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.

Driven singularities

In this chapter we discuss two examples of singularities whose motion is controlled by a balance of external driving on a large scale and energy dissipation on a microscopic scale: the first is contact line motion and the second is crack propagation; see Fig. 15.1. As these problems are very different in their physical manifestations they are rarely discussed together, yet their mathematical structures are very similar. In the contact line problem the singularity occurs at the edge of a drop, the so-called contact line. In the crack problem the singularity is at the tip of the crack.

The core of the problem is an intermediate region, which represents the universal singularity and which connects the small and large scales. Toward large scales the intermediate solution matches to an outer solution, which represents the specific geometry of the problem, for example the shape of the elastic body and its loading (in the crack problem) or the shape of the drop (in the contact line problem). Toward small scales the singularity is cut off by phenomena that take place on a microscopic scale and are therefore dependent on the specific system under study.

In both cases it is instructive to consider the problem from the point of view of the energy flux through the system. The energy input is supplied from the large scale and is consumed on the microscopic scale. The balance between the two determines the propagation of the singularity.

A spreading drop

Three-phase contact lines occur very commonly, for example when a drop of water is attached to a windowpane. In this case the drop is bounded by a line where water, glass, and air meet. Very often these contact lines are observed to move; the drop runs down the windowpane or a drop placed on a flat surface spreads. It would seem a straightforward exercise to describe this motion using the equations of fluid motion: inside the fluid drop we solve the Navier–Stokes or Stokes equations, subject to a boundary condition of vanishing shear stress on the free surface and a no-slip condition on the solid. However, solutions to the continuum equations in which the contact line is moving do not exist.

This difficulty arises because at the contact line the velocity field becomes multivalued.

Numerical methods play an important role in our understanding of singularities. Simulations often provide crucial pointers to the local structure of a singularity, and they reveal which physical effects dominate near a singularity. In addition, the analytical descriptions that we are able to obtain encompass only the local structure of the solution. One often has to rely on numerics in order to capture how local singular solutions are connected on the global scale.

Regrettably, the development of numerical codes is often considered a pursuit best left to the specialist. Our aim is to highlight some of the fundamental ideas that go into the numerical description of singular behavior. As an example, we take the description of a capillary bridge of liquid collapsing under gravity (see Fig. 8.1), which we will describe in some technical detail. Two aspects are of particular importance:

Stability Solutions which are close to singular involve a wide range of time scales. As a result, great demands are placed on the stability of the numerical scheme being used. This issue is addressed by using so-called implicit numerical schemes.

Adaptability As the singularity is approached, the solution evolves on smaller and smaller length and time scales. It is crucial that the numerical scheme adapts to these changes, by adjusting the time step and by refining the computational grid in a small region around the singularity.

Finite-difference scheme

A liquid drop is held between two endplates of radius r0 which are a distance L apart; cf. Example 6.4 in Section 6.2. We solve equations (6.59), (6.57) on a grid zi, i= 1, . . ., k, which divides up the computational domain; see Fig. 8.2. Since the total length of the bridge is L, we have z1 = 0 and zk = L. The grid spacing between two points is denoted as below we discuss in more detail how is chosen to represent a given problem accurately. The radius h(z) is represented by its values on this grid.

The word “singularity” is used popularly to describe exceptional events at which something changes radically or where a new structure emerges. In the mathematical language of this book, we speak of a singularity when some quantity goes to infinity. This is usually related to the solution of a differential equation which loses smoothness in that either the unknown itself or its derivatives become unbounded at some point or region of their domain.

Very often a singularity understood in the strict mathematical sense justifies the popular use of the word, since it represents a situation or structure of special interest. For example, a singularity of the curvature lies at the center of a black hole, which is formed after the collapse of a supermassive star, and the universe itself is generally believed to have begun at a singularity. Unfortunately, the real difficulty here lies with the correct physical interpretation of the mathematical solution, which is the reason we have not been able to include examples from general relativity.

Examples of singularities discussed in this book are vortices, such as the flow around the center of a tornado, shock waves generated by the motion of a supersonic plane, caustic lines of intense brightness produced by the focusing of light, and the formation of a drop that results from the discontinuous separation of a liquid mass into two or several pieces.

Starting in the nineteenth century with the study of shock waves, singularities have been investigated on an individual basis. They have remained one of the most exciting research topics in both pure and applied mathematics. For example, two of the seven Millennium Prize problems, proposed by the Clay Foundation, were directly or indirectly related to singularities. The sixth problem was to investigate whether the Navier–Stokes equation, which describes the motion of fluids, does or does not produce any singularities. A related and hotly debated problem poses the same question for the Euler equation, which is the Navier–Stokes equation in the absence of viscosity. Both problems are still to be solved.

The third Millennium Problem, known as the Poincaré conjecture, was solved by G. Y. Perelman while studying the singularities of the partial differential equation describing Ricci flow (similar equations will be studied in Chapter 9 of this book).