Energetics and mixing in Rayleigh-Benárd-Darcy convection in Hele-Shaw cells

Thermally driven flows in fractures play a fundamental role in enhancing the heat transfer and fluid mixing across the Earth’s lithosphere. Yet little is known about the energy pathways in such confined environments. Building on the quasi-two dimensional (Q2D) Hele-Shaw model able to characterize flows in permeable media, we introduce and analyse expressions for energy transfer rates—energetics—of geometrically controlled Rayleigh-Bénard-Darcy convection in Hele-Shaw cells considering the Boussinesq limit. Based on the conceptual framework, we investigate theoretically the mechanical energy partition and we derive the efficiencies that allow characterising the transformation of the injected energy into motion and irreversible mixing. The analytical expressions for the energy partitioning are supported by direct numerical simulations of the governing equations. Finally, we discuss the relationship between the global Nusselt number and the Rayleigh number of the system, and we show how strength of the thermal forcing may lead the system to transition from a Darcy-like medium to a three-dimensional environment.