Skip to main content Accessibility help

Sloshing in a Hele-Shaw cell: experiments and theory

  • Francesco Viola (a1), François Gallaire (a1) and Benjamin Dollet (a2)


The response of the free liquid surface in a Hele-Shaw cell subjected to a horizontal oscillation is investigated. We study the low-oscillation-amplitude regime and we show, by varying the fluid viscosity, $\unicode[STIX]{x1D708}$ , and the forcing frequency, $\unicode[STIX]{x1D714}$ , that the ratio between the Stokes viscous length, $\sqrt{2\unicode[STIX]{x1D708}/\unicode[STIX]{x1D714}}$ , and the cell thickness greatly affects the amplitude and phase lag of the gravity waves. In particular, the sloshing system undergoes an underdamped/overdamped transition for sufficiently large viscosities. A consistent theoretical model, based on a modification of Darcy’s law to include unsteadiness, is then introduced to rationalize the experimental observations. Contrary to traditional sloshing wave theory, the viscous flow dissipation comes at leading order in the analysis, rather than as a higher-order asymptotic correction to the inviscid sloshing dynamics. The analytical expression for the resonance curves agrees well with experimental results without tunable parameters.



Hide All
Bensimon, D., Kadanoff, L. P., Liang, S., Shraiman, B. I. & Tang, C. 1986 Viscous flows in two dimensions. Rev. Mod. Phys. 58 (4), 977999.
Cantat, I. 2013 Liquid meniscus friction on a wet plate: bubbles, lamellae, and foams. Phys. Fluids 25 (3), 031303.
Cocciaro, B., Faetti, S. & Nobili, M. 1991 Capillarity effects on surface gravity waves in a cylindrical container: wetting boundary conditions. J. Fluid Mech. 231, 325343.
Davies, J. T. & Vose, R. W. 1965 On the damping of capillary waves by surface films. Proc. R. Soc. Lond. A 286 (1405), 218234.
Dwyer, H. & Erickson, D. 1969 Hele-shaw and porous medium flow for space fuel systems. In 2nd Fluid and Plasma Dynamics Conference, AIAA Paper No. 69-678,
Faltinsen, O. M. & Timokha, A. N. 2009 Sloshing. Cambridge University Press.
Gondret, P. & Rabaud, M. 1997 Shear instability of two-fluid parallel flow in a Hele-Shaw cell. Phys. Fluids 9 (11), 32673274.
Ibrahim, R. A. 2005 Liquid Sloshing Dynamics: Theory and Applications. Cambridge University Press.
Ibrahim, R. A., Pilipchuk, V. N. & Ikeda, T. 2001 Recent advances in liquid sloshing dynamics. Appl. Mech. Rev. 54, 133199.
Kalogirou, A., Moulopoulou, E. E. & Bokhove, O. 2016 Variational finite element methods for waves in a Hele-Shaw tank. Appl. Math. Model. 40 (17), 74937503.
Keulegan, G. H. 1959 Energy dissipation in standing waves in rectangular basins. J. Fluid Mech. 6 (01), 3350.
Mayer, H. C. & Krechetnikov, R. 2012 Walking with coffee: why does it spill? Phys. Rev. E 85, 046117.
Miles, J. M. 1967 Surface-wave damping in closed basins. Proc. R. Soc. Lond. A 297 (1451), 459475.
Plouraboué, F. & Hinch, E. J. 2002 Kelvin–Helmholtz instability in a Hele-Shaw cell. Phys. Fluids 14 (3), 922929.
Royon-Lebeaud, A., Hopfinger, E. & Cartellier, A. 2007 Liquid sloshing and wave breaking in cylindrical and square-base containers. J. Fluid Mech. 577, 467494.
Ruyer-Quil, C. 2001 Inertial corrections to the Darcy law in a Hele-Shaw cell. C. R. Acad. Sci. Paris II 329 (5), 337342.
Saffman, P. G. & Taylor, G. 1958 The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245 (1242), 312329.
Sauret, A., Boulogne, F., Cappello, J. D., Dressaire, E. & Stone, H. A. 2015 Damping of liquid sloshing by foams. Phys. Fluids 27 (2), 022103.
Veldman, A. E. P., Gerrits, J., Luppes, R., Helder, J. A. & Vreeburg, J. P. B. 2007 The numerical simulation of liquid sloshing on board spacecraft. J. Comput. Phys. 224, 8299.
Viola, F., Brun, P. T., Dollet, B. & Gallaire, F. 2016 Foam on troubled water: capillary induced finite-time arrest of sloshing waves. Phys. Fluids 28, 091701.
Womersley, J. R. 1955 Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J. Physiol. 127 (3), 553.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

Sloshing in a Hele-Shaw cell: experiments and theory

  • Francesco Viola (a1), François Gallaire (a1) and Benjamin Dollet (a2)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.