Particle Thermal Inertia Delays the Onset of Convection in Particulate Rayleigh-Bénard System

We investigate the linear stability of a thermally stratified fluid layer confined between horizontal walls and subject to continuous injection of dilute thermal particles at one boundary and extraction at the opposite, forming a particulate Rayleigh-Bénard (pRB) system. The analysis focuses on the influence of thermal coupling between the dispersed and carrier phases, quantified by the specific heat capacity ratio $ε$. Increasing $ε$ systematically enhances stability, with this effect persisting across a wide range of conditions, including heavy and light particles, variations in volumetric flux, injection velocity and direction, and injection temperature. The stabilizing influence saturates when the volumetric heat capacity of the particles approaches that of the fluid, $ε= O(1)$. The physical mechanism is attributed to a modification of the base-state temperature profile caused by interphase heat exchange, which reduces thermal gradients near the injection wall and weakens buoyancy-driven motion.

💡 Research Summary

This paper presents a comprehensive linear stability analysis of a particulate Rayleigh‑Bénard (pRB) system in which dilute thermal particles are continuously injected at one horizontal wall and extracted at the opposite wall. The novelty lies in explicitly accounting for the thermal inertia of the particles through the specific‑heat‑capacity ratio ε = cₚ^particle / cₚ^fluid, a parameter that quantifies the volumetric heat capacity of the dispersed phase relative to the carrier fluid. The governing equations are formulated in an Eulerian two‑fluid framework that includes drag, added‑mass, buoyancy, and interphase heat exchange. A relaxation time τ_T ∝ ε characterizes how quickly particle temperature adjusts to the local fluid temperature.

After nondimensionalisation, the system is described by nine independent parameters: the Rayleigh number Ra, Prandtl number Pr, Galileo number Ga, particle‑to‑domain size ratio Φ, modified density ratio β, the thermal‑capacity ratio ε, the imposed particle flux J, the inlet particle velocity W* (scaled by the terminal velocity), and the inlet particle temperature Θ*p. The conductive base state is obtained by solving the steady‑state equations for the fluid velocity (zero), particle velocity profile W₀(Z), and the coupled temperature fields Θ₀(Z) and Θ{p0}(Z). The key observation is that as ε increases, the interphase heat exchange term α 12 Φ²(Θ‑Θ_p) becomes less effective at equilibrating the two temperatures, leading to a pronounced temperature difference near the injection wall. This modifies the base temperature gradient, flattening it in the region where buoyancy would otherwise be strongest.

Linear perturbations of the form exp