Skip to main content Accessibility help

On the lateral migration of a slightly deformed bubble rising near a vertical plane wall



Deformation-induced lateral migration of a bubble slowly rising near a vertical plane wall in a stagnant liquid is numerically and theoretically investigated. In particular, our focus is set on a situation with a short clearance c between the bubble interface and the wall. Motivated by the fact that numerically and experimentally measured migration velocities are considerably higher than the velocity estimated by the available analytical solution using the Faxén mirror image technique for a/(a + c) ≪ 1 (here a is the bubble radius), when the clearance parameter ϵ(=c/a) is comparable to or smaller than unity, the numerical analysis based on the boundary-fitted finite-difference approach solving the Stokes equation is performed to complement the experiment. The migration velocity is found to be more affected by the high-order deformation modes with decreasing ϵ. The numerical simulations are compared with a theoretical migration velocity obtained from a lubrication study of a nearly spherical drop, which describes the role of the squeezing flow within the bubble–wall gap. The numerical and lubrication analyses consistently demonstrate that when ϵ ≤ 1, the lubrication effect makes the migration velocity asymptotically μVB12/(25ϵγ) (here, VB1, μ and γ denote the rising velocity, the dynamic viscosity of liquid and the surface tension, respectively).


Corresponding author

Email address for correspondence:


Hide All
Amsden, A. A. & Harlow, F. H. 1970 A simplified MAC technique for incompressible fluid flow calculations. J. Comput. Phys. 6, 322325.
Bart, E. 1968 The slow unsteady settling of a fluid sphere toward a flat fluid interface. Chem. Engng Sci. 23, 193210.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Becker, L. E., McKinley, G. H. & Stone, H. A. 1996 Sedimentation of a sphere near a plane wall: weak non-Newtonian and inertial effects. J. Non-Newton. Fluid Mech. 63, 201233.
Callens, N., Minetti, C., Coupier, G., Mader, M.-A., Dubois, F., Misbah, C. & Podgorski, T. 2008 Hydrodynamic lift of vesicles under shear flow in microgravity. Europhys. Lett. 83, 24002.
Chaffey, C. E., Brenner, H. & Mason, S. G. 1965 Particle motions in sheared suspensions. Part XVIII. Wall migration (theoretical). Rheol. Acta 4, 6472 (correction in Rheol. Acta 6, 100).
Chan, P. C.-H. & Leal, L. G. 1979 The motion of a deformable drop in a second-order fluid. J. Fluid Mech. 92, 131170.
Cherukat, P. & McLaughlin, J. B. 1994 The inertial lift on a rigid sphere in a linear shear flow field near a flat wall. J. Fluid Mech. 263, 118.
Correas, J.-M., Bridal, L., Lesavre, A., Méjean, A., Claudon, M. & Hélénon, O. 2001 Ultrasound contrast agents: properties, principles of action, tolerance, and artifacts. Eur. Radiol. 11, 13161328.
Cox, R. G. & Brenner, H. 1968 The lateral migration of solid particles in Poiseuille flow. Part I. Theory. Chem. Engng Sci. 23, 147173.
Cox, R. G. & Hsu, S. K. 1977 The lateral migration of solid particles in a laminar flow near a plane. Intl J. Multiph. Flow 3, 201222.
Dimitrakopoulos, P. 2007 Interfacial dynamics in Stokes flow via a three-dimensional fully-implicit interfacial spectral boundary element algorithm. J. Comput. Phys. 225, 408426.
Fukagata, K. & Kasagi, N. 2002 Highly energy-conservative finite difference method for the cylindrical coordinate system. J. Comput. Phys. 181, 478498.
Garstecki, P., Fuerstman, M. J., Stone, H. A. & Whitesides, G. M. 2006 Formation of droplets and bubbles in a microfluidic T-junction-scaling and mechanism of break-up. Lab on a Chip 6, 437446.
Goldman, A. J., Cox, R. G. & Brenner, H. 1967 Slow viscous motion of a sphere parallel to a plane wall-I motion through a quiescent fluid. Chem. Engng Sci. 22, 637651.
Happel, J. & Brenner, H. 1973 Low Reynolds Number Hydrodynamics, 2nd edn. Martinus Nijhoff.
Harlow, F. H. & Welch, J. E. 1965 Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8, 21822189.
Hibiki, T. & Ishii, M. 2007 Lift force in bubbly flow systems. Chem. Engng Sci. 62, 64576472.
Ho, B. P. & Leal, L. G. 1974 Inertial migration of rigid spheres in two-dimensional unidirectional flows. J. Fluid Mech. 65, 365400.
Hodges, S. R., Jensen, O. E. & Rallison, J. M. 2004 Sliding, slipping and rolling: the sedimentation of a viscous drop down a gently inclined plane. J. Fluid Mech. 512, 95131.
Leal, L. G. 1980 Particle motions in a viscous fluid. Annu. Rev. Fluid Mech. 12, 435476.
Leal, L. G. 1992 Laminar Flow and Convective Transport Processes. Butterworth-Heinemann.
Magnaudet, J., Takagi, S. & Legendre, D. 2003 Drag, deformation and lateral migration of a buoyant drop moving near a wall. J. Fluid Mech. 476, 115157.
Makuta, T., Takemura, F., Hihara, E., Matsumoto, Y. & Shoji, M. 2006 Generation of micro gas bubbles of uniform diameter in an ultrasonic field. J. Fluid Mech. 548, 113131.
McLaughlin, J. B. 1993 The lift on a small sphere in wall-bounded linear shear flows. J. Fluid Mech. 246, 249265.
Olla, P. 1997 The lift on a tank-treading ellipsoidal cell in a shear flow. J. Phys. II France 7, 15331540.
O'Neill, M. E. 1964 A slow motion of viscous liquid caused by a slowly moving solid sphere. Mathematika 11, 6774.
O'Neill, M. E. & Stewartson, K. 1967 On the slow motion of a sphere parallel to a nearby plane wall. J. Fluid Mech. 27, 705724.
Sekimoto, K. & Leibler, L. 1993 A mechanism for shear thickening of polymer-bearing surfaces: elasto-hydrodynamic coupling. Europhys. Lett. 23, 113117.
Serizawa, A., Inui, T., Yahiro, T. & Kawara, Z. 2005 Pseudo-laminarizaion of micro-bubble containing milky bubbly flow in a pipe. Multiph. Sci. Tech. 17, 79101.
Shapira, M. & Haber, S. 1988 Low Reynolds number motion of a droplet between two parallel plates. Intl J. Multiph. Flow 14, 483506.
Shortencarier, M. J., Dayton, P. A., Bloch, S. H., Schumann, P. A., Matsunaga, T. O. & Ferrara, K. W. 2004 A method for radiation-force localized drug delivery using gas-filled lipospheres. IEEE Trans. Ultrason. Ferroelectr. Freq. Control 51, 821830.
Skotheim, J. M. & Mahadevan, L. 2004 Soft lubrication. Phys. Rev. Lett. 92, 245509.
Skotheim, J. M. & Mahadevan, L. 2005 Soft lubrication: the elastohydrodynamics of non-conforming and conforming contacts. Phys. Fluids 17, 092101.
Sugiyama, K. & Sbragaglia, M. 2008 Linear shear flow past a hemispherical droplet adhering to a solid surface. J. Engng Math. 62, 4550.
Takemura, F. 2004 Migration velocities of spherical solid particles near a vertical wall for Reynolds number from 0.1 to 0.5. Phys. Fluids 16, 204207.
Takemura, F., Magnaudet, J. & Dimitrakopoulos, P. 2009 Migration and deformation of bubbles rising in a wall-bounded shear flow at finite Reynolds number. J. Fluid Mech. 634, 463486.
Takemura, F., Takagi, S., Magnaudet, J. & Matsumoto, Y. 2002 Drag and lift force on a bubble rising near a vertical wall in a viscous liquid. J. Fluid Mech. 461, 277300.
Uijttewaal, W. S. J. & Nijhof, E. J. 1995 The motion of a droplet subjected to linear shear flow including the presence of a plane wall. J. Fluid Mech. 302, 4563.
Uijttewaal, W. S. J., Nijhof, E. J. & Heethaar, R. M. 1993 Droplet migration, deformation, and orientation in the presence of a plane wall: a numerical study compared with analytical theories. Phys. Fluids A 5, 819825.
Urzay, J., Smith, S. G. L. & Glover, B. J. 2007 The elastohydrodynamic force on a sphere near a soft wall. Phys. Fluids 19, 103106.
Vasseur, P. & Cox, R. G. 1976 The lateral migration of a spherical particle in two-dimensional shear flows. J. Fluid Mech. 78, 385413.
Vasseur, P. & Cox, R. G. 1977 The lateral migration of spherical particles sedimenting in a stagnant bounded fluid. J. Fluid Mech. 80, 561591.
Wang, Y. & Dimitrakopoulos, P. 2006 A three-dimensional spectral boundary element algorithm for interfacial dynamics in Stokes flow. Phys. Fluids 18, 082106.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

On the lateral migration of a slightly deformed bubble rising near a vertical plane wall



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