Bubble growth and collapse in viscoelastic liquids analyzed

A. C. Papanastasiou, L. E. Scriven, C. W. Macosko

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35 Scopus citations

Abstract

Time-dependent, purely radial, incompressible, irrotational, extensional flow around a spherical bubble growing or collapsing in viscolastic liquid is analyzed by means of a new integral constitutive equation of BKZ type designed for any mixture of extensional and shear flow. It requires the history of the relative deformation Finger tensor, which is readily evaluated in the spherosymmetric flow with its straight pathlines. The continuity equation relates radial velocity simply to bubble wall speed, and the momentum equation reduces to a nonlinear integrodifferential equation for the evolution of the bubble radius. This equation is discretized by means of linear basis functions on finite elements of time and Galerkin's weighted residual method; the set of nonlinear algebraic equations is solved by Newton iteration, which converges quadratically up to a Deborah number (extension rate at bubble wall times average relaxation time) of 60. Beyond that value the convergence rate slows. Given as initial condition, a sharp change in the difference between pressure in the bubble and ambient pressure, can excite an oscillatory solution. With the constitutive equation fitted to independent measurements of stress in small amplitude sinusoidal oscillations and in steady shear, computed time courses agree with Pearson and Middleman's [1977] experiments on bubble collapse in hydroxypro-pylcellulose and polyacrylamide solutions (which provide only a limited test of the constitutive equation, however). The same growth and collapse processes are also analyzed with several commonly used differential constitutive equations. These are found to give different predictions. For a second-order fluid-generalized or not-there is an exact solution: this solution becomes oscillatory beyond a turning point in Deborah number, the location of which depends on the rheological parameters and the pressure difference imposed. Similar results are obtained with a corotational and an upper-convected Maxwell model.

Original languageEnglish (US)
Pages (from-to)53-75
Number of pages23
JournalJournal of Non-Newtonian Fluid Mechanics
Volume16
Issue number1-2
DOIs
StatePublished - 1984

Bibliographical note

Funding Information:
This research was made possible by grants-in-aid from the 3M Company and Xerox Corporation.

Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

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