Energetics and mixing of thermally driven flows in Hele-Shaw cells

Thermally driven flows in fractures play a key role enhancing the heat transfer and fluid mixing across Earth's lithosphere. Yet energy pathways such confined environments have not been characterised. Building on Letelier et al. ( J. Fluid Mech. , vol. 864, 2019, pp. 746–767), we introduce novel expressions for rates – energetics of geometrically constrained Rayleigh–Bénard convection Hele-Shaw cells (HS-RBC) based two different conceptual frameworks. First, derived following well-established framework introduced by Winters 289, 1995, 115–128), which gravitational potential energy, $E_{\textit {p}}$ is decomposed into its available, {ap}}$ background, {bp}}$ components. Secondly, considering new decomposition named dynamic, {dp}}$ reference, {rp}}$ energies; defined as departure system's from reference state determined ‘energy’ scalar fluctuations. For HS-RBC, both frameworks lead to same at steady state, satisfying relationship $\langle E_{\textit {ap}} \rangle _{\tau } = \langle {dp}} + 1/6$ . Consistent with work Hughes 729, 2013) three-dimensional convection, report analytical efficiencies HS-RBC terms Rayleigh number global Nusselt number. Additionally, performed numerical experiments illustrate application analysis HS-RBC. Finally, discuss impact thermal forcing geometrical control exerted development boundary layers, protoplumes self-organisation large-scale flows.