Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-27T23:19:12.410Z Has data issue: false hasContentIssue false
  • English
  • Français

Three-dimensional dynamics of a pair of deformable bubbles rising initially in line. Part 1. Moderately inertial regimes

Published online by Cambridge University Press:  09 June 2021

Jie Zhang Open the ORCID record for Jie Zhang [Opens in a new window] ,
Ming-Jiu Ni Open the ORCID record for Ming-Jiu Ni [Opens in a new window]  and
Jacques Magnaudet Open the ORCID record for Jacques Magnaudet [Opens in a new window]
Jie Zhang
Affiliation:
State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi'an Jiaotong University, Xi'an, PR China
Ming-Jiu Ni*
Affiliation:
State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi'an Jiaotong University, Xi'an, PR China School of Engineering, University of Chinese Academy of Sciences, Beijing, PR China
Jacques Magnaudet*
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, Toulouse, France
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The buoyancy-driven motion of two identical gas bubbles released in line in a liquid at rest is examined with the help of highly resolved simulations, focusing on moderately inertial regimes in which the path of an isolated bubble is vertical. Assuming first an axisymmetric evolution, equilibrium configurations of the bubble pair are determined as a function of the buoyancy-to-viscous and buoyancy-to-capillary force ratios which define the Galilei ($Ga$) and Bond ($Bo$) numbers of the system, respectively. The three-dimensional solutions reveal that this axisymmetric equilibrium is actually never reached. Instead, provided $Bo$ stands below a critical $Ga$-dependent threshold determining the onset of coalescence, two markedly different evolutions are observed. At the lower end of the explored $(Ga, Bo)$-range, the tandem follows a drafting–kissing–tumbling scenario, which eventually yields a planar side-by-side motion. For larger $Ga$, the trailing bubble drifts laterally and gets out of the wake of the leading bubble, barely altering the path of the latter. In this second scenario, the late configuration is characterized by a significant inclination of the tandem ranging from $30^\circ$ to $40^\circ$ with respect to the vertical. Bubble deformation has a major influence on the evolution of the system. It controls the magnitude of vortical effects in the wake of the leading bubble, hence the strength of the attractive force acting on the trailing bubble. It also governs the strength and even the sign of the lateral force acting on this bubble, a mechanism of particular importance when the tandem is released with a small angular deviation.

JFM classification

Drops and Bubbles: Bubble dynamics Multiphase and Particle-laden Flows: Gas/liquid flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 920 , 10 August 2021 , A16
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

Adoua, R., Legendre, D. & Magnaudet, J. 2009 Reversal of the lift force on an oblate bubble in a weakly viscous linear shear flow. J. Fluid Mech. 628, 23–41.CrossRefGoogle Scholar
Anthony, C.R., Kamat, P.M., Thete, S.S., Munro, J.P., Lister, J.R., Harris, M.T. & Basaran, O.A. 2017 Scaling laws and dynamics of bubble coalescence. Phys. Rev. Fluids 2, 083601.CrossRefGoogle Scholar
Aoyama, S., Hayashi, K., Hosokawa, S., Lucas, D. & Tomiyama, A. 2017 Lift force acting on single bubbles in linear shear flows. Intl J. Multiphase Flow 96, 113–122.CrossRefGoogle Scholar
Ardekani, M.N., Costa, P., Breugem, W.P. & Brandt, L. 2016 Numerical study of the sedimentation of spheroidal particles. Intl J. Multiphase Flow 87, 16–34.CrossRefGoogle Scholar
Auton, T.R. 1987 The lift force on a spherical body in a rotational flow. J. Fluid Mech. 183, 199–218.CrossRefGoogle Scholar
Batchelor, G.K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bell, J.B., Colella, P. & Glaz, H.M. 1989 A second-order projection method for the incompressible Navier-Stokes equations. J. Comput. Phys. 85, 257–283.CrossRefGoogle Scholar
Bentwich, M. & Miloh, T. 1978 On the exact solution for the two-sphere problem in axisymmetrical potential flow. Trans. ASME J. Appl. Mech. 45, 463–468.CrossRefGoogle Scholar
Biesheuvel, A. & Van Wijngaarden, L. 1982 The motion of pairs of gas bubbles in a perfect liquid. J. Engng Maths 16, 349–365.CrossRefGoogle Scholar
Blanco, A. & Magnaudet, J. 1995 The structure of the axisymmetric high-Reynolds number flow around an ellipsoidal bubble of fixed shape. Phys. Fluids 7, 1265–1274.CrossRefGoogle Scholar
Bonnefis, P. 2019 Etude des instabilités de sillage, de forme et de trajectoire de bulles par une approche de stabilité linéaire globale. PhD thesis, Institut National Polytechnique de Toulouse, Toulouse. Available at: http://www.theses.fr/2019INPT0070.Google Scholar
Brosse, N. & Ern, P. 2014 Interaction of two axisymmetric bodies falling in tandem at moderate Reynolds numbers. J. Fluid Mech. 757, 208–230.CrossRefGoogle Scholar
Bunner, B. & Tryggvason, G. 2002 Dynamics of homogeneous bubbly flows. Part 1. Rise velocity and microstructure of the bubbles. J. Fluid Mech. 466, 17–52.CrossRefGoogle Scholar
Bunner, B. & Tryggvason, G. 2003 Effect of bubble deformation on the properties of bubbly flows. J. Fluid Mech. 495, 77–118.CrossRefGoogle Scholar
Cano-Lozano, J.C., Martinez-Bazan, C., Magnaudet, J. & Tchoufag, J. 2016 a Paths and wakes of deformable nearly spheroidal rising bubbles close to the transition to path instability. Phys. Rev. Fluids 1, 053604.CrossRefGoogle Scholar
Cano-Lozano, J.C., Tchoufag, J., Magnaudet, J. & Martinez-Bazan, C. 2016 b A global stability approach to wake and path instabilities of nearly oblate spheroidal rising bubbles. Phys. Fluids 28, 014102.CrossRefGoogle Scholar
Cartellier, A. & Riviere, N. 2001 Bubble-induced agitation and microstructure in uniform bubbly flows at small to moderate particle Reynolds numbers. Phys. Fluids 13, 2165–2181.CrossRefGoogle Scholar
Chan, D.Y.C., Klaseboer, E. & Manica, R. 2011 Film drainage and coalescence between deformable drops and bubbles. Soft Matt. 7, 2235–2264.CrossRefGoogle Scholar
Chesters, A.K. 1991 The modelling of coalescence processes in fluid-liquid dispersions: a review of current understanding. Chem. Engng Res. Des. 69, 259–270.Google Scholar
Chesters, A.K. & Hofman, G 1982 Bubble coalescence in pure liquids. Appl. Sci. Res. 38, 353–361.CrossRefGoogle Scholar
Chi, B.K. & Leal, L.G. 1989 A theoretical study of the motion of a viscous drop toward a fluid interface at low Reynolds number. J. Fluid Mech. 201, 123–146.CrossRefGoogle Scholar
Davis, R.H., Schonberg, J.A. & Rallison, J.M. 1989 The lubrication force between two viscous drops. Phys. Fluids A 1, 77–81.CrossRefGoogle Scholar
Debrégeas, G., De Gennes, P.-G. & Brochard-Wyart, F. 1998 The life and death of “bare” viscous bubbles. Science 279, 1704–1707.Google Scholar
Duineveld, P.C. 1994 Bouncing and coalescence of two bubbles in water. PhD thesis, University of Twente, Enschede.CrossRefGoogle Scholar
Duineveld, P.C. 1998 Bouncing and coalescence of bubble pairs rising at high Reynolds number in pure water or aqueous surfactant solutions. Adv. Sci. Res. 58, 409–439.Google Scholar
Ervin, E.A. & Tryggvason, G. 1997 The rise of bubbles in a vertical shear flow. Trans. ASME J. Fluids Engng 119, 443–449.CrossRefGoogle Scholar
Esmaeeli, A. & Tryggvason, G. 1998 Direct numerical simulations of bubbly flows. Part 1. Low Reynolds number arrays. J. Fluid Mech. 377, 313–345.CrossRefGoogle Scholar
Esmaeeli, A. & Tryggvason, G. 1999 Direct numerical simulations of bubbly flows. Part 1. Moderate Reynolds number arrays. J. Fluid Mech. 385, 325–358.CrossRefGoogle Scholar
Esmaeeli, A. & Tryggvason, G. 2005 A direct numerical simulation study of the buoyant rise of bubbles at O(100) Reynolds number. Phys. Fluids 17, 093303.CrossRefGoogle Scholar
Feng, J., Hu, H.H. & Joseph, D.D. 1994 Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 1. Sedimentation. J. Fluid Mech. 261, 95–134.CrossRefGoogle Scholar
Figueroa-Espinoza, B. & Zenit, R. 2005 Clustering in high Re monodispersed bubbly flows. Phys. Fluids 17, 091701.CrossRefGoogle Scholar
Fortes, A.F., Joseph, D.D. & Lundgren, T.S. 1987 Nonlinear mechanics of fluidization of beds of spherical particles. J. Fluid Mech. 177, 467–483.CrossRefGoogle Scholar
Gumulya, M., Utikar, R.P., Evans, G.M., Joshi, J.B. & Pareek, V. 2017 Interaction of bubbles rising inline in quiescent liquid. Chem. Engng Sci. 166, 1–10.CrossRefGoogle Scholar
Hallez, Y. & Legendre, D. 2011 Interaction between two spherical bubbles rising in a viscous liquid. J. Fluid Mech. 673, 406–431.CrossRefGoogle Scholar
Happel, J. & Brenner, H. 1963 Low Reynolds Number Hydrodynamics. Martinus Nijhoff.Google Scholar
Harper, J.F. 1970 On bubbles rising in line at large Reynolds numbers. J. Fluid Mech. 41, 751–758.CrossRefGoogle Scholar
Harper, J.F. 1997 Bubbles rising in line: why is the first approximation so bad? J. Fluid Mech. 351, 289–300.CrossRefGoogle Scholar
Hartland, S. 1968 The approach of a rigid sphere to a deformable liquid/liquid interface. J. Colloid Interface Sci. 26, 383–394.CrossRefGoogle Scholar
Hartland, S. 1969 The profile of the draining film between a rigid sphere and a deformable fluid-liquid interface. Chem. Engng Sci. 24, 987–995.CrossRefGoogle Scholar
Jones, A.F. & Wilson, S.D.R. 1978 The film drainage problem in droplet coalescence. J. Fluid Mech. 87, 263–288.CrossRefGoogle Scholar
Joseph, D.D., Fortes, A., Lundgren, T.S. & Singh, P. 1986 Nonlinear mechanics of fluidization of spheres, cylinders and disks in water. In Advances in Multiphase Flow and Related Problems (ed. G. Papanicolaou), pp. 101–122. SIAM.Google Scholar
Kang, I.S. & Leal, L.G. 1988 The drag coefficient for a spherical bubble in a uniform streaming flow. Phys. Fluids 31, 233–237.CrossRefGoogle Scholar
Katz, J. & Meneveau, C. 1996 Wake-induced relative motion of bubbles rising in line. Intl J. Multiphase Flow 22, 239–258.CrossRefGoogle Scholar
Kok, J.B.W. 1993 a Dynamics of a pair of gas bubbles moving through liquid. Part I: theory. Eur. J. Mech. (B/Fluids) 12, 515–540.Google Scholar
Kok, J.B.W. 1993 b Dynamics of a pair of gas bubbles moving through liquid. Part II: eexperiment. Eur. J. Mech. (B/Fluids) 12, 541–560.Google Scholar
Kong, G., Mirsandi, H., Buist, K.A., Peters, E.A.J.F., Baltussen, M.W. & Kuipers, J.A.M. 2019 Hydrodynamic interaction of bubbles rising side-by-side in viscous liquids. Exp. Fluids 60, 155.CrossRefGoogle Scholar
Kurose, R. & Komori, S. 1999 Drag and lift forces on a rotating sphere in a linear shear flow. J. Fluid Mech. 384, 183–206.CrossRefGoogle Scholar
Kusuno, H. & Sanada, T. 2015 Experimental investigation of the motion of a pair of bubbles at intermediate Reynolds numbers. Multiphase Sci. Technol. 27, 51–66.CrossRefGoogle Scholar
Kusuno, H., Yamamoto, H. & Sanada, T. 2019 Lift force acting on a pair of clean bubbles rising in-line. Phys. Fluids 31, 072105.CrossRefGoogle Scholar
Legendre, D. & Magnaudet, J. 1997 A note on the lift force on a spherical bubble or drop in a low-Reynolds-number shear flow. Phys. Fluids 9, 3572–3574.CrossRefGoogle Scholar
Legendre, D. & Magnaudet, J. 1998 The lift force on a spherical bubble in a viscous linear shear flow. J. Fluid Mech. 368, 81–126.CrossRefGoogle Scholar
Legendre, D., Magnaudet, J. & Mougin, G. 2003 Hydrodynamic interactions between two spherical bubbles rising side by side in a viscous liquid. J. Fluid Mech. 497, 133–166.CrossRefGoogle Scholar
Lighthill, J.M. 1956 Drift. J. Fluid Mech. 1, 31–53.CrossRefGoogle Scholar
Loisy, A., Naso, A. & Spelt, P.D.M. 2017 Buoyancy-driven bubbly flows: ordered and free rise at small and intermediate volume fraction. J. Fluid Mech. 816, 94–141.CrossRefGoogle Scholar
Magnaudet, J. & Mougin, G. 2007 Wake instability of a fixed spheroidal bubble. J. Fluid Mech. 572, 311–337.CrossRefGoogle Scholar
Magnaudet, J., Takagi, S. & Legendre, D. 2003 Drag, deformation and lateral migration of a buoyant drop moving near a wall. J. Fluid Mech. 476, 115–157.CrossRefGoogle Scholar
Miloh, T. 1977 Hydrodynamics of deformable contiguous spherical shapes in an incompressible inviscid fluid. J. Engng Maths 11, 349–372.CrossRefGoogle Scholar
Moore, D.W. 1959 The rise of a gas bubble in a viscous liquid. J. Fluid Mech. 6, 113–130.CrossRefGoogle Scholar
Moore, D.W. 1965 The velocity of rise of distorted gas bubbles in a liquid of small viscosity. J. Fluid Mech. 23, 749–766.CrossRefGoogle Scholar
Mougin, G. & Magnaudet, J. 2002 Path instability of a rising bubble. Phys. Rev. Lett. 88, 014502.CrossRefGoogle ScholarPubMed
Munro, J.P., Anthony, C.R., Basaran, O.A. & Lister, J.R. 2015 Thin-sheet flow between coalescing bubbles. J. Fluid Mech. 773, R3.CrossRefGoogle Scholar
Nemer, M.B., Santoro, P., Chen, X., Blawzdziewicz, J. & Loewenberg, M. 2013 Coalescence of drops with mobile interfaces in a quiescent fluid. J. Fluid Mech. 728, 471–500.CrossRefGoogle Scholar
Paulsen, J.D., Carmigniani, R., Kannan, A., Burton, J.C. & Nagel, S.R. 2014 Coalescence of bubbles and drops in an outer fluid. Nat. Commun. 5, 3182.CrossRefGoogle Scholar
Pigeonneau, F. & Sellier, A. 2011 Low-Reynolds-number gravity-driven migration and deformation of bubbles near a free surface. Phys. Fluids 23, 092102.CrossRefGoogle Scholar
Popinet, S. 2003 Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 190, 572–600.CrossRefGoogle Scholar
Popinet, S. 2009 An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228, 5838–5866.CrossRefGoogle Scholar
Popinet, S. 2015 A quadtree-adaptive multigrid solver for the Serre–Green–Naghdi equations. J. Comput. Phys. 302, 336–358.CrossRefGoogle Scholar
Popinet, S. 2017 Available at: http://basilisk.fr/src/examples/bubble.c.Google Scholar
Popinet, S. 2018 Numerical models of surface tension. Annu. Rev. Fluid Mech. 50, 49–75.CrossRefGoogle Scholar
Ramírez-Muñoz, J., Baz-Rodríguez, S., Salinas-Rodríguez, E., Castellanos-Sahagún, E. & Puebla, H. 2013 Forces on aligned rising spherical bubbles at low-to-moderate Reynolds number. Phys. Fluids 25, 093303.CrossRefGoogle Scholar
Ramírez-Muñoz, J., Gama-Goicochea, A. & Salinas-Rodríguez, E. 2011 Drag force on interacting spherical bubbles rising in-line at large Reynolds number. Intl J. Multiphase Flow 37, 983–986.CrossRefGoogle Scholar
Sanada, T., Sato, A., Shirota, M. & Watanabe, M. 2009 Motion and coalescence of a pair of bubbles rising side by side. Chem. Engng Sci. 64, 2659–2671.CrossRefGoogle Scholar
Sanada, T., Watanabe, M. & Fukano, T. 2006 Interactions and coalescence of bubbles in stagnant liquid. Multiphase Sci. Technol. 18, 155–174.CrossRefGoogle Scholar
Sangani, A.S. & Didwania, A.K. 1993 Dynamic simulations of flows of bubbly liquids at large Reynolds numbers. J. Fluid Mech. 250, 307–337.CrossRefGoogle Scholar
Sankaranarayanan, K. & Sundaresan, S. 2002 Lift force in bubbly suspensions. Chem. Engng Sci. 57, 3521–3542.CrossRefGoogle Scholar
Smereka, P. 1993 On the motion of bubbles in a periodic box. J. Fluid Mech. 254, 79–112.CrossRefGoogle Scholar
Taylor, G.I. 1932 The viscosity of fluids containing small drops of another fluid. Proc. R. Soc. Lond. A 138, 41–48.Google Scholar
Tchoufag, J., Magnaudet, J. & Fabre, D. 2013 Linear stability and sensitivity of the flow past a fixed oblate spheroidal bubble. Phys. Fluids 25, 054108.CrossRefGoogle Scholar
Tchoufag, J., Magnaudet, J. & Fabre, D. 2014 Linear instability of the path of a freely rising spheroidal bubble. J. Fluid Mech. 751, R4.CrossRefGoogle Scholar
Tomiyama, A., Tamai, H., Zun, I. & Hosokawa, S. 2002 Transverse migration of single bubbles in simple shear flows. Chem. Engng Sci. 57, 1849–1858.CrossRefGoogle Scholar
Tryggvason, G., Bunner, B., Esmaaeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han, J., Nas, S. & Jan, Y.J. 2001 A front-tracking method for the computations of multiphase flow. J. Comput. Phys. 169, 708–759.CrossRefGoogle Scholar
Unverdi, S.O. & Tryggvason, G. 1992 A front-tracking method for viscous, incompressible, multi-fluid flows. J. Comput. Phys. 100, 25–37.CrossRefGoogle Scholar
Vakarelski, I.U., Marica, R., Li, E.Q., Basheva, E.S., Chan, E.Y.C. & Thoroddsen, S.T. 2018 Coalescence dynamics of mobile and immobile fluid interfaces. Langmuir 34, 2096–2108.CrossRefGoogle ScholarPubMed
Vakarelski, I.U., Yang, F. & Thoroddsen, S.T. 2020 Free-rising bubbles bounce more strongly from mobile than from immobile water-air interfaces. Langmuir 36, 5908–5918.CrossRefGoogle ScholarPubMed
Vakarelski, I.U., Yang, F., Tian, Y.S., Li, E.Q., Chan, E.Y.C. & Thoroddsen, S.T. 2019 Mobile-surface bubbles and droplets coalesce faster but bounce stronger. Sci. Adv. 5, 4292.CrossRefGoogle ScholarPubMed
Van Hooft, J.A., Popinet, S., van Heerwaarden, C.C., van der Linden, S.J.A., de Roode, S.R. & van de Wiel, B.J.H. 2018 Towards adaptive grids for atmospheric boundary-layer simulations. Boundary-Layer Meteorol. 167, 421–443.CrossRefGoogle ScholarPubMed
Van Wijngaarden, L. 1976 Hydrodynamic interactions between gas bubbles in liquid. J. Fluid Mech. 77, 27–44.CrossRefGoogle Scholar
Voinov, O.V. 1969 On the motion of two spheres in a perfect fluid. PMM J. Appl. Math. Mech. 33, 659–667.CrossRefGoogle Scholar
Voinov, V.V., Voinov, O.V. & Petrov, A.G. 1973 Hydrodynamic interaction of bodies in an ideal incompressible liquid and their movement in inhomogeneous flows. PMM J. Appl. Math. Mech. 37, 680–689.Google Scholar
Watanabe, M. & Sanada, T. 2006 In-line motion of a pair of bubbles in a viscous liquid. JSME Intl J. B-Fluids Therm. Engng 49, 410–418.CrossRefGoogle Scholar
Yiantsios, S.G. & Davis, R.H. 1990 On the buoyancy-driven motion of a drop towards a rigid surface or a deformable interface. J. Fluid Mech. 217, 547–573.CrossRefGoogle Scholar
Yin, X. & Koch, D.L. 2008 Lattice-Boltzmann simulation of finite Reynolds number buoyancy-driven bubbly flows in periodic and wall-bounded domains. Phys. Fluids 20, 103304.CrossRefGoogle Scholar
Yuan, H. & Prosperetti, A. 1994 On the in-line motion of two spherical bubbles in a viscous fluid. J. Fluid Mech. 278, 325–349.CrossRefGoogle Scholar
Zenit, R., Koch, D.L. & Sangani, A. 2001 Measurements of the average properties of a suspension of bubbles rising in a vertical channel. J. Fluid Mech. 429, 307–342.CrossRefGoogle Scholar
Zenit, R. & Magnaudet, J. 2008 Path instability of rising spheroidal air bubbles: a shape-controlled process. Phys. Fluids 20, 061702.CrossRefGoogle Scholar
Zhang, J., Chen, L. & Ni, M.J. 2019 Vortex interactions between a pair of bubbles rising side by side in ordinary viscous liquids. Phys. Rev. Fluids 4, 043604.CrossRefGoogle Scholar