Significance of Mixed Convection and Arrhenius Activation Energy in a Non-Newtonian Third Grade Fluid Flow Containing Gyrotactic Microorganisms towards Stretching Surface

In most scenarios of concern, the bulk of fluids treated by researchers and engineers, such as air, water, and oils, can be considered as Newtonian. The inference of Newtonian action however is not true in many situations and the much more complicated non-Newtonian reaction should be superimposed. Such situations exist in the chemical manufacturing sector and the plastics processing plants. Here, we present the mixed convective flow of non-Newtonian third grade fluid containing gyrotactic microorganisms through a stretching surface. The flow is considered as unsteady, laminar, and incompressible. Furthermore, the flow is magnetized and electrically conducting with the help of applied magnetic field. Chemical reaction along with Arrhenius activation energy and viscous dissipation influences are taken into attention. The governing PDEs are transformed to ODEs through appropriate similarity transformations. Analytical and numerical solutions of the present analysis are done with the help of incorporated codes


Introduction
The inclusion of strong nanoparticles in the heat exchangers significantly increases the fluid's thermal conductivity, which now takes scientists and researchers a few days to draw interest.
They have the ability to use the energy devices. In terms of construction, they are robust, costeffective, and simple. References [1][2][3][4][5] have originate their application in thermal transfer devices like aviation sectors, atomic reactors, heat exchangers, and a lot of more. Numerous scholars and analysts have concentrated on this issue in recent decades and have provided many productive and motivated scientific papers. In the context of nanofluid flow containing Ti6Al4V and AA7075, the thermal qualities of porous fin caused by radiation and natural convection through rectangular profiled longitudinal fin were addressed by Sowmya et al. [6]. The numerical exploration of an electrically conducting nanofluid containing microorganisms through a porous extending surface was addressed by Waqas et al. [7]. The comparative analysis of five different nanofluids like: TiO2, CuO, Ag, AL2O3, and Cu whereas H2O as a base fluid, between an two sheets with the influences of Joule heating, viscous dissipation, thermal radiation, and magnetic field was presented by Ahmad et al. [8]. The 3D radiative flow Oldroyd-B nanofluid with convective conditions through a stretching sheet was explored by Gireesha et al. [9]. The global influence of Williamson parameter on MHD Williamson nanofluid flow through a nonlinear extending surface with Arrhenius activation energy was presented by Dawar et al. [10]. The entropy optimization in MHD water based copper and silver nanofluids between two angularly rotating disks which are kept at constant temperature with the heat generation was determined by Shah et al. [11]. The analytical analysis of electrically conducting MHD micropolar nanofluid flow with secondary slips conditions through an extending surface with nonlinear thermal radiation was discussed by Dawar et al. [12]. Furthermore, the related analyses are mentioned in [13][14][15][16][17][18][19].
The analysis of non-Newtonian fluid flow is imperative because of the technical perspective in nonlinear fluid and has real-world uses in various industries. To reveal the existence of non-Newtonian fluids, many fluid models have been proposed. Both natural stresses and the shear thickening/thinning phenomenon can be modeled by the third-grade fluid model. Since this fluid model have distinct complications, so the flows of third-grade fluid have been analyzed in numerous ways by many scholars. The time dependent flow of a third grade fluid with heat transmission in a channel was numerically determined by Chinyoka and Makinde [20]. The combined effects of thermal transmission and mass transmission on the flow of third grade fluid were taken into account by Hayat et al. [21]. Owing to a stretching sheet subjected to partial slip, the entrained flow and heat transfer in third grade fluid was regarded by Sahoo and Do [22].
Using the homotopy analysis procedure, the analytical solution for unsteady flow of third grade fluid through a rotating surface was observed by Abbasbandy and Hayat [23]. A study was carried out in an area of stagnation point through an extending surface for the properties of melting thermal transmission in the third grade fluid by Hayat et al. [24]. The MHD flow of third grade fluid using porous medium was presented by Shafiq et al. [25]. In the manifestation of magnetic field, the numerical and analytical solutions of the third grade fluid flow in a hollow vessel using porous medium was analyzed by Hatami et al. [26]. They considered gold as nanoparticles which are added to the base third grade fluid. The incompressible third grade fluid flow with mixed convection using porous medium through an extending surface was probed by Hayat et al. [27]. The inquiry of MHD third grade fluid through an inverted cone with radiation impact was presented by Gaffar et al. [28]. The coating process using a simple fixed blade on a moving substrate for a third-grade fluid was discussed by Sajid et al. [29]. The two-dimensional MHD flow of Jeffrey nanofluid containing gyrotactic microorganisms through a stratified stretching sheet was examined by Waqas et al. [30].
In view of the literature survey, we present the bioconvective flow of non-Newtonian third grade fluid containing gyrotactic microorganisms through a stretching surface. The flow is considered as unsteady, laminar, and incompressible. The governing PDEs are transformed to ODEs through appropriate similarity transformations. Sections 2-6 present the mathematical modeling, HAM solution, HAM convergence, results and discussion and conclusion respectively.

Mathematical modeling
The unsteady and incompressible flow of motile gyrotactic microorganisms suspended in a third grade nanofluid through a stretching surface is considered. The nonlinear mixed convective influence is taken into account. Viscous dissipation and activation energy is also considered in the fluid flow. Using these suppositions, the principal equations are defined as: with boundary conditions , 0 The terms KT are defined as: where k  is the ambient fluid conductivity,  is the fluid viscosity parameter, and  is the constant of thermal dependent conductivity.
Considering the similarity variables as: Using the similarity transformations defined in eq. (8), eqs. (2-6) are reduced as

HAM solution
HAM is used to solve the eqs. (9-12) with boundary conditions (13) with the following conducts.
Initial guesses: Linear operators with properties:

HAM convergence
The series solutions of the modeled system of nonlinear equations are treated with HAM. HAM contains an auxiliary parameter h which adjusts and regulates the convergence areas of    Figure 1(a, b)       Thus, the reducing impact is depicted here.                      Velocity function increases for higher material parameters, and mixed convection parameter.
 The velocity function declines for higher bioconvective Rayleight number, magnetic parameter and viscosity parameter, third grade fluid and unsteady parameters.
 Thermal function heightens for higher Eckert number and thermal conductivity constant  The thermal function decays for higher Prandtl number.
 Concentration function heightens for activation energy parameter.
 The concentration function reduces for greater Schmidt number, and chemical reaction parameter.
 Density function of microorganisms escalates for escalating Peclet number.
 The density function of microorganisms reduces for Lewis number and concentration difference of microorganisms.
 A quit change behavior in streamlines between the steady case and unsteady case of the present analysis is found.