The main purpose of this study is to investigate the thermal behavior of power law fluid within a plus-shaped cavity under the influence of natural convection, also taking into account the Darcy number and magnetohydrodynamics (MHD). The problem is formulated as a system of partial differential equations considering the power law fluid’s rheological behavior. The left-side walls are maintained at a specific low temperature while the lower and the right-side walls have uniform maximum temperatures. The boundary condition is designed to enhance heat transfer efficiency within the cavity, utilizing advanced thermal insulation methodologies. Finite element method (FEM) simulations are conducted, and a grid independence test is performed to validate the results. The impact of relevant parameters on the variation in momentum and thermal distributions is investigated using streamline and isothermal contour plots. The results indicate that as the Rayleigh number increases, the kinetic energy also increases, whereas the viscosity and circulation zones expand with an increase in the power law index. The Nusselt number exhibits a higher value in the shear-thinning case (n = 0.7) compared to the Newtonian (n = 1) and shear-thickening (n = 1.2) cases. This empirical observation underscores the vital role that fluid rheology plays in molding the overall heat transfer performance within the cavity. The study concludes that there is a distinct correlation between the heat transfer rate and the Rayleigh number (Ra). As Ra increases, there is a significant improvement in the heat transfer rate within the flow domain. Furthermore, the fluid behavior and heat transfer performance within the cavity are significantly influenced by the presence of magnetohydrodynamics (MHD) and the Darcy effect.

The dynamic vibration absorber (DVA) is widely used in engineering models with complex vibration modes. The research on the stability and periodic motions of the DVA model plays an important role in revealing its complex vibration modes and energy transfer. The aim of this paper is to study the stability and periodic motions of a two-degrees-of-freedom system coupled with an encapsulated nonsmooth dynamic vibration absorber under low-frequency forced excitation. Based on the slow–fast method, the model is transformed into a six-dimensional piecewise smooth system coupling two time scales. The existence and stability of the admissible equilibrium points for the model are discussed under different parameter conditions. Based on the first integrals, the Melnikov vector function of the nonsmooth dynamic vibration absorber model is calculated. The existence and number of periodic orbits bifurcated from a family of periodic orbits under different parameters are discussed. The phase diagram configuration of periodic orbits is given based on numerical simulation. The results obtained in this paper offer a new perspective for vibration analysis and parameter control for nonsmooth dynamic vibration absorbers.