**1. Introduction**

The importance of nanofluids has been increasing with time, and investigators intend to study the behavior of nanofluids subjected to heat transfer systems. Nanofluids and their implications in the industrial sector have grown more because of their homogeneous nature in thermal conductivity and rudimentary heat transfer. Typical fluids such as propylene glycol, water, and ethylene glycol, among others, have poor heat transfer properties. Nanofluid, a homogenous solid–liquid mixture, is applied to enhance the thermal conductivity

**Citation:** Nabwey, H.A.; Khan, W.A.; Rashad, A.M.; Mabood, F.; Salah, T. Power-Law Nanofluid Flow over a Stretchable Surface Due to Gyrotactic Microorganisms. *Mathematics* **2022**, *10*, 3285. https:// doi.org/10.3390/math10183285

Academic Editors: Camelia Petrescu and Valeriu David

Received: 24 July 2022 Accepted: 7 September 2022 Published: 9 September 2022

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of the base fluid. Choi [1] and Das et al. [2] exposed the most remarkable characteristic of nanofluids: thermal conductivity is dependent on temperature. Nanofluids are extensively used in energy and biomedical applications (nano-drug delivery, cancer therapeutics, and nano cryosurgery) [3–5]. Buongiorno [6] examined the effect of convective transport in nanofluid by observing the heat transfer properties of the Brownian motion and thermophoresis. Numerous studies are presented in the literature to demonstrate the uses of nanofluid in various fields [7–10].

The study of heat relocation in power-law (non-Newtonian) fluids has received significant attention because of its application in different branches of modern technologies, industries, and engineering like polymer-thickened oils, liquid crystals, polymeric suspensions, and physiological fluid mechanics. Furthermore, instances showing a diversity of non-Newtonian characteristics contain biological fluids, pharmaceutical formulations, toiletries, synthetic lubricants, cosmetics, paints, and foodstuffs (see Irvine and Karni [11]). Various models have been reported to analyze the non-Newtonian attitude toward fluids. Among these patterns, which are famous for following the empirical Ostwald–de Waele model, in which the shear stress varies according to a power function of the strain rate, gained much acceptance. The theoretical analysis of power-law fluid was first scrutinized by Schowalter [12] and Acrivos et al. [13]. Rashad et al. [14] examined the power-law nanofluid flow across a vertical cone in a porous medium. Later on, several studies were analyzed by many investigators [5,15–17].

Bioconvection occurs as the natural microbe swims upwards. Thus, the microorganism is denser than the base fluids. Because there are so many microbes on the upper surface of the foundation, it becomes weak. As a result, the bacteria decrease and promote bioconvection, and the microbes' return to swimming strengthens the process. This bacterial emigration into the water raises the temperature and mass transfer in the environment. Because of medical advancements, microscopic kinds have significantly contributed to the improvement in human life. Microorganisms are essential for life and it cannot exist without them. Decreasing the length of the cavities and the cell resistance enables the construction of continuum numerical patterns. It is often approved that the nanoparticles of concentration distribution are massive relative to the cell pivot. Bioconvection occurs as combined nanofluids are addressed using heat and mass conversion. Platt [18] further proposed the concept of bioconvection, which characterizes the micro-structural flow brought on by gradation density, in addition to motile gyro-tacticmicro-organisms. The first study of the bioconvection of gyro-tactic motile bacteria, including nanoparticles, was conducted by Kuznetsov and Avramenko [19]. Kuznetsov [20] presented key discoveries of nanofluidbioconvection in a horizontal porous layer, including both nanoparticles and gyro-tactic motile bacteria. Khan and coworkers [7,21–24] investigated the free convection of non-Newtonian nanofluid numerically down a vertical plate in a porous media. They considered that the medium contains gyrotactic microorganisms and that the temperature, nanoparticle concentration, and motile microbe density are all controlled in the plate.

The local Nusselt, Sherwood, and density numbers are found to be strongly influenced by nanofluid and bioconvection parameters. Additionally, they provided a mathematical model to examine the flow of a water-based nanofluid containing gyrotactic microorganisms around a truncated cone with a convective boundary condition at the surface. It has been discovered that, as a surface grows rougher, the densities of the mobile microorganisms, Sherwood number, Nusselt number, and skin friction all increase. Nabwey and collaborators [8–10] investigated the flow of a nanofluid containing gyrotactic bacteria over an isothermal cone surface in the presence of viscous dissipation and Joule heating. Consideration was given to the combined effects of a transverse magnetic field and Navier slip in the flow. They also considered mixed bioconvective flow on a vertical wedge in a Darcy porous media filled with a nanofluid that contains both nanoparticles and gyrotactic bacteria. To combine energy and concentration equations with passively controlled boundary conditions, the effects of thermophoresis and Brownian motion are considered. They found that the buoyancy ratio and magnetic field parameters increase the local skin

friction coefficient, Nusselt number, Sherwood number, and local density of the motile microorganism's number. Ishak et al. [25] analyzed the flow of a two-dimensional stagnation point of an incompressible viscous fluid near a steady mixed convection boundary layer flow across a stretching vertical sheet in its plane. For an aiding flow, they demonstrated that both the skin friction coefficient and the local Nusselt number increase with the buoyancy ratio; however, when the Prandtl number increases, only the local Nusselt number increases and the skin friction coefficient decreases.

However, the current study intends to close the knowledge gap regarding the mixed bioconvective stagnation point flow of a power-law nanofluid towards a stretchable surface. The nanofluid flow will be mathematically modeled using Buongiorno's two-component, including the Brownian movement and thermophoresis aspects. Moreover, modeling of the motile microorganism and nanoparticles is addressed in the governing equations.
