Skip to main content Accessibility help
×
Home
Hostname: page-component-55b6f6c457-4lvx9 Total loading time: 0.225 Render date: 2021-09-26T13:59:12.707Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Douady, S. 1990 Experimental study of the Faraday instability. J. Fluid Mech. 221, 383409.CrossRefGoogle Scholar
Douady, S. & Fauve, S. 1988 Pattern selection in Faraday instability. Eur. Phys. Lett. 6 (3), 221226.CrossRefGoogle Scholar
Edwards, W. S. & Fauve, S. 1993 Parametrically exrefd quasicrystalline surface waves. Phys. Rev. E 47 (2), R788R791.Google ScholarPubMed
Edwards, W. S. & Fauve, S. 1994 Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278, 123148.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
Hoffmann, F. M. & Wolf, G. H. 1974 Excitation of parametric instabilities in statically stable and unstable fluid instefaces. J. Appl. Phys. 45, 3859.CrossRefGoogle Scholar
Kalliadasis, S., Ruyer, C., Scheid, B. & Velarde, M. G. 2012 Falling Liquid Films. Springer.CrossRefGoogle 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.CrossRefGoogle Scholar
Kumar, S. 2000 Mechanism for the Faraday instability in viscous liquids. Phys. Rev. E 62 (1), 14161419.Google ScholarPubMed
Lafuma, A. & Quéré, D. 2011 Slippery pre-suffused surfaces. Eur. Phys. Lett. 96, 56001.CrossRefGoogle 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.CrossRefGoogle Scholar
Périnet, N., Gutiérrez, P., Urra, H., Mujica, N. & Cordillo, L. 2017 Streaming patterns in Faraday waves. J. Fluid Mech. 819, 285.CrossRefGoogle 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.CrossRefGoogle Scholar
Pototsky, A. & Bestehorn, M. 2016 Faraday instability of a two-layer liquid film with a free upper surface. Phys. Rev. Fluids 1, 023901.CrossRefGoogle 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 ScholarPubMed
Rajchenbach, J. & Clamond, D. 2015 Faraday waves: their dispersion relation, nature of bifurcation and wavenumber selection revisited. J. Fluid Mech. 777, R2.CrossRefGoogle Scholar
Rayleigh, L. 1883 On the crispations of fluid resting upon a vibrating support. Phil. Mag. 16 (5), 5058.CrossRefGoogle 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.CrossRefGoogle 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.CrossRefGoogle Scholar
Troyon, F. & Gruber, R. 1971 Theory of the dynamic stabilization of the Rayleigh–Taylor instability. Phys. Fluids 14, 2069.CrossRefGoogle Scholar
Westra, M., Binks, D. & Water, W. 2003 Patterns of Faraday waves. J. Fluid Mech. 496, 132.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
1
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Faraday instability of a liquid layer on a lubrication film
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Faraday instability of a liquid layer on a lubrication film
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Faraday instability of a liquid layer on a lubrication film
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *