Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-04T07:20:35.643Z Has data issue: false hasContentIssue false
  • English
  • Français

Breakage of a drop of inviscid fluid due to a pressure fluctuation at its surface

Published online by Cambridge University Press:  26 April 2006

Shreekumar ,
R. Kumar  and
K. S. Gandhi
Shreekumar
Affiliation:
Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India
R. Kumar
Affiliation:
Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore, India
K. S. Gandhi
Affiliation:
Department of Chemical Engineering, Indian Institute of Science, Bangalore 560012, India
Get access Rights & Permissions [Opens in a new window]

Abstract

In this work, an attempt is made to gain a better understanding of the breakage of low-viscosity drops in turbulent flows by determining the dynamics of deformation of an inviscid drop in response to a pressure variation acting on the drop surface. Known scaling relationships between wavenumbers and frequencies, and between pressure fluctuations and velocity fluctuations in the inertial subrange are used in characterizing the pressure fluctuation. The existence of a maximum stable drop diameter dmax follows once scaling laws of turbulent flow are used to correlate the magnitude of the disruptive forces with the duration for which they act.

Two undetermined dimensionless quantities, both of order unity, appear in the equations of continuity, motion, and the boundary conditions in terms of pressure fluctuations applied on the surface. One is a constant of proportionality relating root-mean-square values of pressure and velocity differences between two points separated by a distance l. The other is a Weber number based on turbulent stresses acting on the drop and the resisting stresses in the drop due to interfacial tension. The former is set equal to 1, and the latter is determined by studying the interaction of a drop of diameter equal to dmax with a pressure fluctuation of length scale equal to the drop diameter. The model is then used to study the breakage of drops of diameter greater than dmax and those with densities different from that of the suspending fluid.

It is found that, at least during breakage of a drop of diameter greater than dmax by interaction with a fluctuation of equal length scale, a satellite drop is always formed between two larger drops. When very large drops are broken by smaller-length-scale fluctuations, highly deformed shapes are produced suggesting the possibility of further fragmentation due to instabilities. The model predicts that as the dispersed-phase density increases, dmax decreases.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 328 , 10 December 1996 , pp. 1 - 17
Copyright
© 1996 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.)

Footnotes

Author to whom correspondence should be addressed.

References

Arai, K., Konno, M., Matunga, Y. & Saito, S. 1977 Effect of dispersed phase viscosity on the maximum stable drop size for break-up in turbulent flow. J. Chem. Engng Japan 10, 325–330.Google Scholar
Aris, R. 1962 Vectors, Tensors and Basic Equations of Fluid Mechanics. Prentice-Hall.
Baker, G. R., Meiron, D. I. & Orszag, S. A. 1980 Vortex simulations of the Rayleigh-Taylor instability. Phys. Fluids 23, 1485–1490.Google Scholar
Baker, G. R., Meiron, D. I. & Orszag, S. A. 1982 Generalized vortex methods for free-surface flow problems. J. Fluid Mech. 123, 477–501.Google Scholar
Baker, G. R., Meiron, D. I. & Orszag, S. A. 1984 Boundary integral methods for axisymmetric and three-dimensional Rayleigh-Taylor instability problems. Physica 12D, 19–31.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bentley, B. J. & Leal, L. G. 1986 An experimental investigation of drop breakup in steady two-dimensional flows. J. Fluid Mech. 167, 241–283.Google Scholar
Calabrese, R. V., Chang, T. P. K. & Dang, P. T. 1986 Drop breakup in turbulent stirred-tank contactors. Part I: effect of dispersed phase viscosity. AIChE J. 32, 657–666.Google Scholar
Coulaloglou, C. A. & Tavlarides, L. L. 1977 Description of interaction process in agitated liquid-liquid dispersions. Chem. Engng Sci. 32, 1287–1297.Google Scholar
Davies, J. T. 1985 Drop sizes of emulsions related to turbulent energy dissipation rates. Chem. Engng Sci. 40, 839–842.Google Scholar
Hinze, O. 1955 Fundamentals of hydrodynamic mechanisms of splitting in dispersion processes. AIChE J. 1, 289–295.Google Scholar
Kolmogorov, A. N. 1949 On the disintegration of drops in turbulent flow. Dokl. Akad. Nauk. 66, 825–828.Google Scholar
Lagisetty, J. S., Das, P. K., Kumar, R. & Gandhi, K. S. 1986 Breakage of viscous and non-newtonian drops in stirred dispersions. Chem. Engng Sci. 41, 65–72.Google Scholar
Lam, A., Sathyagal, A., Kumar, S. & Ramkrishna, D. 1996 On the concept of maximum stable drop diameter in stirred vessels. AIChE J. 42, 1547–1552.Google Scholar
Laso, M., Steiner, L. & Hartland, S. 1987 Dynamic simulation of liquid-liquid dispersions. I. Derivation of a simplified model. Chem. Engng Sci. 42, 2429–2436.Google Scholar
Levich, V. G. 1962 Physicochemical Hydrodynamics, 2nd Edn. Prentice-Hall.
Lundgren, T. S. & Mansour, N. N. 1988 Oscillations of drops with zero gravity with weak viscous effects. J. Fluid Mech. 194, 479–510.Google Scholar
Mansour, N. N. & Lundgren, T. S. 1990 Satellite formation in capillary jet breakup. Phys. Fluids A 2, 1141–1144.Google Scholar
Nambiar, D. K. R., Kumar, R., Das, T. R. & Gandhi, K. S. 1992 A new model for breakage frequency of drops in turbulent stirred dispersions. Chem. Engng Sci. 47, 2989–3002.Google Scholar
Narsimhan, G., Nejfelt, G. & Ramkrishna, D. 1984 Breakage functions for droplets in agitated liquid-liquid dispersions. AIChE J. 30, 457–467.Google Scholar
Oboukhov, A. M. 1949 Pressure pulsations in a turbulent flow. Dokl. Akad. Nauk. SSSR 66, 17.Google Scholar
Stone, H. A., Bentley, B. J. & Leal, L. G. 1986 An experimental study of transient effects in the breakup of viscous drops. J. Fluid Mech. 173, 131–158.Google Scholar
Stone, H. A. & Leal, L. G. 1989 Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid. J. Fluid Mech. 198, 399–427.Google Scholar
Taylor, G. I. 1934 The formation of emulsion in definable fields of flow. Proc. R. Soc. Lond. A 146, 501–523.Google Scholar
Tennekes, H. & Lumley, J. L. 1972 A First Course in Turbulence.. MIT Press.
Tjahjadi, M., Stone, H. A. & Ottino, J. M. 1992 Satellite and subsatellite formation in capillary breakup. J. Fluid Mech. 243, 297–317.Google Scholar