Journal of Fluid Mechanics

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Small-deformation theory for a surfactant-covered drop in linear flows

PETIA M. VLAHOVSKAa1 c1, JERZY BŁAWZDZIEWICZa2 and MICHAEL LOEWENBERGa3

a1 Thayer School of Engineering, Dartmouth College, Hanover, NH 03755, USA

a2 Department of Mechanical Engineering, Yale University, New Haven, CT 06520-8284, USA

a3 Department of Chemical Engineering, Yale University, New Haven, CT 06520-8286, USA

Abstract

A small-deformation perturbation analysis is developed to study the effect of surfactant on drop dynamics in viscous flows. The surfactant is assumed to be insoluble in the bulk-phase fluids; the viscosity ratio and surfactant elasticity parameters are arbitrary. Under small-deformation conditions, the drop dynamics are described by a system of ordinary differential equations; the governing equations are given explicitly for the case of axisymmetric and two-dimensional imposed flows. Analytical results accurate to third order in the flow-strength parameter (capillary number) are derived (i) for the stationary drop shape and surfactant distribution in simple shear and axisymmetric straining flows, and (ii) for the rheology of a dilute emulsion in shear flow which include a shear-thinning viscosity and non-zero normal stresses. For drops with clean interfaces, the small-deformation theory presented here improves the results of Barthès-Biesel & Acrivos (J. Fluid Mech., vol. 61, 1973, p. 1). Boundary integral simulations are used to test our theory and explore large-deformation conditions.

(Received May 07 2008)

(Revised November 27 2008)

Correspondence:

c1 Email address for correspondence: petia.vlahovska@dartmouth.edu

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