Thermodynamic selection of stationary Rayleigh-Bénard flow states with slip and no-slip boundaries

Abstract

Multiple stationary flow states may exist under the same external/boundary conditions, while the hydrodynamic equations alone do not provide a global criterion for determining which of these states is preferentially selected. Here, we formulate and test such a criterion for stationary Rayleigh-Bénard (RB) convection within a global non-equilibrium thermodynamic description (Hołyst et al., EPL 149, 30001 (2025)). We find that each stationary flow state in RB convection is characterized by the state function $\mathcal{F}=-E_k-\Psi$, where $E_k$ is the macroscopic kinetic energy of the flow and $\Psi$ is a thermodynamic state function that emerges from the nonlinear inertial term in the Navier-Stokes equation and is associated with the compression and redistribution of the gas accompanying the formation of the convective state. Changes in $\mathcal{F}$ correspond to path-independent work between stationary states. We show that, among dynamically accessible stationary states, the state with the lower value of $\mathcal{F}$ is more stable and is preferentially selected. Both the macroscopic kinetic energy acquired by the flow and the accompanying thermodynamic reorganization of the gas therefore contribute in the same direction to lowering $\mathcal{F}$ and stabilizing the stationary state. We test the criterion using numerical solutions of the hydrodynamic equations for Rayleigh-Bénard convection with both slip and no-slip boundary conditions. For both types of boundary, the observed selection of competing stationary states is consistent with the thermodynamic prediction. The resulting state function provides a global criterion for the selection and stability of stationary hydrodynamic states. It establishes a direct connection between the nonlinear dynamics of the Navier-Stokes equation, thermodynamic work, and macroscopic flow stability.

arXiv comment19 pages, 7 figures, authors manuscript pre peer review