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Transient dynamics of an elastic Hele-Shaw cell due to external forces with application to impact mitigation

Published online by Cambridge University Press:  12 July 2016

A. Tulchinsky
Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa 3200003, Israel
A. D. Gat*
Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa 3200003, Israel
Email address for correspondence:


We study the transient dynamics of a viscous liquid contained in a narrow gap between a rigid surface and a parallel elastic plate. The elastic plate is deformed due to an externally applied time-varying pressure field. We model the flow field via the lubrication approximation and the plate deformation by the Kirchhoff–Love plate theory. We obtain a self-similarity solution for the case of an external point force acting on the elastic plate. The pressure and deformation field during and after the application of the external force are derived and presented by closed-form expressions. We examine a distributed external pressure, spatially uniform and linearly increasing with time, acting on the elastic plate over a finite region and during a finite time period, similar to the viscous–elastic interaction time-scale. The interaction between elasticity and viscosity is shown to reduce by an order of magnitude the pressure within the Hele-Shaw cell compared with the externally applied pressure. The results thus suggest that elastic Hele-Shaw configurations may be used to achieve significant impact mitigation.

© 2016 Cambridge University Press 

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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.CrossRefGoogle ScholarPubMed
Bailey, W. N. 1972 Generalized Hypergeometric Series. Hafner.Google Scholar
Bateman, H. & Erdelyi, A. 1953 Higher Transcendental Functions, vol. 1. McGraw-Hill.Google Scholar
Chauhan, A. & Radke, C. J. 2002 Settling and deformation of a thin elastic shell on a thin fluid layer lying on a solid surface. J. Colloid Interface Sci. 245 (1), 187197.CrossRefGoogle ScholarPubMed
Duchemin, L. & Vandenberghe, N. 2014 Impact dynamics for a floating elastic membrane. J. Fluid Mech. 756, 544554.CrossRefGoogle Scholar
Fischer, C., Braun, S. A., Bourban, P. E., Michaud, V. & Plummer, C. J. G. 2006 Dynamic properties of sandwich structures with integrated shear-thickening fluids. Smart Mater. Struct. 15 (5), 1467.CrossRefGoogle Scholar
Flitton, J. C. & King, J. R. 2004 Moving-boundary and fixed-domain problems for a sixth-order thin-film equation. Eur. J. Appl. Maths 15 (06), 713754.CrossRefGoogle Scholar
Han, Z., Tao, C., Zhou, D., Sun, Y., Zhou, C., Ren, Q. & Roberts, C. J. 2014 Air puff induced corneal vibrations: theoretical simulations and clinical observations. J. Refract. Surg. 30 (3), 208213.CrossRefGoogle ScholarPubMed
Hosoi, A. E. & Mahadevan, L. 2004 Peeling, healing, and bursting in a lubricated elastic sheet. Phys. Rev. Lett. 93 (13), 137802.CrossRefGoogle Scholar
Lee, B.-W. & Kim, C.-G. 2012 Computational analysis of shear thickening fluid impregnated fabrics subjected to ballistic impacts. Adv. Compos. Mater. 21 (2), 177192.CrossRefGoogle Scholar
Lee, Y. S., Wetzel, E. D. & Wagner, N. J. 2003 The ballistic impact characteristics of Kevlar® woven fabrics impregnated with a colloidal shear thickening fluid. J. Mater. Sci. 38 (13), 28252833.CrossRefGoogle 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.CrossRefGoogle Scholar
Luke, Y. L. 1969 The Special Functions and their Approximations. vol. 53. Academic.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, 512547.CrossRefGoogle 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, 487511.CrossRefGoogle Scholar
Reddy, J. N. 2006 Theory and Analysis of Elastic Plates and Shells. CRC Press.Google Scholar
Satsanit, W. & Kananthai, A. 2009 The operator and its spectrum related to heat equation. Intl J. Pure Appl. Maths 54 (1), 141152.Google Scholar
Slater, L. J. 1966 Generalized Hypergeometric Functions. Cambridge University Press.Google Scholar
Tan, Z. H., Zuo, L., Li, W. H., Liu, L. S. & Zhai, P. C. 2016 Dynamic response of symmetrical and asymmetrical sandwich plates with shear thickening fluid core subjected to penetration loading. Mater. Design 94, 105110.CrossRefGoogle Scholar
Timoshenko, S. P. & Woinowsky-Krieger, S. 1959 Theory of Plates and Shells. McGraw-Hill.Google Scholar
Trinh, P. H., Wilson, S. K. & Stone, H. A. 2014 A pinned or free-floating rigid plate on a thin viscous film. J. Fluid Mech. 760, 407430.CrossRefGoogle Scholar
Vella, D., Huang, J., Menon, N., Russell, T. P. & Davidovitch, B. 2015 Indentation of ultrathin elastic films and the emergence of asymptotic isometry. Phys. Rev. Lett. 114 (1), 014301.CrossRefGoogle Scholar