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.

Several processes exist in which a high-power electron or laser beam is focused on the surface of a body. If the beam is stationary with respect to the body, a small, perhaps axisymmetric, pool of liquefied metal is formed, as shown in Figure 6.1(a). This state may be steady, though the liquid may undergo convective motions caused by buoyancy or thermocapillary convection.

If the beam is now translated at some speed VT, the pool translates as well, as shown in Figure 6.1(b). Now the pool is asymmetric, and melting takes place ahead of the beam whereas solidification occurs behind it; typically VT ≈ 1 − 10 m/s. At the rear, one might regard the front as undergoing unidirectional solidification at speed VT sin θ, where θ measures the angle between the planer solid surface and the front. At such high solidification rates, new, nonequilibrium microstructures are formed in the solid after the liquid freezes.

Boettinger et al. (1984) observed what seem to be two-dimensional bands in Ag–Cu alloys in which layers of cells (or dendrites or eutectics) and segregation-free material alternate in the growth direction. Figure 6.2b is a sketch of the configuration. Bands are not a mode that emerges from Mullins-Sekerka theory. Since this work, bands have been seen in many metallic alloy systems, as discussed by Kurz and Trivedi (1990).

As VT is increased, the bands disappear and only a segregation-free material is produced, which is consistent with a modified version of the Mullins and Sekerka (1964) theory of morphological instability.

The processes of freezing and melting were present at the beginning of the Earth and continue to affect the natural and industrial worlds. These processes created the Earth's crust and affect the dynamics of magmas and ice floes, which in turn affect the circulation of the oceans and the patterns of climate and weather. A huge majority of commercial solid materials were “born” as liquids and frozen into useful configurations. The systems in which solidification is important range in scale from nanometers to kilometers and couple with a vast spectrum of other physics.

The solidification of a liquid or the melting of a solid involves a complex-interplay of many physical effects. The solid–liquid interface is an active free boundary from which latent heat is liberated during phase transformation. This heat is conducted away from the interface through the solid and liquid, resulting in the presence of thermal boundary layers near the interface. Across the interface, the density changes, say, from ρℓ to ρs. Thus, if ρs > ρℓ, so that the material shrinks upon solidification, a flow is induced toward the interface from “infinity.”

If the liquid is not pure but contains solute, preferential rejection or incorporation of solute occurs at the interface. For example, if a single solute is present and its solubility is smaller in the (crystalline) solid than it is in the liquid, the solute will be rejected at the interface.

Materials Science is an extremely broad field covering metals, semiconductors, ceramics, and polymers, just to mention a few. Its study is dominated by the fabrication of specimens and the characterization of their properties. A relatively small portion of the field is devoted to phase transformation, the dynamic process by which in the present context a liquid is frozen or a solid is melted.

This book is devoted to the study of liquid (melt)-solid transformations of atomically rough materials: metals or semiconductors, including model organics like plastic crystals. The emphasis is on the use of instability behavior as a means of understanding those processes that ultimately determine the micro-structure of a crystalline solid. The fundamental building block of this study is the Mullins–Sekerka instability of a front, which gives conditions for the growth of infinitesimal disturbances of a soild–liquid front. This is generalized in many ways: into the nonlinear regime, including thermodynamic disequilibrium, anisotropic material properties, and effects of convection in the liquid. Cellular, eutectic, and dendritic behaviors are discussed. The emphasis is on dynamic phenomena rather than equilibria. In a sense then, it concerns “physiology” rather than “anatomy.”

The aim of this book is to present in a systematic way the field of continuum solidification theory. This begins with the primitive field equations for diffusion and the derivation of appropriate jump conditions on the interface between the solid and liquid. It then uses such models to explore morphological instabilities in the linearized range and gives physical explanations for the phenomena uncovered.

In all the systems discussed heretofore the solidification front was considered to be a mathematical interface of zero thickness endowed with surface properties deemed appropriate to the physics. In this chapter, another approach is taken. The front is allowed to be diffuse, and the fields of interest, such as T and C, are supposed to have well-defined bulk behaviors away from the interfacial region and rapid, though continuous, variations within it. Minimally, one would wish the model to satisfy the laws of thermodynamics, appropriately extended into the nonequilibrium regions, and regain the interfacial properties and jump conditions appropriate to the thin-interface limit when the interfacial thickness approaches zero.

On the one hand one would anticipate that an infinite number of such models likely exists. On the other hand one would anticipate that the thin-interface limit might be taken a number of ways, each giving distinct properties to the front. Nonetheless, conceptually there are two possible virtues of the diffuse-interface approach. If the models are well chosen on the basis of some underlying framework, then there would be a systematic means of generalizing the models to systems such as rapid solidification. In Chapter 6 high rates of solidification were modeled by appending to the standard model variations k = k(Vn), m = m(Vn) with, for example, the equilibrium Gibbs–Thomson undercooling. A systematic generalization could indicate how the relationships for k and m emerge, and what other alterations to the model should simultaneously be included. Call this “model building.”

Introduction

When an air mass blows over a sharp gradient in surface roughness and/or temperature, the air mass adjusts its turbulence structure to the new set of surface boundary conditions. Flow from a low roughness region to a downstream region of higher roughness produces a local increase in turbulence intensities and fluxes, exhibited by a sharp yet relatively small step function. The same pattern is found also for flow from a cold surface to a warmer surface, as occurs in the region of sea surface temperature fronts. Conversely, flow from a warm water surface to a colder one and/or from high roughness to low roughness, produces a drop in turbulence and fluxes. The drop in turbulence and flux levels is much larger than the increase in the first example. These changes in surface fluxes are more local in nature, i.e. generally covering scales of 10 km or less, and are caused by ocean currents, surface waves, and thermodynamic properties of the ocean associated with changes in depth, eddies, and overlying wind fields.

On the larger scale, the horizontal variability of surface fluxes and flux profiles must also be affected by variabilities due to processes within the atmosphere, e.g. transport and redistribution of flux with eddy scales, shear-induced gradients caused by moving storms, advection and circulation associated with sea breezes, and flow over sloping discontinuities associated with atmospheric fronts.

Introduction

Disequilibrium between the fast air stream and sluggish water drives the irreversible process of air–sea momentum transfer, much as temperature difference drives heat transfer. The rate of transfer per unit area and unit time, the momentum flux, equals the tangential force per unit area applied to the air–sea interface, “drag” for short. In laminar flow viscous shear would be the instrument of momentum transfer, an air-side boundary layer the principal resistance, as a solution of the Navier–Stokes equations reveals. If we could magically contrive to sustain laminar flow above and below the sea surface, we would witness momentum transfer rates some two orders of magnitude lower than we actually find. Hydrodynamic instability of laminar shear flow makes this impossible and brings two important ingredients to the air–sea momentum transfer process: Reynolds flux of momentum, which becomes the main route of momentum transport to and from the air–sea interface, and wind-waves on the interface, which play a prominent role in the momentum handover process from air to water.

Pathways of Air–Sea Momentum Transfer

Turbulence in the air and in the water sustains Reynolds fluxes. Anything that tends to suppress turbulence increases the resistance to momentum transfer, as the example of drag-reduction chemicals shows: introduction of high-polymer substances into the viscous sublayer of a streamlined object (such as a submarine) dissipates turbulence energy, thickens the viscous sublayer, and reduces drag.