Numerical study of radiative non-Darcy nanofluid flow over a stretching sheet with a convective Nield conditions and energy activation

Numerous industrial processes such as continuous metal casting and polymer extrusion in metal spinning, include ow and heat transfer over a stretching surface. The theoretical investigation of magnetohydrodynamic thermally radiative non-Darcy Nano uid ows through a stretching surface is presented considering also the in uences of thermal conductivity and Arrhenius activation energy. Buongiorno’s two-phaseNano uidmodel is deployed in order to generate Thermophoresis and Brownian motion e ects [1]. By similarity transformation technique, the transport equations and the respective boundary conditions are normalized and the relevant variable and concerned similarity solutions are presented to summarize the transpiration parameter. An appropriate Matlab software (Bvp4c) is used to obtain the numerical solutions. The graphical in uence of various thermo physical parameters are inspected for momentum, energy and nanoparticle volume fraction distributions. Tables containing the Nusselt number, skin friction and Sherwood number are also presented and well argued. The present results are compared with the previous studies and are found to bewell correlated and are in good agreement. The existingmodelling approach in the presence of nanoparticles enhances the performance of thermal energy thermoplastic devices.


Introduction
Nano uids are maneuvered by suspending nanoparticles with average size below 100 nm in traditional heat transfer uids such as ethylene glycol, oil and water. An important role is simulated by thermal conductivity of water, oil and ethylene glycol with nanoparticles in the heat transfer between the heat and mass transfer medium and the heat and mass transfer surface. The rise in thermal conductivity is substantial in improving the heat transfer behavior of the uids. High heat transfer performance is necessitated by several engineering applications. In several engineering and industrial applications, the thermal conductivity of nano uid is not constant and it di ers linearly with temperature. Nano uid is the combination of nanoparticles with water. The addition of a surfactant to a base uid improves thermal conductivity and convective heat transfer. Nano uid technology has emerged as a modern heat transfer technique that is more e ective. Nano uids are used in solar energy applications such as heat exchanger design, medical applications such as cancer therapy and safer surgery by heat treatment applications. Nano uid technology can be used to create better oil and lubricants for realistic applications [2]. Many scientists and engineers have been working on uid forming for the past few decades in order to improve e ciency for various thermal applications. The term "nano uid" has been proposed to address the new heat transfer problems by applying nanotechnology to heat transfer [3]. Many authors have predicted that nano uids would have greater convective heat transfer capabilities than base uids. Heat and mass transfer of nano uids have integrated many practical applications of thermal and solutal strati cation. Patil et al. [4] investigated the unsteady mixed convection nanoliquid ows through an exponentially stretching surface in the presence of applied transverse magnetic eld with realistic application to solar systems. Khan et al. [5] presented the entropy analysis and gyrotactic microorganisms of Buongiorno's nano uid between two stretchable rotating disks using homotopy analysis. Nainaru et al. [6] studied the e ects of heat transfer characteristics on 3D MHD ows of nano uid induced by stretching surface with thermal radiation. They observed a rise in the uid temperature and velocity with a rise in variable thermal conductivity. Shiriny et al. [7] considered the forced convection ow of nano uid in a horizontal microchannel with cross ow injection and slip velocity on microchannel walls. They observed that an increase in velocity slip and heat transfer rate by increasing the angle of injection to 94 • . Suhail and Siddiqui [8] presented the numerical study the natural convection ows of nano uid within a vertical annulus. They discussed the simulations for various nanoparticle concentrations for various heat ux values. Few recent studies on nano uid includes [9][10][11].
Due of its universal and practical applications, the convective boundary layer plays an important role in the process of heat supply to the uid via a particle surface with a restricted heat e ciency, especially in various manufacturing and advanced processes that include transpiration cooling, fabric cleaning, laser pulse heating and so on. Convective heat transfer is important in mechanisms like thermal insulation, gas turbines and nuclear power stations among others [11]. The analysis of electrically conducting uids like sea water, antioxidants, plasma and metal oxides is known as magnetohydrodynamics (MHD). Alfven [12] was the one who invented the word MHD. The strength of magnetic induction a ects MHD. The MHD uid ows has a many engineering and industrial applications like crystal development, nuclear freezing, magnetic improved drug, MHD detectors, and energy production. The MHD uid ows also has many applications in healthcare and biopharmaceutical elds such as hyperglycemia and emergency medicine, radiation therapy and many more. Khilap et al. [13] assessed the melting heat transfer and non-uniform heat source on magnetic nano uid ows past a porous cylinder using Keller box technique. Jha and Malgwi [14] analyzed the mixed MHD convection ow of viscous uid in a vertical microchannel considering the e ects buoyancy forces, pressure gradient, Hall and Ionslip current. Haque [15] explored the e ects of induced magnetic eld and heat sink on the micropolar uid ows past a semi-in nite vertical porous plate using the explicit nite di erence technique. They observed that both heat sink and thermal di usion have a growing e ect of the species concentration and the lighter particles have a higher uid concentration compared to heavier ones. Alam et al. [16] considered the principles of magnetohydrodynamics and ferrohydrodynamics to examine the biomagnetic ows of blood containing gold nanoparticles past a stretching surface in the presence of Biot number, suction and velocity slip using the bvp4c technique. Khan et al. [17] presented the entropy analysis of MHD radiative ows of Je rey uid past an inclined surface using Keller-Box technique. Vedavati et al. [18] examined the MHD convection ows of nano uid past an inverted cone considering the suction/injection e ects and entropy analysis is also presented.
Thermal radiation is a key aspect in many engineering processes that take place at extreme temperatures. Many industrial applications involving extreme temperatures such as gas turbines, nuclear power stations and various combustion turbines for aircraft, rockets, spacecraft and aerospace engineering have stressed the e ects of radiation on convection. Radiative heat transfer is crucial in oxidation, fossil fuel energy cycles, astrophysical ows, renewable energy engineering are the other applications. Mumtaz et al. [19] studied the mixed convective chemically radiative ows of tangent hyperbolic uid with viscous dissipation in a doubly strati ed medium using BVPh2 scheme. Yusuf et al. [20] investigated the entropy analysis on radiative MHD ows of Williamson nano uid past an inclined porous plate considering the chemical reaction, gyrotactic microorganisms and convective heat transfer effects. Jawad et al. [21] presented the radiative MHD ows of second grade hybrid nano uid past a stretching/shrinking surface using the homotopy analysis. They observed that an increase in the volume fraction of the nanoparticles increases the uid's thermal e ciency. Ge-JiLe et al. [22] discussed the radiative MHD ows of Je rey uid past the horizontal walls in the porous medium and considering the viscous dissipation e ects. Jawad et al. [23] considered the MHD mixed convection ows of Maxwell nano uid past a stretching surface considering the in uences of variable thermal conductivity, gyrotactic microorganisms, radiation, Dufour and Soret.
A surface will uid-lled pores (voids) is termed as porous media. In industry, porous materials are commonly seen in reverse osmosis, steam turbines, heterogeneous catalysts, activated carbon columns, ltration centers and evaporative freezing. The analysis of uid ow processes in a porous media has sparked researcher's interest due to its various implications in scienti c, bio-logical and industrial producing goods such as porous bearings, atomic reservoirs, groundwater contamination, crude oil processing, casting and welding in production processes, porous bearing, hydro power, vapor movement in brous packaging, organic catalytic reactors, renewable energy and recycling devices. In 1856, Darcy formulated a principle that states that the volumetric ux of uid across a medium has a direct relationship with the pressure gradient. Practically, the Darcy term [24] is commonly used in the problems relating to ow saturating porous space modeling and analysis. The Darcy's law is only applicable when the velocity is low and the porosity is small. Inertia and boundary impacts are ignored by this rule. The customized version of classical Darcy's principle results in the non-Darcian porous space which integrates inertial and boundary impacts. As a result, Forchheimer [25] took into account the inertia by using a square velocity term in the momentum term. Non-Darcian versions are extensions of the standard Darcy concept that includes inertial drag, vorticity dispersion and combinations of these impacts. Darcy law in the Darcian medium is criticized as failing at high velocity, high porosity medium and enormous Reynolds number. To take the responsibility of the inertia impacts of pressure drop, the Forchheimer expression is integrated into the square velocity term within the momentum equation during this method. Pop and Ingham [26] and Vafai [27] surveyed temperature change in heat and mass transfer ows of Darcian and non-Darcian porous media. Hayat et al. [28] presented the Darcy-Forchheimer 3D ows of nano uid past a rotating surface considering the e ects of activation energy and heat generation/absorption using NDsolve technique. Asma et al. [29] examined the convective heat transfer analysis of Darcy-Forchheimer 3D ows of nano uid past a rotating disk in the presence of Arrhenius activation energy using shooting technique. Ramzan et al. [30] investigated the melting heat transfer e ects of unsteady nano uid ow between two parallel disks considering the Darcy-Forchheimer permeable media, Cattaneo-Christov heat ux ad homogeneous-heterogeneous reactions using bvp4c technique. Sohail et al. [31] studied the MHD convection ows of hybrid nano uid past a stretching porous surface with viscous dissipation impacts using successive over relaxation technique. Kareem and Abdulhadi [32] investigated the axisymmetric MHD Darcy-Forchheimer ows of third grade uid past a stretching cylinder considering the Cattaneo-Christov e ects using homotopy analysis technique. Jawad et al. [33] presented the entropy analysis of MHD radiative Darcy-Forchheimer 3D Casson nano uid ows past a turning disk considering the Arrhenius activation energy using homotopy analysis scheme.
Zhang et al. [34] MHD Darcy-Brinkman-Forchheimer ows of third-grade uid between two parallel plates considering the in uences of Joule heating and viscous dissipation.
Many scientists and engineers have considered the activation energy, which was originally proposed by Svante Arrhenius in 1889 and is de ned as the minimum amount of energy needed to carry out a reaction phase. The activation energy is the energy given to the reactants to transform them into products in di erent chemical reactions. The kinetic and potential energy involved with the molecules are of deemed importance to break bonds or stretch and twist bonds. It is observed that molecules rebound with each other without completion of reaction if their movement is detected slowly with low kinetic energy or they smash improperly. However, due to high momentum energy, a chemical reaction is initiated for which minimum activation energy is required. The activation energy concept is more signi cant in suspension of oil, hydrodynamics, oil storage industries and in geothermal. Owing to such interesting applications, this phenomenon is studied by many researchers. For instance, Umair et al. [35] explored the impacts of Soret and Dufour on chemically radiative magnetohydrodynamics ows of Cross liquid through a shrinking/stretching wedge also considering the e ects of activation energy. Aldabesh et al. [36] dealt with the unsteady ow of Williamson nano uid with gyrotactic microorganisms through a rotating cylinder considering the impacts of activation energy, chemical reaction and variable thermal conductivity. Mehboob et al. [37] investigated the MHD thermally radiative 3D ows of Cross uid considering the e ects convective heat transfer, strati cation phenomena, heat source/sink and activation energy using bvp4c technique. Sami et al. [38] explored e ects of velocity slip and Arrhenius activation energy on MHD radiative 3D rheology of Eyring-Powell nano uid past a stretching surface using shooting technique. Muhammad et al. [39] studied the e ects of Brinkman number, magnetic parameter, di usion parameter, Weissenberg number and activation energy on the entropy analysis of Carreau-Yasuda uid past a stretching surface using the homotopy analysis method. Few recent studies on activation energy include [40][41][42].
By keeping the above studies in mind, the main objective of the current analysis is to study the e ects of Arrhenius activation energy of Darcy-Forchheimer Nano uid past a stretching surface in the presence of thermal radiation and magnetohydrodynamics. In addition, the in uences of thermal conductivity, velocity slip, Biot number and Nield boundary condition are also considered. An appropriate bvp4c along with 3-stage Lobatto IIIa method from MATLAB software is maneuvered to solve the two-point boundary value problem using the similarity variables. The in uences of several thermo physical parameters on velocity, temperature, nanoparticle spices concentration elds, skin friction, Nusselt number and Sherwood numbers are presented graphically and numerically. The results of the present study are compared with those of Wang [43], Gorla and Sidawi [44] and Khan and Pop [45] and found to be a good agreement.

Problem formulation
Let us assume a steady and incompressible thermally radiative MHD ow of Nano uid induced by linear movement of stretching sheet embedded in fully saturated non-Darcy porous medium considering the in uences of thermal conductivity, velocity slip, convective heat transfer and Nield boundary condition. The physical ow model and coordinate system is presented in Figure 1. The x-axis is considered along the stretching surface in the direction of the motion and y-axis is taken normal to the surface. The sheet is stretched along the x-axis with a velocity u = U (x) = ax, where a is a positive constant. The acceleration due to gravity, g is assumed to act downwards. A strong magnetic eld having strength B is imposed in the normal direction. Both viscous and ohmic dissipation e ects are neglected. By choosing a small Reynolds number, the aspects of the induced magnetic eld are ignored.
Under these assumptions along with the Boussinesq and boundary layer approximations, the governing equations for Nano uid [9][10][11] are: (4) Using Rosseland approximation, qr can be expressed in non-linear form as: By assuming that the temperature di erences within the ow are su ciently small, using Taylor's series expansion, can be expressed as, Using Eqs. (5) & (6) in (3), we get, The Navier's slip condition, convective condition and Nield boundary conditions are: As y → ∞ (8) Introducing the similarity variables: The continuity equation is satis ed by considering the stream function ψ(x, y) as follows: Using Eq. (6) in Eqs. (2), (7) and (4), we get, Pr The prescribed two-point boundary conditions are redesigned as: Where To calculate the heat and concentration transfer rates, the physical quintiles shear stress rate (C f ), local Nusselt number (Nux) and local Sherwood number (Shx) are de ned as: Nux Here Rex = ax ν is the local Reynolds number.

Numerical procedure
Mathematically, the system of coupled dimensionless Eqs. (7) -(9) subject to boundary conditions Eq. (10) is strongly non-linear and are indeed very di cult to solve analytically. Hence the bvp4c technique from matlab is used to solve this system of equations numerically. The Matlab BVP solver bvp4c from matlab, a nite di erence code which implements the 3stage Lobatto IIIa formula is used to obtain the numerical solutions. In this technique the Eqs. (7) -(9) are rst transformed into a set of coupled rst-order equations as follows: Therefore, Eqs. (7) -(8) can be written as: And then this is set up as s boundary value problem (bvp) and use the bvp solver in matlab to solve the system along with the speci ed boundary conditions numerically using the RK method of order 4(i.e., bvp4c). The iterative process will be terminated when the error involved is < 10 − .
For further information about the algorithm of bvp4c the readers can refer to Shampine et al. [46] and Ibrahim [47].

Results and discussion
The system of Eqs. (7) Table 2 for distinct values of Pr, M, Nr, R, λ, K , E, Fr, Sc, S, Bi, Nb, and Nt. A rise in Pr is noted to increase C f , Nux and Shx. Hence the heat transfer to the surface is raised with lower thermal conductivity (higher Prandtl number) and the nanoparticle di usion (mass transfer) is raised. Further, it is seen that C f is enhanced with a rise n M values whereas both Nux and Shx are reduced. Fluid energizes with the impact of stronger magnetic eld and the heat is expanded from the boundary that leads to further decrease in heat transfer to the wall. This process a ects adversely on the nanoparticles di usion to the wall. The magnetic quantities leads the energy to the uid and improves the heat transportation through the sheet. Hence, the magnetic parameter could be used as a method for plotting the features of ow and heat transport. It is further seen that increasing Nr values reduces C f , Nux and Shx since radiation stimulates the nano uid. However, increment of Nr with high radiation implies elevation of nanoparticles di usion to the boundary layer and diminishes nanoparticles concentration at the boundary layer. Also, an increase in radiation parameter is seen to decrease C f and Shxbut Nux is boosted. Increasing mixed convection parameter (Richardson number) is seen to largely reduce C f however, a signi cant rise is observed in the case of both Nux and Shx. An increase in Darcy number is seen to enhance C f greatly but both Nux and Shx are seen to decrease.
With a rise in activation energy number, a very slight variation is observed in C f , Nux and Shx. A rise in velocity slip parameter is seen to reduce C f , Nux and Shx. A rise in biot number is seen to enhance Nux and Shx. A rise in Schmidt number is seen to reduce Shx. A rise in Brownian motion parameter is seen to reduce Shx. A rise in Thermophoresis parameter is seen to enhance Shx.       ture is noted with a rise in λ values. The uid density varies with temperature variation due to thermal expandability and the ow can be in uenced by the buoyancy force. As a consequence, the more we go from pure force convection (λ = 0) to natural convection under de ned Reynolds, the more cooler the interface becomes. The nanoparticle concentration is noted to increase with an increase in λ. Figures 8 -10 display the e ect of Darcy number, K , on the velocity, temperature and concentration distributions. An increase in K values leads the velocity pro les to decay. The existence of permeable space increases the uid stream safety by lowering uid velocity and the energy layer associated with it. Physically, the presence of porosity causes resistance to uid motion, resulting in a reduction in uid velocity. The temperature pro les are enhanced with a rise in K values. Porousness, in general, increases uid ow resistance, resulting in increased tem-   Figures 11 -13 portrays the e ect of Forchheimer parameter, Fr, on the velocity, temperature and concentration distributions. The velocity pro les are lowered with greater values of Fr since the inertia coe cient is directly proportional to the drag coe cient. Therefore, the drag coe cient increases as Fr increases. As a consequence, the uid's resistance force increases and hence the velocity decreases. An increase in Fr values leads to thickening the thermal boundary layer and doesn't allow the uid to pass easily. The temperature pro les are strongly in-   Figures 14 -16 display the e ect of velocity slip parameter, S, on the velocity, temperature and concentration distributions. A decay in velocity pro le is seen with an increase in S values. As the S values raise, the stretching impacts are partly pass through the uid and hence the velocity decreases. An increase in S values is noted to increase both temperature and concentration pro les. If S = 0, the uid holds to the boundary and the uid slides with no resistance as S → ∞. As the S values increase, the motion of the uid particles declines and hence the temperature and concentration increases. Figures 17 -18 display the e ect of Prandtl number, Pr on the temperature and concentration distributions. Generally, the temperature pro les decline with a rise in Pr   Figure 19 -20 display the e ect of thermal radiation, R, on the temperature and concentration distributions. A strong elevation in temperature and concentration pro les is noted with a rise in R values. This is due to the fact that certain amount of heat energy is emitted during the radiation cycle.  Figures 21 -22 display the e ect of Biot number, Bi on the temperature and concentration distributions. A strong increase in both temperature and concentration pro les is seen with a rise in Bi values. The buoyancy force increases as the Biot number rises and the uid carries the heat energy at a faster rate. Due to thermal expandability, the uid density varies with temperature and the ow is in uenced by the buoyancy force. Biot number is de ned as the relationship between solid conduction and surface convection. Physically, as the Biot number rises, the surface's thermal resistance decreases dramatically. Convection is increased, resulting in a higher surface temperature.  Figure 23 display the e ect of thermophoresis parameter, Nt, on the concentration distributions. The process of moving the particles so that they contract temperature due to the temperature gradient force is called thermophoresis. The parameter Nt, plays a crucial role in nanoparticle volume fraction. A signi cant enhancement is observed in the concentration pro le with an increase in Nt values. As Nt increases, the heat transfer in the boundary layer rises and at the same time exacerbates particle deposition away from the uid region, thereby raising the nanoparticles volume fraction. Figure 24 display the e ect of Brownian motion parameter, Nb, on the concentration distributions. The nanoparticle concentration is substantially decreased with increasing Nb vales. The nanoparticles reorganize to create a new structure because of the spontaneous di usivity. Thus the thermal conductivity of the nano uid is improved. The Brownian motion warms the uid in the boundary layer   Figure 25 display the e ect of Schmidt number, Sc, on the concentration distributions. A signi cant decrease in concentration pro les and the corresponding boundary layer thickness is noted with a rise Sc values. By def., Sc represents the di usion ratio of momentum to mass. Hence, mass di usivity is the reason for the decrease in the concentration. Figure 26 display the e ect of Reaction rate, σ R , on the concentration distributions. A signi cant reduction in concentration pro les is noted with rising σ values. An increase in σ values results in an increase in the Arrhenius expression which eventually damages the chemical reaction. Hence the concentration pro les decay.  Figure 27 illustrates the in uence of Energy activation number, E, on the concentration pro les. It is found that the concentration pro les increase with an increase in E values. Usage of activation energy is more e cient in enhancing the reaction process and hence increasing the concentration. The Arrhenius expression declines with an increase in E values, resulting in the development of the relational chemical reaction leading to an increase in the concentration pro les. Due to the phenomenon of low temperature and greater activation energy results in a lower reaction rate that slows down the chemical reaction. This way the concentration increases. Figures 28 -29 depicts the in uences of magnetic parameter M on skin friction and heat transfer rate. It has been seen that as the values of M increases, the skin friction increases near the wall and as we move away from the wall it decreases and a quite opposite trend is observed  in the case of heat transfer rate. This con rms the earlier observations that the ow is slowed by the magnetic eld but the uid is heated. Higher heat transfer rates and a decrease in uid temperature are associated with heat transfer from the uid to the surface. Figures 30 -31 depicts the in uences of radiation parameter R on heat and mass transfer rates. A greater increase in heat transfer rate is observed with a rise in radiation parameter whereas the mass transfer rate is decreased. The ow is accelerated by the strong radiation, but the heat transfer to the surface is reduced. Species diffusion to the surface is hampered by the greater contribution of thermal conduction heat transfer. Figure 32 -33 depicts the in uences of velocity slip S and Darcy number K on skin friction. A signi cant decrease in skin friction is seen with an increase in velocity slip. With an increase in Darcy number the skin friction is      Figure 34 presents the in uences of Forchheimer number Fr on heat transfer rate. With an increase in Fr, the skin friction is slightly decreased near the wall and as we move away from the wall it is increased signi cantly. Figure 35 illustrates the in uences of Schmidt number Sc and reaction rate, σ R on mass transfer rate. The mass transfer rate is increased with an increase in Sc values. While the mass transfer rate is decreased with an increase in σ R values. An increase in Sc results in a decline in species mass di usivity and hence increases mass transfer rate.

Conclusion
The present analytical study focuses on the thermally radiative electrically conducting Darcy-Forchheimer Nano uid ow characteristics past a stretching sheet considering the e ects of thermal conductivity, velocity slip, convective heat transfer and Arrhenius activation energy. The governing ow equations are converted into ordinary di erential equations (ODE's) with appropriate similarity transformations. Bvp4c technique is utilized in order to obtain the results. The present numerical code is validated with the previous results available in literature. The in uences of various ow controlled parameters on velocity, temperature and nanoparticle volume fraction as well as shear stress rate, local Sherwood and Nusselt numbers presented and numerically and graphically. The observations of as follows: 1. The uid velocity is decreased with increasing values of M whereas the uid temperature and concentration are increased. 2. Increasing mixed convection parameter elevates the uid velocity and nanoparticle concentration slightly whereas temperature is slightly increased. 3. Increasing Darcy number is noted to reduce the uid velocity and nanoparticle concentration whereas temperature is enhanced. A similar behavior is observed with an increase in Forchheimer number and velocity slip parameter. 4. Increasing Pr reduces the temperature and nanoparticles concentration. Conversely, increasing Biot number enhances both temperature and nanoparticles concentration. 5. Increasing radiation parameter enhances the uid temperature and nanoparticle concentration. 6. Increasing thermophoresis parameter strongly accelerates the nanoparticle concentration whereas increasing Brownian motion parameter and Schmidt number reduces nanoparticle concentration. 7. The nanoparticles concentration is decreased with increasing reaction rate parameter whereas increased slightly with increasing activation energy parameter. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

Con ict of interest:
The authors state no con ict of interest.