Impact of temperature dependent viscosity on thermo-bioconvective flow of Jeffrey fluid containing gravitactic microorganism in a permeable medium

This study investigates how linear and exponential variations in temperature-dependent viscosity influence the initiation of thermo-bioconvective flow in a non-Newtonian Jeffrey fluid with gravitactic microorganisms present in a permeable medium, which has not been addressed in the existing literature. The threshold of the thermo-bioconvective flow, taking into account the absence of microorganism flux at the boundaries, is determined through linear stability theory, and the corresponding eigenvalue problem is resolved analytically using the Galerkin method. The findings indicate that when viscosity changes linearly with temperature, the critical Rayleigh number$R_{D,c}^{Ex}$ at which the system starts convection is approximately 11 % higher than when viscosity changes exponentially with temperature. This shows that the system is more unstable when viscosity changes exponentially with temperature compared to a linear change. The stability of the arrangement decreases as the bio Rayleigh-Darcy number$R_{BD}$, the bio Péclet number$P{e}_B$, the Jeffrey factor$\gamma$, and the viscosity deviation parameter$F$ increase. In instances of exponential viscosity variation with temperature, the size of the convective cells grows with the viscosity deviation parameter$F$, while it remains constant in the case of linear viscosity variation. Additionally, it is important to emphasize that oscillatory convective motion is not relevant to the current analysis.