Deformation and breakup of a single slender drop in an extensional flow

Abstract

The deformation and conditions for breakup of a single slender drop placed symmetrically in a uniaxial extensional flow are examined theoretically. For the case of an inviscid drop in zero-Reynolds-number flow, Buckmaster (1972) showed, using slender-body analysis, that the shape of the drop is given by r≡εR(z)=ε(1−|z|^ν)/2ν, where εεγ/Gμl and γ is the interfacial tension, G the strength of the extensional flow, μ the viscosity of the suspending fluid and l the drop half-length; also v = ½P − 1, where P is the unknown constant pressure inside the drop rendered dimensionless with respect to Gμ. By requiring that R be analytic at z = 0, Buckmaster then concluded that v had to be an even integer and thereby obtained a countably infinite set of slender profiles for any (large) value of the flow strength G. In the present work, the expression for R(z) shown above is obtained readily using the method of inner and outer expansions, the method failing when |z|[les ]O(ε) and ν is not an even integer. Thus, in general, a new solution is needed to describe the shape within the ‘singular’ region |z|[les ]O(ε). The requirement that the two solutions match in their domain of overlap then leads to the conclusion that v can be either equal to 2 or greater than or equal to 3. However, a stability analysis reveals that only the solution with v = 2 is stable, and hence a unique shape exists.

Next, drops of low viscosity μ_i = O(ε²μ) are examined in zero-Reynolds-number flow. Here, again, a unique solution is obtained according to which a steady shape cannot exist if (Gμa/γ) (μ_i/μ)^⅙ > 0·148, where $a \equiv(3V4\pi)^{\frac{1}{3}}$ and V is the volume of the drop. This breakup criterion is identical to that found by Taylor (1964). A similar analysis for the case of an inviscid drop in a flow with non-zero Reynolds number shows that drop breakup will occur if $(G\mu a/\gamma) (\rho a\gamma/\mu^2)^{\frac{1}{5}} > 0.284$, where ρ is the density of the suspending fluid. Finally, when μ_i = O(ε²μ) and inertial effects are neglected within the drop but retained in the surrounding fluid, the critical value of $(G\mu a/\gamma) (\rho a\gamma/\mu^2)^{\frac{1}{5}}$ required for drop breakup is found as a function of the dimensionless group \[ (\rho a\gamma/\mu^2)^{\frac{1}{5}}(\mu/\mu_i)^{\frac{1}{6}}, \] which depends only on the physical properties of the system and the size of the drop. These last two results are the first which take into account inertial effects in determining the deformation and breakup conditions of a drop placed in a shear field.