Introduction

If a fluid's potential density decreases continuously with height then it can support internal waves that, like interfacial waves, move up and down due to buoyancy forces but which are not confined to an interface: they can move vertically through the fluid. This chapter focuses upon the dynamics of small-amplitude internal waves in uniformly stratified, stationary fluid. It also examines some effects of shear in non-uniform stratification, with a more general treatment given in Chapter 6.

The key quantity that determines the temporal evolution and spatial structure of internal waves is the buoyancy frequency. Heuristic methods based on fluid-parcel arguments are used to define the buoyancy frequency. These help to anticipate the dispersion relation that describes internal waves that are not influenced by rotation. The dispersion relation for Boussinesq and non-Boussinesq waves is then derived for liquids and gases. The results are then extended by including the influence of boundaries, shear and rotation.

The buoyancy frequency

In our consideration of interfacial waves we assumed that the density changed discontinuously across the interface between, say, warm and cold or fresh and salty fluid. In reality, such interfaces are not infinitesimally thin. They are thick, and the density varies continuously from one side of the interface to the other. We can justifiably assume the interface thickness is negligibly small only if the horizontal scale of interfacial disturbances is relatively large. If this is not the case, then the effects of continuous vertical density variations must be considered.

Introduction

The linear theory calculations performed so far have the advantage that analytic solutions can be found for a broad range of circumstances. However, the solutions are strictly valid only in the limit of infinitesimally small amplitude waves. Waves are said to be weakly nonlinear when their amplitude is sufficiently large that nonlinear effects arising from the advection terms in the material derivative begin to play an important role. Using perturbation theory, simplified equations can be derived that capture the leading-order nonlinear effects, referred to as ‘weak nonlinearity’. Analysis of such equations not only reveals how nonlinear effects change the evolution of the waves, but they serve to establish bounds on the wave amplitude for which linear theory is valid.

As well as modifying the structure of small-amplitude waves, nonlinear effects can give rise to new classes of steady waves, such as solitary waves, and they can result in the breakdown of steady waves through mechanisms such as modulational instability and parametric subharmonic instability.

In this chapter we focus primarily upon the weakly nonlinear dynamics of interfacial and internal waves in otherwise stationary fluid; there is no ambient mean flow. For those not familiar with perturbation theory for differential eigenvalue problems, we begin with a brief review of the mathematics necessary for the treatment of weakly nonlinear waves. This analysis shows that frequency is a function of amplitude as well as wavenumber. We then examine how weakly nonlinear effects modify the evolution of interfacial and internal waves.

Introduction

In this chapter we consider waves at the interface between two fluids of different density and waves in multi-layer fluids. We derive the fully nonlinear equations for waves of arbitrary amplitude and then determine approximate formulae appropriate for small-amplitude waves. These are expressed through coupled linear partial differential equations for which analytic solutions may be found through standard methods. Approximate solutions for nonlinear waves are considered in Chapter 4.

A ‘one-layer fluid’ has uniform density and is bounded above by a free surface that may oscillate up and down, for example, due to gravitational forces. Waves in a one-layer fluid are specifically referred to here as ‘surface waves’. Waves on the ocean surface can be treated as existing in a one-layer fluid if the density variation with depth in the ocean negligibly affects their dynamics.

A ‘two-layer fluid’ describes a fluid of one density underlying a second fluid of smaller density. ‘Interfacial waves’ propagate at the interface between the two fluids. In one sense, surface waves on the ocean are interfacial waves in that they propagate at the interface between water and air. However, in this book we distinguish surface waves from interfacial waves by requiring for the latter that the density difference between the upper and lower layer fluids is a small fraction of the density of either layer. Thus undular displacements of an interface between fresh and salt water are considered for waves in a two-layer fluid.

This chapter describes a general framework for species and charge transport, which assists us in understanding how electric fields couple to fluid flow in nonequilibrium systems. The following sections first describe the basic sources of species fluxes. These constitutive relations include the diffusivity, electrophoretic mobility, and viscous mobility. The species fluxes, when applied to a control volume, lead to the basic conservation equations for species, the Nernst–Planck equations. We then consider the sources of charge fluxes, which lead to constitutive relations for the charge fluxes and definitions of parameters such as the conductivity and molar conductivity. Because charge in an electrolyte solution is carried by ionic species (in contrast to electrons, as is the case for metal conductors), the charge transport and species transport equations are closely related – in fact, the charge transport equation is just a sum of species transport equations weighted by the ion valence and multiplied by the Faraday constant. We show in this chapter that the transport parameters D, µEP, µi, σ, and ∧ are all closely related, and we write equations such as the Nernst–Einstein relation to link these parameters.

These issues affect microfluidic devices because ion transport couples to and affects fluid flow in microfluidic systems. Further, many microfluidic systems are designed to manipulate and control the distribution of dissolved analytes for concentration, chemical separation, or other purposes.

Vector spaces are essentially linear structures. They are so ubiquitous in pure and applied mathematics that their review in a book such as this is fully warranted. In addition to their obvious service as the repository of quantities, such as forces and velocities, that historically gave birth to the notion of a vector, vector spaces appear in many other, sometimes unexpected, contexts. Affine spaces are often described as vector spaces without a zero, an imprecise description which, nevertheless, conveys the idea that, once an origin is chosen arbitrarily, the affine space can be regarded as a vector space. The most important example for us is the Galilean space of Classical Mechanics and, if nothing else, this example would be justification enough to devote some attention to affine spaces.

Vector Spaces: Definition and Examples

One of the most creative ideas of modern mathematics (starting from the end of the nineteenth century) has probably been to consider particularly important examples and, divesting them of as much of their particularity as permitted without loss of essence, to abstract the remaining structure and elevate it to a general category in its own right. Once created and named, the essential commonality between particular instances of this structure, within what may appear to be completely unrelated fields, comes to light and can be used to great advantage.

Having already established our physical motivation, we will proceed to provide a precise mathematical counterpart of the idea of a continuum. Each of the physical concepts of Continuum Mechanics, such as those covered in Appendix A (configuration, deformation gradient, differentiable fields on the body, and so on), will find its natural geometrical setting starting with the treatment in this chapter. Eventually, new physical ideas, not covered in our Continuum Mechanics primer, will arise naturally from the geometric context and will be discussed as they arise.

Introduction

The Greek historian Herodotus, who lived and wrote in the 5th century bce, relates that the need to reconstruct the demarcations between plots of land periodically flooded by the Nile was one of the reasons for the emergence of Geometry in ancient Egypt. He thus explains the curious name of a discipline which even in his time had already attained the status and the reputation of a pure science. For “geometry” literally means “measurement of the Earth,” and in some European languages to this very day the practitioner of land surveying is designated as geometer. In the light of its Earth-bound origins, therefore, it is perhaps not unworthy of notice that when modern differential geometers were looking for a terminology that would be both accurate and suggestive to characterize the notion of a continuum, they found their inspiration in Cartography, that is, in the science of making maps of the Earth.

This text focuses on the physics of liquid transport in micro- and nanofabricated systems. It evolved from a graduate course I have taught at Cornell University since 2005, titled “Physics of Micro- and Nanoscale Fluid Mechanics,” housed primarily in the Mechanical and Aerospace Engineering Department but attracting students from Physics, Applied Physics, Chemical Engineering, Materials Science, and Biological Engineering. This text was designed with the goal of bringing together several areas that are often taught separately – namely, fluid mechanics, electrodynamics, and interfacial chemistry and electrochemistry – with a focused goal of preparing the modern microfluidics researcher to analyze and model continuum fluid-mechanical systems encountered when working with micro- and nanofabricated devices. It omits many standard topics found in other texts – turbulent and transitional flows, rheology, transport in gel phase, Van der Waals forces, electrode kinetics, colloid stability, and electrode potentials are just a few of countless examples of fascinating and useful topics that are found in other texts, but are omitted here as they are not central to the fluid flows I wish to discuss.

Although I hope that this text may also serve as a useful reference for practicing researchers, it has been designed primarily for classroom instruction. It is thus occasionally repetitive and discursive (where others might state results succinctly and only once) when this is deemed useful for instruction. Worked sample problems are inserted throughout to assist the student, and exercises are included at the end of each chapter to facilitate use in classes.