Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-05-17T22:02:08.028Z Has data issue: false hasContentIssue false

Faraday instability of a liquid layer on a lubrication film

Published online by Cambridge University Press:  27 September 2019

Sicheng Zhao
Affiliation:
Institute for Nano- and Microfluidics, TU Darmstadt, Alarich-Weiss-Strasse 10, 64287 Darmstadt, Germany
Mathias Dietzel
Affiliation:
Institute for Nano- and Microfluidics, TU Darmstadt, Alarich-Weiss-Strasse 10, 64287 Darmstadt, Germany
Steffen Hardt*
Affiliation:
Institute for Nano- and Microfluidics, TU Darmstadt, Alarich-Weiss-Strasse 10, 64287 Darmstadt, Germany
*
Email address for correspondence: hardt@nmf.tu-darmstadt.de

Abstract

The Faraday instability in a system of two conjugated immiscible liquid layers with disparate thicknesses is investigated. The top layer is relatively thick and undergoes short-wavelength instabilities, while the bottom layer is thin and undergoes long-wavelength instabilities. The two layers are coupled by the kinematic and dynamic relations at the interface. Through linear stability analysis, a lubrication effect, which significantly reduces the destabilization threshold, is identified. Especially when the vibration frequency is low, the lubrication effect is seen to influence the transition between the harmonic and subharmonic instability modes. It is studied how far the system with two layers can be approximated by a single-layer system with a Navier-slip boundary condition at the bottom. In corresponding experiments it is found that the time-periodic excitation of the system creates a steady-state deformation of the bottom layer. This indicates nonlinear dynamics of the system and the violation of reversibility. The excellent agreement between experimental and theoretical results for the onset of the instability underpins the validity of the linear stability analysis.

Type
JFM Papers
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Benjamin, T. B. & Ursell, F. 1954 The stability of a plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. Lond. A 225, 505515.Google Scholar
Bestehorn, M. 2013 Laterally extended thin liquid films with inertia under external vibrations. Phys. Fluids 25, 114106.Google Scholar
Bestehorn, M. & Pototsky, A. 2016 Faraday instability and nonlinear pattern formation of a two-layer system: A reduced model. Phys. Rev. Fluids 1, 063905.10.1103/PhysRevFluids.1.063905Google Scholar
Beyer, J. & Friedrich, R. 1995 Faraday instability: linear analysis for viscous liquids. Phys. Rev. E 51 (2), 16621668.Google Scholar
Binks, D. & Water, W. 1997 Nonlinear pattern formation of Faraday waves. Phys. Rev. Lett. 78 (21), 40434046.10.1103/PhysRevLett.78.4043Google Scholar
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383409.10.1017/S0022112090003603Google Scholar
Douady, S. & Fauve, S. 1988 Pattern selection in Faraday instability. Eur. Phys. Lett. 6 (3), 221226.10.1209/0295-5075/6/3/006Google Scholar
Edwards, W. S. & Fauve, S. 1993 Parametrically excited quasicrystalline surface waves. Phys. Rev. E 47 (2), R788R791.Google Scholar
Edwards, W. S. & Fauve, S. 1994 Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278, 123148.10.1017/S0022112094003642Google Scholar
Eifert, A., Paulssen, D., Varanakkottu, S. N., Baier, T. & Hardt, S. 2014 Simple fabrication of robust water-repellent surfaces with low contact-angle hysteresis based on impregnation. Adv. Mater. Interfaces 1, 1300138.Google Scholar
Faraday, M. 1831 On a peculiar class of acoustical figures; and on certain forms assumed by a group of particles upon vibrating elastic surfaces. Phil. Trans. R. Soc. Lond. A 52, 299340.Google Scholar
Feng, J., Jacobi, I. & Stone, H. 2016 Experimental investigation of the Faraday instability on a patterned surface. Exp. Fluid 86, 57.Google Scholar
Floquet, G. 1883 Sur les équations différentielles linéaires á coefficients périodiques. Ann. Sci. École Norm. Sup. 12, 4788.10.24033/asens.220Google Scholar
Gluckman, B. J., Marcq, P., Bridger, J. & Gollub, J. P. 1993 Time averaging of chaotic spatiotemporal wave patterns. Phys. Rev. Lett. 71 (13), 2034.Google Scholar
Hoffmann, F. M. & Wolf, G. H. 1974 Excitation of parametric instabilities in statically stable and unstable fluid instefaces. J. Appl. Phys. 45, 3859.10.1063/1.1663876Google Scholar
Kalliadasis, S., Ruyer, C., Scheid, B. & Velarde, M. G. 2012 Falling Liquid Films. Springer.10.1007/978-1-84882-367-9Google Scholar
Kumar, K. 1996 Linear theory of Faraday instability in viscous liquids. Proc. R. Soc. Lond. A 452, 11131126.Google Scholar
Kumar, K. & Tuckerman, L. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 4968.10.1017/S0022112094003812Google Scholar
Kumar, S. 2000 Mechanism for the Faraday instability in viscous liquids. Phys. Rev. E 62 (1), 14161419.Google Scholar
Lafuma, A. & Quéré, D. 2011 Slippery pre-suffused surfaces. Eur. Phys. Lett. 96, 56001.Google Scholar
Nejati, I., Dietzel, M. & Hardt, S. 2015 Conjugated liquid layers driven by the short-wavelength Bénard–Marangoni instability: experiment and numerical simulation. J. Fluid Mech. 783, 4671.Google Scholar
Périnet, N., Gutiérrez, P., Urra, H., Mujica, N. & Cordillo, L. 2017 Streaming patterns in Faraday waves. J. Fluid Mech. 819, 285.Google Scholar
Piriz, A. R., Cortázar, O. D., López Cela, J. J. & Tahir, N. A. 2006 The Rayleigh–Taylor instability. Am. J. Phys. 74 (12), 1095.10.1119/1.2358158Google Scholar
Pototsky, A. & Bestehorn, M. 2016 Faraday instability of a two-layer liquid film with a free upper surface. Phys. Rev. Fluids 1, 023901.10.1103/PhysRevFluids.1.023901Google Scholar
Pototsky, A., Bestehorn, M., Merkt, D. & Thiele, U. 2005 Morphology changes in the evolution of liquid two-layer films. J. Chem. Phys. 122, 224711.Google Scholar
Rajchenbach, J. & Clamond, D. 2015 Faraday waves: their dispersion relation, nature of bifurcation and wavenumber selection revisited. J. Fluid Mech. 777, R2.10.1017/jfm.2015.382Google Scholar
Rayleigh, L. 1883 On the crispations of fluid resting upon a vibrating support. Phil. Mag. 16 (5), 5058.Google Scholar
Rojas, N. O., Argentina, M., Cerba, E. & Tirapegui, E. 2011 Faraday patterns in lubricated thin films. Eur. Phys. J. D 62, 2531.Google Scholar
Schulze, T. P. 1999 A note on subharmonic instabilities. Phys. Fluids 11 (12), 35733576.10.1063/1.870223Google Scholar
Shu, J., Teo, J. B. M. & Chan, W. K. 2017 Fluid velocity slip and temperature jump at a solid surface. Appl. Mech. Rev. 69 (2), 020801.Google Scholar
Sterman-Cohen, E., Bestehorn, M. & Oron, A. 2017 Rayleigh–Taylor instability in thin liquid films subjected to harmonic vibration. Phys. Fluids 29, 052105.Google Scholar
Thiele, U., Vegal, J. M. & Knobloch, E. 2006 Long-wave Marangoni instability with vibration. J. Fluid Mech. 546, 6187.Google Scholar
Troyon, F. & Gruber, R. 1971 Theory of the dynamic stabilization of the Rayleigh–Taylor instability. Phys. Fluids 14, 2069.Google Scholar
Westra, M., Binks, D. & Water, W. 2003 Patterns of Faraday waves. J. Fluid Mech. 496, 132.10.1017/S0022112003005895Google Scholar
Wong, T., Kang, S.H., Tang, S.K.Y., Smythe, E.J., Hatton, B.D., Grinthal, A. & Aizenberg, J. 2011 Bioinspired self-repairing slippery surfaces with pressure-stable omniphobicity. Nature Lett. 477, 443.Google Scholar