Two-dimensional inviscid pinch-off: An example of self-similarity of the second kind

The pinch-off of a two-dimensional region of inviscid fluid is investigated using numerical and analytical techniques. We find that pinch-off occurs when a sufficiently deformed 2D drop is released from rest. The asymptotic collapse of the pinching region is characterized by an anomalous, nonrational similarity exponent , indicating the existence of self-similarity of the second kind. Numerical solutions of the boundary integral equations show that the height of the pinch region shrinks faster than the width, so that the singularity can be described by a slender approximation. The partial differential equations obtained from this approximation are solved and are consistent with the full boundary integral methods. Furthermore, by casting the partial differential equations into similarity form, we solve a nonlinear eigenvalue problem to obtain the value of the similarity exponent, =0.6869±0.0003. © 2007 American Institute of Physics. DOI: 10.1063/1.2800387