Numerical study of temperature dependent thermal conductivity and homogeneous-heterogeneous reactions on Williamson fluid flow

Understanding the flow nature of non-Newtonian fluids need more studies in recent days in terms of their various aspects. Further, heat transfer analysis is an inspiring topic in non-Newtonian flows because of its dominant role in technology. With these incentives, a theoretical investigation is performed to study the time-dependent flow of a non-Newtonian fluid model with focus on heat and mass transfer. In this study, we proposed the mathematical model for the flow of Williamson fluid by incorporating the impact of infinite shear rate of viscosity. With the assistance of rheological expression of Williamson fluid, we construct the governing field equations. For physical relevance we analyzed the impacts of homogeneous-heterogeneous reactions on the flow field. The system of governing partial differential equations is transformed into a set of ordinary differential equations by adopting the non-dimensional variables. This non-dimensional ruling problem together with physical boundary conditions are tackled numerically by utilizing Runge–Kutta Fehlberg scheme via MATLAB software. The physical behavior of friction factor, reduced Nusselt number, dimensionless velocity, temperature and concentration distributions for distinct estimations of leading parameters is examined with the assistance of graphs.

Williamson fluid flow towards a stretchable surfaces allured distinct researchers due to its vital applications in many aspects such as polymer extrusion, plastic films, metal spinning, and metallurgical processes. In 1929, Williamson [1] proposed a model to assign the flow of pseudoplastic liquids and experimentally substantiated the outcomes. Later on, Nadeem and Akram [2] obtained the analytical solution peristaltic flow of a Williamson fluid in an asymmetric channel and reported that pressure gradient is a decreasing function of Weissenberg number. In another paper, Nadeem et al [3] solved analytically the boundary layer flow of Williamson [11] examined the unsteady flow of Williamson fluid induced by a wedge shape geometry. They proposed that the shear stress at the surface is higher for large wedge angle parameter. Hashim et al [12] analyzed the impacts of Williamson fluid flow driven by a wedge geometry and illustrate that the rate of heat transfer rises with larger unsteady parameter. In another study, Hashim et al [13] considered the mixed convection flow of Williamson nanofluid past a radially stretched surface with magnetic field and variable thermal conductivity. They accomplished that the temperature of the fluid and the thickness of thermal boundary layer increases by higher values of thermal conductivity parameter. Bahiraei et al [14] illustrated the mechanism of fluid flow and energy efficiency of a non-Newtonian liquid in the presence of nanoparticles. Makinde et al [15] analytically studied the behavior of electrically conducting nanofluid induced by a nonlinear stretched surface with the presence of heat generation/ absorption and chemical reacting effects. Mabood et al [16] numerically investigated the phenomenon of heat transport of water-based nanofluid in a rotating system with thermophoresis and Brownian motion parameter.
The homogeneous and heterogeneous interactions are the natural processes of chemical reactions. The heterogeneous reaction parameter has the ability to speed the concentration of the catalyst close to the surface and to decline the concentration of bulk liquid. Some investigators at present described the stretchable flows by considering homogeneous and heterogeneous effects. Merkin [20] initially examined the laminar flow of a viscous fluid with homogeneous-heterogeneous reactions along a flat plate. He reported homogeneous reaction for cubic autocatalysis and heterogeneous reaction on the catalyst surface. It can be seen that near the leading edge of the flat plate reaction is dominant. Homogenous-heterogeneous reactions with equal diffusivities have been proposed by Chaudhary and Merkin [21]. Ziabakhsh et al [22] discussed the feature of flow and diffusion of chemically reactive species along a nonlinear stretching surface immersed in a porous medium. Khan and Pop [23] studied the outcomes of homogeneous-heterogeneous reactions in a viscoelastic fluid past a stretched surface. Their study reveals that the concentration at the surface has been reduced by the large viscoelastic parameter. Kameswaran et al [24] analyzed the impact of homogeneous-hetrogeneous reactions in a nanofluid embedded in a porous stretching surface. They found that the strength of heterogeneous reaction diminishes the concentration profile. In the recent past, Hayat et al [25] reported the characteristics of carbon nano-tubes in the stagnation point flow induced by a nonlinear stretched surface with variable thickness. They concluded that the drag surface force decreases with the increment of ratio parameter. Chen et al [26] obtained the numerical solution of homogeneous-heterogeneous interaction on the homogeneous ignition feature in hydrogen-fueled catalytic micro-reactors in the submillimeter to millimeter range. Recently, Hashim and Khan [27] discussed numerically the flow and heat transfer analysis for Carreau fluid with homogeneous-heterogeneous reactions. Khan et al [28] addressed the flow over a non-linear stretched surface with Joule heating and homogeneousheterogeneous reactions. Moreover, Khan et al [29] examined the characteristics of homogeneousheterogeneous processes in the 3D flow of Sisko fluid over a bidirectional stretching surface. Few investigations about homogeneous-heterogeneous reactions are reported in the [30][31][32][33][34].
Based on literature review, the intention of this paper is to formulate the basic conservation equations for the boundary layer flow of non-Newtonian Williamson fluid model by employing the Boussinesq approximations. These types of flows are usually assumed to be steady, nevertheless, in various engineering and industrial applications, unsteadiness turn into an integral part of the physical problem where flow becomes timedependent. The second objective is the numerical investigation of heat transfer mechanism of Williamson fluid flow past a stretching surface with variable thermal conductivity and homogeneous-hetrogeneous effects in the presence of infinite shear rate viscosity. The numerical solutions for the dimensionless stream function, dimensionless temperature and concentration is deduced with the assistance of Runge-Kutta Fehlberg method alongside the shooting technique. The impacts of several included physical parameters on dimensionless profiles of the velocity, temperature and concentration, the local skin friction and local Nusselt number are illustrated with the help of graphs.

Problem development
In current review, we assume a time-dependent laminar flow of Williamson fluid past a stretching surface. The stretching surface has the velocity U x t , , w ( ) which changes with time and distance along the surface x (see figure 1). It is also mentioned that the temperature of the sheet is T x t , w ( ) which is higher than the ambient The boundary layer flow with homogeneous and heterogeneous processes comprising two chemical species A and B is assumed in this study and reported by Merkin [20] and Chaudhary and Merkin [21] and is given as follows: Under the above assumptions the regulating flow equations can be expressed in the following manner [19,35]; with the boundary conditions In the above expressions n represents the kinematic viscosity, K T ( )is the variable thermal conductivity and it is of the form, with e being a small parameter and C p the specific heat, D A and D B are the respective diffusion species coefficients of A and B, a 0 is a positive constant. Further, we assumed that the wall stretching velocity U x t , w ( )and the temperature T x t , w ( )are of the following form: The following suitable transformations are used in the present case: where a 0 is a constant, h g ( ) and h h( ) is the dimensionless concentration. By introducing the above transformations the governing flow equations of this problem are reduced as follows; We a x ct 1 These are respectively defined as in most applications, we expect the diffusion-coefficients of chemically species A ( ) and B ( ) are of comparable sizes, which leads us to further assumption that the diffusion-coefficients D A ( ) and D B ( ) are equal, i.e., to take d = 1 (Chaudhary and Merkin [21]) This assumption gives us; (15) and (16)  where the wall shear stress t w and the wall heat flux q w are given by

In view of equations (22) and (23) the following expressions takes the form
( ) specify the local Reynolds number.

Numerical method
In this study, we employ an efficient numerical technique Runge-Kutta Fehlberg method with shooting scheme as discussed by Pal and Shivakumara [36] to examine the flow model for distinct values of leading parameters. In this method, choosing an appropriate finite value of h  ¥ is a critical step. Consequently, in order to obtain h  ¥ for the boundary value problem described, we start with initial estimated values for the set of a particular physical parameters, so as to obtain  f 0 , ( ) q¢ 0 ( ) and ¢ g 0 .
( ) The first step towards this is to discretize the governing system into a system of five simultaneous differential equations of first order. For this purpose we introduce the new variables. ( ) Hence, we get the following first order system

The boundary conditions now become
In this system, four initial conditions are known and the other three unknown conditions s , 1 s 2 and s 3 are first guessed and afterword fixed with Newton-Raphson's method for each given set of parameters. Later on, a finite value for h ¥ is selected so that the far field boundary conditions hold at highest value of h ¥ . We took h = ¥ 10 to perform our computations. The absolute convergence criteria is assume to be -10 6 to get the desired degree of accuracy.

Validation of numerical scheme
In order to verify the accuracy of our numerical results, a comparison is made between our results and previously published literature. Table 1 indicates a comparison value of the  f 0 ( ) for several values of A (in the absence of Weissenberg number, viscosity ratio parameter, strength of homogeneous reaction, strength of heterogeneous reaction and Schmidt number) determined by Sharidan et al [37] Chamkha et al [38] and Khan and Azam [39]. On the basis of table 1, we found a good agreement in these outcomes.

Results and discussion
In this section, we explore the effects of time dependent flow of Williamson fluid past a stretching surface with variable thermal conductivity and homogeneous-heterogeneous interactions. The system of nonlinear ordinary differential equations (26)-(28) together with boundary conditions (29) and (30) are numerically elucidated by utilizing RK-45 with Newton shooting technique. The impact of some pertinent physical parameters on dimensionless velocity, temperature gradient and concentration profiles along with the skin friction coefficient and local Nusselt number is analyzed and illustrated graphically. Figure 2 highlights the variation of velocity components h ¢ f , ( ) temperature gradient q h ( ) and concentration profile h g ( ) against multiple values of unsteadiness parameter. From figure 2, one can observe that an enhancement in the unsteadiness parameter A results in a decrease in the velocity and temperature profile. We also perceived a rise in the concentration profile for higher unsteadiness parameter. From graphical illustration we also observed that momentum and thermal boundary layer thickness increases with augmented unsteadiness parameter but quiet opposite behavior is seen for solutal concentration. Physically, it is stated that higher unsteadiness parameter relates to a smaller stretching rate in the x-direction and causes in a slight reduction in the velocity field. Further, less amount of heat and mass is transferred from the fluid to the surface, therefore a reduction occurs in the temperature profile while an opposite pattern is seen for concentration distribution.
The impact of the b* on the h ¢ f ( ) and q h ( ) is sketched in the figure 3. Higher values of viscosity ratio parameter causes a growth in the velocity of the fluid and decline in the temperature profile. However, momentum boundary layer thickness expands and thermal boundary layer thickness diminishes with the augmented values of viscosity ratio parameter b . *  The influence of the Weissenberg number We on the velocity h ¢ f ( ) and temperature distributions q h ( ) are displayed in the figure 4. It can be shown that an increase in We reduces the velocity of the fluid as well as momentum boundary layer thickness. For the physical point of view, Weissenberg number is the dimensionless pararmeter that is used to compare the elastic to viscous forces. It is due to the fact that higher values of We reduces the viscous forces which causes a fact that velocity of the fluid decreases. A noteworthy increment in the temperature profile is marked when the We is increased. Figure 5(a) demonstrates the effect of Prandtl number Pr on the temperature profile q h ( ) It is observed that temperature of the fluid and associated thermal boundary layer thickness decreases with larger Prandtl number. Physically means that the Prandtl number possesses low thermal conductivity and fluid particles needs more time to transferred the heat to its surrounding consequently we observed a reduction in the temperature profile q h ( ) Figure 5(b) depicts the influence of thermal conductivity parameter e on dimensionless temperature field q h .
( ) A significant growth in the temperature gradient is noticed when e is enhanced. It is also observed that thermal boundary layer thickness rises significantly when thermal conductivity parameter is increased. This is due to the fact that when e rises then considerable heat transfers from plate to the material which grow the temperature of Williamson fluid.    To exhibit the effects of Sc on concentration gradient h g ( ) against h is displayed in figure 7. On can observe that as Sc rises then there is an enhancement in the concentration profile h g .
( ) As the ratio of momentum to mass diffusivity is defined as Schmidt number. This means that high momentum diffusivity as compared to mass diffusivity increases Schmidt number. In the physical point of view, an increment in the Schmidt number diminishes the molecular diffusivity and that causes in a decrease concentration boundary layer thickness.
The variation of skin friction coefficient Re C fx 1 2 / against We and b* is illustrated through figure 8. From this plot, it is noticed that higher values of We and b* decreases the drag force at the surface of plate.
The influence of Pr and e on Nusselt number -Re Nu x 1 2 / is represented in figure 9. It can be inferred that the heat flux on the boundary experiences a decreasing influence due to Prandtl number Pr and thermal  conductivity parameter e On the other hand, since the increase in Prandtl number leads to decrease in thermal boundary layer thickness, therefore the Nusselt number decreases with increase of Prandtl number. Figure 10 is sketched to exhibits the comparison of two numerical techniques bvp4c and shooting method for velocity, temperature and concentration profiles. Where dots are representing the solution calculated by bvp4c and the solution obtained by shooting method is drawn through straight lines. The solution profiles of both numerical techniques are in a remarkable assertion for velocity, temperature and concentration distribution.

Conclusions
In this article, the numerical solutions have been obtained for homogeneous and heterogeneous reaction on an unsteady, laminar and two-dimensional flow of Williamson non-Newtonian fluid model through a stretching sheet in the presence variable thermal conductivity. The well-known Runge-Kutta Fehlberg Method with  shooting scheme is utilized to solve the reduced momentum and energy equations of the current fluid flow model. Moreover, a comparison between the results obtained by Runge-Kutta method and the Matlab routine bvp4c is also presented for non-dimensional velocity and temperature profiles. Therefore, a very good agreement is seen between these results. The key outcomes of this analysis are summarized as • The dimensionless velocity profiles h ¢ f ( ) as well as temperature profiles q h ( ) are decrease with an increment in the unsteadiness parameter A.
• The augmented values of We decreases the fluid velocity. However, an opposite behavior is noticed for temperature profile.
• Viscosity ratio parameter b* increases the momentum boundary layer thickness and decreases the thermal boundary layer thickness.
• The temperature and thermal boundary layer thickness are decreasing function for Prandtl number Pr.
• The variable thermal conductivity parameter e increases the temperature of the fluid.
• Higher strength of heterogeneous and homogeneous reactions lead to enlarge the concentration of the catalyst at the surface.