Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6, 5236-5253 2024 Publisher: Learning Gate DOI: 10.55214/25768484.v8i6.3147 © 2024 by the authors; licensee Learning Gate © 2024 by the authors; licensee Learning Gate * Correspondence: g.elkahlout@arabou.edu.sa Bio convection heat transfer in Sisko nanofluid past a stretching cylinder with Soret and Dufour effects Khuram Rafique1, Gamal Elkahlout2* Ayesha Rehman1, Shamaila Kanwal1 1Department of Mathematics, University of Sialkot, Sialkot 51040, Pakistan. Email: khurram.rafique1005@gmail.com (K.R.) Ayesharehmankhalid@gmail.com (A.R.) shamaila3355@gmail.com (S.K.) 2School of Business Studies, Arab Open University, Riyadh, Saudi Arabia; g.elkahlout@arabou.edu.sa (G.E.) Abstract: The present flow model provides useful information for the production of nano-biomaterials, medical treatment, materials science current research model has been utilized. The unique thermal mechanisms of nanoparticles have garnered significant attention from researchers in recent years. These versatile materials have numerous applications in various fields, including cooling and heating control processes, solar systems, energy production, nanoelectronics, hybrid-powered motors, cancer treatments, and renewable energy systems. Furthermore, the bioconvection of nanofluids has exciting implications for bioengineering and biotechnology, with potential uses in biofuels, biosensors, and enzymes. The aim of this study is to investigate the flow behaviour of bioconvection Sisko nanofluid flow through a stretching cylindrical surface. Further, the analysis has been modified by including the effects of Soret and Dufour. The highly nonlinear and coupled differential equations were numerically solved using a BVP4c solver to simulate the problem. The effects of different flow parameters on velocity, temperature, and concentration distributions are examined and illustrated through graphical results. It is clearly observed from the results that Soret impacts increases the concentration distribution. Moreover, Dufour impacts increment increases the temperature of the flow. Keywords: Bioconvection, Motile microorganisms, Sisko nanofluid, Soret and dufour effects, Stretching cylinder. 1. Introduction In today's era of rapid scientific advancements, the Sisko model has emerged as a vital tool for understanding the behaviour of non-Newtonian fluids at the nanoscale. As a non-Newtonian fluid, Sisko fluid exhibits unique properties that are crucial for various applications, including drug delivery, tissue engineering, and biomedical devices. Its ability to simulate the flow dynamics of biofluids like blood and mucus has far-reaching implications for the development of targeted therapies and personalized medicine. Moreover, the Sisko model is also being explored for its potential in enhancing the performance of energy storage devices, such as batteries and supercapacitors, and improving the efficiency of industrial processes, like oil recovery and transportation. With its versatility and accuracy, the Sisko model is poised to revolutionize various fields and transform the way we approach complex fluid dynamics challenges. Researchers have explored the behaviour of Sisko fluid in various scenarios, including laminar and turbulent flows, boundary layers, and heat transfer. Adesanya et al. [1] conducted a computational study on reaction-driven magneto-convective flow in Sisko fluid. Hafez et al. [2] investigated the Electrohydrodynamic (EHD) peristaltic flow of Sisko fluid, examining the combined effects of convection and endoscopy. Islam et al. [3] numerically analyzed convective energy transfer in Sisko fluid flow over an extending device, accounting for radiation and heat dissipation effects. Khan et al., [4] inspected the unsteady flow of a Sisko fluid in a moving cylindrical tube. Nisha et al., [5] discovered the effects of electro-osmotic forces, activation energy, and chemical reactions on the flow of Sisko fluid above a Darcy-Forchheimer absorbent enlarging cylinder. 5237 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 5236-5253, 2024 DOI: 10.55214/25768484.v8i6.3147 © 2024 by the authors; licensee Learning Gate Prasannakumara et al., [6] have conducted a study on the numerical analysis of MHD flow and nonlinear radiative heat transfer in Sisko nanofluid over a nonlinear stretching sheet. Upreti et al., [7] investigated the effects of viscous dissipation and suction on convective heat transfer in Sisko fluid flow over a stretching surface, providing valuable insights into the numerical assessment of this phenomenon. Imran et al., [8] investigated the thermal transport properties in Sisko fluid flow under peristaltic motion. Bisht et al., [9] examined the effects of solar radiation on radiative heat transfer in MHD Sisko nanofluid flow. Figure 1. Some common nanoparticles used in Sisko nanofluid. Nanofluid is a modern type of fluid that combines a base fluid with nano-sized particles of various metals, such as copper, aluminium, and silicon. The primary purpose of adding nanoparticles to the base fluid is to enhance its thermal conductivity, as conventional heat transfer fluids like oil, ethylene glycol, water, and engine oil have naturally poor thermal conductivity. To improve thermal conductivity, numerous experiments have been conducted, including changing the geometry of the problem and adding different-sized metallic particles (e.g., milli, micro) to the base fluid. However, these experiments did not yield the desired results. Two decades ago, Choi [10, 11] used nano-sized particles in the base fluid and surprisingly found that the thermal conductivity of nanofluid is significantly greater than that of the base fluid. Following this successful experiment, many theoretical and experimental studies have been conducted to analyze nanofluids, which are now utilized in various thermal engineering processes. For more information on nanofluid, refer to [12-17]. In addition to thermal engineering applications, nanofluids have also shown great promise in biomedical fields. For instance, nanofluids can be used to enhance the thermal conductivity of cancer cells, allowing for more effective thermal ablation treatments. They can also be used as drug delivery agents, with the nanoparticles serving as carriers for targeted drug delivery. Furthermore, nanofluids can be used to create frameworks for tissue engineering applications, promoting faster healing and tissue growth. Other biomedical applications of nanofluids include medical imaging, wound healing, and biosensing. The unique properties of nanofluids make them an exciting area of research, with potential breakthroughs in various fields, including biomedicine. The Role of 5238 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 5236-5253, 2024 DOI: 10.55214/25768484.v8i6.3147 © 2024 by the authors; licensee Learning Gate nanofluids in advancing drug delivery and biomedical innovations was investigated by Sheikhpur et al., [18]. Thermal ablation cancer therapy using nanoparticles and nanomaterials as emerging tools was discovered by Ashikbayeva et al., [19]. A special topic on new improvements of nanofluids related to pharmaceuticals was examined by Ellahi [20]. Medication delivery by ternary Casson hybrid nanofluids in convergent/ divergent channels was probed by Alnahdi et al., [21]. The impact of nanoparticle geometry on peristaltic pumping of medical magnetohydrodynamic nanofluids with energy transfer was studied by Akbar et al., [22]. The flow of MHD hybrid nanoliquid containing medication through a blood artery was inspected by Alghamdi et al., [23]. In many chemical processes, heat and mass transfer occur simultaneously, leading to complex interactions between temperature and concentration gradients. These interactions can cause changes in the concentration of different species in the system. Two important effects to consider in these processes are Soret effect (mass transfer influenced by temperature gradients) and the Dufour effect (heat transfer influenced by concentration gradients). In biological systems, understanding the coupled transport of heat and mass is crucial, as it can lead to cross-diffusion and changes in concentration gradients of essential species. This is particularly important in processes like metabolic reactions, nutrient uptake, and waste removal. The Soret and Dufour effects, which describe the interplay between heat and mass transfer, have been investigated in various fields, including chemical engineering and geoscience. Similarly, in bio sciences, these effects can play a critical role in understanding phenomena like thermoregulation, blood flow, and nutrient transport. For instance, researchers studying cellular metabolism or tissue engineering may need to consider these effects to optimize their designs and experiments. Makinde and Olanrewaju [24] examined the effects of Soret and Dufour on unsteady mixed convection flow over a permeable plate moving through a binary mixture of chemically reacting fluid, shedding light on the complex interactions between energy, mass, and momentum transfer in this dynamic system. Hayat et al., [25] conducted a comprehensive investigation into the Soret and Dufour effects on the peristaltic flow of magnetohydrodynamic (MHD) Jeffrey fluid in a rotating system with porous medium. Mahday [26] studied the energy transfer and liquid flow characteristics of a Casson fluid in the presence of Soret and Dufour effects, which are induced by a stretching cylinder. Srinivasacharya et al., [27] probed the combined impact of Soret and Dufour impects on mixed convection along a vertical surface in a permeable moderate with variable properties. Ahmed et al., [28] research investigated the bioconvective flow of a variable properties hybrid nanoliquid over a rotating disk, examining the combined effects of Arrhenius activation energy, heat and mass transfer and Soret and Dufour impacts on the liquid flow. Balla et al., [29] explored the combined impact of Soret and Dufour effects on the bioconvective flow of a nanofluid in a porous square cavity. Razaq et al., [30] conducted a study on radiative bioconvective flow with a non-uniform heat source, taking into consideration the effects of Soret and Dufour. Bio-convection, a fascinating phenomenon where living particles accumulate and form density gradients, has emerged as a vital area of research with far-reaching implications for various industries. Recent discoveries have unveiled the vast potential of bio-convection in biotechnology and biomedical applications, including biofuels, biosensors, drug development, and tissue engineering. As scientists continue to understand the deep complexities of bio-convection, its significance in understanding complex biological systems and developing innovative solutions for real-world problems becomes increasingly clear. Furthermore, recent breakthroughs in bio-convection research have highlighted its diverse applications and exciting potential for future advancements, paving the way for groundbreaking discoveries and innovative applications that are expected to transform various fields and improve lives. Yin et al., [31] probed the effects of thermal radiation on bioconvection flow of magnetized Sisko nanofluid with swimming microorganisms along a stretching cylinder. Al-Mubaddel et al., [32] studied the double stratification in Sisko nanofluid bioconvection with radiation and generalized fluxes. Puneeth et al., [33] conducted a study on the three-dimensional bioconvective flow 5239 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 5236-5253, 2024 DOI: 10.55214/25768484.v8i6.3147 © 2024 by the authors; licensee Learning Gate of Sisko nanofluid with robin boundary conditions. Raju et al., [34] examined heat and mass transfer in MHD non-Newtonian bio-convection flow over a rotating cone/plate, considering cross diffusion. This paper studies a new type of fluid flow called bioconvection Sisko nanofluid, which occurs when a stretching cylinder is present and microorganisms are moving around. The fluid's thermal properties are examined under various conditions, including nonlinear thermal radiation and nonuniform heat sources. The analysis additionally takes into account the influences of Soret and Dufour effects. The results are obtained by using a numerical method and graphically presented. This research is unique and has important implications for various fields, including heat transfer, solar energy, electronics, biofuels, and more. Using MATLAB software Bvp4c solver, we've generated numerical results that shed light on the crucial role of key physical parameters. Remarkably, this innovative flow model boasts unique thermal properties that have never been explored before, making it a pioneering achievement. Our discoveries have significant implications for various bio sciences applications, including bioprocessing, biomedical devices, bioelectronics, biofuels, enzyme technology, and biomedical research. 2. Description of the Problem This study examined the 2D, incompressible, steady flow of Sisko nanofluid past a stretching cylinder. To expand the study's range, Soret and Dufour effects are incorporated into the analysis. A transverse magnetic field of strength 𝐵0 is applied perpendicularly to the surface of the cylinder, in the radial direction (along the 𝑟 direction) as shown in Figure 2. Figure 2. Diagrammatic representation of the mathematical model using BVP4c. The Sisko nanofluid is assumed to be continuous in nature, and its physical properties are uninterrupted. Furthermore, the effect of viscous dissipation is neglected in this analysis. Figure 3 shows a schematic representation of the flow problem and the associated coordinate system. 5240 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 5236-5253, 2024 DOI: 10.55214/25768484.v8i6.3147 © 2024 by the authors; licensee Learning Gate Figure 3. Geometrical layout. Under the above assumptions, the governing equations for 2D Sisko nanofluid flow through swimming microorganisms on a stretching surface, as described in references [35-36], can be written as follows: 𝜕𝑟𝑢 𝜕𝑥 + 𝜕𝑟𝑣 𝜕𝑟 = 0, (1) 𝑢 𝜕𝑢 𝜕𝑥 + 𝑣 𝜕𝑣 𝜕𝑟 = 𝑎1 𝜌𝑛𝑓 𝜕 𝜕𝑟 (𝑟 𝜕𝑢 𝜕𝑟 ) − 𝑎2 𝑟𝜌𝑛𝑓 (− 𝜕𝑢 𝜕𝑟 ) 𝑚 + 𝑚𝑎2 𝜌𝑛𝑓 (− 𝜕𝑢 𝜕𝑟 ) 𝑚−1 𝜕𝑢2 𝜕2𝑟 − 𝜀𝑣 𝐾𝑝 𝑢 + 1 𝜌𝑛𝑓 [(1 − 𝐶∞)𝜌𝑛𝑓𝛽∗𝑔(𝑇 − 𝑇∞) − (𝜌𝑛𝑝 − 𝜌𝑛𝑓)𝑔(𝐶 − 𝐶∞) − (𝑁 − 𝑁∞)𝑔𝛾∗(𝜌𝑛𝑚 − 𝜌𝑛𝑓)], (2) 𝑢 𝜕𝑇 𝜕𝑥 + 𝑣 𝜕𝑇 𝜕𝑟 = 1 𝑟 𝜕 𝜕𝑟 (𝑎𝑟 𝜕𝑇 𝜕𝑟 ) + 𝜏 [𝐷𝐵 𝜕𝐶 𝜕𝑟 𝜕𝑇 𝜕𝑟 + 𝐷𝑇 𝑇∞ ( 𝜕𝑇 𝜕𝑟 ) 2 ] + 𝑄0 𝜌𝑓𝑐𝑝𝑓 (𝑇 − 𝑇∞) + 𝑄0 ∗ 𝜌𝑓𝑐𝑝𝑓 (𝑇𝑤 − 𝑇∞)𝑒𝑥𝑝 (−𝑛 ( 2𝑅𝑥 𝑟2−𝑅2 1 𝑅𝑒𝑎2 1 (𝑚+1) )) + 1 (𝜌𝑐)𝑓 𝜕(𝑟 16 3 𝜎∗𝑇∞ 3 𝑘∗ 𝜕𝑇 𝜕𝑟 ) 𝜕𝑟 + 𝐷𝑇𝐾𝑇 𝐶𝑠𝐶𝑝 𝜕2𝐶 𝜕𝑟2 , (3) 𝑢 𝜕𝐶 𝜕𝑥 + 𝑣 𝜕𝐶 𝜕𝑟 = 𝐷𝐵 𝜕2𝐶 𝜕𝑟2 + 𝐷𝑇𝐾𝑇 𝑇∞ 𝜕2𝑇 𝜕𝑟2 , (4) 𝑢 𝜕𝑁 𝜕𝑥 + 𝑣 𝜕𝑁 𝜕𝑟 + 𝑏𝑊𝑐 (𝐶𝑤−𝐶∞) [ 𝜕 𝜕𝑟 (𝑁 𝜕𝐶 𝜕𝑟 )]=𝐷𝑚 ( 𝜕2𝑁 𝜕𝑟2 ), (5) The problem is subject to the following boundary conditions: 𝑢 = 𝑐𝑥 + 𝛽∗ 𝑟 𝜕𝑢 𝜕𝑟 , 𝑣 = 0, −𝑘 𝜕𝑇 𝜕𝑟 = ℎ𝑓(𝑇𝑤 − 𝑇), −𝐷𝐵 𝜕𝐶 𝜕𝑟 = ℎ𝑔(𝐶𝑤 − 𝐶), −𝐷𝑚 𝜕𝑁 𝜕𝑟 = ℎ𝑛(𝑁𝑤 − 𝑁) at 𝑟 = 0 (6) 𝑢 → 0, 𝑁 → 0, 𝑇 → 𝑇∞, 𝐶 → 𝐶∞ , 𝑁 → 𝑁∞ as 𝑟 → ∞ (7) The material constants for the Sisko fluid represent the following physical properties and parameters: 5241 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 5236-5253, 2024 DOI: 10.55214/25768484.v8i6.3147 © 2024 by the authors; licensee Learning Gate Table 1. 𝑎1 Viscosity at an extremely high shear rate 𝑎2 Consistency index or viscosity in the power-law region 𝑚 Power-law index, which varies depending on the specific fluid 𝜌𝑛𝑓 Density of Sisko nanofluid 𝑣 Density 𝜌𝑓 Density of fluid 𝜌𝑛𝑚 Density of nano microorganisms 𝜌𝑛𝑝 Density of nanoparticles 𝛽∗ Volume expansion coefficient 𝛾∗ Average volume of microorganisms 𝑇 Temperature 𝑇∞ ambient temperature 𝐶 Concentration 𝐶∞ Ambient concentration 𝑁 Microorganisms 𝑁∞ Ambient microorganisms (𝜌𝑐)𝑓 Heat capacity of the base fluid 𝑔 Gravitational acceleration 𝜏 Parameter defined by the ratio (𝜌𝑐)𝑝 (𝜌𝑐)𝑓 𝐷𝐵 Brownian motion 𝐷𝑇 Thermophoresis diffusion coefficient 𝑄0 Heat generation/ absorption parameter 𝑐𝑛𝑓 Specific heat of nanofluid 𝜎∗ Stefan–Boltzmann constant 𝑘∗ Thermal conductivity 𝑏 Chemotaxis constant 𝑊𝑐 Maximum speed of swimming 𝐵∗ Porosity parameter 𝑘 Thermal conductivity ℎ𝑓 Convection heat transport 𝑁𝑤 Wall microorganism 𝐷𝑚 Diffusion coefficient of microorganisms The current problem's convergence to nonlinear dimensionless ODEs is similar. 𝜁 = 𝑟2−𝑅2 2𝑅𝑥 𝑅𝑒𝑎2 1 (𝑚+1)⁄ , 𝑓(𝜁) = 𝜓 1 𝑅𝑥𝑈𝑤𝑅𝑒𝑎2 −1 (𝑚+1)⁄ , 𝜃(𝜁) = 𝑇−𝑇∞ 𝑇𝑤−𝑇∞ , 𝜙(𝜁) = 𝐶−𝐶∞ 𝐶𝑤−𝐶∞ , 𝜒(𝜁) = 𝑁−𝑁∞ 𝑁𝑤−𝑁∞ , (8) The function of the velocity stream is: 𝑢 = 𝜕𝜓 𝜕𝑟 , 𝑣 = − 𝜕𝜓 𝜕𝑥 (9) 5242 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 5236-5253, 2024 DOI: 10.55214/25768484.v8i6.3147 © 2024 by the authors; licensee Learning Gate The system of coupled partial differential equations (1-5) with boundary conditions (6-7) is reduced to a set of nonlinear dimensionless ordinary differential equations (ODEs) through the similarity transformations (8). 𝐵(1 + 2𝛼𝜁)𝑓′′′ + 𝑚(−𝑓′′)𝑚−1(1 + 2𝛼𝜁) 𝑚+1 2 𝑓′′′ + 2𝛼𝐵𝑓′′ + 2𝑚 𝑚+1 𝑓𝑓′′ − (𝑓′)2 − 𝑀𝑓′ − (−𝑓′′)𝑚(1 + 𝑚)(1 + 2𝛼𝜁) 𝑚+1 2 + 𝛽(𝜃 − 𝑁𝑟𝜙 − 𝑁𝑐𝜒) = 0 (10) (1 + 2𝛼𝜁)𝜃′′ + 2𝛼𝜃′ + [{1 + 𝑅𝑑(1 + (𝜃𝑓 − 1)𝜃)3}(1 + 2𝛼𝜁)𝜃′]𝜃′′ + 𝑃𝑟 ( 2𝑚 𝑚+1 ) 𝑓𝜃′ + (1 + 2𝛼𝜁) Pr(𝑁𝑏𝜃′𝜙′ + 𝑁𝑡(𝜃′)2) + 𝑃𝑟𝑄𝑝𝜃 + 𝑃𝑟𝑄𝐸 exp(−𝑛𝜁) + 𝐷𝑓𝑃𝑟𝜙′′ = 0, (11) (1 + 2𝛼𝜁)𝜙′′ + 2𝛼𝜙′ + 𝑁𝑡 𝑁𝑏 2𝛼𝜃′′ + (1 + 2𝛼𝜁)𝐿𝑏 𝑆𝑟𝜃′′ + 𝐿𝑒𝑃𝑟 ( 2𝑚 𝑚+1 ) 𝑓𝜙′ = 0, (12) (1 + 2𝛼𝜁)𝜒′′ + 2𝛼𝜒′ + 𝐿𝑏 [( 2𝑚 𝑚+1 ) (𝑓𝜒′)] − 𝑃𝑒[𝜙′′(𝜒 + Ω1) + 𝜒′𝜙′] = 0, (13) With associated boundary conditions 𝑓(0) = 0, 𝑓′(0) = 1 + 𝜆𝑓′′(0), 𝜃′(0) = −𝐴1(1 − 𝜃(0)), 𝜙′(0) = −𝐴2(1 − 𝜙(0)), 𝜒′(0) = −𝐴3(1 − 𝜒(0)), (14) 𝑓′(∞) → 0, 𝜃(∞) → 0, 𝜙(∞) → 0, 𝜒(∞) → 0, The physical system is characterized by a set of dimensionless parameters, including the material parameter, mixed convection parameter, buoyancy ratio parameter, bioconvection Rayleigh number, radiation parameter, temperature ratio parameter, Prandtl number , local Reynolds numbers, thermal dependent heat source parameter, exponential space based source parameter, curvature parameter, thermophoresis parameter, magnetic parameter, Brownian motion parameter, Lewis number, bioconvection Lewis number, Peclet number, thermal Biot number, solutal Biot number, microorganisms Biot number, velocity slip parameter, Soret and Dufour effect parameters. These parameters are used to nondimensionalize the governing equations and boundary conditions, allowing for a more concise and meaningful analysis of the system's behavior. 𝐵 = 𝑅𝑒𝑎2 2 (𝑚+1) 𝑅𝑒𝑎1 , 𝛽 = 𝑐(1−𝐶𝑓)(𝑇𝑓−𝑇∞)𝛽∗𝑔∗ 𝑥 , 𝑁𝑟 = (𝜌𝑝−𝜌𝑓)(𝐶𝑓−𝐶∞) 𝛽∗(1−𝐶𝑓)(𝑇𝑓−𝑇∞)𝜌𝑓 , 𝑁𝑐 = (𝜌𝑚−𝜌𝑓)(𝑁𝑓−𝑁∞)𝛾∗∗𝑄𝑝 𝛽∗(1−𝐶𝑓)(𝑇𝑓−𝑇∞)𝜌𝑓 , 𝑅𝑑 = 16𝜎∗𝑇∞ 3 3𝑘𝑘∗ , 𝑃𝑟 = 𝑥𝑈𝑥 𝛼 𝑅𝑒𝑎2 2 (𝑚+1) ,𝑅𝑒𝑎1 = 𝜌𝑛𝑓𝑈𝑤𝑥 𝑎1 , 𝑅𝑒𝑎2 = 𝜌𝑛𝑓𝑥𝑚𝑈𝑤 2−𝑚 𝑎2 , 𝑄𝑝 = 𝑄0 𝜌𝑓𝐶𝑝𝑓 , 𝑄𝐸 = 𝑄0 ∗ 𝜌𝑓𝐶𝑝𝑓 , 𝛼 = 𝑥 𝑅 𝑅𝑒𝑎2 2 (𝑚+1) , 𝑁𝑡 = 𝜏𝐷𝑇(𝑇𝑤−𝑇∞) 𝛼𝑇∞ , 𝑀 = 𝑣𝜀𝑥 𝐾𝑝𝑈𝑤 , 𝑁𝑏 = 𝜏𝐷𝐵(𝐶𝑤−𝐶∞) 𝛼 , 𝐿𝑒 = 𝛼 𝐷𝐵 , 𝐿𝑏 = 𝑣 𝐷𝑚 , 𝑃𝑒 = 𝑏𝑊𝑒 𝐷𝑚 , 𝐴1 = 𝑥𝑅ℎ𝑓 𝑘𝑅𝑒𝑏 − 1 𝑚+1 , 𝐴2 = 𝑥𝑅ℎ𝑔 𝐷𝐵𝑅𝑒𝑏 − 1 𝑚+1 , 𝐴3 = 𝑥𝑅ℎ𝑛 𝐷𝑚𝑅𝑒𝑏 − 1 𝑚+1 , 𝜆 = 𝛽∗ 𝑥𝑅𝑅𝑒𝑏 − 1 𝑚+1 , 𝑆𝑟 = 𝐷𝑇𝐾𝑇 𝑇∞𝜗 (𝑇𝑤−𝑇∞) (𝐶𝑤−𝐶∞) , 𝐷𝑟 = 𝐷𝑇𝐾𝑇 𝐶𝑝𝐶𝑠 (𝐶𝑤−𝐶∞) 𝑣(𝑇𝑤−𝑇∞) , (15) The important engineering values, such as the dimensional Nusselt number 𝑁𝑢𝑥 and heat transfer rate is given as: 𝑁𝑢𝑥 = 𝑥𝑞𝑤 𝑘(𝑇𝑤−𝑇∞) , (16) Herer 𝑞𝑤 = −𝑘 ( 𝜕𝑇 𝜕𝑟 ) 𝑟=0 is heat flux. The non-dimensional form is 𝑁𝑢𝑥𝑅𝑒𝑎2 − 1 𝑚+1 = −𝜃′(0). (17) 5243 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 5236-5253, 2024 DOI: 10.55214/25768484.v8i6.3147 © 2024 by the authors; licensee Learning Gate 3. Numerical Method In this section, we investigate the bio-convection effects of Sisko nanoliquid flow above an extending cylinder, taking into consideration the Soret and Dufour effects. We numerically solve the coupled nonlinear differential equations (10-13) with boundary conditions (14) using the built-in MATLAB function bvp4c. This solver utilizes a finite difference method to convert the fourth-order differential equations into linear form. To implement this solver, we rewrite the higher-order differential equations as a system of first-order differential equations by introducing auxiliary variables, as shown below. Step 1: For the following coupled nonlinear ODEs, add new parameters: 𝑦(1) = 𝑓, 𝑦(2) = 𝑓′, 𝑦(3) = 𝑓′′, 𝑦′(3) = 𝑓′′′, 𝑦(4) = 𝜃, 𝑦(5) = 𝜃′, 𝑦′(5) = 𝜃′′ 𝑦(6) = 𝜙, 𝑦(7) = 𝜙′, 𝑦′(7) = 𝜙′′ 𝑦(8) = 𝜒, 𝑦(9) = 𝜒′, 𝑦′(9) = 𝜒′′ Step 2: The following parameters of step 1 should be written into the 1st order system of equations: 𝑦(1) = 𝑓, 𝑦(2) = 𝑓′, 𝑦(3) = 𝑓′′ 𝑦′(3) = (𝑦(2))2+𝑀𝑦(2)−(−𝑦(3)) 𝑚 (1+𝑚)(1+2𝛼𝜁) 𝑚+1 2 −𝛽(𝜃−𝑁𝑟𝑦(6)−𝑁𝑐𝑦(8))−2𝛼𝐵𝑦(3)− 2𝑚 𝑚+1 𝑦(1)𝑦(3) [𝐵(1+2𝛼𝜁)+𝑚(−𝑦(3))𝑚−1(1+2𝛼𝜁) 𝑚+1 2 ] (18) 𝑦(4) = 𝜃 , 𝑦(5) = 𝜃′ 𝑦′(5) = −[𝑃𝑟( 2𝑚 𝑚+1 )𝑦(1)𝑦(5)+(1+2𝛼𝜉) Pr(𝑁𝑏𝑦(5)𝑦(7)+𝑁𝑡(𝑦(5))2)+𝑃𝑟𝑄𝑝𝑦(4)+𝑃𝑟𝑄𝐸 exp(−𝜂𝜁)−2𝛼𝑦(5)+𝐷𝑓𝑃𝑟𝑦′(7)] (1+2𝛼𝜁)+[{1+𝑅𝑑(1+(𝜃𝑓−1)𝑦(4)) 3 }(1+2𝛼𝜁)𝑦(5)] (19) 𝑦(6) = 𝜙, 𝑦(7) = 𝜙 𝑦′(7) = −[2𝛼𝑦(7)+{ 𝑁𝑡 𝑁𝑏 2𝛼+(1+2𝛼𝜁)𝐿𝑏 𝑆𝑟}𝑦′(5)+𝐿𝑒𝑃𝑟( 2𝑚 𝑚+1 )𝑦(1)𝑦(7)] (1+2𝛼𝜁) (20) 𝑦(8) = 𝜒 , 𝑦(9) = 𝜒′ 𝑦′(9) = 𝑃𝑒[𝑦′(7)(𝑦(8)+Ω1)+𝑦(9)𝑦(7)]−𝐿𝑏[( 2𝑚 𝑚+1 )(𝑦(1)𝑦(9))]−2𝛼 𝑦(9) (1+2𝛼𝜁) (21) Step 3: The boundary conditions of (6-7) are rewritten in the term of new variable as: 𝑦𝑎(1) = 0, 𝑦𝑎(2) = 1 + 𝜆 𝑦𝑎(3), 𝑦𝑎(3) = −𝐴1(1 − 𝑦𝑎(4)), 𝑦𝑎(7) = −𝐴2(1 − 𝑦𝑎(6)), 𝑦𝑎(9) = −𝐴3(1 − 𝑦𝑎(8)) (22) 𝑦𝑏(2) → 0, 𝑦𝑏(4) → 0, 𝑦𝑏(6) → 0, 𝑦𝑏(8) → 0 Consequently, the variable 𝑎 represents the conditions on the sheet, which is denoted by 𝜉 = 0, and the variable 𝑏 represents the conditions off the sheet, for example, 𝜉 = 1. Step 4: In MATLAB, use the bvp4c solver to solve the system of first-order ODEs (18-21) with boundary conditions (22). 4. Results and Discussion This section provides a numerical analysis of various engineering quantities in response to changes in different parameters. By solving the governing differential equations numerically, we gain flexibility in selecting suitable flow parameter values and a deeper understanding of the physical problem. The results are presented in tables as follows: Table 2 shows the numerical results for local skin friction coefficients, while Table 3 provides the local Nusselt numbers, Table 4 compares the Sherwood numbers across different parameters, and Table 5 details the density of local microorganisms. Additionally, we have graphically examined the impects of Magnetic parameter 𝑀, buoyancy ratio factor 𝑁𝑟, mixed convection parameter 𝛽, thermal Biot number 𝐴1, Dufour effect parameter 𝐷𝑓, 5244 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 5236-5253, 2024 DOI: 10.55214/25768484.v8i6.3147 © 2024 by the authors; licensee Learning Gate Prandtl number 𝑃𝑟, Brownian motion factor 𝑁𝑏, Soret effect parameter 𝑆𝑟, microorganisms Biot number 𝐴3 and Peclet number 𝑃𝑒 on velocity, temperature, concentration and microorganism profiles. All graphical findings were developed using the properties of shear-thinning (𝑚 < 1) and shear- thickening (𝑚 > 1) fluids, which are effectively represented by the Sisko fluid model. These tables provide a complete summary of the numerical results, highlighting the impacts of different parameters on the engineering quantities. Table 2. Numerical analysis of local skin friction coefficient using different parameter values. 𝑴 𝜷 𝑵𝒓 𝑵𝒄 𝑩 α −[𝑩𝒇′′(𝟎) − {𝒇′′(𝟎)} 𝟏 𝟐] −(𝑩 + 𝟏)𝒇′′(𝟎) −[𝑩𝒇′′(𝟎) − 𝒇′′𝟐(𝟎)] 0.5 0.7 1.3 0.2 0.2 0.2 0.5 0.2 0.4712 0.4722 0.4856 0.9565 1.0001 1.0211 0.0987 0.1102 0.1214 0.7 1.2 1.7 0.4771 0.4753 0.4614 0.9368 0.8956 0.8576 0.0787 0.0475 0.0425 0.7 1.4 2.5 0.4992 0.4959 0.4978 0.9987 1.1101 1.1365 0.1141 0.1260 0.1412 0.7 1.4 2.5 0.4806 0.4832 0.4878 0.9835 1.0314 1.0510 0.1045 0.1278 0.1393 0.2 0.5 0.9 0.4772 0.4821 0.4845 0.9612 0.9765 1.0042 0.0865 0.09989 0.1132 0.5 0.9 1.3 0.4841 0.4887 0.4914 1.0096 1.0662 1.1164 0.1121 0.1496 0.1823 Table 3. Numerical analysis of local Nusselt number using different parameter values. Parameters −𝜽′(𝟎) 𝑷𝒓 𝑵𝒕 𝑹𝒅 𝑵𝒓 𝑵𝒄 𝝀 𝑸𝑬 𝒎 = 𝟎. 𝟓 𝒎 = 𝟐. 𝟎 2.5 3.5 4.5 0.3 0.5 0.1 0.1 1.0 0.3 0.1922 0.1966 0.2001 0.1897 0.1935 0.1966 2.0 0.1 0.4 0.8 0.1935 0.1876 0.1791 0.1914 0.1853 0.1766 1.0 1.5 2.0 0.1742 0.1626 0.1536 0.1719 0.1604 0.1516 0.5 2.0 3.5 0.1889 0.1854 0.1803 0.1756 0.1607 0.1521 0.5 2.0 3.5 0.1877 0.1851 0.1801 0.1762 0.1734 0.1791 2.0 0.1829 0.1809 5245 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 5236-5253, 2024 DOI: 10.55214/25768484.v8i6.3147 © 2024 by the authors; licensee Learning Gate 3.0 4.0 0.1794 0.1773 0.1777 0.1757 0.5 1.2 2.4 0.1887 0.1765 0.1753 0.1760 0.1678 0.1597 Table 4. Numerical analysis of Sherwood number using different parameter values. Parameters −𝝓′(𝟎) 𝑷𝒓 𝑵𝒕 𝑵𝒃 𝑵𝒓 𝑵𝒄 𝝀 𝑳𝒆 𝑨𝟐 𝒎 = 𝟎. 𝟓 𝒎 = 𝟐. 𝟎 2.4 3.4 4.4 0.3 0.2 0.1 0.1 1.0 2.0 0.4 0.2702 0.2852 0.2965 0.2676 0.2835 0.2946 2.0 0.15 0.45 0.85 0.2851 0.2464 0.2045 0.1905 0.1842 0.1764 0.2 0.4 0.6 0.2703 0.2864 0.2924 0.2712 0.2854 0.2806 0.4 1.9 3.4 0.2571 0.2545 0.2536 0.2574 0.2548 0.2527 0.3 1.8 3.3 0.2576 0.2547 0.2539 0.2548 0.2538 0.2528 1.5 2.5 3.5 0.2538 0.2509 0.2500 0.2521 0.2504 0.2478 2.5 3.5 4.5 0.2804 0.2937 0.3032 0.2778 0.2919 0.3021 0.5 1.0 1.5 0.4748 0.5818 0.6562 0.4663 0.5744 0.6463 Table 5. Numerical analysis of local microorganism density number using different parameter values. Parameters −𝝌′(𝟎) 𝑷𝒆 𝑳𝒃 𝑵𝒓 𝑵𝒄 𝝀 𝑨𝟑 𝒎 = 𝟎. 𝟓 𝒎 = 𝟐. 𝟎 0.5 1.2 1.3 2.0 0.1 0.1 1.0 0.4 0.2024 0.2398 0.2654 0.1954 0.2340 0.2627 0.1 2.5 4.0 5.0 0.2232 0.2502 0.2656 0.2154 0.2432 0.2582 0.3 1.5 3.0 0.1912 0.1800 0.1598 0.1904 0.1778 0.1582 0.3 1.5 3.0 0.1875 0.1865 0.1802 0.1865 0.1856 0.1808 5246 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 5236-5253, 2024 DOI: 10.55214/25768484.v8i6.3147 © 2024 by the authors; licensee Learning Gate 1.5 2.5 3.5 0.1735 0.1612 0.1547 0.1658 0.1545 0.1485 1.5 2.0 2.5 0.2753 0.3132 0.3216 0.2701 0.2910 0.3098 Figure 4 shows the impact of magnetic parameter 𝑀 on velocity profile for both conditions shear- thinning and shear-thickening fluids (𝑚 = 0.5 𝑎𝑛𝑑 𝑚 = 2.0). The Sisko fluid's velocity decreases with increasing magnetic parameter. Growing values of the magnetic parameter reduce the fluid's velocity by acting as a resistive force in the flow known as the Lorentz force. From a physical perspective, the magnetic parameter affects the Lorentz force, which defines the characteristics of fluid motion in the flow system. Figure 4. Variation in 𝑓′(𝜍) with 𝑀 at shear rates 𝑚 = 0.5 and 𝑚 = 2.0. Figure 5 illustrates the impact of the buoyancy ratio factor 𝑁𝑟 on the velocity profile for both shear- thinning and shear-thickening fluids (𝑚 = 0.5 𝑎𝑛𝑑 𝑚 = 2.0).The Sisko fluid velocity decreases as the values of the buoyancy ratio parameter 𝑁𝑟 increase. 5247 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 5236-5253, 2024 DOI: 10.55214/25768484.v8i6.3147 © 2024 by the authors; licensee Learning Gate Figure 5. Variation in 𝑓′(𝜍) with 𝑁𝑟 at shear rates 𝑚 = 0.5 and 𝑚 = 2.0. Figure 6 demonstrates the effect of the mixed convection parameter 𝛽 on the velocity field, showing that an increase in this parameter improves the velocity field for both values (𝑚 = 0.5 𝑎𝑛𝑑 𝑚 = 2.0). Figure 6. Variation in 𝑓′(𝜍) with 𝛽 at shear rates 𝑚 = 0.5 and 𝑚 = 2.0. Figure 7 depicts the effect of the thermal stratification Biot number 𝐴1 on temperature distribution. It is shown that both shear-thinning and shear-thickening fluids (m=0.5 and m=2.0) have a better temperature distribution when the thermal stratification Biot number rises. 5248 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 5236-5253, 2024 DOI: 10.55214/25768484.v8i6.3147 © 2024 by the authors; licensee Learning Gate Figure 7. Variation in 𝜃(𝜍) with 𝐴1 at shear rates 𝑚 = 0.5 and 𝑚 = 2.0. Figure 8 represents the behaviour of Dufour effect parameter 𝐷𝑓 on temperature profile for both cases (𝑚 = 0.5 𝑎𝑛𝑑 𝑚 = 2.0).Higher Dufour effect parameter boosts the energy flux caused by concentration differences. This effect is most noticeable when irreversible processes are present, leading to changes in the temperature profile due to variations in concentration. Figure 8. Variation in 𝜃(𝜍) with 𝐷𝑓 at shear rates 𝑚 = 0.5 and 𝑚 = 2.0. Figure 9 signifies the features of Brownian motion parameter 𝑁𝑏 on the temperature profile for both shear-thinning and shear-thickening fluids (𝑚 = 0.5 𝑎𝑛𝑑 𝑚 = 2.0). As the Brownian motion parameter 𝑁𝑏 increases, the random motion of fluid particles intensifies, leading to greater layer thickness. Brownian motion, which refers to the random movement of suspended particles in a fluid, causes the temperature of the Sisko fluid to rise due to interaction between these particles. This results in an enhanced temperature profile. 5249 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 5236-5253, 2024 DOI: 10.55214/25768484.v8i6.3147 © 2024 by the authors; licensee Learning Gate Figure 9. Variation in 𝜃(𝜍) with 𝑁𝑏 at shear rates 𝑚 = 0.5 and 𝑚 = 2.0. Figure 10 examine the behaviour of Prandtl number 𝑃𝑟 on the concentration profile for both shear- thinning and shear-thickening fluids (𝑚 = 0.5 𝑎𝑛𝑑 𝑚 = 2.0). It is observed that the concentration distribution decreases as the Prandtl number increases in both types of fluids. Figure 10. Variation in 𝜙(𝜍) with 𝑃𝑟 at shear rates 𝑚 = 0.5 and 𝑚 = 2.0. Figure 11 shows the Soret effect parameter 𝑆𝑟 on the concentration profile for both cases (𝑚 = 0.5 𝑎𝑛𝑑 𝑚 = 2.0). The concentration profile turns up for higher values of 𝑆𝑟, because mass flux caused by temperature differences. 5250 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 5236-5253, 2024 DOI: 10.55214/25768484.v8i6.3147 © 2024 by the authors; licensee Learning Gate Figure 11. Variation in 𝜙(𝜍) with 𝑆𝑟 at shear rates 𝑚 = 0.5 and 𝑚 = 2.0. Figure 12 reflect the results of the Biot number of microorganism stratification 𝐴3 on the microorganism concentration field for both shear-thinning and shear-thickening fluids (𝑚 = 0.5 𝑎𝑛𝑑 𝑚 = 2.0). The findings indicate that when the microorganism stratification Biot number increases, the microorganism concentration field becomes progressively more significant. Figure 12. Variation in 𝜒(𝜍) with 𝐴3 at shear rates 𝑚 = 0.5 and 𝑚 = 2.0. Figure 13 sketch to examine the nature of Peclet number 𝑃𝑒 on the microorganism’s concentration of nanoparticles. The graph shows that increased Peclet number leads to decrease the dispersion of microorganisms in both shear-thinning and shear-thickening fluids. 5251 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 8, No. 6: 5236-5253, 2024 DOI: 10.55214/25768484.v8i6.3147 © 2024 by the authors; licensee Learning Gate Figure 13. Variation in 𝜒(𝜍) with 𝑃𝑒 at shear rates 𝑚 = 0.5 and 𝑚 = 2.0. 5. Conclusion This research explores the bio-convection flow of Sisko nanofluid containing microorganisms over a stretchy cylinder, taking into account Soret and Dufour effects. Bvp4c method is utilized to attain numerical results. The Buongiorno model is used to examine thermophoresis and Brownian motion factors. Main findings are: 1. Velocity profile declines as the magnetic parameter rises. 2. Velocity profile decreases with increasing values of buoyancy ratio factor and mixed convection parameter. 3. Temperature profile boosts with higher Biot number, Dufour parameter and Brownian motion factor. 4. Concentration profile turns up as the Prandtl number and Soret effect parameter increase. 5. Microorganisms profile expands with higher values of microorganisms Biot number while contradictory behaviour is noticed for Peclet number. Acknowledgement: The authors extend their appreciation to the Arab Open University for funding this work through AOU research fund No. (AOURG-2023-016). Copyright: © 2024 by the authors. 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/). 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