Radiative Flow of Powell-Eyring Magneto-Nanofluid over a Stretching Cylinder with Chemical Reaction and Double Stratification near a Stagnation Point

This exploration addresses MHD stagnation point Powell Eyring nanofluid flow with double stratification. The effects of thermal radiation and chemical reaction are added in temperature and nanoparticle concentration fields respectively. Furthermore, appropriate transformations are betrothed to obtain nonlinear differential equations from the system of partial differential equations and an analytical solution of system of coupled differential equations is obtained by means of the renowned Homotopy Analysis method. Through graphical illustrations, momentum, energy and concentration distributions are conversed for different prominent parameters. Comparison in limiting case is also part of present study to validate the obtained results. It is witnessed that nanoparticle concentration is diminishing function of chemical reaction parameter. Moreover, mounting values of thermal and solutal stratification lowers the temperature and concentration fields respectively.


Introduction
Interest of scientists and researchers towards non-Newtonian fluids has immensely increased during the last few decades due to extensive role of these fluids in industrial and engineering applications. Examples of non-Newtonian fluids may include sugar solution, soaps, emulsions, shampoos, apple sauce, paints, cheese, muds, different cosmetic products, asphalt and ice cream etc. Since many non-Newtonian fluids with numerous characteristics are present in nature, subsequently different mathematical models are proposed to represent these fluids. The equations of different non-Newtonian fluid models are more complex and challenging in comparison to Navier-Stokes equations. That is why mathematical modelling and solutions of these fluids' equations are of great importance. The development in mathematical modelling of non-Newtonian fluids is of great interest even today [1][2][3][4][5][6]. Eyring Powell's model has a key role in many chemical engineering processes and has advantages over other non-Newtonian fluid models due its simplicity and ease in computations. This fluid model is obtained from a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 Kinetic theory of liquids instead of empirical equation. Moreover, for high and low shear stresses, it decorously converts to Newtonian fluids [7]. A reasonable number of communications can be quoted highlighting Eyring-Powell fluid model in the existing literature. Akbar et al. [8] examined Eyring Powell magneto fluid flow numerically past a stretching sheet using finite difference method. Hayat et al. [9] discussed Eyring Powell fluid flow past a stretched cylinder in attendance of magneto hydrodynamic and Newtonian heating. Hayat et al. [10] investigated mixed convective stagnation point Powell Eyring fluid flow with the impact of heat generation/absorption, thermal radiation and Newtonian heating. Hayat et al. [11] studied mixed convective Eyring Powell flow with effects of Soret and Dufour. The whole scenario is deliberated in attendance of convective heat and mass boundary conditions over an exponential stretched surface. Khan et al. [12] presented numerical and analytical solution of MHD Powell Eyring fluid flow with effects of joule heating, thermophoresis and chemical reaction.
The problem of low thermal conductivity of fluids like ethylene glycol, engine oil or water is solved by the introduction of nanofluids. Excellent heat transfer features of Nanofluids as compared to ordinary base fluids have made them more beneficial to many engineering and technological processes. Applications of nanofluids include heat exchangers, cancer therapy, thermal engineering, bio medicine and thermal engineering etc. Furthermore, in physics, medicine and engineering, abundant applications related to magnetic field may be found. Magnetic field with electrically conducting fluid has many applications, e.g., boundary layer control, pumps, bearings and MHD generators. Flow behavior highly depends on intensity of applied magnetic field and orientation of fluid molecules. Moreover, suspended particles of the fluid are rearranged by the applied magnetic field. This will ultimately change the concentration of the fluid with heat transfer characteristics. A fluid possessing characteristics of liquid and magnetic properties is known as magnetic nanofluid. Abundant applications of magnetic nanofluids, e. g., optical switches, nonlinear optical materials, magneto-optical wavelength filters and optical modulators etc. can be cited in this regard. All such interesting applications have motivated scientists and researchers to look for more avenues in the field of nanofluid flows and magnetic properties [13][14][15][16][17][18][19][20].
Many researchers are attracted to Stratification because of its important role in heat and mass transfer. Stratification occurs with fluids having varied densities and in flow fields with concentration variances and differences in temperature. Practically, it is imperative to examine the impact of double stratification whenever heat and mass transfer arise collectively. In the case of lakes and ponds, stratification is imperative to keep balance in the ratio of hydrogen and oxygen so that the growth rate of species is not disturbed. Moreover, in Solar engineering too, stratification plays a vital role in obtaining enhanced energy efficiency. Many researches have been conducted in this area. To name a few, Hayat et al. [21] examined series solution of time dependent nanofluid flow with viscous dissipation, double stratification and thermal radiation effects. Abbasi et al. [22] studied Maxwell nanofluid flow with effects of mixed convection, double stratification and magnetohydrodynamic. Hayat et al. [23] explored Jeffrey fluid flow under the influence of double stratification, heat generation/absorption and mixed convection past an inclined stretched cylinder. Hayat et al. [24] found series solution of Oldroyd-B fluid flow with double stratification and chemical reaction. Kaladhar et al. [25] investigated flow of couple stress fluid with mixed convection and double stratification using Keller box method.
Motivation from the above discussion, it is of paramount interest in this article to inspect the impact of double stratification, thermal radiation and magnetohydrodynamic on Powell-Eyring nanofluid flow past a stretched cylinder. None of the above quoted references simultaneously analyzed all such effects even in viscous fluid past a stretching cylinder near a stagnation point. Homotopy Analysis method [26][27][28][29][30] is engaged in order to find the analytical solution of the problem.

Mathematical formulation
We assume a situation in which Eyring Powell nanofluid flows past a stretching cylinder with impact of double stratification, thermal radiation and magnetohydrodynamic. This study also considers chemical reaction near a stagnation point. Moreover, cylindrical coordinates with zaxis are along the stretched cylinder whereas r-axis upright to it as shown in Fig 1 Ref. [31]. In an Eyring Powell fluid [32], we have The equations representing this assumed system are given by with the boundary conditions where u, w are velocity components in r and z directions respectively. Moreover, β and c, b and d, U 0 , l, ν, ρ, c p , k, T, T 1 , w e , D B and D T are fluid parameters, dimensionless constants, reference velocity, characteristic length, kinematic viscosity, density, specific heat, thermal conductivity, fluid temperature, ambient temperature, stretching velocity, Brownian diffusion coefficient and thermophoretic diffusion coefficient respectively.
Here, q r is Rosseland radiative heat flux and is given by using value of q r in Eq (4), we get Using the transformations of the form [33] Requirement of Eq (2), is automatically fulfilled, however, Eqs (3), (5), (6) and (8) take the form given below Here, γ, M, and λ, Le, Pr, Ha, Q 0 , Nb, P 0 and Nt are curvature parameter, fluid parameters, Lewis number, Prandtl number, Hartmann number, chemical reaction parameter, Brownian motion, velocity ratio and thermophoresis parameter respectively and are given by followings: The relations of Skin friction, local Nusselt and Sherwood numbers are as follows: and in non dimensional form where Re z = w e z/ν is the Reynolds number.

Homotopic solutions
Homotopy analysis method necessitates initial guesstimates (f 0 , θ 0 , ϕ 0 ) with auxiliary linear operators ðL f ; L y ; L 0 Þ in the forms The auxiliary linear operators have the following properties where A i (i = 1 − 7) are the arbitrary constants. Through boundary conditions, the values of these constants are given by the equations  Table 1, are also calculated and found in good agreement.

Results and Discussion
The results obtained for velocity, temperature and concentration distributions are presented graphically for prominent arising physical parameters. A comparison of flow behavior past a stretched cylinder versus a flat plat is also portrayed. Moreover, graphical illustrations depicting impact of prominent parameters on skin friction, local Nusselt and Sherwood numbers are also added to the present exploration. From Fig 3, it is perceived that the velocity profile is diminishing function of Hartmann number Ha. This upsurge in Hartmann number results in increase in Lorentz force. Lorentz force is a resistive force therefore attenuation in velocity profile is perceived. Furthermore, it is observed that impact of magnetic parameter is weaker in plate as compared to cylinder. Fig 4 illustrates that velocity distribution results in the mounting function of stagnation point parameter P 0 in both cases when P 0 < 1 and P 0 > 1. This is because of the fact that cylinder's stretching velocity is much smaller in comparison to free stream velocity. It is also noted that there is no boundary layer for P 0 = 1 due to the fact that both fluid and cylinder are moving with similar velocity. Fig 5 portrays the concentration field for varied values of chemical reaction parameter Q 0 . Reduction in solute nanoparticle concentration and its allied boundary layer thickness is seen because of heavy disturbance in fluid's molecules. The influence of radiation parameter Rd on temperature profile is examined in Fig 6. It shows that an increase in radiative parameter enhances the temperature distribution. Actually, more heat is transferred to the fluid due to higher values of radiation parameter. Moreover, it is observed that radiation effects are stronger in case of the cylinder as compared to the plate.     In Table 2, a comparison to the previous exploration in the limiting case for the Sherwood number is presented. All obtained results are found in excellent agreement.

Final remarks
The present study explores the effects of double stratification and magnetohydrodynamic flow of Eyring Powell nanofluid past a stretched cylinder near stagnation point. The analysis is done in the presence of chemical reaction and thermal radiation. Homotopy analysis method is used in order to obtain the series solutions of the said problem. The salient features of the present study are as following.
• Increasing the values of chemical reaction parameter results in reducing the solute nanoparticle concentration.
• Velocity profile decreases when radiation parameter increases.
• Sherwood number is increasing and decreasing function of Nb and Nt respectively.
• Gradual growth in thermal and solutal stratification results in decline in the temperature and concentration distributions respectively.
• Higher values of Brownian motion parameter Nb increase the temperature field but decrease the nanoparticle concentration profile.