A Numerical Simulation of Silver–Water Nanofluid Flow with Impacts of Newtonian Heating and Homogeneous–Heterogeneous Reactions Past a Nonlinear Stretched Cylinder

The aim of the present study is to address the impacts of Newtonian heating and homogeneous–heterogeneous (h-h) reactions on the flow of Ag–H2O nanofluid over a cylinder which is stretched in a nonlinear way. The additional effects of magnetohydrodynamics (MHD) and nonlinear thermal radiation are also added features of the problem under consideration. The Shooting technique is betrothed to obtain the numerical solution of the problem which is comprised of highly nonlinear system ordinary differential equations. The sketches of different parameters versus the involved distributions are given with requisite deliberations. The obtained numerical results are matched with an earlier published work and an excellent agreement exists between both. From our obtained results, it is gathered that the temperature profile is enriched with augmented values radiation and curvature parameters. Additionally, the concentration field is a declining function of the strength of h-h reactions.


Introduction
In copious engineering processes, the role of poor thermal conductivity of certain base fluids is considered to be a big hurdle to shape a refined product. To overcome such snag, numerous practices such as clogging, abrasion, and pressure loss were engaged but the results were not very promising. The novel idea of nanofluid, which is an engineered amalgamation of metallic particles with size (<100 nm) and some traditional fluids like ethylene glycol, presented by Choi [1], has revolutionized the modern world. Many heat transfer applications [2] such as domestic refrigerators, microelectronics, hybrid-power engines, and fuel cells possess numerous characteristics that make them valuable because of nanofluids. In all aforementioned applications, enriched thermal conductivity is observed whenever some metallic particles are added to the ordinary base fluid [3]. The idea of a nanofluid with multiple here is the mixture of water and silver. The numerical simulations are conducted for the proposed problem using the Runge-Kutta method by shooting technique. To corroborate the presented results, a comparison with an already published article is done and an excellent correlation between the two results is found.

Mathematical Modeling
Here, we assume a silver-water nanofluid incompressible flow with impacts of h-h reactions, nonlinear thermal radiation and Newtonian heating over a horizontal cylinder which is stretched in a nonlinear way. The magnetic field B = B 0 x (n−1)/2 is applied in the radial direction. Owing to our assumption of a small Reynolds number, the induced magnetic field is overlooked (Figure 1). corroborate the presented results, a comparison with an already published article is done and an excellent correlation between the two results is found.

Mathematical Modeling
Here, we assume a silver-water nanofluid incompressible flow with impacts of h-h reactions, nonlinear thermal radiation and Newtonian heating over a horizontal cylinder which is stretched in a nonlinear way. The magnetic field is applied in the radial direction. Owing to our assumption of a small Reynolds number, the induced magnetic field is overlooked ( Figure 1). The homogeneous reaction for cubic autocatalysis can be written as: The reaction rate dies out in the outer boundary layer. The system of boundary layer equations of the subject model is given as: Accompanied by the conditions: The homogeneous reaction for cubic autocatalysis can be written as: The reaction rate dies out in the outer boundary layer. The system of boundary layer equations of the subject model is given as: Accompanied by the conditions: where U w (x) = U 0 x n , and q r = 4σ * 3k * The numerical values of specific heat, density, and thermal conductivity of silver (Ag) and water (H 2 O) are given in Table 1. Table 1. Thermo-physical characteristics of the base fluid and nanoparticles [1,37].

Physical Properties
Water Ag With the following characteristics: The use of under-mentioned similarity transformations Satisfy the Equation (3) and convert the Equations (4)-(7) in non-dimensional form (1 + 2γη) Supported by the boundary conditions where Here, it is expected that A 1 and B 1 are equivalent. From this assumption, it is inferring that D A and D B (diffusion coefficients) are equal i.e., δ = 1, and on account of this supposition, we have Equations (17) and (18) after the use of Equation (20) and the relevant boundary conditions take the shape 1 The physical quantities like Skin friction factor and Local Nusselt number in non-dimensional form are labelled as With τ w and q w given by Equation (23), after the use of Equations (14) and (24), takes the form

Numerical Scheme
The numerical solution of Equations (15), (16) and (21) supported by the boundary conditions (19) and (22) is found by the Shooting scheme. In the calculation of the numerical solution of the problem, the second and third order differential equations are transformed to first order with the help of new parameters. The selection of the initial guess estimate is pivotal in the Shooting scheme as it needs to satisfy the equation and the boundary conditions asymptotically. We have selected the tolerance as 10 −7 for this specific problem. The first order system obtained in this regard is appended below: Symmetry 2019, 11, 295 6 of 13 and the boundary condition becomes y 0 (1) = 0, y 0 (2) = 1, y 0 (5) = −λ(1 + y 0 (4)), y 0 (7) = k 2 y 0 (6), We have chosen η ∞ = 6, that guarantees every numerical solution's asymptotic value accurately.
Here, Table 2 depicts the comparative estimates of the present model with Qasim et al. [38] in the limiting case. Both results are found in an excellent correlation.

Results and Discussion
In this section, we have plotted the  Figure 2. For growing values of φ, the axial velocity also augments. In Figure 3, the influence of M versus the velocity field is debated. It is perceived that the velocity field deteriorates for escalated values of M. The Lorentz force is enforced by the strong magnetic field that hinders the fluid's velocity. Figure 4 illustrates the impression of n on the axial velocity. Reduced velocity is witnessed for larger values of n. This is because higher values of n create more collision between the particles of the fluid that hinder the fluid flow and feeble velocity if perceived. In Figures 5 and 6, the behavior of the velocity and temperature fields for increasing values of γ is given. It is seen that the velocity and temperature of the fluid augment for growing estimates of γ. Larger values of γ mean a smaller radius, comparatively minimum contact region of the cylinder with the fluid and increased heat transport. That is why augmented velocity and temperature are witnessed. Figure 7 is drawn to depict the relation between the temperature of the fluid and the λ. It is detected that temperature enhances for improved values of λ. In fact, the sturdier heat transfer process is observed for larger values of λ as more heat is moved from the cylinder to the fluid. Remembering that λ = 0 means the insulated walls and λ → ∞ constant wall temperature. Figures 8 and 9 are sketched to elaborate the influences of K * and M on the temperature field. It is clearly perceived that temperature enhances when both K * and M increase. The decrease in the mean absorption coefficient represents an enriched heat transfer rate and ultimately temperature is enhanced. Similarly, a stronger Lorentz force hinders the movement of the fluid, thus causing more collision between the molecules of the fluid that turns into the improved temperature. In Figures 10 and 11, the behavior of the concentration profile versus h-h reactions is depicted. The concentration diminishes for growing values of h-h reactions.
sketched to elaborate the influences of K and M on the temperature field. It is clearly perceived that temperature enhances when both K * and M increase. The decrease in the mean absorption coefficient represents an enriched heat transfer rate and ultimately temperature is enhanced. Similarly, a stronger Lorentz force hinders the movement of the fluid, thus causing more collision between the molecules of the fluid that turns into the improved temperature. In Figures 10 and 11, the behavior of the concentration profile versus h-h reactions is depicted. The concentration diminishes for growing values of h-h reactions.

Final Comments
In the present exploration, we have pondered over the nanofluid flow (with base fluid water and nanoparticles as silver) past a nonlinear stretching cylinder with impacts of h-h reactions and nonlinear thermal radiation. Additional effects of Newtonian heating and magnetohydrodynamics have also been taken into account. A numerical solution of the dimensionless mathematical model is achieved via the shooting scheme. The core outcomes of the present study are as follows:

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The temperature profile is a growing function of radiation and magnetic parameters.

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For larger values of the curvature parameter, augmented velocity is observed.

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The concentration of the fluid decreases for growing values of homogeneous-heterogeneous reactions.

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For escalated values of the magnetic parameter, velocity and temperature distributions show the opposite trend.

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The skin friction and local Nusselt number show opposite behavior for curvature and nonlinearity parameters.