Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-18T01:10:01.705Z Has data issue: false hasContentIssue false
  • English
  • Français

Stretching and breakup of droplets in chaotic flows

Published online by Cambridge University Press:  26 April 2006

M. Tjahjadi  and
J. M. Ottino
M. Tjahjadi
Affiliation:
Department of Chemical Engineering, University of Massachusetts, Amherst, MA 01003, USA
J. M. Ottino
Affiliation:
Department of Chemical Engineering, Northwestern University, Evanston, IL 60208, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the stretching and breakup of a drop freely suspended in a viscous fluid undergoing chaotic advection. Droplets stretch into filaments acted on by a complex flow history leading to exponential length increase, folding, and eventual breakup; following breakup, chaotic stirring disperses the fragments throughout the flow. These events are studied by experiments conducted in a time-periodic two-dimensional low-Reynolds-number chaotic flow. Studies are restricted to viscosity ratios p such that 0.01 < p < 2.8.

The experimental results are highly reproducible and illustrate new qualitative aspects with respect to the case of stretching and breakup in linear flows. For example, breakup near folds is associated with a change of sign in stretching rate; this mode of breakup leads to the formation of rather large drops. The dominant breakup mechanism, however, is capillary wave instabilities in highly stretched filaments. Other modes of breakup, such as necking and end-pinching occur as well.

We find that drops in low-viscosity-ratio systems, p < 1, extend relatively little, O (101−102), before they break, resulting in the formation of large droplets that may or may not break again; droplets in systems with p > 1, on the other hand, stretch substantially, O (102–104), before they break, producing very small fragments that rarely break again. This results in a more non-uniform equilibrium drop size distribution than in the case of low-viscosity-ratio systems where there is a succession of breakup events. We find as well that the mean drop size decreases as the viscosity ratio increases.

The experimental results are interpreted in terms of a simple model assuming that moderately extended filaments behave passively; this is an excellent approximation especially for low-viscosity-ratio drops. The repetitive nature of stretching and folding, as well as of the breakup process itself, suggests self-similarity. We find that, indeed, upon scaling, the drop size distributions corresponding to different viscosity ratios can be collapsed into a master curve.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 232 , November 1991 , pp. 191 - 219
Copyright
© 1991 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

Acrivos, A. 1983 The breakup of small drops and bubbles in shear flows. Ann. NY Acad. Sci. 404, 1–11.Google Scholar
Abef, H. & Balachandar, S. 1986 Chaotic advection in a Stokes flow. Phys. Fluids 29, 3515–3521.Google Scholar
Bentley, B. J. & Leal, L. G. 1986 An experimental investigation of drop deformation and breakup in steady two-dimensional linear flows. J. Fluid Mech. 167, 241–283.Google Scholar
Chaiken, J., Chevray, R., Tabor, M. & Tan, Q. M. 1986 Experimental study of Lagrangian turbulence in a stokes flows. Proc. R. Soc. Land. A 408, 165–174.Google Scholar
Franjione, J. G. & Ottino, J. M. 1987 Feasibility of numerical tracking of material lines and surfaces in chaotic flows. Phys. Fluids 30, 3641–3643.Google Scholar
Grace, H. P. 1971 Dispersion phenomena in high viscosity immiscible fluid systems and application of static mixers as dispersion devices in such systems. 3rd Engng Found. Res. Conf. Mixing, Andover, NH. (Published in Chem. Engng Commun. 14 (1982), 225–277.)Google Scholar
Hohenberg, P. C. & Halperin, B. I. 1977 Theory of dynamical critical phenomena. Rev. Mod. Phys. 49, 435–479.Google Scholar
Khakhar, D. V. & Ottino, J. M. 1987 Breakup of cylindrical fluid threads in linear flows. Intl J. Multiphase Flow 13, 71–86.Google Scholar
Kuhn, W. 1953 Spontane Aufteilung von Flüssigkeitszylindern in kleine Kugeln. Kolloid Z. 132, 84–99.Google Scholar
Lee, W. K. & Flumerfelt, R. W. 1981 Instability of stationary and uniformly moving cylindrical fluid bodies — I. Newtonian systems. Intl J. Multiphase Flow 7, 363–384.Google Scholar
Leong, C. W. & Ottino, J. M. 1989 Experiments on mixing due to chaotic advection in a cavity. J. Fluid Mech. 209, 463–499.Google Scholar
Mansour, N. N. & Lundgren, T. S. 1990 Satellite formation in capillary jet breakup. Phys. Fluids A 2, 1141–1144.Google Scholar
Meakin, P. 1988 Fractal aggregates and their fractal measures. In Phase Transitions and Critical Phenomena (ed. C. Domb & J. L. Lebowitz), vol. 12, pp. 335–489. Academic.
Mikami, T., Cox, R. & Mason, R. G. 1975 Breakup of extending liquid threads. Intl J. Multiphase Flow 2, 113–118.Google Scholar
Muzzio, F. J. & Ottino, J. M. 1989 Dynamics of a lamellar system with diffusion and reaction: scaling analysis and global kinetics. Phys. Rev. A 40, 7182–7192.Google Scholar
Muzzio, F. J., Swanson, P. D. & Ottino, J. M. 1991 The statistics of stretching and stirring in chaotic flows. Phys. Fluids A 3, 822–834.Google Scholar
Muzzios, F. J., Tjahjadi, M. & Ottino, J. M. 1991 Self-similar drop size distributions produced by breakup in chaotic flows. Phys. Rev. Lett. 67, 54–57.Google Scholar
Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press.
Ottino, J. M. 1990 Mixing, chaotic advection, and turbulence. Ann. Rev. Fluid Mech. 22, 207–253.Google Scholar
Rallison, J. M. 1984 The deformation of small viscous drops and bubbles in shear flows. Ann. Rev. Fluid Mech. 16, 45–66.Google Scholar
Rallison, J. M. & Acrivos, A. 1978 A numerical study of the deformation and burst of a viscous drop in an extensional flow. J. Fluid Mech. 89, 191–209.Google Scholar
Rumscheidt, F. D. & Mason, S. G. 1961 Particle motions in sheared suspensions. XII. Deformation and burst of fluid drops in shear and hyperbolic flow. J. Colloid Sci. 16, 238–261.Google Scholar
Romscheidt, F. D. & Mason, S. G. 1962 Breakup of stationary liquid threads. J. Colloid Sci. 17, 260–269.Google Scholar
Stone, H. A. & Leal, L. G. 1989a Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid. J. Fluid Mech. 198, 399–427.Google Scholar
Stone, H. A. & Leal, L. G. 1989b The influence of initial deformation on drop breakup in subcritical time-dependent flows at low Reynolds number. J. Fluid Mech. 206, 223–263.Google Scholar
Swanson, P. D. & Ottino, J. M. 1990 A comparative computational and experimental study of chaotic mixing of viscous fluids. J. Fluid Mech. 213, 227–249.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
Tjahjadi, M. & Ottino, J. M. 1990 Breakup and dispersion of highly stretched droplets in a chaotic flow field. Phys. Fluids (Gallery of Fluid Motion) A 2, 1524.Google Scholar
Tomotika, S. 1935 On the stability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid. Proc. R. Soc. Lond. A 150, 322–337.Google Scholar
Tomotika, S. 1936 Breaking up of a drop of viscous liquid immersed in another viscous fluid which is extending at a uniform rate. Proc. R. Soc. Lond. A 153, 302–318.Google Scholar
Wannier, G. H. 1950 A contribution to the hydrodynamics of lubrication. Q. Appl. Maths 8, 1–32.Google Scholar