This paper numerically investigates the heat transport and bifurcation of natural convection in a differentially heated cavity filled with entangled polymer solution combined with the boundary layer and kinetic energy budget analysis. The polymers are described by the Rolie-Poly model, which effectively captures the rheological response of entangled polymers. The results indicate that the competition between its shear-thinning and elasticity dominates the flow structures and heat transfer rate. The addition of polymers tends to enhance the heat transfer as the polymer viscosity ratio ($\beta$) decreases or the relaxation time ratio ($\xi$) increases. The amount of heat transfer enhancement (HTE) behaves non-monotonically, which first increases significantly and then remains almost constant or decreases slightly with the Weissenberg number ($Wi$). The critical $Wi$ gradually increases with the increasing $\xi$, where the maximum HTE reaches approximately $64.9\,\%$ at $\beta = 0.1$. It is interesting that even at low Rayleigh numbers, the flow transitions from laminar to periodic flows in scenarios with strong elasticity. The bifurcation is subcritical and exhibits a typical hysteresis loop. Then, the bifurcation routes driven by inertia and elasticity are examined by direct numerical simulations. These results are illustrated by time histories, Fourier spectra analysis and spatial structures observed at varying time intervals. The kinetic energy budget indicates that the stretch of the polymers leads to great energy exchange between polymers and flow structures, which plays a crucial role in the hysteresis phenomenon. This dynamic behaviour contributes to the strongly self-sustained and self-enhancing processes in the flow.

In this paper, a finite element method is proposed to investigate multiple solutions of the Navier-Stokes equations for an unsteady, laminar, incompressible flow in a porous expanding channel. Dual or triple solutions for the fixed values of the wall suction Reynolds number R and the expansion ratio α are obtained numerically. The computed multiple solutions for the symmetric flow are validated by comparing them with approximate analytic solutions obtained by the similarity transformation and homotopy analysis method. Unlike previous works, our method deals with the Navier-Stokes equations directly and thus has no similarity and other restrictions as in previous works. Finally we use the method to study multiple solutions for three cases of the asymmetric flow (which has not been studied before using the similarity-type techniques).