On the early evolution in the transient Bénard–Marangoni problem

We focus on the early evolution of small (linear) perturbations following the sudden (step-function) exposure of a liquid layer to a cold adjacent atmosphere. On a time scale short relative to that characterizing thermal relaxation across the liquid layer, the temperature distribution is nonlinear and highly transient. Thus, the conduction reference state may not be regarded quasi steady. We accordingly consider the initial-value problem and obtain a Volterra-type integral equation governing the evolution of surface-temperature perturbations. Assuming an O(1) Biot number we study the effects on perturbations evolution of the wavenumber, the Prandtl number (Pr) and the effective Marangoni number (⁠$Ma¯$⁠, which, from the dominant balance in the thermal perturbation equation, is based on the current values of the width of the thermal boundary layer and the temperature difference across it, respectively). Explicit results are first presented in the limit of $Pr>>1$ wherein the hydrodynamic perturbation problem is quasi steady. The dominant perturbations correspond to large wavenumbers and convection effectively confined to the thermal boundary layer. Typical of the results is the nonmonotonic temporal evolution of perturbations which initially diminish and only after some finite delay time start to grow. These trends are rationalized in terms of the Marangoni mechanism by observing that while the magnitude of the reference-state temperature gradients remain essentially constant, they extend over a widening thermal boundary layer allowing for enhanced convective effects. Thus, since perturbations may be introduced at all positive times, those evolving on the favourable background of further developed (wider) thermal boundary layers may take over perturbations introduced earlier. This suggests the existence of a nonzero “optimal” appearance time of perturbations which will eventually dominate the instability process. We study the effects of finite O(1) values of Pr on the evolution of the hydrodynamic perturbation problem which is no longer quasi steady. Increasing Pr at a constant $Ma¯$ is destabilizing which reflects the dual role of liquid viscosity in the transient Bénard-Marangoni problem. It is further demonstrated that with increasing wavenumber convergence to the asymptotic $Pr>>1$ limit is practically achieved at smaller Pr.