A numerical study of the deformation and burst of a viscous drop in an extensional flow

Abstract

We study the deformation and conditions for breakup of a liquid drop of viscosity λμ freely suspended in another liquid of viscosity μ with which it is immiscible and which is being sheared. The problem at zero Reynolds number is formulated exactly as an integral equation for the unknown surface velocity, which is shown to reduce to a particularly simple form when Δ = 1. This equation is then solved numerically, for the case in which the impressed shear is a radially symmetric extensional flow, by an improved version of the technique used, for Δ = 0, by Youngren & Acrivos (1976) so that we model the time-dependent distortion of an initially spherical drop. It is shown that, for a given Δ, a steady shape is attained only if the dimensionless group Ω ≡4πGμa/γ lies below a critical value Ωc(Δ), where G refers to the strength of the shear field, a is the radius of the initial spherical drop and γ is the interfacial tension. On the other hand, when Ω > Ωc the drop extends indefinitely along its long axis. The numerical results for Δ = 0·3, 0·5, 1, 2, 10 and 100 are in good agreement with the predictions of the small deformation analysis by Taylor (1932) and Barthès-Biesel & Acrivos (1973) and, at the smaller Δ, with those of slender-body theory (Taylor 1964; Acrivos & Lo 1978).