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Dynamics of viscous liquid within a closed elastic cylinder subject to external forces with application to soft robotics

  • S. B. Elbaz (a1) and A. D. Gat (a1)


Viscous flows in contact with elastic structures apply both pressure and shear stress at the solid–liquid interface and thus create internal stress and deformation fields within the solid structure. We study the interaction between the deformation of elastic structures, subject to external forces, and an internal viscous liquid. We neglect inertia in the liquid and solid and focus on viscous flow through a thin-walled slender elastic cylindrical shell as a basic model of a soft robot. Our analysis yields an inhomogeneous linear diffusion equation governing the coupled viscous–elastic system. Solutions for the flow and deformation fields are obtained in closed analytical form. The functionality of the viscous–elastic diffusion process is explored within the context of soft-robotic applications, through analysis of selected solutions to the governing equation. Shell material compressibility is shown to have a unique effect in inducing different flow and deformation regimes. This research may prove valuable to applications such as micro-swimmers, micro-autonomous systems and soft robotics by allowing for the design and control of complex time-varying deformation fields.


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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.
Antkowiak, A., Audoly, B., Josserand, C., Neukirch, S. & Rivetti, M. 2011 Instant fabrication and selection of folded structures using drop impact. Proc. Natl Acad. Sci. USA 108, 1040010404.
Canic, S. & Mikelic, A. 2003 Effective equations modeling the flow of a viscous incompressible fluid through a long elastic tube arising in the study of blood flow through small arteries. SIAM J. Appl. Dyn. Syst. 2 (3), 431463.
Chandra, D., Yang, S., Soshinsky, A. A. & Gambogi, R. J. 2009 Biomimetic ultrathin whitening by capillary-force-induced random clustering of hydrogel micropillar arrays. ACS Appl. Mater. Interfaces 1, 16981704.
De Volder, M. F. L., Park, S. J., Tawfick, S. H., Vidaud, D. O. & Hart, A. J. 2011 Fabrication and electrical integration of robust carbon nanotube micropillars by self-directed elastocapillary densification. J. Micromech. Microengng 21, 045033.
Dugdale, D. S. & Ruiz, C. 1971 Elasticity for Engineers. McGraw-Hill.
Duprat, C., Aristoff, J. M. & Stone, H. A. 2011 Dynamics of elastocapillary rise. J. Fluid Mech. 679, 641654.
Elwenspoek, M., Abelmann, L., Berenschot, E., van Honschoten, J., Jansen, H. & Tas, N. 2010 Self-assembly of (sub-)micron particles into supermaterials. J. Micromech. Microengng 20, 064001.
Gat, A. D. & Gharib, M. 2013 Elasto-capillary coalescence of multiple parallel sheets. J. Fluid Mech. 723, 692705.
Gibson, J. E. 1965 Linear Elastic Theory of Thin Shells. Pergamon Press.
Heil, M. 1996 The stability of cylindrical shells conveying viscous flow. J. Fluids Struct. 10, 173196.
Heil, M. 1997 Stokes flow in collapsible tubes – computation and experiment. J. Fluid Mech. 353, 285312.
Heil, M. 1998 Stokes flow in an elastic tube – a large-displacement fluid–structure interaction problem. Intl J. Numer. Meth. Fluids 28, 243265.
Heil, M. & Pedley, T. J. 1995 Large axisymmetric deformations of a cylindrical shell conveying a viscous flow. J. Fluids Struct. 9, 237256.
Huang, X., Zhou, J., Sansom, E., Gharib, M. & Haur, S. C. 2007 Inherent opening controlled pattern formation in carbon nanotube arrays. Nanotechnology 18, 305301.
Ilievski, F., Mazzeo, A. D., Shepherd, R. F., Chen, X. & Whitesides, G. M. 2011 Soft robotics for chemists. Angew. Chem. 123 (8), 19301935.
Kang, S. H., Wu, N., Grinthal, A. & Aizenberg, J. 2011 Meniscus lithography: evaporation-induced self-organization of pillar arrays into Moiré patterns. Phys. Rev. Lett. 107, 177802.
Love, A. E. H. 1888 The small free vibrations and deformations of a thin elastic shell. Phil. Trans. R. Soc. Lond. A 179, 491546.
Lowe, T. W. & Pedley, T. J. 1995 Computation of Stokes flow in a channel with a collapsible segment. J. Fluids Struct. 9 (8), 885905.
Marchese, A. D., Onal, C. D. & Rus, D. 2014 Autonomous soft robotic fish capable of escape maneuvers using fluidic elastomer actuators. Soft Robot. 1 (1), 7587.
Martinez, R. V., Branch, J. L., Fish, C. R., Jin, L., Shepherd, R. F., Nunes, R., Suo, Z. & Whitesides, G. M. 2013 Robotic tentacles with three-dimensional mobility based on flexible elastomers. Adv. Mater. 25 (2), 205212.
Martinez, R. V., Fish, C. R., Chen, X. & Whitesides, G. M. 2012 Elastomeric origami: programmable paper-elastomer composites as pneumatic actuators. Adv. Funct. Mater. 22 (7), 13761384.
Mollmann, H. 1981 Introduction to the Theory of Thin Shells. John Wiley & Sons.
Morin, S. A., Shepherd, R. F., Kwok, S. W., Stokes, A. A., Nemiroski, A. & Whitesides, G. M. 2012 Camouflage and display for soft machines. Science 337 (6096), 828832.
Païdoussis, M. P. 1998 Fluid–Structure Interactions, Slender Structures and Axial Flow. Academic Press.
Pineirua, M., Bico, J. & Roman, B. 2010 Capillary origami controlled by an electric field. Soft Matt. 6, 44914496.
Pokroy, B., Kang, S. H., Mahadevan, L. & Aizenberg, J. 2009 Self-organization of a mesoscale bristle into ordered, hierarchical helical assemblies. Science 323, 237240.
Py, C., Bastien, R., Bico, J., Roman, B. & Boudaoud, A. 2007 3D aggregation of wet fibers. Europhys. Lett. 77, 44005.
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. USA 108 (51), 2040020403.
Shepherd, R. F., Stokes, A. A., Freake, J., Barber, J., Snyder, P. W., Mazzeo, A. D., Cademartiri, L., Morin, S. A. & Whitesides, G. M. 2013 Using explosions to power a soft robot. Angew. Chem. 125 (10), 29642968.
Steltz, E., Mozeika, A., Rodenberg, N., Brown, E. & Jaeger, H. M. 2009 JSEL: jamming skin enabled locomotion. In IEEE/RSJ International Conference Intelligent Robots and Systems, IROS 2009, pp. 56725677. IEEE.
Stokes, A. A., Shepherd, R. F., Morin, S. A., Ilievski, F. & Whitesides, G. M. 2013 A hybrid combining hard and soft robots. Soft Robot. 1 (P), 7074.
Toppaladoddi, S. & Balmforth, N. J. 2014 Slender axisymmetric stokesian swimmers. J. Fluid Mech. 746, 273299.
Zhao, Z., Tawfick, S. H., Park, S. J., De Volder, M., Hart, A. J. & Lu, W. 2010 Bending of nanoscale filament assemblies by elastocapillary densification. Phys. Rev. E 82, 041605.
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Dynamics of viscous liquid within a closed elastic cylinder subject to external forces with application to soft robotics

  • S. B. Elbaz (a1) and A. D. Gat (a1)


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