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Contactless rebound of elastic bodies in a viscous incompressible fluid

Published online by Cambridge University Press:  24 May 2022

G. Gravina Open the ORCID record for G. Gravina [Opens in a new window] ,
S. Schwarzacher Open the ORCID record for S. Schwarzacher [Opens in a new window] ,
O. Souček Open the ORCID record for O. Souček [Opens in a new window]  and
K. Tůma Open the ORCID record for K. Tůma [Opens in a new window]
G. Gravina
Affiliation:
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovska 83, 186 75 Prague, Czech Republic Department of Mathematics, Temple University, Philadelphia, PA 19122, USA
S. Schwarzacher*
Affiliation:
Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovska 83, 186 75 Prague, Czech Republic
O. Souček
Affiliation:
Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovska 83, 186 75 Prague, Czech Republic
K. Tůma
Affiliation:
Mathematical Institute, Faculty of Mathematics and Physics, Charles University, Sokolovska 83, 186 75 Prague, Czech Republic
*
†Email address for correspondence: [email protected]
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Abstract

In this paper, we investigate the phenomenon of particle rebound in a viscous incompressible fluid environment. We focus on the important case of no-slip boundary conditions, where it is now well established that collisions cannot occur in finite time under certain assumptions. In a simplified framework, we provide conditions which allow us to prove that rebound is possible even in the absence of a topological contact. Our results lead to the conjecture that a qualitative change in the shape of the solid is necessary to obtain a physically meaningful rebound in fluids. We support the conjecture by comparing numerical simulations performed for the reduced model with finite element solutions obtained for corresponding well-established partial differential equation systems describing elastic solids interacting with incompressible fluids.

JFM classification

Low-Reynolds-number flows: Lubrication theory Low-Reynolds-number flows: Stokesian dynamics Multiphase and Particle-laden Flows: Particle/fluid flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 942 , 10 July 2022 , A34
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

REFERENCES

Alnæs, M., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M.E. & Wells, G.N. 2015 The FEniCS project version 1.5. Arch. Numer. Softw. 3 (100).Google Scholar
Barnocky, G. & Davis, R.H. 1989 The influence of pressure-dependent density and viscosity on the elastohydrodynamic collision and rebound of two spheres. J. Fluid Mech. 209, 501–519.CrossRefGoogle Scholar
Bassani, R. & Piccigallo, B. 1992 Hydrostatic Lubrication. Tribology Series, vol. 22. Elsevier Science.Google Scholar
Benesova, B., Kampschulte, M. & Schwarzacher, S. 2020 A variational approach to hyperbolic evolutions and fluid–structure interactions. arXiv:2008.04796.Google Scholar
Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Engng Sci. 16 (3), 242–251.CrossRefGoogle Scholar
Casanova, J.-J., Grandmont, C. & Hillairet, M. 2021 On an existence theory for a fluid-beam problem encompassing possible contacts. J. Éc. polytech. Math. 8, 933–971.CrossRefGoogle Scholar
Cooley, M.D.A. & O'Neill, M.E. 1969 On the slow motion generated in a viscous fluid by the approach of a sphere to a plane wall or stationary sphere. Mathematika 16 (1), 37–49.CrossRefGoogle Scholar
Cox, R.G. & Brenner, H. 1967 The slow motion of a sphere through a viscous fluid towards a plane surface-II small gap widths, including inertial effects. Chem. Engng Sci. 22 (12), 1753–1777.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
Donea, J., Huerta, A., Ponthot, J.-P. & Rodríguez-Ferran, A. 2004 Arbitrary Lagrangian–Eulerian Methods, chap. 14. American Cancer Society.CrossRefGoogle Scholar
Dunne, T. & Rannacher, R. 2006 Adaptive finite element approximation of fluid–structure interaction based on an Eulerian variational formulation. In Fluid–Structure Interaction (ed. H.-J. Bungartz & M. Schäfer), pp. 110–145. Springer.CrossRefGoogle Scholar
Feireisl, E. 2003 On the motion of rigid bodies in a viscous compressible fluid. Arch. Ration. Mech. Anal. 167, 281–308.CrossRefGoogle Scholar
Feireisl, E. 2004 On the Motion Of Rigid Bodies in a Viscous Incompressible Fluid, pp. 419–441. Birkhäuser.Google Scholar
Gérard-Varet, D. & Hillairet, M. 2010 Regularity issues in the problem of fluid structure interaction. Arch. Ration. Mech. Anal. 195 (2), 375–407.CrossRefGoogle Scholar
Gérard-Varet, D. & Hillairet, M. 2014 Existence of weak solutions up to collision for viscous fluid–solid systems with slip. Commun. Pure Appl. Maths 67 (12), 2022–2075.CrossRefGoogle Scholar
Gérard-Varet, D., Hillairet, M. & Wang, C. 2015 The influence of boundary conditions on the contact problem in a 3D Navier–Stokes flow. J. Math. Pure Appl. 103 (1), 1–38.CrossRefGoogle Scholar
Glowinski, R., Pan, T.-W., Hesla, T.I. & Joseph, D.D. 1999 A distributed lagrange multiplier/fictitious domain method for particulate flows. Intl J. Multiphase Flow 25 (5), 755–794.CrossRefGoogle Scholar
Grandmont, C. 2002 Existence for a three-dimensional steady state fluid–structure interaction problem. J. Math. Fluid Mech. 4 (1), 76–94.CrossRefGoogle Scholar
Grandmont, C. & Hillairet, M. 2016 Existence of global strong solutions to a beam-fluid interaction system. Arch. Ration. Mech. Anal. 220, 1283–1333.CrossRefGoogle Scholar
Hagemeier, T., Thévenin, D. & Richter, T. 2021 Settling of spherical particles in the transitional regime. Int. J. Multiph. Flow 138, 103589.CrossRefGoogle Scholar
Hesla, T.I. 2004 Collisions of smooth bodies in viscous fluids: a mathematical investigation. PhD thesis, University of Minnesota.Google Scholar
Hillairet, M. 2007 Lack of collision between solid bodies in a 2D incompressible viscous flow. Commun. Part. Diff. Equ. 32 (7–9), 1345–1371.CrossRefGoogle Scholar
Hillairet, M., Seck, D. & Sokhna, L. 2018 Note on the fall of an axisymmetric body in a perfect fluid over a horizontal ramp. C. R. Math. Acad. Sci. Paris 356 (11–12), 1156–1166.CrossRefGoogle Scholar
Hillairet, M. & Takahashi, T. 2009 Collisions in three-dimensional fluid structure interaction problems. SIAM J. Math. Anal. 40 (6), 2451–2477.CrossRefGoogle Scholar
Janela, J., Lefebvre, A. & Maury, B. 2005 A penalty method for the simulation of fluid – rigid body interaction. ESAIM: Proc. 14, 115–123.CrossRefGoogle Scholar
Joseph, G.G. 2003 Collisional dynamics of macroscopic particles in a viscous fluid. PhD thesis, California Institute of Technology.Google Scholar
Joseph, G.G., Zenit, R., Hunt, M.L. & Rosenwinkel, A.M. 2001 Particle–wall collisions in a viscous fluid. J. Fluid Mech. 433, 329–346.CrossRefGoogle Scholar
Leal, L.G. 1992 Chapter 7 – thin films, lubrication, and related problems. In Laminar Flow and Convective Transport Processes (ed. L. Gary Leal), pp. 345–448. Butterworth-Heinemann.CrossRefGoogle Scholar
Liu, C. & Walkington, N.J. 2001 An Eulerian description of fluids containing visco-elastic particles. Arch. Ration. Mech. Anal. 159 (3), 229–252.CrossRefGoogle Scholar
Olsson, E. & Kreiss, G. 2005 A conservative level set method for two phase flow. J. Comput. Phys. 210, 225–246.CrossRefGoogle Scholar
Richter, T. 2017 Fluid–Structure Interactions: Models, Analysis and Finite Elements, vol. 118. Springer.CrossRefGoogle Scholar
Scovazzi, G. & Hughes, T. 2007 Lecture notes on continuum mechanics on arbitrary moving domains. Tech. Rep. SAND-2007-6312P. Sandia National Laboratories.Google Scholar
Starovoitov, V.N. 2004 Behavior of a rigid body in an incompressible viscous fluid near a boundary. In Free boundary problems (Trento, 2002), Internat. Ser. Numer. Math., vol. 147, pp. 313–327. Birkhäuser.CrossRefGoogle Scholar
Sugiyama, K., Ii, S., Takeuchi, S., Takagi, S. & Matsumoto, Y. 2011 A full Eulerian finite difference approach for solving fluid–structure coupling problems. J. Comput. Phys. 230 (3), 596–627.CrossRefGoogle Scholar
Vázquez, J.L. & Zuazua, E. 2006 Lack of collision in a simplified 1D model for fluid–solid interaction. Math. Models Meth. Appl. Sci. 16 (5), 637–678.CrossRefGoogle Scholar
von Wahl, H., Richter, T., Frei, S. & Hagemeier, T. 2020 Falling balls in a viscous fluid with contact: comparing numerical simulations with experimental data. Preprint arXiv:2011.08691.CrossRefGoogle Scholar