Theory of Multicomponent Fluids

In this book, we give a rational treatment of multicomponent materials as interacting continua. We offer two derivations of the equations of motion for the interacting continua; one which uses the concepts of continua for the components, and one which applies an averaging operation to the continuum equations for each component. Arguments are given for constitutive equations appropriate for dispersed multicomponent flow. The forms of the constitutive equations are derived from the principles of continuum mechanics applied to the components and their interactions. The solutions of problems of hydromechanics of ordinary continua are used as motivation for the forms of certain constitutive equations in multicomponent materials. The balance of the book is devoted to the study of problems of hydrodynamics of multicomponent flows.

One view of physical reality is that the matter commonly perceived as filling space in fact consists principally of empty space, with an occasional bit of matter. Such a bit may be called an elementary particle. The distance between the bits of matter and the structure in which they are arranged, as well as their particular arrangement, dictates whether the gross material is perceived as a solid, as a liquid, or as a gas. A bit of matter may itself have a very rich structure, in which case the bit is no longer an elementary particle, but rather is in itself composed of elementary particles. This kind of structure is worthy of study. The scale of such structures is such that they are not accessible for purposes of describing the behavior of materials normally encountered in everyday experience. Neither is such structure interesting for most such purposes. The exceptions are, for example, that a solid is perceived as amorphous or as having a particular crystalline structure based on the symmetry of the arrangement of the matter in it. This affects its gross symmetry, that is, the mechanics of the macroscopic body reflects the arrangement of the bits of matter. The difficulty of scaling such theories up to the size of a body that interests us here makes it desirable to search for an alternative to description at this level of detail. Moreover, theories of such small structures are often revised, because they are not yet completely understood, and both deeper theoretical understanding and more sophisticated experimental work continue to appear. On the other hand, our experience of the gross behavior of the materials of everyday life, such as steel or water, is quite constant. Thus it is extremely rare for us to revise its mathematical description. The result is, for the purposes of modeling phenomena on the scales ordinarily perceived, it is appropriate to devise models that are independent of modern theories of atomic and subatomic physics. Two types of such models are corpuscular models and continuum models.

A body B is a set of particles X that can be mapped into a closed region of three-dimensional physical space at each time. Places in physical space are denoted by x, and times by t. Usually a preferred configuration of the body is chosen, and the location of the particles in that configuration is denoted by X. This configuration is called the reference configuration. It need not be one ever occupied by the body, but it may be. Here for convenience in visualizing results, we let the reference configuration be the configuration of the body at time 0. The symbols X and x do not denote coordinates. Rather, they are positions. Nonetheless they are often called, respectively, material coordinates and spatial coordinates. Popular usage also often assigns them the respective names Lagrangian and Eulerian. Though this terminology is not historically accurate [92], we do sometimes follow the popular usage.

In this chapter, we shall summarize the governing partial differential equations for inviscid fluids, Stokes flow, and elastic solids.

A popular model for gases is based on a corpuscular theory where the gas molecules are modeled as rigid spheres. Random motions of the particle around the mean motion are caused by collisions, which in turn cause transfers of momentum and energy. We also note that the model can be applied to granular materials. However, for practical models of granular materials, the effects of inelastic collisions are important [50].

The major focus of this book is multicomponent materials. The mathematical theory appropriate to such mixtures of materials is still in a stage of development. A simpler type of material is the classical mixture, or solution. In such a material, the components are not physically distinct, that is, the mixing of the materials is at a molecular level. A kinetic theory for such materials was given by Maxwell [62], and was transposed into a form appropriate to continuum theory by Truesdell [91].¹

The mixtures described in Chapter 5 are based on the concept that a mixture may be represented by “a sequence of bodies B _k , all of which ...occupy regions of space ... simultaneously” [89, p. 81]. Examples of such materials are air (a mixture of nitrogen, oxygen, and other materials in small amounts), and whisky (a mixture of water, alcohol, and other materials in small amounts). However, it was commonly recognized at an early stage that so strong an assumption of intermiscibility was not appropriate to all physical situations. For example, soils, porous rock, suspensions of coal particles in water, packed powders, granular propellants, etc., consist of identifiable solid particles surrounded by one or more continuous media, or an identifiable porous matrix through which one or more of the continua are dispersed. Motions of the individual components are possible and, as long as there are no chemical reactions, each constituent retains its integrity¹. We call such materials multicomponent mixtures. They are more complicated than classical mixtures in the sense that they have geometrical structure, but less complicated in the sense that the constituents are not intimately intermixed. A theory describing them should reflect these facts.

At each point x and at each time t, the agglomeration of configurations of B _k forms a composite body B. However, it is useful to construct the body from the viewpoint of physical motivation, because it is the entity most easily observed and measured with conventional instruments. For example, much of the older work on the rheometry of suspensions [24] is based on measurements of B alone. It is also convenient to construct B from the viewpoint of mathematics, principally because boundary conditions are often more easily stated or understood in terms of B than in terms of B _k .

A prime characteristic of many flows of multicomponent materials is that there is uncertainty in the exact locations of the particular constituents at any particular time. For some predictions, this is not important. Often, we are concerned with more gross features of the motion. This means that, for equivalent macroscopic flows, there will be uncertainty in the locations of particular constituents for all times. For instance, consider a suspension of small particles in a liquid. If such a suspension is to be used in, for example, a falling ball viscometer, it might be mixed outside the viscometer, then placed in that device prior to the conduction of an experiment with a falling ball. In a properly conducted experiment, the particles would be approximately uniformly distributed in the fluid. However, there would be no assurance that a particular point in the fluid would contain a particle or not. Experiments conducted with nominally identical fluids in identical viscometers would yield identical gross results, even though the exact initial locations of the suspended particles in the two experiments were noticeably different. As another example, in a sedimenting suspension, the exact distribution of the locations of the particles is immaterial as long as they are reasonably “spread out.” We would not allow the particles to be lumped in some way that is not consistent with the initial conditions appropriate for the flow; neither would we allow the particles to be “paired” so that very near each particle is exactly one neighbor. A more interesting case is the set of all experiments with the same boundary conditions, and initial conditions with some (undefined) properties that we would like to associate with the mean and distribution of the particles and their velocities. We call this set an ensemble. Such ensembles are reasonable sets over which to perform averages because variations in the details of the flows are assured in all situations, while at the same time variations in the gross flows cannot occur.

Here, we define the ensemble average, and give some results pertaining to its application to multicomponent flows.

In many multicomponent flows, the variability that justifies the use of an averaging process to make predictions is manifested in a particular realization as rapid variations in space or time. For example, a time trace of some field may show large fluctuations due to the passing of dispersed units in a fluid, or a photograph may show many similar units within a distance which is small compared to the size of the flow domain.

Averages of balance equations are obtained by taking the product of the balance equations with X _k , then performing the averaging process. We have: .

It is physically plausible that the three-dimensional, unsteady two-fluid model given by (6.12), (6.25), and (6.41)¹, must be supplemented with state equations, constitutive equations, and boundary and initial conditions. That is mathematically plausible also, because of the experience that a physical system must be described by the same number of equations as it has unknowns. Such a system is called a determined system. Underdetermined and overdetermined then have the obvious meaning.

This chapter is devoted to deriving and using microstructural information to find constitutive equations, giving relationships between certain average quantities and the average fields. The physical situation is one in which corpuscular models can be highly useful in motivating the forms of the constitutive equations. In effect, the study of the mechanics of multicomponent fluids necessitates the study of the form of the momentum exchange term and the stress in the momentum equation. The study of these terms is aided by a physically appropriate corpuscular model, consisting of solutions for flows in this model, which are sometimes exact but usually approximate. Once these solutions are found, they can be averaged to find the force of interaction in the continuum equations.

Kinetic theory modeling can be applied to the dispersed component in a dispersed multicomponent flow. It motivates the use of an entropy associated with the dispersed component, and leads to constitutive equations for the dispersed component Reynolds stress.

Multicomponent mechanics is a description of a variety of phenomena, many of which involve the evolution of interfaces separating the components. In some processes the interface may evolve smoothly, remaining approximately planar throughout. In others, the interface may assume complicated shapes such as dimples, dendrites, ripples, and irregular shapes that resemble fractals.

Some general properties of constitutive equations were discussed in Section 14.3. It is clear that any properly formulated theory of multicomponent fluids will be complex, and even in the simplest case will involve a large number of constants. Thus, the accustomed process for, e.g., finding the viscosity of a linearly viscous fluid by simple experimentation, must be superceded by a much more complex process. The current state of the art involves exploration of numerical solutions, extrapolation, exploration of analogies, and often not a small amount of guessing. For instance, additional guidance in formulating realistic constitutive equations can be obtained by assuming that the quantities computed from the “exact” solutions for single large particles in a flow can be taken as forms for constitutive equations. For example, the dynamics of the irrotational flow of an inviscid fluid about a sphere (see Section 15.3) can be used to compute the average force on a sphere and the average interfacial pressure. Constitutive equations arrived at in this way should reduce to the appropriate limit for dilute flows. As is evident from the state of our abilities to calculate the flow around assemblages of bodies of arbitrary shape, we cannot hope to obtain solutions that can be averaged to give appropriate constitutive equations for general motions. The calculated forms from “exact” solutions provide a guide to the appropriate forms for empirical testing. In order to attempt to obtain a system of equations that will predict the flow of a dispersed multicomponent material, it is helpful to find experiments that isolate appropriate phenomena and give (relatively) direct measurements of unknown coefficients involved therein. For best results, the physical experiment should be simple, and should correspond to a simple solution of the equations that depends mostly on the particular constitutive form being evaluated. Examples of such flows are sometimes called “viscometric,” or “separate effects” experiments. We give some of the solutions in Part V. It is clear that simple exact solutions are desirable.

A model that is not properly formulated mathematically cannot describe physical phenomena correctly.¹ Our confidence that our equations, along with our boundary and initial conditions, can correspond with physical experience, is increased if their solutions satisfy the following three prerequisites:

It is common physical experience that when a nondilute multicomponent fluid is placed in a viscometric testing device, the flow that is produced is not a viscometric flow, that is, the velocity gradient is not necessarily constant, and the concentration is not uniform. Instead, relatively thin layers of fluid with relatively few suspended particles accumulate near the boundaries. These layers exhibit high shear rates. The suspended particles accumulate at some distance from the boundaries, giving a high local viscosity and a low shear rate. These phenomena are so mathematically robust that they are exhibited by a large number of theories for suspensions. Here, we consider two properly invariant theories of multicomponent fluids. We investigate the simplest type of viscometric test, steady flow between parallel plates with one plate stationary and the other plate moving parallel to it at constant speed. We also consider steady flow in a channel, with the flow forced by a pressure gradient. For one theory, it is possible to demonstrate exact solutions to the field equations. For the other, plausible approximate solutions are found. Both types of solutions exhibit the phenomenon of a core of concentrated suspension surrounded by a layer of clear fluid.

The study of waves [73] represents a rich and useful branch of mechanics. For fluids, the simplest branch of this study is the investigation of infinitesimal waves in homogeneous compressible fluids with no surfaces.