Capillary pinch-off in inviscid fluids

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Abstract
The axisymmetric pinch-off of an inviscid drop of density ρ 1 immersed in an ambient inviscid fluid of density ρ 2 is examined over a range of the density ratio D=ρ 2 /ρ 1 . For moderate values of D, time-dependent simulations based on a boundary-integral representation show that inviscid pinch-off is asymptotically self-similar with both radial and axial length scales decreasing like τ 2/3 and velocities increasing like τ −1/3 , where τ is the time to pinch-off. The similarity form is independent of initial conditions for a given value of D. The similarity equations are solved directly using a modified Newton’s method and continuation on D to obtain a branch of similarity solutions for 0⩽D⩽11.8. All solutions have a double-cone interfacial shape with one of the cones folding back over the other in such a way that its internal angle is greater than 90°. Bernoulli suction due to a rapid internal jet from the narrow cone into the folded-back cone plays a significant role near pinching. The similarity solutions are linearly stable for 0⩽D⩽6.2 and unstable to an oscillatory instability for D⩾6.2. Oscillatory behavior is also seen in the approach to self-similarity in the time-dependent calculations. Further instabilities are found as D increases and the steady solution branch is lost at a stationary bifurcation at D=11.8.