Stability analysis of thermo-bioconvection flow of Jeffrey fluid containing gravitactic microorganism into an anisotropic porous medium

Abstract

This paper presents a novel study on the stability of thermo-bioconvection due to gravitactic microorganisms into an anisotropic porous fluid layer saturated with Jeffrey liquid. A Jeffrey-Darcy model along with Boussinesq approximations is utilized. The field equations are treated with non-dimensionalization, linear stability analysis, and the normal mode technique to formulate a set of ordinary differential equations. These equations along with Robin boundary conditions are then analytically solved by employing the weighted residual Galerkin method utilizing the trigonometric trial functions. The traditional thermal Rayleigh-Darcy number Ra,c is obtained as a well-compiled function of the mechanical anisotropy parameter ξ, Jeffrey parameter γ, the thermal anisotropy parameter η, bioconvection Rayleigh-Darcy number Rb, and Péclet number Q, while it is independent from bioconvection Lewis number Lb. It is observed that mounting ξ and γ in between 0 to 1 hasten the formulation of bioconvection patterns and also enlarges the size of convective cells. The results demonstrate that increasing bioconvection Péclet number and microorganism concentration constitute an unstable system. η ranged between 0 to 1 has shown dual effect which is dominated by the concentration of gravitactic microorganism. For small microorganism concentration, augmenting thermal anisotropy strength stabilizes the system and increases the size of the convective cells. Mathematically, the stabilizing nature of η is bounded by the feasibility of the inequality (π2+δc2ξ)(1+γ)ξ>2Rbδc2(eQ−1)π2Q2(π2+δc2)(π2+Q2)(4π2+Q2). This study may find relevance in applications related to pharmaceutical, bio-mechanics, and in microbial enhanced oil recovery (MEOR). The experimental Ra,c value of measure 4π2 at critical wave number value δc=π is also regained as a limiting case from this study.