Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-10T17:46:34.171Z Has data issue: false hasContentIssue false
  • English
  • Français

Axial creeping flow in the gap between a rigid cylinder and a concentric elastic tube

Published online by Cambridge University Press:  10 October 2016

S. B. Elbaz  and
A. D. Gat
S. B. Elbaz
Affiliation:
Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel
A. D. Gat*
Affiliation:
Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Haifa 32000, Israel
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We examine transient axial creeping flow in the annular gap between a rigid cylinder and a concentric elastic tube. The gap is initially filled with a thin fluid layer. We employ an elastic shell model and the lubrication approximation to obtain governing equations for the elastohydrodynamic interaction. At long axial length scales viscous forces are balanced by elastic tension, while at shorter length scales the viscous–elastic balance is achieved by means of an interplay between elastic bending, tension and shear stresses. Based on a viscous gravity current analogy in the tensile–viscous regime, we devise propagation laws for displacement flows which are induced by a variety of boundary conditions and examine different limits of the prewetting thickness. Next we focus on the moving elastohydrodynamic contact line at the edge of a penetrating film. A uniform matched asymptotic solution connecting the interior tension-based region with a boundary layer region near the propagation front is presented. Finally, a constructive example is shown in which isolated moving deformation patterns are created and superimposed to form a travelling wave displacement field. The presented interaction between viscosity and elasticity may be applied to fields such as soft robotics and micro-scale or larger swimmers by allowing for the time-dependent control of an axisymmetric compliant boundary.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 806 , 10 November 2016 , pp. 580 - 602
Check for updates
Copyright
© 2016 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
Balmforth, N. J., Craster, R. V. & Hewitt, I. J. 2015 The speed of an inclined ruck. Proc. R. Soc. Lond. A 471, 20140740.Google Scholar
Barenblatt, G. I. 1952 On some unsteady fluid and gas motions in a porous medium. Prikl. Mat. Mekh. 16 (1), 67–78; in Russian.Google Scholar
Buckmaster, J. 1977 Viscous sheets advancing over dry beds. J. Fluid Mech. 81 (04), 735–756.Google Scholar
Carlson, A., Mandre, S. & Mahadevan, L.2015 Elastohydrodynamics of contact in adherent sheets. arXiv:1508.06234.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), 187–197.CrossRefGoogle ScholarPubMed
Dugdale, D. S. & Ruiz, C. 1971 Elasticity for Engineers. McGraw-Hill.Google Scholar
Elbaz, S. B. & Gat, A. D. 2014 Dynamics of viscous liquid within a closed elastic cylinder subject to external forces with application to soft robotics. J. Fluid Mech. 758, 221–237.Google Scholar
Gad-el Hak, M. 2002 Compliant coatings for drag reduction. Prog. Aerosp. Sci. 38 (1), 77–99.Google Scholar
Halpern, D. & Grotberg, J. B. 1992 Fluid-elastic instabilities of liquid-lined flexible tubes. J. Fluid Mech. 244, 615–632.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. D., Robinson, J. & Stone, H. A. 2013 Gravity-driven thin-film flow on a flexible substrate. J. Fluid Mech. 732, 190–213.Google Scholar
Huppert, H. E. 1982 The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58.Google Scholar
Lister, J. R. & Kerr, R. C. 1989 The propagation of two-dimensional and axisymmetric viscous gravity currents at a fluid interface. J. Fluid Mech. 203, 215–249.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
Love, A. E. H. 1888 The small free vibrations and deformations of a thin elastic shell. Phil. Trans. R. Soc. Lond. A 179, 491–546.Google Scholar
Marchese, A. D. & Rus, D. 2015 Design, kinematics, and control of a soft spatial fluidic elastomer manipulator. Intl J. Robotics Res. doi:10.1177/0278364915587925.Google Scholar
McEwan, A. D. & Taylor, G. I. 1966 The peeling of a flexible strip attached by a viscous adhesive. J. Fluid Mech. 26 (01), 1–15.CrossRefGoogle Scholar
Mollmann, H. 1981 Introduction to the Theory of Thin Shells. Wiley.Google Scholar
Momoniat, E. 2006 Axisymmetric spreading of a thin drop under gravity and time-dependent non-uniform surface tension. J. Math. Anal. Appl. 322 (1), 41–50.Google Scholar
Paidoussis, M. P. 1998 Fluid-Structure Interactions: Slender Structures and Axial Flow, vol. 1. Academic.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
Rus, D. & Tolley, M. T. 2015 Design, fabrication and control of soft robots. Nature 521 (7553), 467–475.Google Scholar
Setter, E., Bucher, I. & Haber, S. 2012 Low-Reynolds-number swimmer utilizing surface traveling waves: analytical and experimental study. Phys. Rev. E 85 (6), 066304.Google Scholar
Tolley, M. T., Shepherd, R. F., Mosadegh, B., Galloway, K. C., Wehner, M., Karpelson, M., Wood, R. J. & Whitesides, G. M. 2014 A resilient, untethered soft robot. Soft Robot. 1 (3), 213–223.Google Scholar
Toppaladoddi, S. & Balmforth, N. J. 2014 Slender axisymmetric stokesian swimmers. J. Fluid Mech. 746, 273–299.CrossRefGoogle Scholar
Trinh, P. H., Wilson, S. K. & Stone, H. A. 2014a A pinned or free-floating rigid plate on a thin viscous film. J. Fluid Mech. 760, 407–430.Google Scholar
Trinh, P. H., Wilson, S. K. & Stone, H. A.2014b An elastic plate on a thin viscous film. arXiv:1410.8558.Google Scholar
White, J. P. & Heil, M. 2005 Three-dimensional instabilities of liquid-lined elastic tubes: a thin-film fluid-structure interaction model. Phys. Fluids 17 (3), 031506.Google Scholar
Zel’dovich, Y. B. & Kompaneets, A. S. 1950 Towards a theory of heat conduction with thermal conductivity depending on the temperature. In Collection of Papers Dedicated to 70th Anniversary of A. F. Ioffe, vol. 1, pp. 61–72. Izvestiya Akademii Nauk SSSR.Google Scholar