Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-19T23:30:39.451Z Has data issue: false hasContentIssue false
  • English
  • Français

Motion of a tightly fitting axisymmetric object through a lubricated elastic tube

Published online by Cambridge University Press:  10 September 2021

Bhargav Rallabandi Open the ORCID record for Bhargav Rallabandi [Opens in a new window] ,
Jens Eggers Open the ORCID record for Jens Eggers [Opens in a new window] ,
Miguel Angel Herrada Open the ORCID record for Miguel Angel Herrada [Opens in a new window]  and
Howard A. Stone Open the ORCID record for Howard A. Stone [Opens in a new window]
Bhargav Rallabandi*
Affiliation:
Department of Mechanical Engineering, University of California, Riverside, CA 92521, USA
Jens Eggers
Affiliation:
School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol BS8 1UG, United Kingdom
Miguel Angel Herrada
Affiliation:
Departamento de Mecánica de Fluidos e Ingeniería Aeroespacial, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092, Seville, Spain
Howard A. Stone
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email address for correspondence: bhargav@engr.ucr.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the translation of a rigid, axisymmetric, tightly fitting object through a cylindrical elastic tube filled with viscous fluid, using a combination of theory and direct numerical simulations. The intruding object is assumed to be wider than the undeformed tube radius, forcing solid–solid contact in the absence of relative motion. The motion of the object establishes a thin fluid film that lubricates this contact. Our theory couples lubrication theory to a geometrically nonlinear membrane description of the tube's elasticity, and applies to a slender intruding object and a thin tube with negligible bending rigidity. We show using asymptotic and numerical solutions of the theory, that the thickness of the thin fluid film scales with the square root of the relative speed for small speeds, set by a balance of hoop stresses, membrane tension and fluid pressure. While membrane tension is relatively small at the entrance of the film, it dominates near the exit and produces undulations of the film thickness, even in the limit of vanishing speeds and slender objects. We find that the drag force on the intruding object depends on the slope of its surface at the entrance to the thin fluid film, and scales as the square root of the relative speed. The predictions of the lubricated membrane theory for the shape of the film and the force on the intruder are in quantitative agreement with three-dimensional direct numerical simulations of the coupled fluid–elastic problem.

JFM classification

Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 926 , 10 November 2021 , A27
Check for updates
Copyright
© The Author(s), 2021. Published by 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

REFERENCES

Audoly, B. & Pomeau, Y. 2010 Elasticity and Geometry: From Hair Curls to the Non-Linear Response Of Shells. Oxford University Press.Google Scholar
Barakat, J.M. & Shaqfeh, E.S.G. 2018 a The steady motion of a closely fitting vesicle in a tube. J. Fluid Mech. 835, 721–761.CrossRefGoogle Scholar
Barakat, J.M. & Shaqfeh, E.S.G. 2018 b Stokes flow of vesicles in a circular tube. J. Fluid Mech. 851, 606–635.CrossRefGoogle Scholar
Batchelor, G.K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Beaucourt, J., Biben, T. & Misbah, C. 2004 Optimal lift force on vesicles near a compressible substrate. Europhys. Lett. 67 (4), 676.CrossRefGoogle Scholar
Bradfield, J.R.G. 1951 Radiographic studies on the formation of the hen's egg shell. J. Expl Biol. 28, 125–140.CrossRefGoogle ScholarPubMed
Bretherton, F.P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166–188.CrossRefGoogle Scholar
Budiansky, B. 1968 Notes on nonlinear shell theory. Trans. ASME J. Appl. Mech. 35 (2), 393–401.CrossRefGoogle Scholar
Chatkoff, M.L. 1975 A biophysicist's view of ovum transport. Gynecol. Obstet. Invest. 6 (3–4), 105–122.CrossRefGoogle ScholarPubMed
Davies, H.S., Débarre, D., El Amri, N., Verdier, C., Richter, R.P. & Bureau, L. 2018 Elastohydrodynamic lift at a soft wall. Phys. Rev. Lett. 120 (19), 198001.CrossRefGoogle Scholar
Dimakopoulos, Y. & Tsamopoulos, J. 2003 A quasi-elliptic transformation for moving boundary problems with large anisotropic deformations. J. Comput. Phys. 192 (2), 494–522.CrossRefGoogle Scholar
Dowson, D. & Higginson, G.R. 1959 A numerical solution to the elasto-hydrodynamic problem. J. Mech. Engng Sci. 1 (1), 6–15.CrossRefGoogle Scholar
Dowson, D. & Higginson, G.R. 1966 Elastohydrodynamic Lubrication, the Fundamentals of Roller and Gear Lubrication. Pergamon Press.Google Scholar
Essink, M.H., Pandey, A., Karpitschka, S., Venner, C.H. & Snoeijer, J.H. 2021 Regimes of soft lubrication. J. Fluid Mech. 915, A49.CrossRefGoogle ScholarPubMed
Fitz-Gerald, J.M. 1969 Mechanics of red-cell motion through very narrow capillaries. Proc. R. Soc. Lond. B 174 (1035), 193–227.Google ScholarPubMed
Freund, J.B. 2014 Numerical simulation of flowing blood cells. Annu. Rev. Fluid Mech. 46, 67–95.CrossRefGoogle Scholar
Greenwood, J.A. 2020 Elastohydrodynamic lubrication. Lubricants 8 (5), 51.CrossRefGoogle Scholar
Herrada, M.A. & Montanero, J.M. 2016 A numerical method to study the dynamics of capillary fluid systems. J. Comput. Phys. 306, 137–147.CrossRefGoogle Scholar
Holmes, D.P. 2019 Elasticity and stability of shape-shifting structures. Curr. Opin. Colloid Interface Sci. 40, 118–137.CrossRefGoogle Scholar
Johnson, K.L. 1970 Regimes of elastohydrodynamic lubrication. J. Mech. Engng Sci. 12 (1), 9–16.CrossRefGoogle Scholar
Kamrin, K., Rycroft, C.H. & Nave, J.C. 2012 Reference map technique for finite-strain elasticity and fluid-solid interaction. J. Mech. Phys. Solids 60 (11), 1952–1969.CrossRefGoogle Scholar
Koiter, W.T. 1966 On the nonlinear theory of thin elastic shells. Proc. Koninkl. Ned. Akad. van Wetenschappen B 69, 1–54.Google Scholar
Landau, L.D. & Lifshitz, E.M. 1984 Elasticity. Pergamon.Google Scholar
Lighthill, M.J. 1968 Pressure-forcing of tightly fitting pellets along fluid-filled elastic tubes. J. Fluid Mech. 34, 113–143.CrossRefGoogle Scholar
Moin, P. 2010 Fundamentals of Engineering Numerical Analysis. Cambridge University Press.CrossRefGoogle Scholar
Rallabandi, B., Marthelot, J., Jambon-Puillet, E., Brun, P.-T. & Eggers, J. 2019 Curvature regularization near contacts with stretched elastic tubes. Phys. Rev. Lett. 123, 168002.CrossRefGoogle ScholarPubMed
Rallabandi, B., Oppenheimer, N., Zion, M.B.Z. & Stone, H.A. 2018 Membrane-induced hydroelastic migration of a particle surfing its own wave. Nat. Phys. 14 (12), 1211–1215.CrossRefGoogle Scholar
Rallabandi, B., Saintyves, B., Jules, T., Salez, T., Schönecker, C., Mahadevan, L. & Stone, H.A. 2017 Rotation of an immersed cylinder sliding near a thin elastic coating. Phys. Rev. Fluids 2 (7), 074102.CrossRefGoogle Scholar
Saintyves, B., Jules, T., Salez, T. & Mahadevan, L. 2016 Self-sustained lift and low friction via soft lubrication. Proc. Natl Acad. Sci. 113 (21), 5847–5849.CrossRefGoogle ScholarPubMed
Saintyves, B., Rallabandi, B., Jules, T., Ault, J., Salez, T., Schönecker, C., Stone, H.A. & Mahadevan, L. 2020 Rotation of a submerged finite cylinder moving down a soft incline. Soft Matt. 16 (16), 4000–4007.CrossRefGoogle Scholar
Salamon, A. & Kent, J.P. 2014 Orientation of the egg at laying - is the pointed or the blunt end first? J. Poultry Sci. 13, 316–318.CrossRefGoogle Scholar
Sanders, J.L. Jr. 1963 Nonlinear theories for thin shells. Q. J. Appl. Maths 21 (1), 21–36.CrossRefGoogle Scholar
Secomb, T.W., Skalak, R., Özkaya, N. & Gross, J.F. 1986 Flow of axisymmetric red blood cells in narrow capillaries. J. Fluid Mech. 163, 405–423.CrossRefGoogle Scholar
Sekimoto, K. & Leibler, L. 1993 A mechanism for shear thickening of polymer-bearing surfaces: elasto-hydrodynamic coupling. Europhys. Lett. 23 (2), 113–117.CrossRefGoogle Scholar
Skotheim, J.M. & Mahadevan, L. 2004 Soft lubrication. Phys. Rev. Lett. 92, 245509.CrossRefGoogle ScholarPubMed
Snoeijer, J.H., Eggers, J. & Venner, C.H. 2013 Similarity theory of lubricated Hertzian contacts. Phys. Fluids 25, 101705.CrossRefGoogle Scholar
Snoeijer, J.H., Pandey, A., Herrada, M.A. & Eggers, J. 2020 The relationship between viscoelasticity and elasticity. Proc. R. Soc. A 476, 20200419.CrossRefGoogle ScholarPubMed
Suo, Z. 2013 Finite deformation: general theory. See https://imechanica.org/node/538.Google Scholar
Szeri, A.Z. 2010 Fluid Film Lubrication. Cambridge University Press.CrossRefGoogle Scholar
Takagi, D. & Balmforth, N.J. 2011 Peristaltic pumping of rigid objects in an elastic tube. J. Fluid Mech. 672, 219–244.CrossRefGoogle Scholar
Tani, M., Cambau, T., Bico, J. & Reyssat, E. 2017 Motion of a rigid sphere through an elastic tube with a lubrication film. Bull. Am. Phys. Soc. 62, L20-012.Google Scholar
Urzay, J., Llewellyn Smith, S.G. & Glover, B.J. 2007 The elastohydrodynamic force on a sphere near a soft wall. Phys. Fluids 19, 103106.CrossRefGoogle Scholar
Vann, P.G. & Fitz-Gerald, J.M. 1982 Flow mechanics of red cell trains in very narrow capillaries. I. Trains of uniform cells. Microvasc. Res. 24 (3), 296–313.CrossRefGoogle ScholarPubMed
Vlahovska, P.M., Podgorski, T. & Misbah, C. 2009 Vesicles and red blood cells in flow: from individual dynamics to rheology. C. R. Phys. 10 (8), 775–789.CrossRefGoogle Scholar
Vurgaft, A., Elbaz, S.B. & Gat, A.D. 2019 Forced motion of a cylinder within a liquid-filled elastic tube–a model of minimally invasive medical procedures. J. Fluid Mech. 881, 1048–1072.CrossRefGoogle Scholar