Emergence of natural convection beneath a fluid-supported sheet

Understanding the interplay between thermal, elastic and hydrodynamic effects is crucial for a variety of applications, including the design of soft materials and microfluidic systems. Motivated by these applications, we investigate the emergence of natural convection in a fluid layer that is supported from below by a rigid surface, and covered from above by a thin elastic sheet. The sheet is laterally compressed and is maintained at a constant temperature lower than that of the rigid surface. We show that for very stiff sheets, and below a certain magnitude of the lateral compression, the system behaves as if the fluid were confined between two rigid walls, where the emergent flow exhibits a periodic structure of vortices with a typical length scale proportional to the depth of the fluid, similar to patterns observed in Rayleigh-Bénard convection. However, for more compliant sheets, and above a certain threshold of the lateral compression, a new local minimum appears in the stability diagram, with a corresponding wavenumber that depends solely on the bending modulus of the sheet and the specific weight of the fluid, as in wrinkling instability of thin sheets. The emergent flow field in this region synchronises with the wrinkle pattern. We investigate the exchange of stabilities between these two solutions, and construct a stability diagram of the system.