Natural convection in a hemispherical soap bubble

Abstract

An experimental, numerical and theoretical study for the thermal convection in a hemispherical soap bubble is presented. The bubble is heated at the equator promoting a temperature gradient with the polar region. Buoyancy forces induce thermal plumes at the equator that move towards the pole. This thermal convection cell was firstly addressed by Seychelles et al. (2008). Experimentally, the time-dependent flows were explored with a Prandtl number $Pr=6.85$ and different temperature gradients, which resulted in Rayleigh numbers $Ra$ ranging from $4.4\times 10^4$ to $1.6\times 10^6$. Additionally, the thermal plumes tilt and reach smaller latitudes if the hemisphere rotates at constant angular speed. Two rotating angular velocities were explored, resulting in two Ekman numbers, namely, $Ek=4.7\times 10^{-4}$ and $Ek=2.4\times 10^{-4}$. The flows on the curved surface of the soap bubble are time-dependent flows with Reynolds numbers $Re$ ranging from 270 to 380, denoting a laminar regime. Experimental results obtained through dye visualization, particle image velocimetry, and thermal imaging show that the number of initial thermal plumes at the equatorial region depends significantly on the Rayleigh number. These observations helped to develop a two-dimensional analytical solution of the phenomenon. The analytical solution reproduces qualitatively various aspects of the flow. To the best knowledge of the authors, this is the first time an analytical solution of the phenomenon under study is reported. A full three-dimensional numerical simulation validates the analytical solution. Additionally, the numerical results agree with the experimental observations.