Overstability for surface tension and coupled buoyancy-driven instability in a horizontal liquid layer - Toward the understanding of thermal lens oscillations

Abstract

The overstability for surface tension and coupled buoyancy -driven instability in a horizontal liquid layer, with very general conditions, is studied. A linear formulation to compute the critical quantities is established. Numerical results are given and compared with overstability experiments in which a free surface is heated by a controlled hot-wire located near and below it. When correctly presented in terms of well chosen reduced quantities, theoretical and experimental results agree very well, showing that there is an analogy between the theoretical problem (horizontal liquid layer, basic conductive state) and the experimental situation (hot-wire heating, hasic convective state). Disagreements are pointed out, to stress the limitations of the analogy. The original motivation of the work is the understanding of thermal lens oscillations produced when heating below the free surface is carried out using a laser beam. Bo Bond number Cri Crispation number dhr Distance between the hot-wire and the free surface (HWE) or between solid wall and free surface (model) D Operator d/dz f Planform function g Gravity acceleration (buoyancy) gs Gravity acceleration (gravity waves) H Mean curvature of the free surface or heat transfer coefficient at the free surface kr KT I PI P Pr Ma ni Nu Ra t T TO Adverse temperature gradient K ~ . t t Thermal and efficient thermal diffusivities pressure in the medium i pressure perturbation Prandtl number Marangoni number Unit vector Nusselt number Rayleigh number Time Liquid temperature Reference temperature Professor, member AIAA. v Copyright @ American IoIlIlule of A C I O ~ U ~ I C S aod ASI~O~~UI IC I . Inc , 1987. All rights reserved. Ts Free surface temperature in the nonperturbed state Ut Liquid velocity UI Liquid velocity perturbation uf8.i Free surface velocity perturbation Vi Viscosity number x,y,z Cartesian coordinate system z f . free surface location in the basic c( Wave number of the normal made 0, Expansion factor E Time constant 6 . z dependent part of the perturbation 6T. Critical temperature difference 6 z t . perturbation of the free surface All Surface Laplatian operator AT* Critical temperature difference r Rate of variation of the surface 5 z dependent part of the velocity e z dependent part of the temperature state of the free surface location