Global non-equilibrium thermodynamics of stationary states applied to the Rayleigh-Bénard convection

TL;DR

The paper extends global non-equilibrium thermodynamics to stationary states of fluid systems by incorporating macroscopic kinetic energy into a global energy balance. It introduces a gauge-dependent state function $\Psi$, derived from the inertial term via Helmholtz-Hodge decomposition, and shows that the external work along a stationary trajectory satisfies $W_{ext} = -\Delta\Psi$, with $-\Psi$ minimized for spontaneous transitions. Applying this to Rayleigh-Bénard convection, the authors derive first- and second-law relations that govern transitions between stationary convection states and verify them with OpenFOAM simulations, including scenarios with changing gravity and geometry. The framework links energy, entropy, and kinetic effects in a unified global description, offering a principled way to predict directionality of evolution in driven flows and suggesting broad applicability to complex hydrodynamic systems.

Abstract

Classical thermodynamics describes physical systems in thermodynamic equilibrium, characterized in particular by the absence of macroscopic motion. Global non-equilibrium thermodynamics extends this framework to include physical systems in stationary states (Hołyst et al., EPL 149, 30001 (2025)). Here, we demonstrate that this extended theory captures macroscopic motion in stationary states, thereby providing a unified framework for global thermodynamics and fluid mechanics. We apply the theory to stationary Rayleigh-Bénard convection and show how the second law of global non-equilibrium thermodynamics determines the direction of changes in fluid motion.

Figures (9)

• Figure 1: Square domain filled with a perfect gas under gravity. The bottom wall is kept at $T_0$ and the top wall at $T_L < T_0$. The grey map corresponds to the temperature distribution. The lighter shade indicates a higher temperature. Panel a) illustrates the quiescent solution that supports a heat flux without macroscopic fluid motion. Panel b) illustrates a single Rayleigh-Bénard convection cell, which supports a heat flux with the occurrence of macroscopic fluid motion. Additionally, we present the flow lines for clockwise rotation with boundary slip.
• Figure 2: Convection patterns appearing for $\textrm{Ra}\approx 1.6 \times 10^4$. a) Flow lines. The flow directions are indicated by the large arrows. The resulting convection patterns are directly correlated with the specific form of the initial perturbation. The perturbations were introduced as small $0.02 L \times 0.02 L$ areas of elevated velocity $\mathbf{u}=0.01$ m/s in the places and directions indicated by the small arrows. b) Change in internal energy with respect to the quiescent state. c) Change in potential and kinetic energy with respect to the quiescent state. d) Work that the system performed against inertial forces. In panels b), c), and d), the vertical dashed line marks the time at which the velocity perturbation was applied and time is normalized by thermal relaxation scale $\tau_D = L^2 \rho_0 c_v/k \approx 88$ s.
• Figure 3: Changes in work against inertial forces during changes in $g$ normalized with $g_0$. a) Decreasing from $g_0$ to 0. b) Increasing from $g_0$ to $100g_0$.
• Figure 4: Spontaneous transitions during changes in $L_x$. a) Typical features of states 1 and 2. The temperature profile as a heat map varying from cold (blue) to hot (red), and flow lines. b) The value of $\varPsi$ for different sizes of the system. The direction of spontaneous changes of state and changes in $\varPsi$ is indicated by the arrows. A movie (supplemantary information 1) illustrates complete expansion and compression cycle.
• Figure 5: Toy model consisting of a rotating stick and a point mass attached to a nonlinear spring. The model has two stationary states: (1) an unstable state with the spring compressed, and (2) a stable state with the spring extended.