Hostname: page-component-77c89778f8-vsgnj Total loading time: 0 Render date: 2024-07-19T22:11:41.088Z Has data issue: false hasContentIssue false
  • English
  • Français

Elastohydrodynamics of a pre-stretched finite elastic sheet lubricated by a thin viscous film with application to microfluidic soft actuators

Published online by Cambridge University Press:  14 January 2019

Evgeniy Boyko Open the ORCID record for Evgeniy Boyko [Opens in a new window] ,
Ran Eshel ,
Khaled Gommed ,
Amir D. Gat Open the ORCID record for Amir D. Gat [Opens in a new window]  and
Moran Bercovici Open the ORCID record for Moran Bercovici [Opens in a new window]
Evgeniy Boyko
Affiliation:
Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa 3200003, Israel
Ran Eshel
Affiliation:
Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa 3200003, Israel
Khaled Gommed
Affiliation:
Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa 3200003, Israel
Amir D. Gat*
Affiliation:
Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa 3200003, Israel
Moran Bercovici*
Affiliation:
Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa 3200003, Israel Department of Mechanical Engineering, The University of Texas at Austin, Austin, Texas, TX 78712, USA
*
†Email addresses for correspondence: amirgat@technion.ac.il, mberco@technion.ac.il
†Email addresses for correspondence: amirgat@technion.ac.il, mberco@technion.ac.il
Get access Rights & Permissions [Opens in a new window]

Abstract

The interaction of a thin viscous film with an elastic sheet results in coupling of pressure and deformation, which can be utilized as an actuation mechanism for surface deformations in a wide range of applications, including microfluidics, optics and soft robotics. Implementation of such configurations inherently takes place over finite domains and often requires some pre-stretching of the sheet. Under the assumptions of strong pre-stretching and small deformations of the lubricated elastic sheet, we use the linearized Reynolds and Föppl–von Kármán equations to derive closed-form analytical solutions describing the deformation in a finite domain due to external forces, accounting for both bending and tension effects. We provide a closed-form solution for the case of a square-shaped actuation region and present the effect of pre-stretching on the dynamics of the deformation. We further present the dependence of the deformation magnitude and time scale on the spatial wavenumber, as well as the transition between stretching- and bending-dominant regimes. We also demonstrate the effect of spatial discretization of the forcing (representing practical actuation elements) on the achievable resolution of the deformation. Extending the problem to an axisymmetric domain, we investigate the effects arising from nonlinearity of the Reynolds and Föppl–von Kármán equations and present the deformation behaviour as it becomes comparable to the initial film thickness and dependent on the induced tension. These results set the theoretical foundation for implementation of microfluidic soft actuators based on elastohydrodynanmics.

JFM classification

Low-Reynolds-number flows: Hele-Shaw flows Micro-/Nano-fluid dynamics: MEMS/NEMS Micro-/Nano-fluid dynamics: Microfluidics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 862 , 10 March 2019 , pp. 732 - 752
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

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
Arutkin, M., Ledesma-Alonso, R., Salez, T. & Raphaël, E. 2017 Elastohydrodynamic wake and wave resistance. J. Fluid Mech. 829, 538–550.Google Scholar
Bjorck, A. 1996 Numerical Methods for Least Squares Problems. SIAM.Google Scholar
Christov, I. C., Cognet, V., Shidhore, T. C. & Stone, H. A. 2018 Flow rate–pressure drop relation for deformable shallow microfluidic channels. J. Fluid Mech. 841, 267–286.Google Scholar
Chronis, N., Liu, G. L., Jeong, K. H. & Lee, L. P. 2003 Tunable liquid-filled microlens array integrated with microfluidic network. Opt. Express 11 (19), 2370–2378.Google Scholar
Dendukuri, D., Gu, S. S., Pregibon, D. C., Hatton, T. A. & Doyle, P. S. 2007 Stop-flow lithography in a microfluidic device. Lab on a Chip 7 (7), 818–828.Google Scholar
Gervais, T., El-Ali, J., Günther, A. & Jensen, K. F. 2006 Flow-induced deformation of shallow microfluidic channels. Lab on a Chip 6 (4), 500–507.Google Scholar
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121–147.Google Scholar
Grover, W. H., Ivester, R. H. C., Jensen, E. C. & Mathies, R. A. 2006 Development and multiplexed control of latching pneumatic valves using microfluidic logical structures. Lab on a Chip 6 (5), 623–631.Google Scholar
Hardy, B. S., Uechi, K., Zhen, J. & Kavehpour, H. P. 2009 The deformation of flexible PDMS microchannels under a pressure driven flow. Lab on a Chip 9 (7), 935–938.Google Scholar
Hewitt, I. J., Balmforth, N. J. & De Bruyn, J. R. 2015 Elastic-plated gravity currents. Eur. J. Appl. Maths 26 (01), 1–31.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
Howell, P., Kozyreff, G. & Ockendon, J. 2009 Applied Solid Mechanics. Cambridge University Press.Google Scholar
Hunter, R. J. 2000 Foundations of Colloid Science. Oxford University Press.Google Scholar
Jeong, K. H., Liu, G. L., Chronis, N. & Lee, L. P. 2004 Tunable microdoublet lens array. Opt. Express 12 (11), 2494–2500.Google Scholar
Kim, S., Laschi, C. & Trimmer, B. 2013 Soft robotics: a bioinspired evolution in robotics. Trends Biotechnol. 31 (5), 287–294.Google Scholar
Kodio, O., Griffiths, I. M. & Vella, D. 2017 Lubricated wrinkles: Imposed constraints affect the dynamics of wrinkle coarsening. Phys. Rev. Fluids 2 (1), 014202.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Theory of Elasticity. Pergamon.Google Scholar
Leal, L. G. 2007 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press.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.Google Scholar
Mukherjee, U., Chakraborty, J. & Chakraborty, S. 2013 Relaxation characteristics of a compliant microfluidic channel under electroosmotic flow. Soft Matt. 9 (5), 1562–1569.Google Scholar
Panda, P., Yuet, K. P., Dendukuri, D., Hatton, T. A. & Doyle, P. S. 2009 Temporal response of an initially deflected PDMS channel. New J. Phys. 11 (11), 115001.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.Google Scholar
Pihler-Puzović, D., Juel, A. & Heil, M. 2014 The interaction between viscous fingering and wrinkling in elastic-walled Hele-Shaw cells. Phys. Fluids 26 (2), 022102.Google Scholar
Pihler-Puzović, D., Périllat, R., Russell, M., Juel, A. & Heil, M. 2013 Modelling the suppression of viscous fingering in elastic-walled Hele-Shaw cells. J. Fluid Mech. 731, 162–183.Google Scholar
Prosperetti, A. 2011 Advanced Mathematics for Applications. Cambridge University Press.Google Scholar
Rubin, S., Tulchinky, A., Gat, A. D. & Bercovici, M. 2017 Elastic deformations driven by non-uniform lubrication flows. J. Fluid Mech. 812, 841–865.Google Scholar
Shepherd, R. F., Ilievski, F., Choi, W., Morin, S. A., Stokes, A. A., Mazzeo, A. D., Chen, X., Wang, M. & Whitesides, G. M. 2011 Multigait soft robot. Proc. Natl Acad. Sci. 108 (51), 20400–20403.Google Scholar
Thorsen, T., Maerkl, S. J. & Quake, S. R. 2002 Microfluidic large-scale integration. Science 298 (5593), 580–584.Google Scholar
Timoshenko, S. & Woinkowsky-Krieger, S. 1987 Theory of Plates and Shells. McGraw-Hill.Google Scholar
Tulchinsky, A. & Gat, A. D. 2016 Transient dynamics of an elastic Hele-Shaw cell due to external forces with application to impact 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.Google Scholar
Zheng, Z., Griffiths, I. M. & Stone, H. A. 2015 Propagation of a viscous thin film over an elastic membrane. J. Fluid Mech. 784, 443–464.Google Scholar

Boyko et al. supplementary movie 1

The evolution of pressure and deformation fields, resulting from two square-shaped regions with opposite signs of zeta potential, subjected to an electric field suddenly applied at t=0.

Download Boyko et al. supplementary movie 1(Video)
Video 9.4 MB

Boyko et al. supplementary movie 2

The steady-state oscillations of pressure and deformation fields, resulting from a single square region with a fixed zeta potential, subjected to an oscillating electric field.

Download Boyko et al. supplementary movie 2(Video)
Video 9.3 MB
Supplementary material: PDF

Boyko et al. supplementary material

Supplementary material

Download Boyko et al. supplementary material(PDF)
PDF 1.7 MB