Vibrational thermocapillary instabilities

We study vibrational instabilities of the thermocapillary return flow driven by a constant temperature gradient along the free surface of an infinite layer that vibrates in its normal direction with acceleration of amplitude g(1) and frequency omega(1). The layer is unstable to hydrothermal waves in the absence of vibrations beyond a critical Marangoni number M. Modulated gravitational instabilities with M = 0 are also possible beyond a critical Rayleigh number R based on g(1). We employ two-time-scale high-frequency asymptotics to derive the equations governing the mean field. The influence of vibrations on the hydrothermal waves is found to be characterized by a dimensionless parameter G that is proportional to R-2. The return flow at G = 0 is also a mean field basic flow and we study its linear instability at different Prandtl numbers P. The hydrothermal waves are stabilized with increasing G and reverse their direction of propagation at particular values of G that decrease with increasing P. At finite frequencies, a time-periodic base state exists and we study its linear instability by calculating the Floquet exponents. The stability boundaries in the (R, M)-plane are found to be composed of two intersecting branches emanating from the points of pure thermocapillary or buoyant instabilities. Three-dimensional modes are always preferred and the region of stability, while anchored at the point of hydrothermal waves corresponding to R = 0, is found to grow without bound along the R-axis with increasing frequencies. Results from the two approaches are shown to be in asymptotic agreement at large frequencies.