Using non-Fourier’s heat flux and non-Fick’s mass flux theory in the Radiative and Chemically reactive flow of Powell-Eyring fluid

: The behavior of convective boundary conditions is studied to delineate their role in heat and mass relegation in the presence of radiation, chemical reaction, and hydromagnetic forces in three-dimensional Powell-Eyring nanofluids. Implications concerning non-Fourier’s heat flux and non-Fick’s mass flux with respect to temperature nanoparticle concentration were examined to discuss the graphical attributes of the principal parameters. An efficient optimal homotopy analysis method is used to solve the transformed partial differential equations. Tables and graphs are physically interpreted for significant parameters                              (5) 2 2 2


Introduction
Heat transfer is mainly observed due to variation in temperature of bodies. This process has a vital role in mechanization and industry like climate engineering, cooling of devices, nuclear power plants, and energy acquirement. The well-known Fourier law for heat transfer [1] and Fick's law for mass transfer have been widely used in literature. The fragility of Fourier law is that the initial bugging is immediately perceived by the medium, which is unpractical. Classical Fourier law was amended by adding relaxation time to heat flux by Cattaneo [2]. Christov [3] further modified the Cattaneo's law by incorporating a Lie derivative for the heat flux. The Cattaneo-Christov theory has been applied to both Newtonian and non-Newtonian fluids with various physiological effects.
The Cattaneo-Christov model was discussed by Straughan for the thermal convection of a viscous fluid [4]. Salahuddin et al. [5] applied theory given in [3] to Williamson fluid.
printing, paint suspensions, and biological flows. Thus, the analysis of such fluids is of substantial research and significance importance. Powell and Eyring presented an integral mathematical model for a non-Newtonian fluid well known as Eyring-Powell fluid model [12]. The model is dominant to the other nonlinear models as it can depreciate to visco fluids for immense and limited shear rates. Moreover, the model is deduced from the kinetic theory of fluids rather than empirical relations. Considering the importance of this non-Newtonian fluid, Hayat et al. [13] analyzed the steady flow of [12].
The peristaltic flow of Eyring-Powell nanofluid in an endoscope was investigated by Akbar and Nadeem [14]. Boundary layer flow of [12] was examined numerically by Jalil et al. [15]. Malik et al. [16] investigated boundary layer flow in a stretching cylinder with variable viscosity for the fluid characterized by constitutive equations due to Powell and Eyring. Mixed convected flow of [12] along a rotating cone was presented by Nadeem and Saleem [17]. The role of hybrid Eyring-Powell nano fluid [18] is observed for peristaltic transport. Hayat et al. [19] presented results for the axisymmetric radial flow of [12] over an impermeable stretching surface and the heat transfer process was analyzed through convective boundary conditions. Most Recently, Ibrahim [20] proposed the numerical solution for the rotating Eyring Powell fluid flow in three dimensions with theory [3].
Magneto fluid dynamics has become an important topic in recent years. The study of magneto hydrodynamics has won real life applications, for instance, electromagnetic forces can be used to pump liquid metals without the need for any moving parts. MHD has significant importance in stellar and planetary processes as well. The concept of MHD has also boosted the engineering applications. For example, the direct conversion generator and flow problems of ionized gasses. The MHD unsteady flow in a porous channel with convective heat conditions at the surface has been explored by Makinde [21] and the study concludes that the presence of magnetic field strengthens the flow control.
The boundary layer flow of MHD Maxwell nanofluid was discussed with numerical aid [22]. Ellahi et al. [23] numerically inquired the Coutte flow of [12] and heat transfer in magnetohydrodynamics. Thermal radiation is a ruling factor in the thermo dynamic analysis of high temperature systems like boilers and solar connectors. The heat and mass relocation analysis with thermal radiation play a vital role in manufacturing industries. For instance, the design of flippers, gas motors, cooling towers and various propulsion devices for aircraft, energy utilization, food processing as well as diverse agricultural, military, and health applications. As a result, much work has been done on fluid flow considering radiation in thermal radiation. Analysis has been carried out for viscoelastic fluid in the effect of thermal radiation and the Rosseland approximation is applied to characterize the heat flux in the heat equation by Qasim et al. [24]. Ayub et al. [25] discuss holdings of wall shield on the radiation of the transverse electromagnetic wave. The solution is obtained by the Wiener-Hopf technique. A short time ago, Raju et al. [26] address the buoyancy accommodating heat and mass transfer considering thermal radiation and Buongiorno's model. The results were presented for the flow behavior over a paraboloid, a cone, and a cylinder adopting numerical method.
Many of the modern propositions in technology are intended on making small devices.
This can improve the efficiency scale and enhance the productivity. Similarly, advances are also occurring in fluid dynamics at a rapid pace known as micro fluidics and nanofluidics. The application includes the designing of electronic gadgets, polyphase flows in lab-on-a-chip, and basic procedures in individual biological cells. Due to the enormous applications of nanofluids, many researchers have shown their interest in the studing effects of nanoparticles in non-Newtonian fluids on different physiological aspects [27][28][29][30][31][32].
Above literature motivated us to target the analytical solutions of three-dimensional rotating Eyring Powell fluid inclusive of magnetohydrodynamics, radiation effects, and convective boundary conditions with [3]. Convergent series solutions by the optimal homotopy approach are constructed [33][34][35][36][37]. The impact of important parameters on velocity components, temperature, and concentration are illustrated graphically.
However, skin friction, Nusselt number, and Sherwood numbers are tabulated numerically.

Mathematical Formulation
The mathematical design of the non-Newtonian fluid called Powell-Eyring fluid is investigated. For this purpose, the stress tensor of the fluid is taken from [12].
Here μ is the dynamic viscosity, are the rheological Powell-Eyring fluid model parameters. ̇= and = ∇ + (∇ ) so that the second-order approximation of sinh after using Taylor series expansion is Hence Eq. (1) takes the form The chemical reaction effect of MHD steady and incompressible Powell-Eyring nano fluid in the presence of thermal radiation over two-sided stretching sheet with [3] is Cattaneo-Christov theory is incorporated inplace of classical Fourier's heat flux law and Fick's mass flux law of diffusion [3]. The equations involving heat flux and mass flux are given.
where is the thermal relaxation time and is the concentration relaxation time. To proceed further, we use the Rosseland approximation for the radiative heat flux .
Expanding into Taylor series about and neglecting higher order terms In accordance with the above supposition, the heat and mass transfer equation will reduce to where is the Stefan-Boltzmann constant and is the mean absorption coefficient.
The boundary conditions associated with the study are Selecting the following similarity transformations The continuity equation is satisfied and the rest of nondimensional governing equations The corresponding boundary conditions are where surface heat flux, ℎ surface mass flux, wall shear stress along -axis and wall shear stress along -axis are given by By incorporating the above equations, we get , are local Reynolds numbers.

Method of solution
Optimal homotopy method [33][34][35][36][37] is adapted to obtain the solutions for the nonlinear Eqs. (16) to (19) together with boundary conditions. The initial guesses are given as The concept of minimizing the average square residual errors is utilized [33] to find the ideal values of nonzero auxiliary parameters ℎ , ℎ , ℎ and ℎ which are actually responsible for defining the convergence region of homotopy series solutions.
where is the total of the square of residual error, = 0.5, = 20.
Mathematica package BVPh2.0 has been used to reduce the average residual error. Table   1 is aligned to show the minimized values of the total residual error at various iterations.
In Table 2, the residual errors for , , , are given at three distinct iterations with the 8th-order optimal convergence control parameters. It is evident that the residual errors are reduced by raising iterations. Therefore, OHAM provides a procedure to select any set of local convergence control parameters to find convergent outcomes.

Effect of parameters on Eyring Powell fluid flow
The plots of velocities along and direction with coordinate are executed for the important parameters involved in the flow.          In Table 3a, the skin friction coefficient on the surface is approximated for various values of and . We observed that the local skin friction coefficient along -axis is reduced for sufficiently large values of , hence the smooth flow along -direction and reverse is the behavior of skin friction co-efficient if is raised. Skin friction coefficient along -direction abates for leading values of . In addition, the skin friction co-efficient exceeds for greater values of ( ≤ 0) and recedes for greater values of ( > 0) as indicated in Table 4a. Tables 3b and 4b are given to analyze the special behavior of local skin friction coefficients in the absence of MHD, radiation and chemical reaction.

Effects of parameters on temperature and concentration
The plots for temperature with coordinate are executed for the important parameters involved in the heat/ mass transfer.            Table 5, we can see that the heat transfer and mass transfer are enhanced significantly by strengthening the magnetic effect. The same outcomes can be observed by mounting the radiation parameter, heat and mass transfer Biot numbers.

Conclussion
Radiative and chemically reactive flow of Eyring Powell nanofluid with non-Fourier's heat flux and non-Fick's mass flux is studied with the consideration of magnetohydrodynamics and convective boundary conditions. The developments of the present paper can be compiled as: The amplitude of velocity along the abscissa decreases, while the velocity along the ordinate increases with the escalating magnetic effect. Temperature profile increases with the consideration of nanoparticles in the fluid, but the opposite is true for the concentration profile.
Rotational parameter decreases the velocity boundary layer thickness in both directions.
The Eyring Powell fluid parameter has the property of reducing the Nusselt number in both directions.