In Chaps. 10 and 11 we introduced the notion of an approximate-solution sequence to the 2D Euler equations. The theory of such sequences is important in understanding the kinds of small-scale structures that form in the zero-viscosity limiting process and also for modeling the complex phenomena associated with jets and wakes. One important result of Chap. 11 was the use of the techniques developed in this book to prove the existence of solutions to the 2D Euler equation with vortex-sheet initial data when the vorticity has a fixed sign.

To understand the kinds of phenomena that can occur when vorticity has mixed sign and is in three dimensions, we address three important topics in this chapter. First, we analyze more closely the case of concentration by devising an effective way to measure the set on which concentration takes place. In Chap. 11 we showed that a kind of “concentration–cancellation” occurs for solution sequences that approximate a vortex sheet when the vorticity has distinguished sign. This cancellation property yielded the now-famous existence result for vortex sheets of distinguished sign (see Section 11.4). In this chapter we show that for steady approximate-solution sequences with L1 vorticity control, concentration–cancellation occurs even in the case of mixed-sign vorticity (DiPerna and Majda, 1988).

We go on to discuss what kinds of phenomena can occur when L1 vorticity control is not known. This topic is especially relevant to the case of 3D Euler solutions in which no a priori estimate for L1 vorticity control is known.

In the first half of this book we studied smooth flows in which the velocity field is a pointwise solution to the Euler or the Navier–Stokes equations. As we saw in the introductions to Chaps. 8 and 9, some of the most interesting questions in modern hydrodynamics concern phenomena that can be characterized only by nonsmooth flows that are inherently only weak solutions to the Euler equation. In Chap. 8 we introducted the vortex patch, a 2D solution of a weak form of the Euler equation, in which the vorticity has a jump discontinuity across a boundary. Despite this apparent singularity, we showed that the problem of vortex-patch evolution is well posed and, moreover, that such a patch will retain a smooth boundary if it is initially smooth.

In Chap. 9 we introduced an even weaker class of solutions to the Euler equation, that of a vortex sheet. Vortex sheets occur when the velocity field forms a jump discontinuity across a smooth boundary. Unlike its cousin, the vortex patch, the vortex sheet is known to be so unstable that it is in fact an ill-posed problem. We saw this expicitly in the derivation of the Kelvin–Helmholtz instability for a flat sheet in Section 9.3. This instability is responsible for the complex structure observed in mixing layers, jets, and wakes. We showed that an analytic sheet solves self-deforming curve equation (9.11), called the Birkhoff–Rott equation. However, because even analytic sheets quickly develop singularities (as shown in, e.g., Fig. 9.3), analytic initial data are much too restrictive for practical application.

Vorticity is perhaps the most important facet of turbulent fluid flows. This book is intended to be a comprehensive introduction to the mathematical theory of vorticity and incompressible flow ranging from elementary introductory material to current research topics. Although the contents center on mathematical theory, many parts of the book showcase a modern applied mathematics interaction among rigorous mathematical theory, numerical, asymptotic, and qualitative simplified modeling, and physical phenomena. The interested reader can see many examples of this symbiotic interaction throughout the book, especially in Chaps. 4–9 and 13. The authors hope that this point of view will be interesting to mathematicians as well as other scientists and engineers with interest in the mathematical theory of incompressible flows.

The first seven chapters comprise material for an introductory graduate course on vorticity and incompressible flow. Chapters 1 and 2 contain elementary material on incompressible flow, emphasizing the role of vorticity and vortex dynamics together with a review of concepts from partial differential equations that are useful elsewhere in the book. These formulations of the equations of motion for incompressible flow are utilized in Chaps. 3 and 4 to study the existence of solutions, accumulation of vorticity, and convergence of numerical approximations through a variety of flexible mathematical techniques. Chapter 5 involves the interplay between mathematical theory and numerical or quantitative modeling in the search for singular solutions to the Euler equations. In Chap. 6, the authors discuss vortex methods as numerical procedures for incompressible flows; here some of the exact solutions from Chaps. 1 and 2 are utilized as simplified models to study numerical methods and their performance on unambiguous test problems.

So far we have discussed classical smooth solutions to the Euler and the Navier–Stokes equations. In the first two chapters we discussed elementary properties of the equations and exact solutions, including some intuition for the difference between 2D and 3D and the role of vorticity. In Chaps. 3 and 4 we established the global existence of smooth solutions from smooth initial data in two dimensions (e.g., Corollary 3.3) and global existence in three dimensions, provided that the maximum of the vorticity is controlled (see, e.g., Theorem 3.6 for details). However, many physical problems possess localized, highly unstable structures whose complete dynamics cannot be described by a simple smooth model.

The remaining chapters of this book deal with mathematical issues related to non-smooth solutions of the Euler equations. This chapter addresses a type of weak solution appropriate for modeling an isolated region of intense vorticity, such as what one might use to model the evolution of a hurricane. In particular, we consider problems that have vorticity that is effectively discontinuous, exhibiting a strong eddylike motion in one region while being essentially irrotational in an adjacent region. To treat this problem mathematically, we must derive a formulation of the Euler equation that makes sense when the vorticity is discontinuous but bounded. We also assume that vorticity can be decomposed by means of a radial-energy decomposition (Definition 3.1) and in particular that it has a globally finite integral.

In this book we study incompressible high Reynolds numbers and incompressible inviscid flows. An important aspect of such fluids is that of vortex dynamics, which in lay terms refers to the interaction of local swirls or eddies in the fluid. Mathematically we analyze this behavior by studying the rotation or curl of the velocity field, called the vorticity. In this chapter we introduce the Euler and the Navier–Stokes equations for incompressible fluids and present elementary properties of the equations. We also introduce some elementary examples that both illustrate the kind of phenomena observed in hydrodynamics and function as building blocks for more complicated solutions studied in later chapters of this book.

This chapter is organized as follows. In Section 1.1 we introduce the equations, relevant physical quantities, and notation. Section 1.2 presents basic symmetry groups of the Euler and the Navier–Stokes equations. In Section 1.3 we discuss the motion of a particle that is carried with the fluid. We show that the particle-trajectory map leads to a natural formulation of how quantities evolve with the fluid. Section 1.4 shows how locally an incompressible field can be approximately decomposed into translation, rotation, and deformation components. By means of exact solutions, we show how these simple motions interact in solutions to the Euler or the Navier–Stokes equations. Continuing in this fashion, Section 1.5 examines exact solutions with shear, vorticity, convection, and diffusion. We show that although deformation can increase vorticity, diffusion can balance this effect.

In all the earlier discussions of morphological instabilities, the driver of the morphological changes was the gradient at the front of a diffusive quantity. For the pure material it was an adverse temperature gradient, and for the binary alloy in directional solidification it was an adverse concentration gradient. It seems clear if fluid flow is present in the liquid (melt) that the heat or solute can be convected and hence that its distribution can be altered. Such alterations would necessarily change the conditions for instability. What makes the situation more subtle is that flow over a disturbed (i.e., corrugated) interface will not only alter “vertical” distributions but will also create means of lateral transport, which can stabilize or destabilize the front. The allowance of fluid flow can create new frontal instabilities, flow-induced instabilities, that can preempt the altered morphological instability and dominate the behavior of the front.

It has been known since time immemorial that fluid flow can affect solidification; cases range from the stirring of a partially frozen lake to the agitation of “scotch on the rocks.” Crystal growers are keenly aware of the importance of fluid flows. Rosenberger (1979) states that “non-steady convection is now recognized as being largely responsible for inhomogeneity in solids.”

When one attempts to grow single crystals, the state of pure diffusion rarely exists. Usually, flow is present in the melt; it may be created by direct forcing or it may be due to the presence of convection. Brown (1988) has given a broad survey of the processing configurations and the types of flows that occur.

In Chapter 9 the discussion of the interaction of fluid flows with solidification fronts focused on how individual cells or dendrites are affected by the motion of the melt. The discussion was confined to the onset, or near onset, of morphological instability and the influence of laminar flows because the systems had small scale and the rate of solidification could be readily controlled. There are many situations in industrial or natural situations in which the systems have large scale and the freezing rates are externally provided.

When the freezing rate is not carefully controlled near M = Mc, the typical morphology present is dendritic (or eutectic) and strongly nonlinear in the parameter space V – C∞ of Figure 3.6. The region within which there are both dendrites and interstitial liquid is called a mushy zone. Such zones should not be described pointwise in the same sense that one would not want to describe flow pointwise in a porous rock. Instead, the zone is treated as a porous region that is reactive in the sense that the matrix melts and the liquid freezes and whose properties are described in terms of quantities averaged over many dendrite spacings. There still is a purely liquid region and a purely solid region, but now they are separated by an intermediate layer, the mushy zone.

When the length scale of the system is large and the fluid is subjected to gravity, the fluid in the fully liquid system will undergo buoyancy-driven convection. The large scale may imply that the Rayleigh number is large enough that the convection is unsteady, laminar, or turbulent.

In Chapter 2 we saw that a spherical front in a pure material in an undercooled liquid is either unstable or not, depending on the size of the sphere. There is no secondary control that allows one to mediate the growth. In Chapter 3, we saw that, in directional solidification of a binary liquid, the concentration gradient at the interface creates an instability; however, there is a secondary parameter, the temperature gradient, that opposes the instability and thus can be used to control the local growth beyond the linearized stability limit. Thus, much attention has been given to directional solidification both experimentally and theoretically, and it is in this chapter that nonlinear theory will be discussed.

There are two approaches to the nonlinear theory, depending upon whether the critical wave number ac of linear theory is of unit order, or is small (i.e., asymptotically zero). In the former case one can construct Landau, Ginzburg–Landau, or Newell–Whitehead–Segel equations to study (weakly nonlinear) bifurcation behavior. This gives information regarding the nature of the bifurcation (sub- or supercritical), the question of wave number selection, the preferred pattern of the morphology, and hence the resulting microstructure. If ac is small one must use a longwave theory that generates evolution equations governing the nonlinear development. Such longwave theories can be weakly or strongly nonlinear, depending on the particular situation. They, too, can then be analyzed to discover the nature of the bifurcation and the selection of preferred wave number and pattern.

Chapter 2 addresses nucleate growth. It was found that a spherical nucleus in an undercooled liquid will melt and disappear if its radius R is smaller than the critical nucleation radius R*. In this case, the curvature is so large that surface energy effects dominate those of undercooling. When R > R*, the sphere will continue to grow, and, as time increases, the effects of surface energy will decrease. When R reaches Rc, a morphological instability causes the spherical interface to become unstable to spatially periodic disturbances, leading to the growth of “bumps” on the interface. Experimental observation shows that the “bumps” grow, become dendritic, and continue to grow until they impact each other or a system boundary. Figure 7.1 shows a single bump that has become dendritic.

The term dendrite does not seem to have an accepted definition in the literature though it does refer to a treelike structure. Here it will be used to denote a two- or three-dimensional structure with side arms. Cells can, as well, be either two- or three-dimensional.

Dendritic growth is likely the most common form of microstructure, being present in all macroscopic castings. In fact, unless limitations of speed (or undercooling) are taken, a melt will usually freeze dendritically. If a sample of dendritically structured material having coarse microstructure is reprocessed, it will crack or otherwise produce defects. However, if the microstructure were fine enough, the reprocessing could proceed without ill effects. In either case, the “ghost” of the dendrites will remain after reprocessing.