Weakly Nonlinear Bio-Thermal Convection in a Porous Media Layer Under Rotation, Gravity Modulation, and Heat Source

Abstract

In this paper, the influence of gravitational modulation on weakly nonlinear biothermal convection in a porous rotating layer is investigated. We consider a layer of porous medium saturated with Newtonian fluid, containing gyrotactic microorganisms, and subject to gravitational modulation, rotation, and internal heating. To analyze linear stability, it is sufficient to represent disturbances in the form of normal modes, while nonlinear analysis includes a truncated Fourier series containing a harmonic of the nonlinear interaction. A six-dimensional nonlinear Lorentz-type model is constructed, exhibiting both reflection symmetry and dissipation. We determined heat and mass transfer using a weakly nonlinear theory based on the representation of a truncated Fourier series. Additionally, the behavior of nonstationary Nusselt and Sherwood numbers was investigated by numerically solving finite amplitude equations. Applying the expansion of regular perturbations in a small parameter to a six-dimensional model of Lorentz equations with periodic coefficients, we obtained the Ginzburg-Landau (GL) equation. This equation describes the evolution of the finite amplitude of the onset of convection. The amplitude of convection in the unmodulated case is determined analytically and serves as a standard for comparison. The study examines the effect of various parameters on the system, including the Vadasz number, modified Rayleigh-Darcy number, Taylor number, cell eccentricity, and modulation parameters such as amplitude and frequency. By varying these parameters, in different cases, we analyzed heat and mass transfer, quantitatively expressed by the Nusselt and Sherwood numbers. It has been established that the modulation amplitude has a significant effect on the enhancement of heat and mass transfer, while the modulation frequency has a decreasing effect.