Numerical Investigation of Aligned Magnetic Flow Comprising Nanoliquid over a Radial Stretchable Surface with Cattaneo–Christov Heat Flux with Entropy Generation

The influence of entropy generation on aligned magnetic flow-including nanoparticles through a convectively heated radial stretched surface in the existence of Cattaneo–Christov heat flux is inspected. The highly nonlinear leading PDE’s via the similar scaling transformation are developed. The resulting system via the bvp4c technique from Matlab is computed. The impacts of rising constraints on the liquid velocity, nanoparticles concentration and temperature profile are argued and showed via portraits and table. In addition, the performance of liquid flow is inspected through the friction factor, the mass and heat transfer rate. With the rise in the thermal relaxation constraint, the thermal boundary layer is appreciably altered. Due to an aligned angle, the velocity of nanoliquid declines, while the concentration and temperature of nanofluid augment. It is also observed that the values of friction factor increase, whereas the values of heat and mass transfer decline due to an aligned angle. Entropy generation profiles developed due to magnetic parameters and the aligned angle. Lastly, a comparative scrutiny is composed via the previous studies which lead to support for our presently developed model.


Introduction
Regular heat transport liquids similar to ethylene-glycol, water and oil have lesser thermal conductivity are characterized as poor materials and consequently cannot accomplish the lofty claim of technological and industrial fields. Modern progresses in nanotechnology permit to develop a novel group of fluids express as a nanofluid, which is initially exploited by Choi [1]. Thereafter, this topic of research has magnetized the interest of several researchers globally in light of its delightful thermal characteristics and prospective applications in several areas, for instance, biomedical, microelectronics, and transportation. Buongiorno [2] suggested the convective-transport model for nanoliquid. He scrutinized that the movement of thermophoresis and Brownian constraints of nanoparticles bestow the immense development in fluids thermal conductivity. Nield and Kuznetsov [3] utilized the model of Buongiorno to investigate the problem of Cheng-Minkowycz for free convective flow containing nanoparticles through a vertical surface by porous media. Nanofluid towards a stretched sheet through heat transfer was considered by Khan and Pop [4]. Turkyilmazoglu [5] discussed the single model as well as multi-phase model containing nanofluids and obtained the analytic solutions. He scrutinized that the water-based Ag nanoparticle is the finest mixture of heat transfer. Pour and Nassab [6] discussed the single phase 2D laminar forced convective flow comprising nanoliquids through a backward facing step by employing bleeding conditions. Goodarzi et al. [7] utilized the model of two-phase to scrutinize the turbulent and laminar mixed convective flow of water-based copper nanoparticles through a rectangular shallow cavity. An analytical and numerical investigation on axisymmetric flow comprising nanoliquid was explored by Mustafa et al. [8]. They detected that development in Schmidt number transmits to a slimmer nanoparticles volume fraction of boundary-layer. Akbari et al. [9] discussed the laminar mixed convective flow comprising water-based water copper nanoparticle inside a micro-tube through a 90° angle of curvature angle by utilizing a finite volume technique. The unsteady flow of a Carreau fluid containing nanofluids over a stretched surface was scrutinized by Khan and Azam [10]. They explored that Nusselt number as well as Sherwood number are deflating due to thermophoresis parameter. Hussanan et al. [11] investigated the characteristics of inertial and microstructure containing magnetic ferrofluid towards a stretching/shrinking surface via an efficient model of thermal conductivity with a suction/injection. They scrutinized that a notable modification happens to the micro rotation velocity due to suction/injection. Rafique et al. [12] examined the impacts of Dufour and Soret numbers on magneto flow comprising nanoliquid through an inclined non-linear radiative surface with micropolar fluid. They showed that the velocity declines due to the nonlinear stretchable factor. Yousefzadeh et al. [13] studied the laminar flow through an open cavity by taking three distinct areas of heat transfer with mixed convection and different geometries. Faraz et al. [14] discussed the magnetic impact on time dependent flow involving Casson nanofluid through a radially stretched surface with mixed convection. Bagherzadeh et al. [15] investigated the laminar magneto flow of water based Al2O3 nanoparticle through a rectangular micro-channel with slip condition. They observed that the magnetic force squeezes the liquid to the bottom wall. Recently, Tian et al. [16] examined the impact of different surfactants on sediment of surface and coefficient of pool boiling heat transfer of waterbased DI/silica nanoparticles and obtained the results experimentally.
Entropy generation which has been utilized to measure the importance of irreversibility connected to friction, heat transport and additional non-ideal processes in a thermal system. To enhance the presentation of any heat transport procedure, entropy generation has been widely used as a useful tool. Spasojević et al. [17] used minimization of entropy generation in column of diabetic refinement through trays, where they considered the exchanged heat as a control variable rather than temperature. Abu-Nada [18] scrutinized the influence of suction/injection about a backward facing step below bleeding conditions with entropy generation. Finite volume technique is utilized to obtain the solution of leading equations. Entropy generation under the impact of magnetic field over an inclined permeable planar channel was scrutinized by Komurgoz et al. [19]. They noticed that the highest entropy generation was acquired in the absence of porosity and magnetic parameters. Further, Butt and Ali [20] explored the influences of thermal radiation and entropy generation on hydro magnetic free convective flow towards a vertical surface entrenched in permeable media. Rashidi et al. [21] examined the entropy generation on magneto flow towards a rotated disk entrenched in a permeable medium through slip impact and variable properties. Goodarzi et al. [22] scrutinized the impact of thermal radiation on turbulent and laminar mixed convective flow through an enclosure by semitransparent medium with entropy generation. The time dependent magneto flow of four water-based nanoparticles (Cu, Ag, Al2O3 and TiO2) through an accelerating stretched sheet with the convective boundary condition and entropy generation was discussed by Das et al. [23]. Shateyi et al. [24] used a relaxation spectral technique to inspect the entropy generation on magneto flow involving Maxwell liquid from a stretched surface. The influence of entropy generation using two significant slip mechanisms of Jeffery fluid including nanofluid was examined by Rehman et al. [25]. Almakki et al. [26] explored the impact of unsteady MHD axisymmetric flow of a nanoliquid towards a nonlinear stretched surface with entropy generation. The minimization of entropy generation on magneto flow induced by the slandering stretched surface with Joule and frictional heating was scrutinized by Afridi et al. [27]. Hayat et al. [28] discussed the importance of entropy generation on flow through a stretched surface with mixed convection, non-linear heat flux and source/sink. Aghaei et al. [29] examined the entropy generation of a water based multi wall carbon nanotube with vertical as well as horizontal baffles inside an enclosure with mixed convection. El-Aziz and Afify [30] explored the influence of Hall current on the magneto flow of Casson liquid from a stretched sheet with slip effect and entropy generation. They found that entropy generation is increased due to magnetic number and decreased due to slip, Casson parameter, Eckert number, and Hall parameter. Recently, Pordanjani et al. [31] scrutinized the combined impacts of thermal radiation and magnetic field on free convective flow comprising nanofluid inside a cavity with entropy generation. They observed that the heat transfer along with entropy generation decline due to angle of magnetic parameter.
In heat conduction, the Fourier's model is very significant for the mechanism of heat transfer. For the increment of heat transfer, a thermal time relaxation in Fourier's law was suggested by Cattaneo [32]. However, several thermal relaxation times have dissimilar materials. Taking into account, Christov [33] modified a time derivative known as a model of Cattaneo-Christov heat flux which is extremely important for analysis of convective heat transfer. Straughan [34] discussed the importance of Cattaneo-Christov heat flux on Newtonian incompressible liquids with downward gravity. He examined that effect of thermal relaxation significantly increased due to Cattaneo number. Li et al. [35] considered the Cattaneo-Christov model of heat flux on viscoelastic fluid over a vertical stretched surface with magnetic field and partial slip. The impact of homogenousheterogeneous on magneto flow involving third grade fluid with Cattaneo-Christov with the convective boundary condition was scrutinized by Ramzan et al. [36]. Hashim and Khan [37] examined the homogenous-heterogeneous influence on flow involving Carreau liquid from a slandering stretched sheet with Cattaneo-Christov effect. They observed that temperature of fluid is lower in case of Cattaneo-Christov model compared to Fourier's heat model. After the fundamental work of Christov, many researchers recently were attempted in this perspective are presented in the [38][39][40][41].
The literature discussed above exposes that the impact of aligned magnetic field along with Cattaneo-Christov heat flux comprising nanofluid from a radial stretched surface is not yet considered. Therefore, in this research, the influence of aligned magnetic flow containing nanoparticles through a convectively heated radial stretched sheet with Cattaneo-Christov heat flux is discussed. This assessment added a narrative approach for researchers and scientists to scrutinize the characteristics of nanofluid along with characteristics of heat and mass transfer. Similarity equations are built and acquired the computational solution via the bvp4c solver. The impacts of the significant restrictions are discussed in detail through graphs and table. The present analysis explains the significant features in the skin burns, heat propagation in tissues, laser surgery in biomedical engineering and food technology.

Problem Formulation
A steady magnetic aligned flow towards a radial stretched surface comprising nanoparticles is scrutinized. The surface is stretchable in the radially way with velocity The physical boundary conditions are The similarity transformation is used as: In light of this relation (6), Equations (2)-(4) are transmuted to The converted boundary restrictions are   (0) 0, '(0) 1 0, '(0) 1 (0) 0, (0) 1, where ' points out the differentiation to , magnetic parameter, Brownian parameter, thermal relaxation parameter, thermophoresis parameter, Prandtl number, convective parameter, the Schmidt number.
The significant physical quantities which are well-defined as   (11) where w  , w q , w m the shear stress, the heat and mass flux, respectively are described as That is, where Re w r ru      is the local Reynolds number.

Analysis of Entropy Generation
Entropy generation in the existence of MHD aligned flow containing nanofluid is engraved as The entropy generation volumetric has two factors, (i) Irreversibility (HTI) of heat transport and (ii) Irreversibility (FFI) of liquid friction (iii) Diffusive Irreversibility. It is considered as , the difference of temperature, the Brinkman number, the characteristic length based Reynolds number, the difference of concentration and the diffusive constant parameter, respectively. To check whether the entropy generation through dominates of heat transfer by entropy generation due to the liquid friction, the magnetic field and the diffusive irreversibility, or vice versa. The Bejan number is described

Results and Discussion
The transmuted nonlinear ODE's (7)-(9) through the converted boundary restrictions (10) are deciphered numerically via the bvp4c solver. The influences of relevant constraints concerning in the liquid flow problem are argued via tables and graphs. The range of the physical parameters are  [42] and Soid et al. [43] in restrictive cases are closely harmonized, which guaranteed the soundness of the present methodology. Geometry of the problem is given in Figure 1.         Figures 8 and 9 are set to monitor the motion of Brownian Nb and thermophoresis parameter Nt on the distribution of temperature profile and nanoparticles concentration profile. The thermal boundary layer mounting for the temperature distribution as we augmented the values of Nb and graphically behavior is captured in Figure 8. Physically, the nanoparticles kinetic energy enhances owing to the strength of the chaotic motion and thus, the fluid temperature raises. It is inspected from Figure 8 that the nanoparticles concentration decays because of enhancing Nb . Therefore, the motion of Brownian parameter builds the warm liquid inside the boundary and intensifies particle deposition left from the fluid regime to sheet at that time that causing in a decline in nanoparticles concentration and the thickness of the boundary-layer. The higher Brownian motion values involve the strong performance for the minor particle, while the lesser values of Nb pertained for stronger particle. Figure 9 discloses that the behavior of the solutions boosts up for both the temperature distribution and as well as for the concentration of nanoparticles for bigger values of Nt . Physically, diffusion enters into the fluid deeper by enhancing Nt which grounds to increase in concentration and thermal boundary layers. Further, the impact of Nt being further marked on the volume fraction contrast in temperature.  The outcome of convective parameter  on     and     are portrayed in Figure 10. It is analyzed from these figures that the profiles of     and     elevated as  raises. The strong convective heat at the sheet appearing due to mounting values of  , which growing the temperature and concentration. It authorized the species and as well as the thermal effect to pierce deeper into the quiescent fluid. As a result of the outcome they augmented the boundary layer with widening  for both the temperature and as well as the concentration. Figure 11 depicts the behavior of thermal relaxation constraint  on     .The results indicate that the temperature of fluid declines with augmenting thermal relaxation parameter. Physically, enhancing the thermal relaxation parameter increases the gap between the liquid molecules and due to these effects, the temperature of fluid depreciates.   Figure 12 predicts that the profile of entropy generation decreases with growing . Figure 13 reveals that the entropy generation augments due to larger M . It perceives that the magnetic field is a resource of entropy generation to the heat transfer and friction of fluid. It can be finalized that the entropy generation can be controlled by taking a small magnetic parameter which is an important issue in the propulsion of MHD nuclear. It is perceived from Figures 14 and 15 that entropy profile accelerates by growing either Bror Reynolds number. As the entropy generation created from machines of all irreversibility and consequently the entropy generation rises with augmenting R e L . The higher amount in the entropy generation formed via the irreversibility of liquid friction happens owing to growing Br.     Figures. 17 and 18, respectively, while these values appreciably decline with increasing values of M . These values are also tabulated in Table 2.

Final Remarks
The impacts of entropy generation and Cattaneo-Christov heat flux on MHD aligned flow past a radial convectively-heated stretched sheet comprising nanoparticles were studied numerically. The governing equations are developed into ODE's via the similar scaling transformation and these equations worked out numerically via the bvp4c solver. The subsequent attention-grabbing results can be ported from this research:


The fluid flow velocity diminishes due to aligned angle, while the temperature and concentration of nanoparticles show escalating behavior.  The velocity profile decays due to magnetic parameter, whereas concentration of nanoparticles and temperature display opposite behavior.  The impacts of Brownian and thermophoresis parameter on temperature distribution are identical.  The influences of motion of Brownian and thermophoresis on the volume fraction are opposite.  Temperature distribution as well as the concentration of nanofluid improves due to convective parameter.  Due to the thermal relaxation parameter, the temperature distribution declines.