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The constitutive equation for a dilute emulsion

Published online by Cambridge University Press:  29 March 2006

N. A. Frankel  and
Andreas Acrivos
N. A. Frankel
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, California Present address: Sun Oil Co., Marcus Hook, Pennsylvania.
Andreas Acrivos
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, California
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Abstract

A constitutive equation for dilute emulsions is developed by considering the deformations, assumed infinitesimal, of a small droplet freely suspended in a time-dependent shearing flow. This equation is non-linear in the kinematic variables and gives rise to ‘fluid memory’ effects attributable to the droplet surface dynamics. Furthermore, it has the same form as the corresponding expression for a dilute suspension of Hookean elastic spheres (Goddard & Miller 1967), and reduces to a relation previously proposed by Schowalter, Chaffey & Brenner (1968) when time-dependent effects become small.

Numerical solutions are also presented for the case of a small bubble in a steady extensional flow for the purpose of estimating the range of validity of the small deformation analysis. It is shown that, unlike the drag of a bubble which, in creeping motion, is known to be relatively insensitive to its exact shape, the macroscopic stress field in an emulsion is not well described by the present analysis unless the shapes of the deformed bubbles agree closely with those given by the first-order theory. Thus, the present rheological equation should prove of value in a qualitative rather than a quantitative sense.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 44 , Issue 1 , 21 October 1970 , pp. 65 - 78
Copyright
© 1970 Cambridge University Press

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