Weakly nonlinear bio-convection in a porous media under temperature modulation and internal heating

This study examines how Darcy-Brinkman bio-convection is affected by internal heating and thermal modulation in a porous media saturated with a Newtonian fluid and contains gyrotactic microorganisms. The stationary mode with low thermal modulation of bioconvection is investigated using a weak nonlinear stability analysis. With the use of the Ginzburg-Landau equation (GLE), the Nusselt number a measure of heat transfer is calculated. The solvability condition is applied at the third order of the perturbed system to determine the GLE. The results are presented graphically to demonstrate how heat transfer is impacted by system parameters. The results show that the modulation amplitude and internal Rayleigh number have a progressive effect on heat transfer. However, as cell eccentricity and the modified bioconvection Rayleigh number rise, heat transfer decreases. On mean Nu, the effects of internal Rayleigh number \(\text {R}_\text {i}\) ${\text{R}}_{\text{i}}$ and bioconvection Rayleigh number \(\text {R}_\text {b}\) ${\text{R}}_{\text{b}}$ are opposing. The system becomes unstable and transfers heat more quickly due to internal heating. It is also found that OPM/LBM is the only approach that effectively controls heat transfer compared to IPM. Additionally, it is found that only OPM/LBM can effectively control the transfer of heat in comparison to IPM. Furthermore, it is found that irregularities ( \(\alpha \ne 0\) $\alpha \ne 0$ ) and asymmetries in microorganisms other than spherical-shaped ones ( \(\alpha = 0\) $\alpha =0$ ) may cause a delay in the convective heat transfer mechanism.