In the preceding chapters, we studied integrable systems and their perturbations. We noted that integrability is rare among dynamical systems, and that, while the perturbative approach is quite successful in any finite order, the perturbation series cannot be counted on to converge in the generic case. As we shall soon see, the perturbative convergence problem can be overcome if the perturbation is small enough and certain other hypotheses are satisfied, thanks to the famous theorem of Kolmogorov, Arnol'd, and Moser (KAM) [26, 27, 28]. There are several approaches (none of them easy!) to the statement and proof of this theorem. In this chapter we will rely mainly on that of [28]. A helpful discussion of the theorem, without detailed proofs, can be found in [29].

Perhaps the main message of the KAM theorem is that if we label the invariant n-tori of the unperturbed integrable model by the n oscillation frequencies ω1, …, ωn, and if the perturbation is weak enough, then a fraction, arbitrarily close to unity, of the tori will be preserved. This is the main result concerning “order” in Hamiltonian systems. No comparably strong statement exists concerning what replaces those tori which break up under the perturbation. Here we rely mainly on numerical investigations in a variety of models. These suggest certain universal features, principally island chains and deterministic chaos.

In the present chapter we will introduce the KAM theorem in the context of nonlinear stability of equilibrium states.

This is a textbook on classical Hamiltonian dynamics designed primarily for students commencing graduate studies in physics. The aim is to cover all essential topics in a relatively concise format, without sacrificing the intellectual coherence of the subject, or the conceptual precision which is the sine qua non of advanced education in physics.

Encouraged by my colleagues at New York University, I have taken it as a pedagogical challenge to create a textbook suitable for a twenty-first-century course of duration no more than one semester (at NYU, the material is covered in about two-thirds of a semester). To do so, I have chosen to limit the scope of the book in certain important ways. It is assumed that the student has already had a course in which Newtonian mechanics, in both F = ma and Lagrangian versions, has been systematically developed and applied to a standard array of soluble examples: the harmonic oscillator, the simple pendulum, the Kepler problem, small oscillations (normal modes), and rigid-body motion. In the present book, the Hamiltonian formulation in phase space is introduced at the outset and applied directly to the same familiar systems.

Topics usually found in more encyclopedic textbooks, but omitted from the present treatment, include dissipative systems, nonholonomic constraints, special and general theories of relativity, continuum mechanics, and classical field theory. A further choice I have made is to limit the use of advanced differential geometry.

In this chapter, our primary purpose is to go beyond Stokes flow to tackle the very difficult problem of understanding the influence of fluid inertia on particle-laden flows. Specifically, the issue of interest is the effect of inertia at the particle scale. Following the structure of the preceding two chapters, we consider first the influence of inertia on sedimentation, and then on shear flows of particle-laden fluid, where we will also consider the rheological consequences of inertia. Inclusion of inertia changes the form of the equation of motion, and even weak inertia can have singular effects when large domains are considered; for both sedimentation and shear, we provide a sketch of results obtained using the singular perturbation method of matched asymptotic expansions in the limit of weak inertia, i.e. at small Reynolds number. While Stokes flow is a good approximation near the particle, a pronounced change in symmetry of the disturbance flow caused by the particle is seen if we are far enough away, as the fore–aft symmetry of Stokes flow is completely lost in this “far-field” region.

We can only give an outline of the subject of inertial suspension flow, as most issues are far from completely resolved. In the previous two chapters, the issues which remain unclear are primarily collective, whereas the microhydrodynamic theory is well-established. For inertial suspensions, the level of understanding at the microscopic, i.e. single and pair, level is incomplete. Hence understanding of collective phenomena based on the microscopic physics is not well-developed and may expand rapidly.

Finally in this book, we would like to broaden the discussion to topics where the understanding is less clear. The future of the subject will involve study of these open questions, but we do not intend to suggest that the list of topics that we are discussing is all-inclusive, or even to suggest these topics as priorities. Instead we seek to provide some indication of the scope of activities for which the concepts developed in this book may find future use.

Moving toward open questions While some of the issues discussed in the last chapters are mostly settled (or perhaps will be resolved soon), there remain greater challenges in many areas of suspension flows. We are perhaps touching on the more obvious of issues which come to mind following the exposition in the preceding chapters, and thus we likely miss novel avenues of study. Nonetheless, a list of issues in suspensions where many open questions remain includes:

Dense suspensions: Flow of suspensions approaching the maximum packing limit is often referred to as “dense suspension flow” and this condition raises special issues which we have only noted briefly in this book. In particular, for such mixtures, the particle surfaces are likely to make enduring contacts, and the details of surface roughness and friction coefficient will play a role in the behavior. How such contact forces interact with hydrodynamic lubrication forces in dense suspensions, and the relation of dense suspension flow to dense granular flow in which the interstitial fluid is a gas, are open questions of interest.

In Part I, we have presented the basis of microhydrodynamics. In its broad definition, microhydrodynamics represents the theory of viscous fluid flows at small spatial scales. For the purposes of the present book on suspensions, we have specifically considered the single- and pair-body dynamics of small particles immersed in viscous fluid. In Part II, we will describe macroscopic phenomena encountered in flows involving a large number of particles interacting through viscous fluid. We will also provide an introduction to methods developed for understanding and (hopefully) predicting certain macroscopic phenomena in suspensions in terms of the microscopic concepts described in Part I, combined with ideas that fall in the realms of statistical physics and dynamical systems.

In this transitional chapter, we are concerned primarily with introducing statistical techniques and concepts from stochastic processes which we will apply in the following chapters. We will also briefly consider the related issue of chaotic dynamics.

Statistical physics

The theoretical framework for relating microscopic mechanics to macroscopic or bulk properties is statistical physics or statistical mechanics. Understanding systems made up of many interacting particles is far from trivial, and the difficulty involved is not just a mere question of solving the hydrodynamic equations with better, faster computers. The collective interactions between the particles can give rise to quite unexpected qualitative behavior, often much simpler than the microscopic motions seem to suggest.