Stability theory of nano-fluid over an exponentially stretching cylindrical surface containing microorganisms

www.nature.com/scientificreports OPEN Stability theory of nano‑fluid over an exponentially stretching cylindrical surface containing microorganisms M. Ferdows1,2,3*, Amran Hossan4, M. Z. I. Bangalee2, Shuyu Sun3 & Faris Alzahrani1 This research is emphasized to describe the stability analysis in the form of dual solution of the flow and heat analysis on nanofluid over an exponential stretching cylindrical surface containing microorganisms. The research is also implemented to manifest the dual profiles of velocity, w temperature and nanoparticle concentration in the effect of velocity ratio parameter (s = UU∞ ). Living microorganisms’ cell are mixed into the nanofluid to neglect the unstable condition of nano type particles. The governing equations are transformed to non-linear ordinary differential equations with respect to pertinent boundary conditions by using similarity transformation. The significant differential equations are solved using build in function bvp4c in MATLAB. It is seen that the solution is not unique for vertical stretching sheet. This research is reached to excellent argument when found results are compared with available result. It is noticed that dual results are obtained demanding on critical value ( sc ), the meanings are indicated at these critical values both solutions are connected and behind these critical value boundary layer separates thus the solution are not stable. The flow profile conveyed by viscous fluid between to coaxially rotated disk is surveyed in Majeed et al.1. Heat and mass transfer profile of second grad fluid over an inclined cylinder with diffusion heat flux described by Bilal et al.2. The features of Carreau flow and important aspect of thermal stratification are concluded by Bilal et al.3. The momentum and heat transfer of electro-magneto hydrodynamics boundary layer flow are incorporated in Bilal et al.4 over stretching sheet with slip. Analysis of Newtonian flow and the flow of power law fluid are manifested by Mahmood et al.5 with the feature of shear thinning and shear thickening. The flow features of power law materials with channel driven cavity configuration investigated by Mahmood et al.6. The pseudo plastic and dilatant materials have extensive applications on metallurgical processes. Characteristics of power law fluid were addressed in Mahmood et al.7 with the attributes of pseudo plastic and dilatant materials in channel driven cavity. The aspects of temperature dependent dynamic viscosity of Maxwell fluid are obtained over a variable thicken surface by Khan et al.8. Bio convection MHD Carreau Nano fluid flow and thermo physical aspects of MHD were focused respectively on Khan et al.9 and Hussain et al.10 and model have been constructed by Fourier’s and Fick’s laws. The features of MHD Prandtl-Eyring Nano fluid over stretching surface introduced by Rehman et al.11 with the effect of Navier slip and convective boundary condition. The Maxwell fluid flow of heat and mass transfer over stretching sheet were explicitly drawn with solar radiation and viscous desperation by Khan et al.12. The heat and mass diffusion of Maxwell Nano fluid over stretching surface near stagnation point incorporated in Khan et al.13 and are implied by Fourier’s and Fick’s laws. The Bio-convection boundary layer flow and nanofluid model were introduced in Buongiorno14. Several similar works of nanofluid and heat rate were done on14–18. Advance analysis of applications of nanofluid have been reviewed in Refs.19–22 and many others have been analyzed to enhance nanofluid effect over heat transfer with the use of parameters. Buongiorno’s model14 and the Tiwari-Das model23 are two familiar method for the analysis of nanofluid which have been worked by researchers. In Buongiorno model, the total fluid velocity and the relative/slip velocity were counted as the nanofluid velocity. This model also scrutinized the effect of parameter as 1 Department of Mathematics, King Abdulaziz University, Jeddah 21589, Kingdom of Saudi Arabia. 2Research Group of Fluid Flow Modeling and Simulation, Department of Applied Mathematics, University of Dhaka, Dhaka 1000, Bangladesh. 3Applied Mathematics and Computational Science Division, King Abdullah University of Science and Technology, Thuwal 23955‑6900, Kingdom of Saudi Arabia. 4Department of Mathematics, College of Arts and Science, International University of Business and Agriculture and Technology, Uttara, Dhaka 1230, Bangladesh. *email: Scientific Reports | (2020) 10:17004 | https://doi.org/10.1038/s41598-020-72545-y 1 Vol.:(0123456789) www.nature.com/scientificreports/ Figure 1.  Physical Significant Model and fluid coordination. Brownian diffusion and thermophoresis. In opposition of Buongiorno’s model, the solid volume fractions of the nanoparticles were introduced by Tiwari-Das model et al.23. By the characteristic of nanofluid flows Brownian diffusion and thermophoresis are the most important parameters implies by Ref.14. Present study of Refs.24–29 add nanofluid in the convective boundary layer flow. Nano-polymer stretching flows with radioactive magneto hydrodynamics were investigated by Ferdows et al.30. Numerical studies of magnetic Nano-bio-polymerswere done by Uddin et al.31. To create the bio convection process add microorganism with the base fluid. Bioconvection could cause an unstable density profile of the fluid, if the density of microorganism is seen to be greater than the free stream fluid which is followed by Raees et al.32. Microorganisms survive to base fluid if the base fluid is water and remain stable in the nanofluid suspension for a few of weeks by Anoop et al.33. Nanoparticles could multiply the nanofluid’s viscosity and tends to accelerate bio convection instability34. Nano-fluidon boundary layer flow, stretching cylinder, containing microorganism and bio-convection have been described respectively by Refs.35–38. The existence of dual solutions for conducting flow and mixed convection boundary layer flow with suction and injection are analyzed by Ishak et al.39. In study on Newtonian fluids boundary layer is contrasting to the free stream flow then multiple solutions would be found on Ishak et al.40. Najib et al.41 is analyzed the dual solutions exists over stretching cylinder along with mass suction. The researchers such as Refs.42–44 have scrutinized the stability theory into their research to ensure the flow is stable and have meaningful solution. The article is mainly cover on expending research work by Refs.42–48 to verify the existence of the dual (first solution and second solution) solutions with the consideration of different parameter. The stable solution will be noticed when we get the dual or multiple solutions and with the help of numerical analysis to verify which solution is stable or not. Problem formulation. Consider circular cylinder of radius a which is stretching exponentially along with velocity Uw . Tw , Cw , and mw are the constant temperature, c (...truncated)