# American Institute of Mathematical Sciences

June  2011, 15(4): 1045-1064. doi: 10.3934/dcdsb.2011.15.1045

## Boundary element approach for the slow viscous migration of spherical bubbles

 1 LadHyX. Ecole Polytechnique, 91128 Palaiseau, France

Received  May 2010 Published  March 2011

This paper examines the slow viscous migration of a collection of $N\geq 1$ spherical bubbles immersed in a bounded Newtonian liquid under the action of prescribed uniform gravity field and/or arbitrary ambient Stokes flow. The liquid domain is either open or closed with fixed boundary(ies) where the ambient Stokes flow vanishes. The incurred translational velocity of each bubble is obtained by resorting to a well-posed boundary formulation which requires to invert $3N$ boundary-integral equations. Depending upon the selected Green tensor, these integral equations holds on the cluster's surface plus the boundaries or solely on the cluster's surface. The advocated numerical strategy resorts to quadratic triangular curvilinear boundary elements on each encoutered surface and enables one to accurately compute at a reasonable cpu time cost each bubble velocity. A special attention is paid, both theoretically and numerically, to the case of $N-$bubble cluster located near a solid and motionless plane wall with numerical results given and discussed for a few clusters subject to gravity effects and ambient linear or quadratic shear flows.
Citation: Antoine Sellier. Boundary element approach for the slow viscous migration of spherical bubbles. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 1045-1064. doi: 10.3934/dcdsb.2011.15.1045
##### References:
 [1] N. O. Young, J. S. Goldstein and M. J. Block, The motion of bubbles in a vertical temperature gradient,, J. Fluid Mech., 6 (1959), 350. doi: doi:10.1017/S0022112059000684. [2] R. S. Subramanian, The Stokes force on a droplet in an unbounded fluid medium due to capillary effects,, J. Fluid Mech., 153 (1985), 389. doi: doi:10.1017/S0022112085001306. [3] A. Sellier, On the capillary motion of arbitrary clusters of spherical bubbles. Part 1. General theory,, J. Fluid Mech., 197 (2004), 391. doi: doi:10.1017/S0022112004008146. [4] G. Hetsroni, E. Wacholder and S. Haber, The hydrodynamic resistance of a fluid sphere submerged in Stokes flows,, Z. Angew. Math. Mech. \textbf{15} (1971), 15 (1971), 45. doi: doi:10.1002/zamm.19710510105. [5] A. Sellier, Slow viscous migration of bubbles near a plane solid wall under a gravity and/or an external flow field,, Far East J. Appl. Math., 23 (2006), 349. [6] J. Happel and H. Brenner, "Low Reynolds Number Hydrodynamics,'', Martinus Nijhoff, (1973). [7] S. Kim and S. J. Karrila, "Microhydrodynamics: Principles and Selected Applications,'', Applications, (1991). [8] C. Pozrikidis, "Boundary Integral and Singularity Methods for Linearized Viscous Flow,'', Cambridge University Press, (1992). doi: doi:10.1017/CBO9780511624124. [9] C. A. Brebbia, J. C. L. Telles and L. C. Wrobel, "Boundary Element Techniques,'', Springer-Verlag, (1984). [10] D. E. Beskos, "Introduction to Boundary Element Methods,'', In Computational Methods in Mechanics (ed. D. E. Beskos). Elsevier Science Publishers, (1987). [11] M. Bonnet, "Boundary Integral Equation Methods for Solids and Fluids,'', John Wiley & Sons Ltd, (1999). [12] J. R. Blake, A note on the image system for a Stokeslet in a no-slip boundary,, Proc. Camb. Phil. Soc., 70 (1971), 303. doi: doi:10.1017/S0305004100049902.

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##### References:
 [1] N. O. Young, J. S. Goldstein and M. J. Block, The motion of bubbles in a vertical temperature gradient,, J. Fluid Mech., 6 (1959), 350. doi: doi:10.1017/S0022112059000684. [2] R. S. Subramanian, The Stokes force on a droplet in an unbounded fluid medium due to capillary effects,, J. Fluid Mech., 153 (1985), 389. doi: doi:10.1017/S0022112085001306. [3] A. Sellier, On the capillary motion of arbitrary clusters of spherical bubbles. Part 1. General theory,, J. Fluid Mech., 197 (2004), 391. doi: doi:10.1017/S0022112004008146. [4] G. Hetsroni, E. Wacholder and S. Haber, The hydrodynamic resistance of a fluid sphere submerged in Stokes flows,, Z. Angew. Math. Mech. \textbf{15} (1971), 15 (1971), 45. doi: doi:10.1002/zamm.19710510105. [5] A. Sellier, Slow viscous migration of bubbles near a plane solid wall under a gravity and/or an external flow field,, Far East J. Appl. Math., 23 (2006), 349. [6] J. Happel and H. Brenner, "Low Reynolds Number Hydrodynamics,'', Martinus Nijhoff, (1973). [7] S. Kim and S. J. Karrila, "Microhydrodynamics: Principles and Selected Applications,'', Applications, (1991). [8] C. Pozrikidis, "Boundary Integral and Singularity Methods for Linearized Viscous Flow,'', Cambridge University Press, (1992). doi: doi:10.1017/CBO9780511624124. [9] C. A. Brebbia, J. C. L. Telles and L. C. Wrobel, "Boundary Element Techniques,'', Springer-Verlag, (1984). [10] D. E. Beskos, "Introduction to Boundary Element Methods,'', In Computational Methods in Mechanics (ed. D. E. Beskos). Elsevier Science Publishers, (1987). [11] M. Bonnet, "Boundary Integral Equation Methods for Solids and Fluids,'', John Wiley & Sons Ltd, (1999). [12] J. R. Blake, A note on the image system for a Stokeslet in a no-slip boundary,, Proc. Camb. Phil. Soc., 70 (1971), 303. doi: doi:10.1017/S0305004100049902.
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