Abstract
The nonlinear free-boundary problem of finding the equilibrium shapes of two equal-sized two-dimensional inviscid bubbles with surface tension situated in a polynomially-singular slow viscous flow is solved in terms of closed-form formulae. The singular flow is taken to be within the class of those realizable at the centre of a four-roller mill apparatus. The associated flow field is also found explicitly. These solutions allow investigation of the bubble shapes and associated streamline patterns as functions of the far-field asymptotic conditions. In certain regimes, the bubbles are found to exhibit both near-cusps and a characteristic dimpling as they draw closer together. The results provide the first instances of exact solutions involving two interacting bubbles in an unbounded Stokes flow.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A.K. Chesters, The modelling of coalescence processes in fluid-liquid dispersions. Chem. Eng. Res. Des. 69 (1991) 259–270.
J.M. Rallison, The deformation of small viscous drops and bubbles in shear flows. Ann. Rev. Fluid. Mech. 16 (1984) 45–66.
H.A. Stone, Dynamics of drop deformation and breakup in viscous fluids. Ann. Rev. Fluid. Mech. 26 (1994) 65–102.
S. Richardson, Two-dimensional bubbles in slow viscous flows. J. Fluid Mech. 33 (1968) 475–493.
S. Richardson, Two-dimensional bubbles in slow viscous flows. Part 2. J. Fluid Mech. 58 (1973) 115–127.
R.W. Hopper, Plane Stokes flow driven by capillarity on a free surface. J. Fluid Mech. 213 (1990) 349–375.
R.W. Hopper, Plane Stokes flow driven by capillarity on a free surface, Part 2. Further developments. J. Fluid Mech. 230 (1991) 355–364.
S. Tanveer and G.L. Vasconcelos, Time-evolving bubbles in two dimensional Stokes flow. J. Fluid Mech. 301 (1995) 325–344.
L.K. Antanovskii, Quasi-steady deformation of a two-dimensional bubble placed within a potential viscous flow. Eur. J. Mech. B/Fluids 13 (1994) 73–92.
L.K. Antanovskii, A plane inviscid incompressible bubble placed within a creeping viscous flow: formation of a cusped bubble. Eur. J. Mech. B 13 (1994) 491–509.
L.K. Antanovskii, Formation of a pointed drop in Taylor's four-roller mill. J. Fluid Mech. 327 (1996) 325–341.
D.G. Crowdy and S. Tanveer, A theory of exact solutions for plane viscous blobs. J. Nonlin. Sci. 8 (1998) 261–279.
D.G. Crowdy and S. Tanveer, A theory of exact solutions for annular viscous blobs. J. Nonlin. Sci. 8 (1998) 375–400.
S. Richardson, Plane Stokes flow with time-dependent free boundaries in which the fluid occupies a doublyconnected region. Eur. J. Appl. Math. 11 (2000) 249–269.
D.G. Crowdy, Viscous sintering of unimodal and bimodal cylindrical packings with shrinking pores. Eur. J. Appl. Math. (2002) to appear.
D.G. Crowdy, Exact solutions for the viscous sintering of multiply-connected fluid domains. J. Eng. Math. 42 (2002) 225–242.
G.I. Taylor, The formation of emulsions in definable fields of flow. Proc. R. Soc. London A 146 (1934) 501–523.
H. Yang, C.C. Park, Y.T. Hu, L.G. Leal, The coalescence of two equal-sized drops in a two-dimensional linear flow. Phys Fluids, 13 (2001) 1087–1106.
C. Pozrikidis, Numerical studies of singularity formation at free surfaces and fluid interfaces in twodimensional Stokes flow. J. Fluid Mech. 331 (1997) 145–167.
C. Pozrikidis, Numerical studies of cusp formation at fluid interfaces in Stokes flow. J. Fluid Mech. 357 (1998) 29–57.
M. Siegel, Cusp formation for time-evolving bubbles in two-dimensional Stokes flow, J. Fluid Mech. 412 (2000) 227–257.
D. Crowdy and M. Siegel, Exact solutions for the evolution of a bubble in Stokes flow: a Cauchy transform approach. SIAM J. Appl. Math. (2002) to appear.
W.E. Langlois, Slow Viscous Flow. New York: Macmillan (1964) 229 pp.
Z. Nehari, Conformal Mapping, New York: McGraw-Hill (1966) 396pp.
M. Abramovitz and I.A. Stegun, Handbook of Mathematical Functions. New York: Dover, (1972) 1046pp.
P. Henrici, Applied and computational complex analysis, Volume III. New York: Wiley and Sons (1986) 637pp.
J.T. Jeong and H.K. Moffatt, Free surface cusps associated with flow at low Reynolds number. J. Fluid Mech. 241 (1992) 1–22.
M.A. Rother, A.Z. Zinchenko and R.H. Davis, Buoyancy-driven coalescence of slightly deformable drops. J. Fluid Mech. 346 (1997) 117–148.
M. Loewenberg and E.J. Hinch, Collision of two deformable drops in shear flow. J. Fluid Mech. 338 (1997) 299–315.
C. Pozrikidis, The deformation of a liquid drop moving normal to a plane wall. J. Fluid Mech. 215 (1990) 331–363.
M. Siegel, Influence of surfactant on rounded and pointed bubbles in two-dimensional Stokes flow. SIAM. J. Appl. Math. 59 (1999) 1998–2027.
M. Siegel, Cusp formation for time-evolving bubbles in two-dimensional Stokes flow. J. Fluid Mech. 412 (2000) 227–257.
D.G. Crowdy, Compressible bubbles in Stokes flow. J. Fluid Mech. (to appear).
C. Pozrikidis, Expansion of a compressible gas bubble in Stokes flow. J. Fluid Mech. 442 (2001) 171–189.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Crowdy, D. Exact solutions for two steady inviscid bubbles in the slow viscous flow generated by a four-roller mill. Journal of Engineering Mathematics 44, 311–330 (2002). https://doi.org/10.1023/A:1021267512989
Issue Date:
DOI: https://doi.org/10.1023/A:1021267512989