtl number Pr. Since the thermal diffusivity decreases by increasing Pr so the temperature decreases. Fig 7 indicates that

Dimensionless pressure profile
Figs 12-14 elucidate the variation in pressure profile for increasing values of Deborah number λ, ratio of relaxation to retardation times λ 1 and curvature parameter K. Here an enhancement in pressure distribution is noted for increasing λ. Also pressure is decreasing function of λ 1 and K. It is noted that skin friction coefficient enhances for larger K and it reduces when M is increased. Impact of ratio of relaxation to retardation times λ 1 via Deborah number λ on surface drag force is illustrated in Fig 16. Here surface drag force decreases for increasing λ 1 while it increases for larger λ. Computed results of skin friction coefficient are compared with previously published articles in limiting cases and found in very good agreement (see Table 2). K. Here surface heat transfer rate decreases as curvature parameter K is enhanced while opposite effect is observed for increasing Pr.

Surface concentration
Variation of homogeneous reaction parameter k 1 on surface concentration F(0) against Schmidt number Sc is shown in Fig 19. One can see that surface concentration decreases with the increase of k 1 and Sc. It is in view of the fact that surface concentration reduces due to the

Concluding Remarks
Effects of homogeneous-heterogeneous reactions in convective flow of Jeffrey fluid due to a curved stretching sheet are studied. The following outcomes are noticed: • Increase in the values of Deborah number and curvature parameter has similar effects on the velocity in a qualitative sense • Fluid velocity and temperature enhance for larger curvature parameter.
• The strength of heterogeneous reaction enhances the fluid concentration.
• Pressure distribution has direct relationship with Deborah number.
• Opposite behavior of curvature parameter and Hartman number is seen on the surface drag force.
• Increasing values of Biot number correspond to an enhancement in temperature and Nusselt number.

Author Contributions
Conceptualization: MI TH AA.
Data curation: MI TH AA.
Formal analysis: MI TH AA.
Investigation: MI TH AA.
Methodology: MI TH AA.
Project administration: MI TH AA.
Resources: MI TH AA.
Software: MI TH AA.
Supervision: MI TH AA.
Validation: MI TH AA.
Visualization: MI TH AA.