Dual solution for double-diffusive mixed convection opposing flow through a vertical cylinder saturated in a Darcy porous media containing gyrotactic microorganisms

www.nature.com/scientificreports OPEN Dual solution for double‑diffusive mixed convection opposing flow through a vertical cylinder saturated in a Darcy porous media containing gyrotactic microorganisms Abdulaziz Alsenafi1* & M. Ferdows2 The steady mixed convection flow towards an isothermal permeable vertical cylinder nested in a fluidsaturated porous medium is studied. The Darcy model is applied to observe bioconvection through porous media. The suspension of gyrotactic microorganisms is considered for various applications in bioconvection. Appropriate similarity variables are opted to attain the dimensionless form of governing equations. The resulting momentum, energy, concentration, and motile microorganism density equations are then solved numerically. The resulting dual solutions are graphically visualized and physically analyzed. The results indicate that depending on the systems’ parameters, dual solutions exist in opposing flow beyond a critical point where both solutions are connected. Our results were also compared with existing literature. The study of mixed convection, which is the combination of free and forced convection flow, has become of great interest for many researchers over the last few decades because of its wide range of technological and industrial applications that have been reviewed in Refs.1–3. This includes heat exchanges placed in a low velocity environment, solar collectors exposed to wind currents, atmospheric boundary layer flows, nuclear reactors when cooled during emergency shutdowns, and various electronic equipment. Convection heat transfer in porous medium has many theoretical and practical studies, such as in Refs.4–7, where the effects of buoyancy phenomena on flow and temperature fields through porous media were studied. In a porous medium, the pores are typically filled with fluid (liquid or gas), which causes enhance heat transfer in fluid flow. Several studies were performed on convection heat sources that are based on the Darcy model8–12. For instance, in Ref.8, Lai et al. applied the Darcy model to observe mixed convection in porous media. Abbas et al.9 studied natural convection using the Darcy–Brinkman–Forcheimer model in a vertical cylinder. Srinivasacharya and R eddy10,11 studied the problem of natural and mixed convection for a power-law fluid in a Darcy porous media. In Ref.12, Naveen et al. studied the velocity term impacts of both the maximum density and the momentum equation on the stability of a natural convection through a vertical layer in a Darcy porous media. Furthermore, other studies were done on convection flow through porous media, such as those in Refs.13–17. Very recently, Mondal et al.18 observed internal heat generation and thermal radiation for mixed convection flow over a porous vertical plate. Mixed convection through porous media with heat generation is also studied by Abu-hamdeh et al.19 and Maleque20. Additionally, Shankar and Shivakumara studied the natural convection in a non-Newtonian Oldroyd-B fluid that is saturated in a vertical porous layer that is maintained at varying uniform temperatures21–23. By analyzing the systems’ stability, the authors found that the system is unconditionally stable for Newtonian fluids, and is unstable with viscoelastic fluids. Flow over vertical cylinder has become of great interest to authors due to its numerous applications. For example, it is used to insulate vertical porous pipes, connect with oil/gas lines, underground electrical power 1 Department of Mathematics, Kuwait University, Kuwait City, Kuwait. 2Present address: Research Group of Fluid Flow Modeling and Simulation, Department of Applied Mathematics, University of Dhaka, Dhaka, Bangladesh. *email: Scientific Reports | (2021) 11:19918 | https://doi.org/10.1038/s41598-021-99277-x 1 Vol.:(0123456789) www.nature.com/scientificreports/ transmission lines, radioactive waste disposal, polymer process, and heating or cooling of sheets and films. In Ref.24, Sankar and Do investigated the effects of discrete heating on convection heat transfer in a vertical cylindrical annulus. Several works have also been done on free convection heat transfer in vertical cylinder a nnulus25–28. Moreover, free convective boundary layer flows over a vertical porous cylinder have been investigated by Totala et al.29, Paul et al.30, Minkowyez and C heng31. In Ref.32, Popiel observed free convection heat transfer from the vertical slender cylinder, and Loganathan et al.33 observed natural convection flow for a vertical moving cylinder. Several researchers, such as r eferences34–36, studied mixed convection flow over a vertical cylinder. Very recently, Girish et al.37 studied mixed convection in vertical double annular passages through three coaxial cylinders, Rihan38 observed mixed convection over a short vertical cylinder, and Mkhatshwa et al.39 studied mixed convection nanofluid flow over a vertical slender cylinder. Bioconvection can be classified as a development process in the field of fluid flow, which deals with the steps of self-propelled up swimming microorganisms, such as algae and bacteria that contain oxytaxis, gyrotaxis, and gravitaxis organisms. Motile microorganisms are heavier than their encompassing liquid and usually swim in the upward direction, which brings about producing different flow profiles into the system, as described briefly in Refs.40–46. The advantages of adding motile microorganisms to the suspension include improved mass transfer and microscale mixing. In Refs.47,48, Ghorai et al. observed the stability and development of gyrotactic microorganisms in an in-depth cavity. Mixed convection nanofluid flow containing gyrotactic microorganisms is observed by several r esearchers49–51. Moreover, Mahdy49 studied gyrotactic microorganism mixed convection flow along with isothermal vertical wedge. Khan et al.50 observed mixed convection in a gravity-driven thin film for non-Newtonian nanofluid with microorganisms, and Saleem et al.51 presented the behavior of magneto Jeffrey nanofluid with gyrotactic microorganisms over a rotating cone. Recently, Rashad et al.52,53 and Sudhagar et al.54 explored mixed convection nanofluid flow over a vertical circular cylinder containing gyrotactic microorganisms. In convective heat transfer, there exist complex nonlinear problems. For highly nonlinear problems, multiple (dual) solutions can sometimes be obtained. It is important to compute unstable states along with stable ones as the unstable solutions often interact with stable solutions, which produce unexplainable phenomena, as observed by Rohni et al.55. The study on the existence of dual solutions in mixed convective boundary layer flows may bring a new outlook on engineering applications described in Ref.56. In Ref.57, Ridha et al. showed the existence of a dual solution for opposing flow. After that, Ref.58 extended on that research for assisting flow. Dual solutions for mixed convection boundary laye (...truncated)