Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-12-01T02:22:24.408Z Has data issue: false hasContentIssue false
  • English
  • Français

Elastic deformations driven by non-uniform lubrication flows

Published online by Cambridge University Press:  05 January 2017

Shimon Rubin ,
Arie Tulchinsky ,
Amir D. Gat  and
Moran Bercovici
Shimon Rubin
Affiliation:
Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa, Israel
Arie Tulchinsky
Affiliation:
Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa, Israel
Amir D. Gat*
Affiliation:
Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa, Israel
Moran Bercovici*
Affiliation:
Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa, Israel
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The ability to create dynamic deformations of micron-sized structures is relevant to a wide variety of applications such as adaptable optics, soft robotics and reconfigurable microfluidic devices. In this work, we examine non-uniform lubrication flow as a mechanism to create complex deformation fields in an elastic plate. We consider a Kirchhoff–Love elasticity model for the plate and Hele-Shaw flow in a narrow gap between the plate and a parallel rigid surface. Based on linearization of the Reynolds equation, we obtain a governing equation which relates elastic deformations to gradients in non-homogeneous physical properties of the fluid (e.g. body forces, viscosity and slip velocity). We then focus on a specific case of non-uniform Helmholtz–Smoluchowski electro-osmotic slip velocity, and provide a method for determining the zeta-potential distribution necessary to generate arbitrary static and quasi-static deformations of the elastic plate. Extending the problem to time-dependent solutions, we analyse transient effects on asymptotically static solutions, and finally provide a closed form solution for a Green’s function for time periodic actuations.

JFM classification

Low-Reynolds-number flows: Hele-Shaw flows Micro-/Nano-fluid dynamics: Microfluidics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 812 , 10 February 2017 , pp. 841 - 865
Check for updates
Copyright
© 2017 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

Ajdari, A. 1995 Electro-osmosis on inhomogeneously charged surfaces. Phys. Rev. Lett. 75 (4), 755–759.CrossRefGoogle ScholarPubMed
Ajdari, A. 1996 Generation of transverse fluid currents and forces by an electric field: electro-osmosis on charge-modulated and undulated surfaces. Phys. Rev. E 53 (5), 4996.Google Scholar
Al-Housseiny, T. T., Christov, I. C. & Stone, H. A. 2013 Two-phase fluid displacement and interfacial instabilities under elastic membranes. Phys. Rev. Lett. 111 (3), 034502.Google Scholar
Baudry, J., Charlaix, E., Tonck, A. & Mazuyer, D. 2001 Experimental evidence for a large slip effect at a nonwetting fluid–solid interface. Langmuir 17 (17), 5232–5236.Google Scholar
Bazant, M. Z., Kilic, M. S., Storey, B. D. & Ajdari, A. 2009 Nonlinear electrokinetics at large voltages. New J. Phys. 11 (7), 075016.Google Scholar
Boyko, E., Rubin, S., Gat, A. D. & Bercovici, M. 2015 Flow patterning in Hele-Shaw configurations using non-uniform electro-osmotic slip. Phys. Fluids 27 (10), 102001.Google Scholar
Brotherton, C. M. & Davis, R. H. 2004 Electroosmotic flow in channels with step changes in zeta potential and cross section. J. Colloid Interface Sci. 270 (1), 242–246.Google Scholar
Cheng, Q., Sun, Z., Meininger, G. A. & Almasri, M. 2010 Note: mechanical study of micromachined polydimethylsiloxane elastic microposts. Rev. Sci. Instrum. 81 (10), 106104.Google Scholar
Chronis, N., Liu, L. L., Jeong, K.-H. & Lee, L. P. 2003 Tunable liquid-filled microlens array integrated with microfluidic network. Opt. Express 11 (11), 2370–2378.CrossRefGoogle ScholarPubMed
Feuillebois, F., Bazant, M. Z. & Vinogradova, O. I. 2009 Effective slip over superhydrophobic surfaces in thin channels. Phys. Rev. Lett. 102 (2), 026001.Google Scholar
Gabor, D. 1946 Theory of communication. Part 1: the analysis of information. J. Inst. Electr. Engrs: Radio Commun. Engng 93 (26), 429–441.Google Scholar
Hosoi, A. E. & Mahadevan, L. 2004 Peeling, healing, and bursting in a lubricated elastic sheet. Phys. Rev. Lett. 93 (13), 137802.Google Scholar
Hunter, R. J. 2001 Foundations of Colloid Science. Oxford University Press.Google Scholar
Khair, A. S. & Squires, T. M. 2008 Surprising consequences of ion conservation in electro-osmosis over a surface charge discontinuity. J. Fluid Mech. 615, 323–334.Google Scholar
Landau, L. D., Bell, J. S., Kearsley, M. J., Pitaevskii, L. P., Lifshitz, E. M. & Sykes, J. B. 1984 Electrodynamics of Continuous Media, vol. 8. Elsevier.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1986 Theory of elasticity. Course of Theoretical Physics 3, 109.Google Scholar
Lister, J. R., Peng, G. G. & Neufeld, J. A. 2013 Viscous control of peeling an elastic sheet by bending and pulling. Phys. Rev. Lett. 111 (15), 154501.Google Scholar
Michaut, C. 2011 Dynamics of magmatic intrusions in the upper crust: theory and applications to laccoliths on Earth and the Moon. J. Geophys. Res. 116 (B5), B05205.CrossRefGoogle Scholar
Minor, M., Van der Linde, A. J., Van Leeuwen, H. P. & Lyklema, J. 1997 Dynamic aspects of electrophoresis and electroosmosis: a new fast method for measuring particle mobilities. J. Colloid Interface Sci. 189 (2), 370–375.Google Scholar
Moraes, C., Labuz, J. M., Shao, Y., Fu, J. & Takayama, S. 2015 Supersoft lithography: candy-based fabrication of soft silicone microstructures. Lab on a Chip 15 (18), 3760–3765.Google Scholar
Olver, F. W. J. 2010 NIST Handbook of Mathematical Functions. Cambridge University Press.Google Scholar
Panton, R. L. 2006 Incompressible Flow. Wiley.Google Scholar
Peng, G. G., Pihler-Puzović, D., Juel, A., Heil, M. & Lister, J. R. 2015 Displacement flows under elastic membranes. Part 2. Analysis of interfacial effects. J. Fluid Mech. 784, 512–547.Google Scholar
Pihler-Puzović, D., Illien, P., Heil, M. & Juel, A. 2012 Suppression of complex fingerlike patterns at the interface between air and a viscous fluid by elastic membranes. Phys. Rev. Lett. 108 (7), 074502.CrossRefGoogle Scholar
Pihler-Puzović, D., Juel, A., Peng, G. G., Lister, J. R. & Heil, M. 2015 Displacement flows under elastic membranes. Part 1. Experiments and direct numerical simulations. J. Fluid Mech. 784, 487–511.Google Scholar
Trinh, P. H., Wilson, S. K. & Stone, H. A.2014 An elastic plate on a thin viscous film. arXiv:1410.8558.Google Scholar
Trivedi, D., Rahn, C. D., Kier, W. M. & Walker, I. D. 2008 Soft robotics: biological inspiration, state of the art, and future research. Appl. Bionics Biomech. 5 (3), 99–117.Google Scholar
Tulchinsky, A. & Gat, A. D. 2015 Transient dynamics of elastic Hele-Shaw cell due to external forces with application to impulse mitigation. J. Fluid Mech. 800, 517–530.Google Scholar
Unger, M. A., Chou, H. P., Thorsen, T., Scherer, A. & Quake, S. R. 2000 Monolithic microfabricated valves and pumps by multilayer soft lithography. Science 288 (5463), 113–116.CrossRefGoogle ScholarPubMed
Van Der Wouden, E. J., Heuser, T., Hermes, D. C., Oosterbroek, R. E., Gardeniers, J. G. E. & Van Den Berg, A. 2005 Field-effect control of electro-osmotic flow in microfluidic networks. Colloids Surf. A 267 (1), 110–116.Google Scholar
Vinogradova, O. I. 1999 Slippage of water over hydrophobic surfaces. Intl J. Miner. Process. 56 (1), 31–60.Google Scholar
Yariv, E. 2004 Electro-osmotic flow near a surface charge discontinuity. J. Fluid Mech. 521, 181–189.Google Scholar