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1996-2010
J.A. Sethian
• # Pinching Drops

## Overview

While waiting for the plumber to show up (or before fixing it yourself), you might pass the time staring at the drop stretching away from the faucet. Exactly what is happening?

It seems pretty clear that the bottom part stretches and pulls away, and gravity that separates distends and pulls the drop away. But as the little neck of fluid separating the water at the faucet head from the drop stretches, what exactly happens?

In fact, microbubbles can be so small that gravity is not really the key effect in the splitting of a tube. Instead, it is the key role of surface tension and curvature that causes the bubble to break, similar to what is showing in the dynamics of inkjet plotters.

## Zooming In: What happens at the pinch off point?

We want to look at this little tube of fluid, and follow it as it collapses. Some important comments:
• We will look at potential flow for water inside the fluid tube, and air outside of it.
• To make life easier, we will assume that the structure is periodic: you are just looking at one piece of a structure of a fluid tube that repeats itself.
• Surface tension, which is proportional to curvature, is what makes this tube pinch off in the middle. Here, it is mean curvature which matters.
• We interested in devising a numerical method which can pass through the pinch off (change topology), and keep going.
• And once that pinch off happens, surface capillary waves create wild oscillations.

## How will we solve this?

A Level Set Method will allow us to track this boundary of this fluid tube as it moves, all the way through pinch off and beyond. We embed the interface, as well as the velocity and the potential function, all in a higher dimensional implicit framework, and solve the resulting system of equations in a time-stepping procedure.

## Results

The resulting numerical method smoothly tracks the evolution of a fluid tube as it pinches off. The results are unchanged under numerical convergence studies, and match the scaling exponents for asymptotic studies for self-similar pinch.

And they nicely match experiment:

Numerical simulation of droplet pinchoff

Experimental portrait, from Thoroddsen (S.T. Thoroddsen, Micro-droplets and micro-bubbles imaging motion at small scales, Nus. Eng. Res. News 22 (1) (2007)

## Movies

Simulation:
 Large-scale Structure Zoom-in motion

Experiment Movies: (From Thoroddsen)
S.T. Thoroddsen, Micro-droplets and micro-bubbles imaging motion at small scales, Nus. Eng. Res. News 22 (1) (2007)
 Large-scale Structure Zoom-in motion Zoom-in motion

## Details

The calculations were made by coupling a level set method to a boundary integral formulation. The key idea is to start with the equations for potential flow, and to emebed each of the interface, the velocity, and the potential as higher dimensional functions which smoothly default to the correct values on the interface. Thus, as a time-stepping sequence, we execute the following in order: