Self-similar solutions for viscous capillary pinch-off

The axisymmetric capillary pinch-off of a viscous fluid thread of viscosity (lambda)(mu) and surface tension (gamma) immersed in a surrounding fluid of viscosity (mu) is studied. Similarity variables are introduced (with lengthscales decreasing like (tau), the time to pinch-off, in a rapidly translating frame) and the self-similar shape is determined directly by a combination of modified Newton iteration and a standard boundary-integral method. A large range of viscosity ratios is studied (0.002<= (lambda)<= 500) and asymmetric profiles are observed for all (lambda), with conical shapes far from the pinching point, in agreement with previous time-dependent studies. The stability of the steady solutions is investigated and oscillatory instability is found for (lambda)>= 32. For (lambda)<< 1 an asymptotic scaling of (lambda)1/2 is suggested for the slopes of the far-field conical shapes. These compare well with the quantitative predictions of a onedimensional theory based on Taylorʹs (1964) analysis of a slender bubble.