**1. Introduction**

It is well known that magnetohydrodynamics (MHD) is a concept common to both Physics and Mathematics that deals with the study of the interactions of magnetic fields in conducting fluids. The involvement of magnetic fields results in forces that in turn affect the fluid. The structure and intensity of the magnetic fields themselves are therefore possibly altered. The relative performance of the advective movements in the fluid is a key question for a certain conducting fluid experiencing a diffusive impact induced by the resistivity. It also has several application areas such as aerodynamics, life sciences, polymer or fiberglass, cooling systems, exchangers, metallurgy, etc. Hady et al. [1] examined the MHD flow of nanofluid-having gyrotactic micro-organisms with viscous dissipation effects. Pal and Mondal [2] investigated the nanofluid MHD flow with gyrotactic micro-organisms including thermal radiation effects. Yasmin et al. [3] discussed MHD micropolar fluid flows

**Citation:** Ragupathi, P.; Ahammad, N.A.; Wakif, A.; Shah, N.A.; Jeon, Y. Exploration of Multiple Transfer Phenomena within Viscous Fluid Flows over a Curved Stretching Sheet in the Co-Existence of Gyrotactic Micro-Organisms and Tiny Particles. *Mathematics* **2022**, *10*, 4133. https:// doi.org/10.3390/math10214133

Academic Editors: Camelia Petrescu, Valeriu David and Efstratios Tzirtzilakis

Received: 16 September 2022 Accepted: 2 November 2022 Published: 5 November 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

due to a curved stretching sheet. Nagaraja and Gireesha [4] addressed the MHD flow of Casson fluid. Additionally, the exponential heat generation and chemical reaction effects were also investigated. Few attempts on the topic can be mentioned in the studies [5–8].

Bio-convection results from to the upward-swimming of an average number of microorganisms that are heavier than water. The swimming micro-organisms are collected at the upper water surface. When the collected layer of micro-organisms becomes thicker and thicker, the surface becomes unstable. As a result, a large portion of the gathering falls deeper into the water. This process is repeated by the micro-organisms, and eventually results in bio-convection. Kessler [9] was the person who first noticed and studied the gyrotaxis of microbes. Recently, it has been discovered that several significant phenomena are dependent on the gyrotaxis of microorganisms. These physical phenomena include accumulation at free water surfaces [10], turbulent channel flows in photobioreactors [11], the formation of a thin phytoplankton layer brought on by gyrotactic trapping [12], and microscale patches of motile phytoplankton [13]. Alharbi et al. [14] scrutinized the bio-convection caused by gyrotactic micro-organisms present in the magnetic hybrid nanofluid. This study attempted to support the Targeted Drug Delivery (TDD) system. Modal and Pal [15] inspected the influence of variable viscosity in the bio-convection of micro-organisms present in the nanofluid. Bio-convection in the existence of the Marangoni thermo-solutal effect was considered by Kairi et al. [16]. Khan and Nadeem [17] studied the bio-convection of Maxwell nanofluid. The notion of variability in the thermal conductivity model to analyze the bio-convective Williamson nanofluid was looked over by Abdelmalek et al. [18]. Similarly, numerous investigations on the bio-convection of micro-organisms have been addressed by researchers [19–24].

Due to its various applications in the area of research and production, boundary layer flow over-stretching surfaces is an attractive topic for researchers (Wang [25], Noghrehabadi et al. [26], Rauf et al. [27]). Flows generated by fiber spinning, injection molding, glass molding, spray coating, and pulling of rinsed wires, paper, rubber, glass-fiber, polymer sheet production, etc., are some of the useful applications. As an extension of the stretching surface, the current work is connected with the fluid flow through a curved stretching surface grabbing the attention of numerous researchers. Hayat et al. [28] studied the entropy optimization of CNTs (Carbon Nano Tubes) over a curved stretching sheet. Similarly, Raza et al. [29] examined the entropy optimization of Carreau fluid over a curved stretching sheet. A non-Fourier heat flux model was taken up by Madhukesh et al. [30] to investigate the hybrid nanofluid flow. Darcy-Forchheimer flows of CNTs driven by a curved stretching sheet were considered by Gireesha et al. [31]. Stagnation point flow, involving MHD and Joule heating, was demonstrated by Zhang et al. [32]. The latest developments concerning curved stretching sheets are mentioned in Refs [33–38].

The distribution and swimming properties of gyrotactic microorganisms, in a variety of flows, including horizontal shear flow [11], density stratified flow [39], steady vertical flow [12,40], free surface flow [9], Poiseuille flow [9], as well as the flow past a single vertical circular cylinder [41], have all been the subject of extensive research. The findings demonstrate that gravitational torque and the viscous torque, caused by flow shear in a fluid flowing with non-zero vorticity, have an impact on the swimming of gyrotactic phytoplankton species. Although they are crucial for the prediction of the corresponding concentration distribution, the nanofluid flow behavior relating to the swimming characteristics of gyrotactic microorganisms is currently poorly understood, particularly when Brownian diffusion and thermophoresis are combined.

Ultimately, the goal of this research is to investigate the MHD bio-convective heat transfer caused by gyrotactic micro-organism swimming within the nanofluid past a curved stretched sheet. Using the BVP4c integrated MATLAB package, the solutions to the nonlinear system of ODEs are solved. Differences in motile microorganisms, temperature, velocity, and concentration profiles are explained in terms of various influencing parameters, via graphs and tables. The manuscript is prepared in such a way that: Section 2 presents the problem formulation. The numerical scheme's validation is explained in

Section 3. Section 4 contains comments on the collected results, while Section 5 has a list of significant observations.
