Bifurcations patterns and heat transmissions in couple-stress fluid layer with isothermal rigid boundaries

In this work, we investigate the dynamical patterns of thermo-convective loops in a horizontal shallow layer of couple-stress fluid confined between isothermal rigid boundaries and heated from beneath. The novelty of this study lies in examining the influences of couple-stresses on the dynamical patterns of fluid convection in the presence of rigid boundaries. Both the linear and non-linear stability analyses are performed. It is found that the critical Rayleigh number for the onset of convection increases significantly with the couple-stress parameter ($C$). Using low-order Galerkin approximations within the framework of non-linear stability analysis, a three-dimensional, non-linear, dissipative system governed by four control parameters is derived. Studies reveal that the transitions to the stationary and oscillatory convections (through pitchfork and Hopf bifurcations, respectively) detain with the enhancement of $C$, consistent with observations for free-isothermal boundaries. Notably, a striking outcome of this research is that, unlike the stress-free case, the subcritical Hopf bifurcation evolves to supercritical one as $C$ exceeds a critical threshold of approximately 0.1022708, and it fundamentally alters the flow dynamics. At this threshold value of the parameter $C$, the system experiences a codimension-2 Bautin bifurcation, which is not likely to appear in the classical Lorenz system for realistic parameters values. Furthermore, the mode of heat transport stabilizes from convection to conduction with increasing $C$. Variations in the stream function and isotherm function with respect to $C$ are analyzed and depicted. Additionally, the effect of couple-stresses on the chaotic regime at a high reduced normalized Rayleigh number ($r_1=200$) exhibits intermittent behavior, and chaos is entirely suppressed for a suitable value of $C$, indicating the stabilization of the system.