Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-29T10:09:00.942Z Has data issue: false hasContentIssue false
  • English
  • Français

Soft-lubrication interactions between a rigid sphere and an elastic wall

Published online by Cambridge University Press:  23 December 2021

Vincent Bertin Open the ORCID record for Vincent Bertin [Opens in a new window] ,
Yacine Amarouchene Open the ORCID record for Yacine Amarouchene [Opens in a new window] ,
Elie Raphaël Open the ORCID record for Elie Raphaël [Opens in a new window]  and
Thomas Salez Open the ORCID record for Thomas Salez [Opens in a new window]
Vincent Bertin
Affiliation:
Univ. Bordeaux, CNRS, LOMA, UMR 5798, 33405 Talence, France UMR CNRS Gulliver 7083, ESPCI Paris, PSL Research University, 75005 Paris, France
Yacine Amarouchene
Affiliation:
Univ. Bordeaux, CNRS, LOMA, UMR 5798, 33405 Talence, France
Elie Raphaël
Affiliation:
UMR CNRS Gulliver 7083, ESPCI Paris, PSL Research University, 75005 Paris, France
Thomas Salez*
Affiliation:
Univ. Bordeaux, CNRS, LOMA, UMR 5798, 33405 Talence, France Global Station for Soft Matter, Global Institution for Collaborative Research and Education, Hokkaido University, Sapporo, Hokkaido 060-0808, Japan
*
†Email address for correspondence: thomas.salez@u-bordeaux.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

The motion of an object within a viscous fluid and in the vicinity of a soft surface induces a hydrodynamic stress field that deforms the latter, thus modifying the boundary conditions of the flow. This results in elastohydrodynamic interactions experienced by the particle. Here, we derive a soft-lubrication model, in order to compute all the forces and torque applied on a rigid sphere that is free to translate and rotate near an elastic wall. We focus on the limit of small deformations of the surface with respect to the fluid-gap thickness, and perform a perturbation analysis in dimensionless compliance. The response is computed in the framework of linear elasticity, for planar elastic substrates in the limiting cases of thick and thin layers. The EHD forces are also obtained analytically using the Lorentz reciprocal theorem.

JFM classification

Low-Reynolds-number flows: Lubrication theory Complex Fluids: Colloids
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 933 , 25 February 2022 , A23
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

Abkarian, M., Lartigue, C. & Viallat, A. 2002 Tank treading and unbinding of deformable vesicles in shear flow: determination of the lift force. Phys. Rev. Lett. 88 (6), 068103.CrossRefGoogle ScholarPubMed
Abramowitz, M. & Stegun, I.A. 1964 Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, vol. 55. US Government Printing Office.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
Bertin, V., Zhang, Z., Boisgard, R., Grauby-Heywang, C., Raphael, E., Salez, T. & Maali, A. 2021 Contactless rheology of finite-size air-water interfaces. Phys. Rev. Res. 3, L032007.CrossRefGoogle Scholar
Bocquet, L. & Charlaix, E. 2010 Nanofluidics, from bulk to interfaces. Chem. Soc. Rev. 39 (3), 1073–1095.CrossRefGoogle ScholarPubMed
Bouchet, A.-S., Cazeneuve, C., Baghdadli, N., Luengo, G.S. & Drummond, C. 2015 Experimental study and modeling of boundary lubricant polyelectrolyte films. Macromolecules 48, 2244.CrossRefGoogle Scholar
Brodsky, E.E. & Kanamori, H. 2001 Elastohydrodynamic lubrication of faults. J. Geophys. Res. 106 (B8), 16357–16374.CrossRefGoogle Scholar
Campbell, C.S. 1989 Self-lubrication for long runout landslides. J. Geol. 97 (6), 653–665.CrossRefGoogle Scholar
Chan, D.Y.C., Klaseboer, E. & Manica, R. 2009 Dynamic deformations and forces in soft matter. Soft Matt. 5 (15), 2858–2861.CrossRefGoogle Scholar
Chandler, T.G.J. & Vella, D. 2020 Validity of Winkler's mattress model for thin elastomeric layers: beyond Poisson's ratio. Proc. R. Soc. A 476 (2242), 20200551.CrossRefGoogle ScholarPubMed
Chaoui, M. & Feuillebois, F. 2003 Creeping flow around a sphere in a shear flow close to a wall. Q. J. Mech. Appl. Maths 56 (3), 381–410.CrossRefGoogle Scholar
Daddi-Moussa-Ider, A., Rallabandi, B., Gekle, S. & Stone, H.A. 2018 Reciprocal theorem for the prediction of the normal force induced on a particle translating parallel to an elastic membrane. Phys. Rev. Fluids 3 (8), 084101.CrossRefGoogle Scholar
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
Davis, R.H., Serayssol, J.-M. & Hinch, E.J. 1986 The elastohydrodynamic collision of two spheres. J. Fluid Mech. 163, 479–497.CrossRefGoogle Scholar
Dowson, D. & Higginson, G.R. 2014 Elasto-Hydrodynamic Lubrication: International Series on Materials Science and Technology. Elsevier.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
Goldman, A.J., Cox, R.G. & Brenner, H. 1967 Slow viscous motion of a sphere parallel to a plane wall – II. Couette flow. Chem. Engng Sci. 22 (4), 653–660.CrossRefGoogle Scholar
Gondret, P., Lance, M. & Petit, L. 2002 Bouncing motion of spherical particles in fluids. Phys. fluids 14 (2), 643–652.CrossRefGoogle Scholar
Gong, J.P. 2006 Friction and lubrication of hydrogels – its richness and complexity. Soft Matt. 2 (7), 544–552.CrossRefGoogle ScholarPubMed
Grandchamp, X., Coupier, G., Srivastav, A., Minetti, C. & Podgorski, T. 2013 Lift and down-gradient shear-induced diffusion in red blood cell suspensions. Phys. Rev. Lett. 110 (10), 108101.CrossRefGoogle ScholarPubMed
Guan, D., Charlaix, E., Qi, R.Z. & Tong, P. 2017 Noncontact viscoelastic imaging of living cells using a long-needle atomic force microscope with dual-frequency modulation. Phys. Rev. Appl. 8 (4), 044010.CrossRefGoogle Scholar
Happel, J. & Brenner, H. 2012 Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media, vol. 1. Springer Science & Business Media.Google Scholar
Hui, C.-Y., Wu, H., Jagota, A. & Khripin, C. 2021 Friction force during lubricated steady sliding of a rigid cylinder on a viscoelastic substrate. Tribol. Lett. 69 (2), 1–17.CrossRefGoogle Scholar
Jahn, S., Seror, J. & Klein, J. 2016 Lubrication of articular cartilage. Annu. Rev. Biomed. Engng 18, 235–258.CrossRefGoogle ScholarPubMed
Jones, M.B., Fulford, G.R., Please, C.P., McElwain, D.L.S. & Collins, M.J. 2008 Elastohydrodynamics of the eyelid wiper. Bull. Math. Biol. 70 (2), 323–343.CrossRefGoogle ScholarPubMed
Karan, P., Chakraborty, J. & Chakraborty, S. 2018 Small-scale flow with deformable boundaries. J. Ind. Inst. Sci. 98 (2), 159–183.CrossRefGoogle Scholar
Kargar-Estahbanati, A. & Rallabandi, B. 2021 Lift forces on three-dimensional elastic and viscoelastic lubricated contacts. Phys. Rev. Fluids 6 (3), 034003.CrossRefGoogle Scholar
Lainé, A., Jubin, L., Canale, L., Bocquet, L., Siria, A., Donaldson, S.H. & Niguès, A. 2019 Micromegascope based dynamic surface force apparatus. Nanotechnology 30 (19), 195502.CrossRefGoogle ScholarPubMed
Leroy, S. & Charlaix, E. 2011 Hydrodynamic interactions for the measurement of thin film elastic properties. J. Fluid Mech. 674, 389–407.CrossRefGoogle Scholar
Leroy, S., Steinberger, A., Cottin-Bizonne, C., Restagno, F., Léger, L. & Charlaix, É. 2012 Hydrodynamic interaction between a spherical particle and an elastic surface: a gentle probe for soft thin films. Phys. Rev. Lett. 108 (26), 264501.CrossRefGoogle Scholar
Masoud, H. & Stone, H.A. 2019 The reciprocal theorem in fluid dynamics and transport phenomena. J. Fluid Mech. 879, P1.CrossRefGoogle Scholar
Meeker, S.P., Bonnecaze, R.T. & Cloitre, M. 2004 Slip and flow in soft particle pastes. Phys. Rev. Lett. 92 (19), 198302.CrossRefGoogle ScholarPubMed
Mow, V.C., Holmes, M.H. & Lai, W.M. 1984 Fluid transport and mechanical properties of articular cartilage: a review. J. Biomech. 17 (5), 377–394.CrossRefGoogle ScholarPubMed
Moyle, N., Wu, H., Khripin, C., Bremond, F., Hui, C.-Y. & Jagota, A. 2020 Enhancement of elastohydrodynamic friction by elastic hysteresis in a periodic structure. Soft Matt. 16 (6), 1627–1635.CrossRefGoogle Scholar
O'Neill, M.E. 1964 A slow motion of viscous liquid caused by a slowly moving solid sphere. Mathematika 11 (1), 67–74.CrossRefGoogle Scholar
O'Neill, M.E. & Stewartson, K. 1967 On the slow motion of a sphere parallel to a nearby plane wall. J. Fluid Mech. 27 (4), 705–724.CrossRefGoogle Scholar
Pandey, A., Karpitschka, S., Venner, C.H. & Snoeijer, J.H. 2016 Lubrication of soft viscoelastic solids. J. Fluid Mech. 799, 433–447.CrossRefGoogle Scholar
Rallabandi, B., Oppenheimer, N., Zion, M.Y.B. & 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, 4000.CrossRefGoogle Scholar
Salez, T. & Mahadevan, L. 2015 Elastohydrodynamics of a sliding, spinning and sedimenting cylinder near a soft wall. J. Fluid Mech. 779, 181–196.CrossRefGoogle Scholar
Sekimoto, K. & Leibler, L. 1993 A mechanism for shear thickening of polymer-bearing surfaces: elasto-hydrodynamic coupling. Europhys. Lett. 23 (2), 113.CrossRefGoogle Scholar
Skotheim, J.M. & Mahadevan, L. 2004 Soft lubrication. Phys. Rev. Lett. 92 (24), 245509.CrossRefGoogle ScholarPubMed
Skotheim, J.M. & Mahadevan, L. 2005 Soft lubrication: the elastohydrodynamics of nonconforming and conforming contacts. Phys. Fluids 17 (9), 092101.CrossRefGoogle Scholar
Snoeijer, J.H., Eggers, J. & Venner, C.H. 2013 Similarity theory of lubricated hertzian contacts. Phys. Fluids 25 (10), 101705.CrossRefGoogle Scholar
Tan, M.R., Wang, Y. & Frechette, J. 2019 Criterion for particle rebound during wet collisions on elastic coatings. Phys. Rev. Fluids 4 (8), 084305.CrossRefGoogle Scholar
Urzay, J. 2010 Asymptotic theory of the elastohydrodynamic adhesion and gliding motion of a solid particle over soft and sticky substrates at low Reynolds numbers. J. Fluid Mech. 653, 391–429.CrossRefGoogle Scholar
Urzay, J., Llewellyn Smith, S.G. & Glover, B.J. 2007 The elastohydrodynamic force on a sphere near a soft wall. Phys. Fluids 19 (10), 103106.CrossRefGoogle Scholar
Vakarelski, I.U., Manica, R., Tang, X., O'Shea, S.J., Stevens, G.W., Grieser, F., Dagastine, R.R. & Chan, D.Y.C. 2010 Dynamic interactions between microbubbles in water. Proc. Natl Acad. Sci. 107 (25), 11177–11182.CrossRefGoogle ScholarPubMed
Vialar, P., Merzeau, P., Giasson, S. & Drummond, C. 2019 Compliant surfaces under shear: elastohydrodynamic lift force. Langmuir 35 (48), 15605–15613.CrossRefGoogle ScholarPubMed
Villey, R., Martinot, E., Cottin-Bizonne, C., Phaner-Goutorbe, M., Léger, L., Restagno, F. & Charlaix, E. 2013 Effect of surface elasticity on the rheology of nanometric liquids. Phys. Rev. Lett. 111 (21), 215701.CrossRefGoogle ScholarPubMed
Vinogradova, O.I. & Feuillebois, F. 2000 Elastohydrodynamic collision of two spheres allowing slip on their surfaces. J. Colloid Interface Sci. 221 (1), 1–12.CrossRefGoogle ScholarPubMed
Wang, Y., Dhong, C. & Frechette, J. 2015 Out-of-contact elastohydrodynamic deformation due to lubrication forces. Phys. Rev. Lett. 115 (24), 248302.CrossRefGoogle ScholarPubMed
Wang, Y., Feng, Z. & Frechette, J. 2020 Dynamic adhesion due to fluid infusion. Curr. Opin. Colloid Interface Sci. 50, 101397.CrossRefGoogle Scholar
Wang, Y., Pilkington, G.A., Dhong, C. & Frechette, J. 2017 a Elastic deformation during dynamic force measurements in viscous fluids. Curr. Opin. Colloid Interface Sci. 27, 43–49.CrossRefGoogle Scholar
Wang, Y., Tan, M.R. & Frechette, J. 2017 b Elastic deformation of soft coatings due to lubrication forces. Soft Matt. 13 (38), 6718–6729.CrossRefGoogle ScholarPubMed
Wang, Y., Zeng, B., Alem, H.T., Zhang, Z., Charlaix, E. & Maali, A. 2018 Viscocapillary response of gas bubbles probed by thermal noise atomic force measurement. Langmuir 34 (4), 1371–1375.CrossRefGoogle ScholarPubMed
Wu, H., Moyle, N., Jagota, A. & Hui, C.-Y. 2020 Lubricated steady sliding of a rigid sphere on a soft elastic substrate: hydrodynamic friction in the Hertz limit. Soft Matt. 16 (11), 2760–2773.CrossRefGoogle ScholarPubMed
Zhang, Z., Bertin, V., Arshad, M., Raphael, E., Salez, T. & Maali, A. 2020 Direct measurement of the elastohydrodynamic lift force at the nanoscale. Phys. Rev. Lett. 124 (5), 054502.CrossRefGoogle Scholar