Temperature and Concentration Stratification Effects in Mixed Convection Flow of an Oldroyd-B Fluid with Thermal Radiation and Chemical Reaction

This research addresses the mixed convection flow of an Oldroyd-B fluid in a doubly stratified surface. Both temperature and concentration stratification effects are considered. Thermal radiation and chemical reaction effects are accounted. The governing nonlinear boundary layer equations are converted to coupled nonlinear ordinary differential equations using appropriate transformations. Resulting nonlinear systems are solved for the convergent series solutions. Graphs are plotted to examine the impacts of physical parameters on the non-dimensional temperature and concentration distributions. The local Nusselt number and the local Sherwood number are computed and analyzed numerically.


Introduction
Analysis of non-Newtonian fluids has great importance due to its several industrial and engineering applications. In particular these fluids are encountered in the material processing, chemical and nuclear industries, bioengineering, oil reservoir engineering, polymeric liquids and foodstuffs. Several fluids like paints, paper pulp, shampoos, ketchup, apple sauce, slurries, certain oils and polymer solutions are the non-Newtonian fluids. The characteristics of all the non-Newtonian fluids cannot be explained via one constitutive relationship. Hence various fluid models are proposed in the literature for the properties of non-Newtonian fluids. Generally non-Newtonian materials are classified under three categories namely (i) differential type (ii) rate type and (iii) integral type. The Maxwell fluid model is the simplest subclass of rate type fluids. This model describes only the properties of relaxation time. The characteristics of retardation time cannot be predicted by the Maxwell fluid. An Oldroyd-B fluid model was developed to examine both the relaxation and retardation times characteristics. Instabilities in viscoelastic liquids were studied by Larson [1]. In this investigation, he discussed the instabilities in Taylor-Couette flows, instabilities in cone-and-plate and plate-and-plate flows, instabilities in parallel shear flows, instabilities in external and multi-dimensional flows. The instabilities in the flows occurring in the absence of inertial forces were investigated by Shaqfeh [2]. Laso and Ottinger [3] presented a study to examine the numerical simulation of viscoelastic liquids based on molecular models. Rajagopal and Bhatnagar [4] computed the asymptotically decaying solution of an Oldroyd-B fluid past an infinite porous plate. Thermodynamic properties of rate type non-Newtonian fluids were investigated by Rajagopal and Srinivasa [5]. Numerical solutions of Oldroyd-B and PTT-fluids with both the linear and exponential stress functions were developed by Alves et al. [6]. Some recent investigations on non-Newtonian fluids can be seen in the references [7][8][9][10][11][12][13][14].
Heat and mass transfer analysis in the boundary layer flow over a stretching surface has key role in the industrial and engineering applications, for example, manufacturing of plastic and rubber sheets, annealing and thinning of copper wires, drawing on stretching sheets through quiescent fluids, boundary layer along a liquid film condensation process, damage of crops due to freezing, desalination, refrigeration and air conditioning, compact heat exchangers, solar power collectors, human transpiration and many others (see refs. [15,16]). Heat and mass transfer effects in boundary layer flow of viscoelastic fluid with thermal slip condition were investigated by Turkyilmazoglu [17]. Hayat and Alsaedi [18] carried out a study to examine the heat and mass transfer phenomena in buoyancy driven flow of an Oldroyd-B fluid. Thermophoresis and Joule heating effects are further considered. Hayat et al. [19] presented the series solutions of magnetohydrodynamic (MHD) flow of Casson fluid with heat and mass transfer. Soret and Dufour effects are present in this investigation. Mixed convection flow of Jeffrey fluid in the presence of heat and mass transfer is investigated by Shehzad et al. [20]. Gupta et al. [21] discussed the effect of cadmium on growth and active constituents of bacopa monnieri. Induced magnetic field effect in mixed convection peristaltic flow of third order fluid with nanoparticles is discussed by Noreen [22]. Bachok et al. [23] studied the boundary layer flow of viscous fluid in presence of mixed convection and viscous dissipation. Su et al. [24] developed a lattice Boltzmann method coupled with the Oldroyd-B constitutive equation to stimulate flow of viscoelastic fluid. Here the numerical results of 2D channel flow agree well with the analytical and some experimental results reported in the previous studies. Slip effects in peristaltic flow of generalised Oldroyd-B fluids is explored by Tripathi et al. [25]. They computed the homotopic solutions of the modelled differential system.
Effect of stratification is an important aspect in heat and mass transfer analyses. Stratification of fluids occurs due to temperature variations, concentration differences or the presence of different fluids of different densities. When the heat and mass transfer are present simultaneously then it is important to analyze the effect of double stratification on the convective flows. The analysis of mixed convection in a doubly stratified medium is an important problem. It is because of its occurrence in geophysical flows (see ref. [26]). Such flows involve in the rivers, lakes and seas, thermal energy storage systems and solar ponds etc. Chang and Lee [27] investigated the free convection flow by a vertical plate with uniform and constant heat flux in a thermally stratified micropolar fluid. Cheng [28] examined the combined heat and mass transfer effect in natural convection flow from a vertical wavy surface in a power-law fluid saturated porous medium. Both thermal and mass stratification effects were present. Srinivasacharya and Reddy [29] discussed the effect of double stratification in mixed convection flow of micropolar fluid. Effect of double stratification on MHD free convection flow of micropolar fluid is investigated by Srinivasacharya and Upendar [30]. Non-Darcy mixed convection flow in a doubly stratified medium under Soret and Dufour effects is studied by Srinivasacharya and Surender [31]. Srinivasacharya and Surender [32] addressed the effect of double stratification on mixed convection boundary layer flow of a nanofluid past a vertical plate in porous medium.
The basic theme of present study is to investigate the effects of thermal radiation, chemical reaction, thermal and solutal stratification in the mixed convection boundary layer flow of an Oldroyd-B fluid over a stretching surface. The studies available in the literature on this topic mostly dealt with the thermal stratification effect. Some recent aforementioned studies investigated the effects of both thermal and concentration stratification in viscous fluid flow. This is the first attempt to study such effects for non-Newtonian fluids. Mathematical modelling is developed under the consideration of thermal and concentration stratification effects. The series solutions to the resulting nonlinear differential systems are constructed via homotopy analysis method (HAM) [33][34][35][36][37][38][39][40][41]. The effects of various emerging parameters on the temperature and concentration fields are presented through plots and tables. The local Nusselt and the local Sherwood numbers are computed numerically and analyzed.

Mathematical Modeling
We consider the steady two-dimensional doubly stratified mixed convection flow of an incompressible Oldroyd-B fluid. The flow is caused by a linearly stretching surface at y = 0. The flow occupies the domain y > 0. Boundary layer flow is considered in the presence of thermal radiation and first order chemical reaction. The governing boundary layer equations for incompressible flow of an Oldroyd-B fluid with heat and mass transfer are given below (see Appendix for detailed derivation): À @u @x @ 2 u @y 2 À @u @y @ 2 v 1 r @p @y ¼ À l 1 r À @v @y @p @x þ u @ 2 p @x@y ; ð3Þ The appropriate boundary conditions are where u and v are the velocity components in the x-and y-directions respectively, λ 1 the relaxation time, ν = μ / ρ the kinematic viscosity, μ the dynamic viscosity, ρ the density of fluid, λ 2 the retardation time, g the gravitational acceleration, β T the thermal expansion coefficient, T the temperature, β C the concentration expansion coefficient, C the concentration, α = k / ρc p the thermal diffusivity of the fluid, k the thermal conductivity, c p the specific heat at constant pressure, q r the radiative heat flux, D the diffusion coefficient, K 1 the reaction rate, T w and T 1 the temperatures of the surface and far away from the surface and C w and C 1 the concentrations at the surface and far away from the surface. The subscript w denotes the wall condition. This study assumes that the surface stretching velocity, wall temperature and wall concentration are where a, A 1 , B 1 , M 1 , N 1 , T 1,0 and C 1,0 are the positive constants. The radiative heat flux q r via Rosseland's approximation can be expressed as follows: in which σ 1 is the Stefan-Boltzman constant and m is the mean absorption coefficient. We assume that the difference in temperature within the flow is such that T 4 can be written as a linear combination of temperature. By employing Taylor's series and neglecting higher order terms we have [18]: Substituting Eq (10) in Eq (9) we get @q r @y ¼ À Using Eq (11) in Eq (4) we have The dimensionless variables can be defined as follows: Incompressibility condition is now identically satisfied and Eqs (2)- (8) and (12) become In above expressions β 1 and β 2 are the Deborah numbers in terms of relaxation and retardation times respectively, λ is the mixed convection parameter, Gr x is the Grashof number, Re x is the local Reynolds number, N is the buoyancy ratio parameter, Rd is the thermal radiation parameter, Pr is the Prandtl number, ε 1 is the thermal stratification parameter, Sc is the Schmidt number, γ is the chemical reaction parameter, ε 2 is the solutal stratification parameter and prime stands for differentiation with respect to η. Note that when β 2 = 0, this analysis reduced to the Maxwell fluid flow case. Eq (15) indicates that P is constant in the y-direction. The involved variables can be defined as follows: The local Nusselt number Nu x and the local Sherwood number Sh x are given by

Convergence Analysis
The series solutions (40) Table 1 shows that the 11th order of approximations are sufficient for the convergent series solutions. Here the temperature and thermal boundary layer thickness are decreased while concentration and its related boundary layer thickness are increased when we increase in thermal stratification parameter. When the thermal stratification effect is taken into account, the effective temperature difference between the surface and the ambient fluid is decreased while opposite behavior is observed for concentration profile [28]. Influence of solutal stratification parameter ε 2 on the temperature profile θ(η) and concentration profile ϕ(η) is shown in       • Temperature and concentration fields are increased when we increase the values of β 1 . Here the relaxation time is enhanced when we give rise to Deborah number that leads to the higher temperature and concentration fields. This observation is similar that obtained in [18].
• Temperature and thermal boundary layer thickness are decreased when Prandtl number increases. Prandtl number is used to control the heat transfer rate in industrial process [29]. The proper value of Prandtl number is quite essential to control the heat transfer rate in industrial and engineering processes.
• Temperature and thermal boundary layer thickness are increasing functions of thermal radiation parameter. An increase in thermal radiation parameter provides more heat to fluid due to which larger temperature and thicker thermal boundary layer thickness are achieved.
• Influence of thermal stratification parameter on the temperature and concentration fields are opposite [30]. Here temperature is decreased while concentration is enhanced for the higher values of thermal stratification parameter. • Temperature and concentration fields show opposite behavior for solutal stratification parameter. It is also observed that the thermal and concentration boundary layer thicknesses are reverse for the larger solutal stratification parameter.
• Concentration field and associated boundary layer thickness are reduced when we increase the values of chemical reaction parameter. For γ = 0 our analysis reduces to the case when there is no chemical reaction.
• Table 4 shows that our solutions have an excellent agreement with the previous published numerical results in limiting sense.
• The used technique for the solutions development and analysis has advantages over the other in the sense of following points: a. It is independent of small/large physical parameters.
b. It provides a simple way to ensure the convergence of series solutions.
c. It provides a large freedom to choose the base functions and related auxiliary linear operators.

Temperature and Concentration Stratification Effects
• Besides this the presented analysis is capable of describing relaxation and retardation times feature which many polymers show. Such analysis is particularly useful in polymer extrusion coating process, blood related viscoelastic effects in hemodynamic [43][44][45] etc. Such analysis provides a stimulus for future investigations on the topic in regimes of magnetohydrodynamics and convective conditions of heat transfer at the surface. It should be pointed out that presented analysis is not able to describe the rheological fluid properties in terms of normal stress effects, shear thinning and shear thickening features. Further the present analysis just deals with the hydrodynamic flow situation due to which the effects of viscous dissipation and Joule heating are ignored. In future one can discuss the hydromagnetic flow case in the presence of Joule heating and viscous dissipation.
components of identity tensor and the components of an extra stress tensor S ij is defined as follows: where μ denotes the dynamic viscosity, λ 1 and λ 2 are the relaxation and retardation times respectively, A ij 1 the components of first Rivlin-Ericksen tensor and D Dt the contravariant convective derivative which can be written below in the forms The above expressions represent a contravariant vector and a contravariant tensor having rank 1 and 2 respectively, (where v i denote the components of velocity and "," represents the covariant derivative). In case of Cartesian coordinates the covariant derivative reduces to the where ρ denotes the fluid density, g the gravitational field and the definition of acceleration 'a' is By applying the operator 1 þ l 1 D Dt À Á ; Eq (A6) gives or Assuming to derive the above equation that D Dt À Á ;j ¼ 0 and Eq (A2), we get the following equation:  For an incompressible steady and two-dimensional flow one can write ¼ u @ @x @p @x þ v @ @y @p @x À @u @x @p @x À @u @y @p @x ; ¼ u @ 2 p @x 2 þ v @ 2 p @x@y À @u @x @p @x À @u @y @p @x ; À u @ 2 v @x 2 þ v @ 2 v @y 2 @u @y À @ 2 u @x 2 þ @ 2 u @y 2 @u @x : ðA19Þ