Stability of rolls in finite-amplitude Rayleigh–Bénard convection in a high-Prandtl-number fluid between a perfectly conducting boundary and a slab of finite thickness and finite conductivity

Non-linear solutions in the form of two-dimensional rolls are investigated for Rayleigh–Bénard convection in the limit of an infinite Prandtl number fluid in an infinitely wide horizontal fluid layer. The temperature is kept constant at the upper boundary of a slab of finite thickness and thermal conductivity placed on top of the fluid layer. The convection is driven by the temperature difference between this upper boundary and a higher temperature kept constant at the rigid lower fluid boundary. The dependency of the heat transfer on the thickness and conductivity of the slab is reported. Similar to the classical case of two perfectly heat conducting, rigid boundaries the stability region of the rolls is restricted by the cross-roll instability and the zigzag instability. Stability regions relevant for experimental conditions are presented for upper slabs covering a wide range of thicknesses and thermal conductivities. It is shown, how the uppermost Rayleigh number within the stability region varies as a function of the slab properties.