Instability of a gravity-modulated fluid layer with surface tension variation (Q2731195)

The linear stability of a differentially heated fluid layer with a rigid bottom and an undeformable thermocapillary top surface is considered in the presence of harmonic gravity modulation. The direction of the time-dependent gravity vector is perpendicular to the top and bottom surfaces. The perturbations are represented by normal modes in two horizontal directions. In the vertical coordinate, a Chebyshev-collocation discretization is used. The effects of the Prandtl number, Marangoni number, modulation amplitude, modulation frequency, and mean gravity level on the critical conditions are computed and displayed graphically. Apart from a synchronous response, also subharmonic response is found in closed regions near unit Prandtl number. The author discusses the relation of the present modulated Marangoni-Bénard problem to the modulated Rayleigh-Bénard problem. In particular, the critical modulation amplitude for the modulated Marangoni-Bénard problem depends strongly on the Prandtl number. A good qualitative picture is obtained by a one-term Galerkin approximation of the problem which leads to Mathieu equation. Using classical averaging method, the author obtains an analytical stability criterion valid for small damping and modulation-amplitude coefficients of Mathieu equation. Finally, dimensional examples are given for mercury and two silicone oils, and the limitations of the analysis regarding the flat-interface assumption is discussed.