Buoyancy-Driven Rayleigh–Taylor Instability in a Vertical Channel

Abstract

Abstract Suppose that a vertical tube is composed of two chambers that are separated by a retractable thermally insulated thin membrane. The upper and lower chambers are filled with an incompressible fluid and maintained at temperatures T c {T_{c}} and T h > T c {T_{h}}>{T_{c}} , respectively. Upon removal of the membrane, the two fluid masses form an unstably stratified Rayleigh–Taylor-type configuration with cold and heavy fluid overlying a warmer and lighter fluid and separated by an interface across which there is a discontinuity in the density. Due to the presence of an initial discontinuity between two homogeneous states, this problem is mathematically homologous to that of the shock tube problem with the thermal expansion playing the role of pressure. When the two fluid regions are brought directly into contact with each other and the transient interfacial fluctuations have subsided, we show the emergence of a stationary state of convection through a supercritical bifurcation provided a threshold value for the temperature difference is exceeded. We suggest a possible way for the experimental testing of the theoretical results put forth in this paper.