Drop Shapes in Shear from a Second-Order Theory

Abstract

The recent second-order theory of Chaffey and Brenner for the deformation of a drop suspended in an immiscible liquid undergoing slow steady shear flow has a narrow range of validity. The deformation parameter DI (proportional to the product of the velocity gradient, the drop radius, the suspending liquid’s viscosity and the reciprocal of the interfacial tension) must be less than 0.24 if the predicted deformation of drops of low viscosity in Couette flow is to be realistic; for highly viscous drops DI must not exceed 0.1. For all drops in hyperbolic flow and hyperbolic-radial flow DI must be less than 0.22 and 0.24, respectively. The second-order approximation, DII, to the observable deformation ratio D (the difference between the drop’s length and width, divided by their sum) exceeds DI for viscous drops in Couette flow but is slightly smaller than DI for drops of low viscosity. Calculated values of DII deviate from experimental data on D. The second-order theory does predict the lengthening in hyperbolic flow of one drop axis and the shortening of the other two. A new first-order theory by Cox has a much wider range of validity but does not conflict with the second-order theory.