Analytical Framework of Forced Convective Two-dimensional MHD Flow of Jeffrey Nano Liquid With Impact of Thermal Radiation and Activation Energy


 The present communication owns to address the mathematical framework of two-dimensional electrically conductive and thermally radiative Jeffrey nanofluid flow by a curved surface. The interaction of a periodic magnetic field with the suspended nanoparticles and mixed convection are critically important due to its application in a broad spectrum. Buongiorno’s model, incorporates the effect of thermophoretic force and Brownian movement, describes the nature of Jeffrey nanofluid. The influence of activation energy, viscous dissipation, and thermal radiation effects are reserved. The dimensionless system of differential equations has been diminished from the modeled equations via transformation framework which is solved analytically versus homotopic algorithm. The stability and convergence analysis has been carried out to optimize system parameters and accuracy of the system. The effect of physical constraints on flow field, energy, and concentration of nanoparticles are portrayed via plotted graphs and debated.


Introduction
The advancements in nanofluid technology increased largely during the past few years due to their higher thermal performances. Nanofluid is a suspension of small size (1-100 nm) particles into a base liquid. Normally such fluids are made by using a mixture of solid particles into base fluids like water, engine oil, ethylene glycol (EG) and many other base liquids. The structure of nanoparticles consists of metal carbide, nitride, metal oxide, and even nanoscale liquid droplets [1]. Such particles have shape of sphere, tubular, rod-like etc. Choi's investigation [2] shows that the implementation of the nanofluids is utilized in vast range of industrial manufacturing processes like textile, transportation, paper production, electronic equipment, energy production and many others. The basic characteristics of nanofluids are their thermal conductivities which are much higher than the other base fluids. The combination of such substances offers us a medium for heat transfer that behaves as a fluid. In fact, addition of nano size particles in the base fluids develops thermal behavior due to which heat transfer characteristics are increased substantially. Further, the magnetohydrodynamic (MHD) nanofluids are significant in engineering and industrial processes. Such fluids are used in optical modulators, optical gratings, opticallswitches, tunable optical fiber filters, stretching of plasticcsheets and metallurgy, polymerrindustry, and other applications. Many metallurgic procedures involve drawing continuous strips/filaments through a nanofluid to cool them. Such strips are sometimes stretched processes the drawing and thinning of copper processes. In suchhsituations, the quality and wanted features of the final product is obtained by drawing such strips into electricallyyconducting fluid. Magneticcnanoparticles have a key role in the construction of loudspeakers, magnetic cell separation, hyperthermia, druggdelivery etc. The general applications of nanofluids include cooling of vehicle, making new types of fuel, saving of fuel in electric powerrgeneration units, cancer therapy, imaging and sensing and other useful applications. References [3][4][5][6][7][8][9][10] show some recent attempts on this topic. Ahmad et al. [11] investigated MHD flow of three-dimensional unsteady Jeffreyynanofluid with thermal radiation and Joule heating in view of porous medium. Fiza et al. [12] studied homotopic solution 3D steady state Jeffrey liquid in rotating channel subject constant magnetic field with the application of Hall current. Ahmad et al. [13] proposed analytical solution of unsteady 3D flow of Jeffrey nanofluid past a bi-directional oscillatory surface. The influence of Brownian movement and thermophoretic force are analyzed by incorporating Buongiorno's model. Rasheed et al. [14] discussed the consequences of unsteady hydromagnetic thermally radiative Jeffrey nanofluid flow over a vertical surface with viscous dissipation and Joule heating. Naidu et al. [15] debated numerical implication of partial slip effect on MHD Jeffrey nanofluid flow including microorganisms over a stretching surface. Ge-JiLe et al. [16] examine numerically the implication of slip flow of Jeffrey nanofluid by a stretching surface with entropy generation and activation energy. Noor et al. [17] studied heat mass transfer effect on thermally radiative and chemically reactive time dependent Jeffrey nanofluid flow in a permeable channel with slip condition and heat reservoir source. Shahzada et al. [18] scrutinized numerical solution of steady state 2-dimensional Jeffrey nanofluid by an elongated surface. Hayat et al. [19] studied entropy generation in flow with Ag and Cu nanoparticles over a rotating circular disk with variable thickness and thermal radiation. Krishnamurthy et al. [20] investigated numerical solution of 2-dimensional slip flow by a curved surface in view of porous matrix. Kumar et al. [21] examined thermally radiative Marangoni flow of Casson nanofluid in view of extended surface with chemical reaction, heat reservoir source and heat generation mechanism. Roja et al. [22] debated numerical solution of thermally radiative hydromagnetic flow of micropolar nanofluid by an inclined curved surface with entropy generation, Joule heating and multiple slip effects. Hamid et al. [23] examined convection and heat transfer features in MHD flow of water based CNTs in a fin-shapeddcavity with thermal radiation and viscous dissipation effect. Khan et al. [24] developed numerical algorithm to analyzed Eyring-Powell nanofluid by nonlinear curved surface by incorporating Brownian thermophoretic effects in the model equations. Khan et al. [25] explored various mechanism to developed heat and mass transfer rates by considering different permeable cavities for innumerable fluid models. Some recent and interesting attempt on this topic are provided in [26][27][28][29][30][31] for innumerable fluid models to understand heat transfer mechanism.

Mathematical model of the problem
Here, we assumed magnetohydrodynamics flow of Jeffrey nanofluid in view of vertically extending sheet enclosure by a uniform magnetic field and thermal radiation. The fluid flow is expected to be incompressible and laminar. The applied magnetic field is taken as B . The Reynolds number is small enough such that the influence of induced field is weaker in comparison to the applied magnetic field. Therefore, induceddmagnetic effect is zero. The electriccfield is absent and physical properties are constant and independent of temperature. Fig. 1 particularizes the physical configuration of modeled flow problem.
The parameters having engineering applications in the present problem are skin-friction and Nusselt numbers elucidated as:

Convergence analysis of homotopic solution
In section, we opted a well-known explicit analytical technique HAM for nonlinear computation. The computational analysis is remained insufficient unless the stability and convergence of the analytical solution are discussed. The derived series solution is mainly depending on the appropriate optimal selection of auxiliary constraints [12][13][14]. The precise choice of these parameters has a substantial importance in controlling and regulating convergence criteria. In this regard, we plotted and outlined curves  h for nonlinear governing Eqs. (8)-(10) in Fig. 2. The plat part of these curves which is parallel to x axis  fixes the allowable region of the convergence. The ranges of acceptable values of the auxiliary constraints for fruitful convergence have been disclosed in Table1. It has been perceived that average squared residual error dwindled via larger order of approximations.

Results and discussion
The discussion segment is devoted to understanding the elementary features of the physical constraints graphically. The nonlinear constitutive flow laws in Eqs. (8)  perceived that higher magnetic field diminished nanofluid flow. In reality when magnetic field is applied to fluid it intensifying viscosity of the fluid. In consequence, fluid particles experience resistance to flow due to which velocity retards. It is of great engineering interest that the yield stress of fluid flow can be measured correctly through a variation in magnetic strength. Figure 5 depicts variations in thermal field   T subjected to larger radiation parameter   R . This figure discloses temperature development for higher thermal radiation factor. In fact, the working fluid gets additional heat energy subject to radiation parameter. In consequence, temperature upsurges. Attributes of   Ec on temperature distribution   T curves perceived in Fig. 7. It is seen clearly from this plot that the temperature upsurges for the escalating values of   Ec constraint effectively. This augmentation in the fluid temperature is largely because of the fact that the frictional viscous heating in nanofluid reasons to allow heat energy into the working nanofluid due to which the temperature augments in the boundary layer section. Similarly, the presence of viscous dissipation effects in the energyyequation, the temperatureeprofile is boosted. When Eckert number is zero it signifies absence of viscous dissipation effects. Fig. 8 unveils the influence of   Nb on temperature field. The arbitrary motion of nanoparticles boosts for larger   Nb value developed the kinetic energy of the fluid particles due to which additional thermal energy is produced. Hence, fluid temperature upsurges. Fig. 9 predicts variations in thermal field curves boosts through larger thermophoresis constraint   Nt . Here, one can perceive that  

Conclusion
The present investigation displays the simulation of periodic magnetic Jeffrey nanofluid flow by a stretching surface. The fundamental flow laws diminished into dimensionless form via transformation and solved analytically with the implementation of homotopic method. One of the novel concerns is to perceive theoretically, how a periodic magnetic field affects nanoparticles. We witnessed following distinguished features through abovementioned investigation:  It is observed that velocity field enhances subject to increment in thermal radiation constraint. On the other hand, velocity profile diminishes due to higher implication of magnetic field.  Consideration of radiative factor and magnetic influence escalates thermal field curves.  One can perceived that heat mass transference coefficient boosts through higher radiation constraint and diminishes with increment of magnetic parameter.  An increment in Brownian motion parameter dwindles concentration profile and oppositive characteristics witnessed for thermophoretic force.  Here, one can noticed that heat mass transference coefficient boosts through higher radiation constraint.  The Nusselt number diminishes for higher values magnetic parameter and increases for larger radiation parameter.  The uprising value of magnetic parameter reducing fluid velocity and heat transfer rate rapidly.

Data availability statement
The data used to support the findings of this study are included within the article.