Vibrational thermocapillary instabilities

• Published in: Journal of fluid mechanics 2005-10, Vol.540 (1), p.353-371
• Main Author: ZEBIB, ABDELFATTAH
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Fluid dynamics; Free surfaces; Fundamental areas of phenomenology (including applications); Hydrodynamic stability; Physics; Return flow; Surface-tension-driven instability; Temperature gradients; Waves
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/S0022112005006014

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.