Numerical study of bio-convection flow of magneto-Cross nanofluid containing gyrotactic microorganisms with effective Prandtl number approach


 In this study, a mathematical model is developed to scrutinize the transient magnetic flow of Cross nanoliquid past a stretching sheet with thermal radiation effects. Binary chemical reactions and heat source/sink effects along with convective boundary condition are also taken into the consideration. Apposite similarity transformations are utilized to transform partial differential equations (PDE’s) into ordinary ones and then numerically tackled by shooting method. The Impacts of different emerging parameters on the thermal, concentration, velocity, and micro-rotation profiles are incorporated and discussed in detail by means of graphs. Results reveals that, the escalation in Magnetic parameter and Rayleigh number slowdowns the velocity and momentum of the fluid. The increase in Biot number, radiation and Heat sink/source parameters upsurges the thermal boundary but, converse trend is seen for escalating Prandtl number. The density number of motile microorganisms acts as a growing function of bioconvection Lewis number and declining function of bioconvection Peclet number.


Introduction
Researchers and scientists nowadays have more research fields due to developments in nanotechnology and nanoscience. Nanofluids are proving to be useful in a variety of situations, including heat transference Technological advances demand efficient heat conveyance processes, and nanoliquids provide a more effective medium for heat passage from one source to the other.
In this regard different fluid models have been presented over time to illustrate fluid properties.
The Cross-viscosity model for generalized Newtonian fluid was proposed by Cross. By using this concept, Khan et .al [1] elucidated the chemically reacting magnetohydrodynamic (MHD) radiative flow of cross nanoliquid towards a stretchy sheet. Ali et .al [2] explicated the radiative stream of cross nanoliquid instigated by a stretchy sheet. Abbas et .al [3] expounded the MHD convective stream of cross nanoliquid with thermal radiation. Waqas et .al [4] investigated the bioconvective stream of cross nanoliquid flow with radiation and melting effect. Xiong et .al [5] illuminated the chemically reacting dissipative flow of cross nanoliquid with mixed convection.
The studies of magnetohydrodynamic (MHD) streams of non-Newtonian and Newtonian liquids over a stretching surface have gotten a lot of interest because of their wide range of uses in chemical engineering and metallurgy. Such MHD flow investigations are critical in industry and have uses in a number of fields, including petroleum processing and metallurgical processes. The rate of cooling, as well as the desired end product properties, can be regulated by using electrically conducting fluids and applying a magnetic field. In the purification of molten metals from nonmetallic inclusions, a magnetic field has been used. Recently, Hayat et .al [6] elucidated the MHD radiative stream of liquid past a porous stretchy sheet. thermal radiation. Radhika et .al [7] explicated the MHD steam of Dusty nanoliquid instigated by an elastic sheet with melting effect.
Senapati et .al [8] expounded the MHD flow of Casson nanoliquid flow above a stretch sheet. Patil et .al [9] deliberated the chemically reacting radiative stream of Powel-Eyring liquid above a stretchy sheet. Naidu et .al [10] elucidated the MHD radiative stream of Jeffrey nanoliquid above a stretchy sheet with Gyrotactic microorganisms.
The mechanism of thermal radiation involves a hot body releasing electromagnetic radiation in both directions. Many objects on the planet emit radiation in the infrared range of the electromagnetic spectrum. Thermal radiation is important in heat nuclear reactor protection, power plants, exchangers, furnace architecture, solar energy, and power technology. Motivated by these uses, Khan et .al [11] explored the radiative flow of Carreau liquid above a stretchy surface. geometry. Ali et .al [13] elucidated the radiative flow of nanoliquid with heat production/absorption. Reddy et .al [14] expounded the radiative flow of nanoliquid past a melting surface. Ijaz et .al [15] explicated the effect of radiation and magnetic dipole on bioconvective flow of Jeffery nanofluid.
The relationships between chemical reactions and mass transport are typically complex, as shown by the synthesis and absorption of reactant species at various rates both within the liquid and during mass transfer. Chemical reactions have plentiful usages in fields engineering and sciences like generator electric power and food processing. The concept of binary chemical reaction was initially elucidated by Merkin [16]. Rasool et .al [17] exemplified the impact of binary chemical reaction on Williamson nanoliquid above an elastic sheet. Wang et .al [18] elucidated the chemically reactive dissipative stream of Carreau nanoliquid above a stretchy surface. Gowda et .al [19] expounded the impact of chemical reaction on nanoliquid stream past a stretchy sheet coiled in a circle. Khan et .al [20] expounded the impact of binary chemical reaction on bioconvective stream of nanoliquid.
Bio-convection is a natural process that occurs as microorganisms move randomly in single-celled or colony-like formation. The directional motion of various forms of microorganisms is the basis for various bio-convection systems. Gyrotactic microorganisms are those that swim upstream against gravity in still water, causing the upper portion of the suspension to be denser than the lower part. Bioconvection's importance can be seen in a variety of bio-microsystems, such as biotechnology related to mass transportation, enzyme biosensors and mixing. Recently, Khan et .al [21] elucidated the MHD bioconvective stream of Newtonian fluid with chemical reaction. Chu et .al [22] explicated the MHD bioconvective stream of third grade liquid past a stretchy sheet by using Buongiorno model. Al-Khaled et .al [23] expounded the radiative bioconvective stream of nanoliquid. Jayadevamurthy et .al [24] elucidated the bioconvection flow of hybrid nanofluid instigated by a moving spinning disk. Arafa et .al [25] explicated MHD bioconvective stream of nanofluid with convective boundary constraint and radiation effect.
It is familiar that there are various approaches that could be contemplated in order to explain few realistic solutions for this specific type of issue. However, to the best of the authors' understanding, no numerical solution has been earlier inspected for magnetic flow of Cross nanofluid past a stretching sheet with thermal radiation effects. The focal point in the current paper is to examine the above-described flow numerically.

Mathematical formulation
In this study, we have considered transient two-dimensional magnetic flow of an incompressible Cross nanofluid past a stretching sheet in the presence of thermal radiation effects. A non-uniform time dependent transverse magnetic field of strength 0 B is applied perpendicular to flow direction (see Fig. 1). Since the magnetic Reynolds number is low, the induced magnetic field can be  Continuity: Energy equation: Bioconvection equation: The physical realistic boundary conditions are We have the similarity transformations is given by , 0 2 Pr 1 3 and transformed boundary conditions are: where prime denotes the derivative with respect to  and the dimensionless physical parameters are defined as follows: It should be remembered that the physical quantities of engineering concern, namely the local skin friction coefficient, local Nusselt quantity, and motile microorganisms, are also essential characteristics of the current investigation.    This deteriorating behaviour is caused by the fact that Rb is related to the buoyancy force caused by bioconvection, which allows the velocity profile to decay. Here, the increased values of Bi increase the heat transference. Physically describes that even more heating is delivered from the surface to the nanoparticles, resulting in an increase in the temperature gradient. Fig. 8 demonstrates the behaviour of 0 Q on temperature profile. The larger 0 Q enhances the thermal profile. In fact, Internal heat generation/absorption either improves or dampens heat transport. A incremental increase in 0 Q increases thermal gradient, showing mechanically that an increase in heat source intensity contributes to a greater thermal diffusion layer, which may increase thermal gradient. Fig. 9 highlights the outcome of the Rd on thermal profile. The increase in radiation parameter upsurges the thermal distribution. Here, more heat is applied to active liquid due to upshot in radiation phenomenon. As a result, heat transfer escalates Fig. 10 describes the behaviour of the Pr on thermal profile. The inclination in Pr declines the thermal gradient. Lower Pr equate to higher thermal diffusivity, although higher values result in higher diffusivity. As a consequence of this justification, the temperature is reduced. Shear thickening activity is characterised by gradual thinning of the boundary layer. Furthermore, by extension, the Pr is inversely related to thermal diffusivity, resulting in a decrease in the thermal profile.

Discussion of results
The graphical results for mass transference for varied values of the Nb is expressed in Fig. 11.
Here, mass distribution diminishes for inclined Nb . Concentration gradient decays after more particles are pushed in the reverse way of the solutal distribution to preserve solution homogeneity.

Data availability Statement
The data that support the findings of this study are available within the article, the data are made by the authors themselves and do not involve references of others.