Numerical simulation for homogeneous–heterogeneous reactions and Newtonian heating in the silver-water nanofluid flow past a nonlinear stretched cylinder

The present exploration aims to deliberate silver-water nanofluid flow with homogeneous–heterogeneous reactions and magnetic field impacts past a nonlinear stretched cylinder. The novelty of the presented work is enhanced with the addition of Newtonian heating, heat generation/absorption, viscous dissipation, nonlinear thermal radiation and joule heating effects. The numerical solution is established via Shooting technique for the system of ordinary differential equations with high nonlinearity. The influences of miscellaneous parameters including nanoparticles volume fraction 0.0 ≤ ϕ ≤ 0.3 , magnetic parameter 1.0 ≤ Μ ≤ 4.0 , nonlinearity exponent 1.0 ≤ n ≤ 5.0 , curvature parameter 0.0 ≤ γ ≤ 0.4 , conjugate parameter 0.4 ≤ λ ≤ 0.7 , heat generation/absorption parameter ( 0.2 ≤ E c ≤ 0.8 ) , radiation parameter 0.7 ≤ K * ≤ 1.0 , Eckert number ( 0.1 ≤ E c ≤ 0.7 ) , strength of homogeneous reaction 0.1 ≤ κ 1 ≤ 1.8 , strength of heterogeneous reaction 0.1 ≤ κ 2 ≤ 1.8 and Schmidt number ( 3.0 ≤ S c ≤ 4.5 ) on axial velocity, temperature profile, local Nusselt number, and skin friction coefficient are discussed via graphical illustrations and numerically erected tabulated values. It is examined that the velocity field diminishes while the temperature profile enhances for mounting values of the magnetic parameter. An excellent concurrence is achieved when our obtained numerical calculations are compared with an already published paper in limiting case; hence dependable results are being presented.

The present exploration aims to deliberate silver-water nanofluid flow with homogeneousheterogeneous reactions and magnetic field impacts past a nonlinear stretched cylinder. The novelty of the presented work is enhanced with the addition of Newtonian heating, heat generation/absorption, viscous dissipation, nonlinear thermal radiation and joule heating effects. The numerical solution is established via Shooting technique for the system of ordinary differential equations with high nonlinearity.

Introduction
The feeble thermal conductivity of certain base fluids in numerous processes has been a big obstacle to shape a refined product. Certain techniques like pressure loss, abrasion and clogging were proposed by the researchers to overcome this deficiency but outcomes were not very encouraging. Nevertheless, the novel concept of nanofluid [1] (an amalgamation of suspended Nano metered sized metallic particles and some ordinary fluid such as oil, water or ethylene glycol) has revolutionized the modern industrial world. These nano-sized (<100 nm) metallic particles are comprised of metals, their oxides, and carbon nanotubes. Nanofluids possess certain unique properties that make them potentially worthwhile in numerous engineering and industrial heat transfer applications The flows of numerous fluids in attendance of magnetohydrodynamic (MHD) have extensive important applications in aerospace engineering, MHD generators, medicine, geothermal field, petroleum processes, nuclear reactors engineering and astrophysics. A reasonable number of explorations have been conducted featuring MHD fluid flows featuring an effort by Ramzan et al [21], they inspected the MHD flow of Jeffery nanofluid with radiation effects. Hayat et al [22] studied the MHD micropolar fluid flow with homogeneous-heterogeneous (h-h) reactions over a curved surface which is stretched in a linear manner. The study of MHD water-based nanofluid thin film using Homotopy analysis method past a stretched cylinder is considered by Khan et al [23]. Ramzan and Bilal [24] deliberated the 3D nanofluid flow in the attendance of MHD and chemical reaction. Ishak et al [25] utilized the stretching cylinder to examine the MHD flow. Qayyum et al [26] inspected the MHD stagnation point nanoliquid flow with Newtonian heat and mass conditions. Haq et al [27] scrutinized the MHD nanofluid flow with thermal radiation via a stretching sheet near a stagnation point. Bhatti and Rashidi [28] studied Hall Effect on an MHD peristaltic flow. Nadeem and Hussain [29] examined the MHD Williamson flow of nanoliquid past the heated surface. Ramzan et al [30] numerically studied the MHD micropolar nanofluid past a rotating disk. Ibrahim [31] used the linearly stretched surface to discuss the MHD nanofluid flow in the occurrence of melting heat near a stagnation point.
A direct proportionate between heat transfer rate and the local temperature is called Newtonian heating. It is also named as conjugate convective flow. It is utilized in many processes like designing of heat exchangers, conjugate heat transfer around fins and convective flows in which heat is absorbed from solar radiators by surrounding bounded surfaces etc. Merkin [32] was the first to consider four distinct categories of heat transfer phenomenon from wall to ambient fluid namely (a) Newtonian heating (b) conjugate boundary conditions (c) constant or prescribed surface heat flux, and (d) constant or prescribed surface temperature. Lately, various researchers have used the impact of Newtonian heating because of its broad practical applications [33][34][35][36][37][38][39][40].
A literature survey indicates that abundant research articles are available pertaining to nanofluid flows with combined impacts of the h-h reactions and MHDs past linear/nonlinear stretching surfaces. Comparatively, less research work is done with nanoliquids past cylinders and this choice gets even narrower if we talk about nanoliquid flows over nonlinear stretching cylinders. As far as our knowledge is concerned no study so far is conducted for the nanoliquid flow (with silver nanoparticles and water) past a nonlinear stretched cylinder with impacts of both h-h reactions, Newtonian heating, and nonlinear thermal radiation. Thus, our prime objective is to examine the nanoliquid flow past a nonlinear stretching cylinder with Newtonian heating, nonlinear thermal radiation, and h-h reactions. This exploration is unique in its own way and will attract a good readership. Numerical solution of the system of equations is acquired with the Runge-Kutta method by shooting technique. A comparative study with an already established result is also made and an excellent concurrence of both results is obtained.

Flow analysis
Consider an incompressible Ag-water nanoliquid flow past a nonlinear stretching cylinder with h-h reactions. In addition, nonlinear thermal radiation and Newtonian heating effects are also considered. It is presumed that a magnetic field = -( )/ B B x n 0 1 2 is operated along the radial direction. The induced magnetic field is overlooked due to our supposition of small Reynold number figure 1.
The homogeneous reaction for cubic autocatalysis can be communicated as given below: These reaction equations guarantee that reaction rate vanishes in the outer tier of the boundary layer.
Usage of the boundary layer approximation, the continuity, momentum, temperature and concentration equations are appended below: with allied boundary conditions  Figure 1. Illustration of the flow geometry. The numerical value of specific heat, density, and thermal conductivity of nanoparticle (Ag) and conventional fluid water is given in table 1.
The measured forms for the thermo-physical properties are given as: , .
Here, it is anticipated that A 1 and B 1 are analogous. This assumption implies that D A and D B (diffusion coefficients) are equivalent i.e. δ=1 and because of this assumption, we have Using equations (16), (13) and (14) with corresponding boundary conditions take the form

Local Nusselt number and Skin friction factor
The dimensional form of the skin friction factor (C f ) and local Nusselt number (Nu x ) are described as where t w and q w are Consuming equations (10) and (20), in equation (19), we get j l q 2 1 x f x x nf f 1 2 2.5 1 2

Numerical technique
Shooting technique is employed to find out the numerical solution of equations (11), (12) and (17) with associated boundary conditions (15) and (18). While finding the numerical solution, the third and second order differential equations are converted to first order by utilizing new parameters. In shooting technique, we select an initial guess that satisfies the boundary conditions and the equation asymptotically. For the present problem, tolerance is taken as -10 . 7 A comparison of the present analysis with already published paper Qasim et al [41] in limiting case is given in table 2 and all numerical calculations depict a good agreement.           reduction in the velocity of the fluid is noticed. The axial velocity diminishes with growing the value of nonlinear exponent n, this effect is depicted in figure 4. This is due to the fact that fluid particles are disturbed for larger values of n. Actually, more collision amongst fluid particles is witnessed that obstructs the movement of the fluid and ultimately a reduction in axial velocity is noticed. The impression of the curvature parameter g on axial velocity is depicted in figure 5. It is seen that the axial velocity is a growing function of g. In fact, increased values of the g result in squeezed radius and ultimately less contact area between the fluid and the cylinder is detected. This is the main reason behind the augmented axial velocity.

Temperature profile for several parameters
Figures 6-11 analyze the effect of curvature parameter, radiation parameter, conjugate parameter, and magnetic parameter on temperature field. Figure 6 exhibits the effects of curvature parameter g on the temperature profile. The fluid's temperature enhances for augmenting values of g. Actually, increase in heat transport is detected for augmented values of g, thus rise in temperature profile is witnessed. The impact of the conjugate parameter l on the temperature field is characterized in figure 7. It is determined that the temperature profile upsurges with mounting values of l. Higher values of l leads to stronger heat transfer coefficient and as a result more heat will transfer from the cylinder to the fluid. It is pertinent to mention that l  ¥ exhibits the constant wall temperature and l = 0 indicates the insulated wall. Figure 8 is drawn to analyze the effect of radiation parameter K * on temperature profile. The temperature field enhances for augmented estimations of K * . In fact, for growing values of K * , the mean absorption coefficient decreases thus growth in radiative heat transfer rate is perceived. Figure 9 is portrayed to examine the impact of magnetic parameter M on temperature field. It is seen that for the growing values of magnetic parameter M, the temperature profile enhances. This is because of the verity that the Lorentz force augments owing to increased estimates of M thus impeding the fluid's movement. In this way, more collision between molecules of the fluid is observed and additional heat is produced thus increasing the temperature of the fluid. The impacts of heat generation/absorption parameter D c and Eckert number E c are illustrated in figures 10 and 11 respectively. The temperature of the fluid escalates for growing estimates of D c and E c , which is an obvious veracity.

Concentration profile for different parameter
Figures 12 and 13 are illustrated to portray the strength of homogeneous and heterogeneous reactions' impact on concentration profile respectively. It is observed that the concentration profile intensifies in both cases for growing values of h, and after a certain estimate of h, no impact on concentration distribution is seen for both cases of the strength of homogeneous and heterogeneous reactions. for various estimates of parameters. It is detected that the numerical value of the Nusselt number and skin friction coefficient are enhanced for growing values of the nonlinearity parameter n, curvature parameter g, and solid volume friction f. While for the M (magnetic parameter) the skin friction coefficient enhances and the local Nusselt number diminishes. Further, for the value of temperature ratio parameter N r and radiation parameter K * , the Nusselt number diminishes, while the skin friction coefficient is constant for temperature ratio parameter and slight change is observed for radiation parameter.

Concluding remarks
The problem of nanoliquid flow with silver nanoparticles and water (base fluid), is discussed with nonlinear thermal radiation with Newtonian heating past a nonlinear stretching  cylinder. The effects of heterogeneous/homogeneous reactions with MHDs are also examined. The shooting technique is engaged to solve the nonlinear ODEs. The key points of the current effort are appended as follows: • For growing values of the solid volume fraction of nanoparticles, escalation in velocity field is observed. • The velocity profile diminishes, and temperature profile enhances for augmented values of the magnetic parameter. • For mounting values of radiation and curvature parameters, the temperature field enhances. • Concentration field decreases versus increasing values of the strength of heterogeneous and homogeneous reactions.