Marangoni waves in a two-layer film under the action of an inclined temperature gradient

Abstract

The development of the longwave deformational instability of a thermocapillary flow in a two-layer film under the action of an inclined temperature gradient is studied in the framework of the lubrication approximation. The stability boundaries with respect to different oscillatory modes are calculated by means of the linear stability theory. In a contradistinction to the case of a vertical temperature gradient, these boundaries strongly depend on the direction of the wave propagation. Numerical simulations of spatially periodic nonlinear regimes are fulfilled. It is shown that because of the anisotropy of the problem, the most typical kind of patterns is a traveling wave. For small inclination of the temperature gradient, temporally quasiperiodic waves are observed. A number of new three-dimensional traveling wave planforms is revealed.

Thermocapillary instabilities in a liquid layer subjected to an oblique temperature gradient

“…In the case of a fixed temperature gradient at the substrate, Davis (1987) showed that there exists a relationship between the critical Marangoni numbers defined using the imposed HTG and VTG, which did not predict the stabilization caused by the imposed HTG. However, the later studies of Nepomnyashchy, Simanovskii & Braverman (2001) and Shklyaev & Nepomnyashchy (2004) and the present work show that a prescribed HTG has a strong stabilizing effect on the modes found by Pearson (1958) which are caused by the prescribed VTG. Furthermore, Davis (1987) showed a negligible effect on the surface-wave instabilities at low Prandtl numbers Pr.…”

“…Thermocapillary instability in the fluid layers subjected to OTG has been previously studied by Nepomnyashchy et al (2001) and Shklyaev & Nepomnyashchy (2004) for a return flow configuration. They considered a two-layer structure of the fluids with the fluid-fluid interface assumed to be non-deformable.…”

“…Inspired by the experimental limitations on imposing and maintaining a purely vertical temperature gradient and practical applications, Nepomnyashchy et al (2001), Shklyaev & Nepomnyashchy (2004), Nepomnyashchy & Simanovskii (2007, 2014 and Kowal et al (2018) investigated the stability of liquid layers subjected to an OTG. In addition to the VTG-related stabilization of instabilities, Nepomnyashchy et al (2001) and Shklyaev & Nepomnyashchy (2004) also showed theoretically the emergence of hydrothermal waves and spanwise rolls, which could become dominant at sufficiently high HTG. The experiments of Schwabe (2007) and Mizev & Schwabe (2009) confirmed these theoretical results.…”

“…It must be noted that Nepomnyashchy et al (2001) and Shklyaev & Nepomnyashchy (2004) studied the return flow in the layer with a non-deformable free surface. However, linear flow is important in thin films and can be experimentally realized (Schwabe 2007).…”

Linear-stability analysis of thermocapillary-buoyancy convection in annular two-layer system with a radial temperature gradient

“…However, up to now, most of the interest about thermocapillary-buoyancy convection in a two-layer liquid system is related to vertical heating. This type of convection is called the Rayleigh-Bénard convection or the Rayleigh-Bénard-Marangoni convection, which correspond to two-layer systems with or without an upper free surface, respectively [3][4][5][6].…”