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

TL;DR

This work analyzes the linear stability of a particulate Rayleigh–Bénard system with finite particle thermal inertia, introducing the specific heat-capacity ratio $\epsilon = c_{Pp}/c_P$ to decouple thermal effects from mechanical coupling. Using an Eulerian two-fluid model and a conductive base state, the authors solve a coupled eigenvalue problem to determine how $Ra_c$ and $k_c$ depend on $\epsilon$, $\beta$, injection conditions, and particle flux. They find that increasing $\epsilon$ consistently delays the onset of convection and raises both $Ra_c$ and $k_c$, with a saturation near $\epsilon = \mathcal{O}(1)$; the stabilizing mechanism arises from the base-state temperature modification caused by interphase heat exchange, which reduces the gradient near the injection wall. The density ratio exhibits an asymmetric influence, with heavier and lighter particles stabilizing relative to the single-phase case, and the injection velocity and flux modulating this stabilization. These results offer a quantitative foundation for nonlinear analyses and experiments in particle-laden convection and underscore the importance of thermal inertia in interphase heat transfer for multiphase flow stability.

Abstract

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.

Figures (10)

• Figure 1: Vertical dependence of the steady-state temperature fields of the (left) fluid $\Theta_0(Z)$ and (right) particulate $\Theta_{p0}(Z)$ phases, for (bottom) light and (top) heavy particles, and different values of the specific heat capacity ratio $\epsilon$. The particle flux is set to $\mathcal{J}_0= 533.3$, $\mathbf{W}^{\ast} = 0.5\,\mathbf{W}_T$, and the particle injection temperature $\Theta_p^{\ast}$ is assumed to match the temperature of the wall from which they are introduced. The arrow indicate the direction of increase of $\epsilon$.
• Figure 2:
• Figure 3:
• Figure 5: Effect of particle injection temperature on the critical threshold. (a) Heavy particles ($\beta = 0.5$) are injected from above with the cold wall temperature $\Theta_p^* = 0$, and inversely with the temperature of the opposite hot wall, $\Theta_p^* = 1$. (b) Light particles ($\beta = 2.5$) are injected from below with the hot wall temperature $\Theta_p^* = 1$, and inversely with the opposite cold wall temperature $\Theta_p^* = 0$.
• Figure 6: