An important issue is what happens after a singularity has occurred. A very simple example was considered in Section 3.5; now we will turn to some physical examples in more detail. First, we consider the breakup of a liquid drop, which marks the transition from one piece to two separate pieces of liquid. To know how the two pieces evolve after breakup one needs to continue across the singularity. In particular, is the continuation unique or are there several possible ways for the solution to evolve after the singularity? As in Section 3.5 we will consider two very different approaches to continuation. In the first approach we construct a similarity solution that applies directly after the singularity, by solving two separate problems corresponding to the two pieces resulting from breakup. We show that they are determined uniquely by the pre-breakup dynamics. In the second approach we regularize the equations on a small scale, so that a true singularity never occurs.

Post-breakup solution: viscous thread

In Chapter 7 we saw that the asymptotics of drop breakup is described by the similarity solution (7.41). We described in some detail how a unique, stable, similarity solution is selected; this is shown in Fig. 7.3. Now we show how the knowledge of the pre-breakup solution permits us to construct a unique post-breakup solution. The strategy has already been laid out and hinges on the matching conditions (3.84), which transfer information about the behave-ior of the pre-breakup solution far from the pinch point to the post-breakup solution.

However, there remains a technical challenge, illustrated in Fig. 10.1. In the immediate neighborhood of the tip, where the slope h′ of the profile diverges, the solution can no longer be considered slender. As a result the derivation of the slender jet equations (6.57), (6.58) is not valid; the equations break down at the tip. Instead, we must consider a separate tip region, whose width tip is comparable with its radial extension. In the language of matched asymptotics the tip is the inner region, while the receding thread, over which the slender jet approximation applies, is the outer region.

General ideas

This book deals for the most part with partial differential equations, whose basis is to view the world as a continuum. This agrees with our perception of liquids, gases, and many solids; they appear to have no characteristic structure and are perfectly isotropic: no property is linked to a particular point or direction in space. We hasten to add that the reality is often more complicated. A liquid may contain components which possess a significant microstructure, e.g. polymers. Solids are often crystalline, which means that their properties depend on the direction of observation relative to the crystal axes.

It is worth remembering the complexity of any microscopic description of a liquid, gas, or solid, even if the system is extremely small. Consider, for example, the computer simulation of a liquid jet emerging from a nozzle only 6 nm in diameter, shown in Fig. 4.1. Each molecule of the liquid is treated as a mathematical point, with a known force between any two molecules. At very small distances, the molecules repel; at larger distances, they attract. The balance between attraction and repulsion causes the molecules to condense into a liquid state. Even on a scale of a few nanometers, the jet appears essentially like a continuum.

The simulation shown in Fig. 4.1 was obtained by solving Newton's equations for each molecule, taking into account the forces exerted on it by the other molecules. This is feasible for a few hundred thousand molecules by only including interactions with molecules close to the target particle. Thus, from a microscopic point of view, we have an incredibly complicated system of nonlinear ODEs. The only hope of gaining analytical insight lies in developing a continuum description. Here we will emphasize the basic physical principles on which the continuum description of fluids and solids relies, and briefly develop the equations of motion on which the rest of this book is based.

The Navier–Stokes equation

The continuum description of liquids and gases is based on the fundamental laws of conservation of matter and of momentum. Here anything that flows is usually called a fluid and comprises both liquids and gases described by the same flow equations.