Computational geometry methods such as Bezier, B-spline, and Non-Uniform Rational B-Splines (NURBS) are widely used in the design of engineering and physics systems (Dierckx, 1993; Farin, 1999; Piegl and Tiller, 1997; Rogers, 2001). At the design stage, the geometry of the systems is defined using computer-aided design (CAD) software. The CAD models then are converted to a finite element (FE) mesh to perform the analysis to determine deformations, stresses, and forces as the result of the applied loads. CAD systems use computational geometry methods that accurately define complex shapes and allow for efficient shape manipulation. These methods have many desirable features and share many of the properties required for the development of accurate analysis methods. Nonetheless, computational geometry methods are used primarily for the system geometric construction.

Because of the limitations of existing FE formulations and the fact that the geometric (i.e., kinematic) descriptions used in them are not equivalent to the geometric description used in CAD systems, there is a recent trend to use computational geometry methods as analysis tools. Although both FE and computational geometry methods are based on polynomial representations, many of the existing FE formulations distort the geometry because of the nature of the nodal coordinates selected. As a result, the geometry of the FE mesh used in the analysis can be different from the geometry defined in the CAD systems. This inconsistency makes the conversion of the CAD model to an analysis model difficult and costly and leads to analysis models that are not consistent with the CAD-geometry models. The two FE formulations discussed in Chapters 6 and 7 were developed to address and remedy this problem. The two formulations can be used as the basis for a successful integration of computer-aided design and analysis (I-CAD-A).

In this chapter, the general kinematic equations for the continuum are developed. The kinematic analysis presented in this chapter is purely geometric and does not involve any force analysis. The continuum is assumed to undergo an arbitrary displacement and no simplifying assumptions are made except when special cases are discussed. Recall that in the special case of an unconstrained three-dimensional rigid-body motion, six independent coordinates are required in order to describe arbitrary rigid-body translation and rotation displacements. The general displacement of an infinitesimal material volume on a deformable body, on the other hand, can be described in terms of twelve independent variables; three translation parameters, three rigid-body rotation parameters, and six deformation parameters. One can visualize these modes of displacements by considering a cube that may undergo an arbitrary displacement. The cube can be translated in three independent orthogonal directions (translation degrees of freedom), it can be rotated as a rigid body about three orthogonal axes, and it can experience six independent modes of deformation. These deformation modes are elongations or contractions in three different directions and three shear deformation modes. It is shown in this chapter that the rotations and the deformations can be completely described using the matrix of the position vector gradients, which in general has nine independent elements. This fact can be mathematically proven using the polar decomposition theorem discussed in the preceding chapter. The deformation at the material points on the body can be described in terms of six independent strain components. These strain components can be defined in the undeformed reference configuration leading to the Green–Lagrange strains or can be defined using the current deformed configuration leading to the Eulerian or Almansi strains. The velocity gradients and the rate of deformation tensor also play a fundamental role in the theory of nonlinear continuum mechanics and for this reason they are discussed in detail in this chapter. The concept of objectivity or frame indifference, which is important in the analysis of large deformations, particularly in formulations that involve the strain rates, is also introduced and will be discussed in more detail in the following chapter of this book. In order to correctly formulate the dynamic equations of the continuum, one needs to develop the relationships between the volume and area of the body in the reference configuration and its volume and area in the current configuration. These relationships as well as the continuity equation derived from the conservation of mass are presented in Section 8 and Section 9 of this chapter. Reynold’s transport theorem, which is used in fluid mechanics, is discussed in Section 10. In Section 11, several examples of simple deformations are presented.

In the preceding chapter, a nonlinear finite element formulation for the large-deformation analysis was presented. This formulation, which is consistent with the motion description used in the theory of continuum mechanics and can be used to correctly describe an arbitrary rigid-body motion, leads to a constant mass matrix and nonlinear vector of elastic forces. The formulation imposes no restrictions on the amount of rotation or deformation within the element, except for the restriction imposed by the order of the interpolating polynomials used. In large-deformation problems, in general, the shape of deformation of the bodies can be complex and this, in turn, necessitates the use of a large number of finite element nodal coordinates in order to be able to correctly capture the geometry of deformation. Therefore, in the analysis of the large-deformation problem using the absolute nodal coordinate formulation discussed in the preceding chapter, one simply selects an adequate number of finite elements and formulates the equations of motion in terms of the element nodal coordinates. There is no need to introduce another reference frame or be concerned with the use of coordinate reduction techniques. The results published in the literature on the absolute nodal coordinate formulation demonstrated that this formulation can be used in modeling very large deformations with relatively small number of finite elements compared to other existing nonlinear finite element formulations.

The use of a full finite element representation to study small-deformation problems is not recommended because such a representation is not the most efficient approach to solve for the small deformations. The geometry of the small deformation of the bodies takes simple forms, and one in this case can develop a lower-dimension model that can be efficiently used to solve this class of problems. Furthermore, in the analysis of small deformations, with a proper selection of the coordinate systems, one can use finite elements, which have smaller number of nodal coordinates. For example, conventional nonisoparametric beam, plate, and shell finite elements, which cannot correctly describe arbitrary rigid-body motion using the element nodal coordinates, can still be used in small-deformation large-displacement formulations. By defining a local linear elasticity problem, linear modes can also be used to further reduce the number of the model degrees of freedom and eliminate high-frequency modes of vibration. The approach that is most widely used to solve the small-deformation, large-rotation problem is called the floating frame of reference formulation. The finite element floating frame of reference formulation, which is discussed in this chapter, was introduced in the early eighties (Shabana and Wehage, 1981; Shabana, 1982) and was the basis for developing new computational algorithms that led to introducing new generation of codes that became known as flexible multibody computer codes, which are widely used in industry, universities, and research institutions.

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.

The purpose of this book is to provide an introduction to suspension dynamics, and so we (the authors) thought it would be good to give some historical (as well as personal) perspective on the study of suspensions. Early development of the subject was largely due to two “schools,” one in England and one in the United States. In England, the subject developed from the fluid mechanical tradition at the University of Cambridge, dating from the work of G. G. Stokes and H. Lamb in the mid- and late-1800s. The subject developed in earnest from the work of George Batchelor and collaborators at Cambridge's Department of Applied Mathematics and Theoretical Physics (DAMTP). In the United States, the development of the discipline took place primarily in chemical engineering departments, largely through the efforts of Andreas Acrivos and a number of his students at the University of California Berkeley, Stanford University, and the City College of New York (CCNY). The authors' approaches to suspensions owe much to these “schools” of suspension dynamics. Élisabeth Guazzelli was introduced to the subject by Bud Homsy at Stanford University and extended interactions with John Hinch of the University of Cambridge. Jeff Morris received his introduction to suspensions as a doctoral student of John Brady (a student of Acrivos) at the California Institute of Technology.