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Formation of a pointed drop in Taylor's four-roller mill

Published online by Cambridge University Press:  26 April 2006

Leonid K. Antanovskii
Leonid K. Antanovskii
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia Present address: Moldflow International Pty Ltd, 259–261 Colchester Road, Kilsyth, Victoria 3137, Australia.
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Abstract

The paper addresses the mathematical modelling of the formation of a pointed drop in a four-roller mill, observed by Taylor (1934) in the Cavendish Laboratory. Since the experiments were carried out with drops of small diameter compared to the mill size, the method of matched asymptotic expansions is applicable. A two-dimensional Stokes flow generated by the rotating rollers in the mill but with no drop effect (outer problem) is computed numerically by a boundary-element method. The local expansion of that flow at the centre of the mill, where the drop is to be positioned, is used as a far field for the flow around the drop in unbounded fluid (inner problem). Employing a plane-flow model and using complex-variable techniques, the explicit solutions previously obtained by the author are adapted to the inner problem. It is proved that, with an increasing rotation rate of the rollers, the drop does develop two apparent cusps on the interface, and its shapes have striking similarities with Taylor's experiments. Response diagrams showing the drop distortion versus the elongational strain demonstrate that these are one-to-one function of each other if the drop diameter is greater than a critical value determined by the size of the mill but cease to be one-to-one otherwise. This behaviour is identified with a sudden transition from a rounded drop to a cusped one at a critical strain.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 327 , 25 November 1996 , pp. 325 - 341
Copyright
© 1996 Cambridge University Press

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References

Acrivos, A. & Lo, T. S. 1978 Deformation and breakup of a single slender drop in an extensional flow. J. Fluid Mech. 86, 641–672.Google Scholar
Antanovskii, L. K. 1994a Influence of surfactants on a creeping free-boundary flow induced by two counter-rotating horizontal thin cylinders. Eur. J. Mech. B/Fluids 13, 73–92.Google Scholar
Antanovskii, L. K. 1994b Quasi-steady deformation of a two-dimensional bubble placed within a potential viscous flow. Meccanica — J. Ital. Assoc. Theor. Appl. Mech. 29, 27–42.Google Scholar
Antanovskii, L. K. 1994c A plane inviscid incompressible bubble placed within a creeping viscous flow: formation of a cusped bubble. Eur. J. Mech. B/Fluids 13, 491–509.Google Scholar
Antanovskii, L. K. 1995 Mathematical modelling of formation of a pointed drop in a four-roll mill. In Proc. 3rd Caribbean Congr. on Fluid Dynamics and the 3rd Latin—American Symp. on Fluid Mechanics, Caracas, Venezuela, 2–9 Feb. 1995 (ed. J. J. Manzano-Ruíz, S. Zarea, A. Tremante, E. Guevara & R. García), vol. 1, pp. 1–6, Universidad Simón Bolívar.
Buckmaster, J. D. 1972 Pointed bubbles in slow viscous flow. J. Fluid Mech. 55, 385–400.Google Scholar
Buckmaster, J. D. 1973 The bursting of pointed drops in slow viscous flow. Trans ASME J. Appl. Mech. E: 40, 18–24.Google Scholar
Buckmaster, J. D. & Flaherty, J. E. 1973 The bursting of two-dimensional drops in slow viscous flow. J. Fluid Mech. 60, 625–639.Google Scholar
Hinch, E. J. & Acrivos, A. 1979 Steady long slender droplets in two-dimensional straining motion. J. Fluid Mech. 91, 401–414.Google Scholar
Hopper, R. W. 1990 Plane Stokes flow driven by capillarity on a free surface. J. Fluid Mech. 213, 349–375.Google Scholar
Hopper, R. W. 1991 Plane Stokes flow driven by capillarity on a free surface. Part 2. Further developments. J. Fluid Mech. 230, 355–364.Google Scholar
Hopper, R. W. 1992 Stokes flow of a cylinder and a half-space driven by capillarity. J. Fluid Mech. 243, 171–181.Google Scholar
Hopper, R. W. 1993 Capillarity-driven plane Stokes flow exterior to a parabola. Q. J. Mech. Appl. Maths 46, 193–210.Google Scholar
Hromadka II, T. V. & Lai, C. 1987 The Complex Variable Boundary Element Method in Engineering Analysis. Springer.
Jeong, J. T. & Moffatt, H. K. 1992 Free surface cusps associated with flow at low Reynolds number. J. Fluid Mech. 241, 1–22.Google Scholar
Joseph, D. D., Nelson, J., Renardy, M. & Renardy, Y. 1991 Two-dimensional cusped interfaces. J. Fluid Mech. 223, 383–409.Google Scholar
Langlois, W. E. 1964 Slow Viscous Flow. Macmillan.
Liu, Y. J., Liao, T. Y. & Joseph, D. D. 1995 Two-dimensional cusp at the trailing edge of an air bubble rising in a viscoelastic liquid. J. Fluid Mech. 304, 321–342.Google Scholar
Moffatt, H. K. 1977 Behaviour of a viscous film on the outer surface of a rõating cylinder. J. Mécanique 16, 651–673.Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.
Richardson, S. 1968 Two-dimensional bubbles in slow viscous flows. J. Fluid Mech. 33, 475–493.Google Scholar
Richardson, S. 1973 Two-dimensional bubbles in slow viscous flows. Part 2. J. Fluid Mech. 58, 115–127.Google Scholar
Richardson, S. 1992 Two-dimensional slow viscous flows with time-dependent free boundaries driven by surface tension. Eur. J. Appl. Maths 3, 193–207.Google Scholar
Rumscheidt, F. D. & Mason, S. G. 1961 Particle motion in sheared suspensions. XII. Deformation and burst of liquid drops in shear and hyperbolic flow. J. Colloid Sci. 16, 238–261.Google Scholar
Sherwood, J. D. 1981 Spindle-shaped drops in a viscous extensional flow. Math. Proc. Camb. Phil. Soc. 90, 529–536.Google Scholar
Sherwood, J. D. 1984 Tip streaming from slender drops in a nonlinear extensional flow. J. Fluid Mech. 144, 281–295.Google Scholar
Stone, H. A. 1994 Dynamics of drop deformation and breakup in viscous fluids. Ann. Rev. Fluid Mech. 26, 65–102.Google Scholar
Tanveer, S. & Vasconcelos, G. L. 1995 Time-evolving bubbles in two-dimensional Stokes flow. J. Fluid Mech. 301, 325–344.Google Scholar
Taylor, G. I. 1932 The viscosity of a fluid containing small drops of another fluid. Proc. R. Soc. Lond. A 138, 41–48.Google Scholar
Taylor, G. I. 1934 The formation of emulsions in definable fields of flow. Proc. R. Soc. Lond. A 146, 501–523.Google Scholar
Youngren, G. K. & Acrivos, A. 1976 On the shape of a gas bubble in a viscous extensional flow. J. Fluid Mech. 76, 433–442.Google Scholar