Multistability and transition to chaos in non-Fourier convection under cross-flow forcing

This study explores the intricate dynamical interplay between thermal relaxation time and vertical through-flow in a shallow, horizontally extended Boussinesq fluid layer heated from below, governed by the Cattaneo–Christov model. Linear stability analysis reveals that the presence of thermal time lag advances the onset of stationary convection under through-flow, while oscillatory convection is delayed. To capture the weak non-linear thermo-fluidic behavior, a reduced five-dimensional non-linear system is derived via a low-order Galerkin projection. Local stability and bifurcation analysis around equilibria confirm the emergence of pitchfork and Hopf bifurcations in the conductive phase, which is in agreement with linear theory. A critical threshold of the time lag also permits overstability, where oscillatory convection precedes the stationary mode. In the convective regime, equilibria undergo Hopf and Bogdanov–Takens bifurcations, further advanced by the through-flow. Hysteresis-induced multistability arises in the steady conductive state, with coexisting conductive and convective attractors forming smooth, organized basins of attraction. At moderate thermal relaxation, gravity-opposing through-flow drives the system through a classical period-doubling route to chaos, while gravity-aligned through-flow triggers a reversed period-halving sequence. Heat transfer analysis via Nusselt number reveals that through-flow enhances convective mode of heat transport in pitchfork states and stabilizes through conduction mode in Hopf states, depending on the time lag. Additionally, the influence of through-flow on Shil'nikov-type chaos is investigated: at lower Rayleigh numbers, through-flow accelerates single-scroll chaos, whereas at higher Rayleigh numbers, it suppresses multi-scroll chaotic spirals. These transitions are characterized using bifurcation diagrams, Lyapunov spectra, phase portraits, and Poincaré sections.