On the importance of material frame-indifference and lift forces in multiphase flows
Introduction
Multiphase flows have become the subject of considerable attention, especially in the last few decades, because of their increasing importance in many industries. Generally speaking, whenever we have more than one phase, whether the phases are of the same material or of different materials, then we need to use general theories appropriate for these conditions. It might therefore be more accurate to refer to these studies as “multicomponent” problems [in this paper we use the terms multiphase and multicomponent interchangeably]. In multicomponent theories, it is not necessary to have all the components in motion; for example, in composite materials, the different components undergo elastic (or plastic) deformation without involving any motion. In this paper, however, we focus our attention on multicomponent theories applicable to flowing conditions. Historically, two distinct approaches have been used. In the first case, the amount of the dispersed phase is so small that the motion of this phase does not greatly affect the motion of the continuous phase. This is generally known as the “dilute phase approach”, sometimes also called the Lagrangean approach. This method is used extensively in applications such as atomization, sprays, and in flows where bubbles, droplets, and particles are treated as the dispersed phase (cf. Lefebvre, 1989; Sirignano, 1999; Crowe, Sommerfeld, & Tsuji, 1998; Clift, Grace, & Weber, 1978; Sadhal, Ayyaswamy, & Chung, 1997. In the second approach, the two phases are interacting with each other to such an extent that each phase (or component) directly influences the motion and the behavior of the other phase. This is known as the “dense phase approach”, sometimes also called the Eulerian (or the two-fluid) approach. This method is used extensively in fluidization (cf. Davidson, Clift, & Harrison, 1985; Gidaspow, 1994), gas–solid flows (cf. Fan and Zhu, 1998), pneumatic conveying (cf. Marcus, Leung, Klinzing, & Rizk, 1990), suspensions (cf. Govier & Aziz, 1982; Ungarish, 1993), and is described for a variety of application in general textbooks such as Soo (1990), Meyer (1983), Roco (1993), Wallis (1969), Hetsroni (1982), Joseph and Schaeffer (1990) and Afgan (1988).
The large number of articles published concerning multiphase flows typically employ one of the two continuum theories developed to describe such situations: Mixture Theory (or the theory of interacting continua) or averaging (cf. Ishii, 1975). Both approaches are based on the underlying assumption that each phase may be mathematically described as a continuum. Mixture Theory was first formulated by Truesdell (1957). It provides a means for studying the motions of bodies made up of several constituents by generalizing the equations and principles of the mechanics of a single continuum. The fundamental assumption in this theory is that at any instant of time, every point in space is occupied by one particle from each constituent. In contrast, the averaging method directly modifies the classical transport equations to account for the discontinuities or ‘jump’ conditions at moving boundaries between the phases (cf. Anderson & Jackson, 1967; Jackson 1971, Jackson 1997; Drew and Segel 1971a, Drew and Segel 1971b or Ishii, 1975). The modified balance equations must then be averaged in either space or time (hence the name averaging) to arrive at an acceptable local form. In this approach point-wise equations of motion, valid for a single fluid or a single particle, are modified to account for the presence of the other components and the interactions between components. These equations are then averaged over time or some suitable volume that is large compared with a characteristic dimension (for example particle spacing or the diameter of the solid particles) but small compared to the dimensions of the whole system. From the mathematical manipulation of the averaged quantities, a number of terms, some of unknown physical origin, arise. These terms are usually interpreted as some form of interaction between the constituents.
Since the approach that we have taken in this study and our other studies (cf. Johnson, Massoudi, & Rajagopal, 1991; Massoudi, Rajagopal, & Phuoc, 1999; Massoudi, 2002) is based on Mixture Theory, a brief review of the notation and basic equations of Mixture Theory is presented in this section. More detailed information, including an account of the historical development, is available in the articles by Adkins 1963a, Adkins 1963b, Green and Naghdi 1965, Green and Naghdi 1967, Green and Naghdi 1968, Atkin and Craine 1976a, Atkin and Craine 1976b, Bowen (1976), and Bedford and Drumheller (1983), and in the books by Truesdell (1984), Dobran (1991), and Rajagopal and Tao (1995). We consider the mixture of fluid and solid particles to be a purely mechanical system in which thermal, electromagnetic, and chemical reactions are ignored [these effects can of course be included in more general theories of mixtures]. The fluid in the mixture will be represented by S1 and the particles by S2. At each instant of time, t, it is assumed that each point in space is occupied by particles belonging to both S1 and S2. Let and denote the positions of particles of S1 and S2 in the reference configuration. The motion of the constituents is represented by the mappingsThese motions are assumed to be one-to-one, continuous, and invertible. The kinematical quantities associated with these motions are:where denotes velocity, is the acceleration, is the velocity gradient, denotes the symmetric part of the velocity gradient, and is the spin tensor. D1/Dt denotes differentiation with respect to t, holding fixed, and D2/Dt denotes the same operation holding fixed. Also, ρ1 and ρ2 are the bulk densities of the mixture components given bywhere ρf is the density of the pure fluid, ρs is the density of the solid grains, ν is the volume fraction of the solid component, and φ is the volume fraction of the fluid. For a saturated mixture φ=1−ν. The mixture density, ρm is given byand the mean velocity of the mixture is defined by
Section snippets
Governing equations
In order to show where and how constitutive relations are needed in multiphase flow problems, we present the basic equations of motion. In the absence of any chemical, thermal, or electro-magnetic effects, these equations are the conservation laws for mass, linear momentum, and angular momentum.
Forces acting on single particle
In trying to describe the behavior of particles suspended or entrained in a fluid, most researchers resort to the equation of motion of a single (spherical) particle in a fluid (Birkhoff, 1960; Robinson, 1956). Tchen1 (1947) synthesizes the work of Basset (1888), Boussinesq, Stokes (1851), and Oseen on the motion of a sphere settling under the force of gravity in a fluid at rest. The
Frame-independence and constitutive equations
The differences among the materials that make up different bodies are reflected in the theory by constitutive relations. In mechanics, a constitutive relation is a restriction on the forces or the motion of the body or both. This means that a body undergoes a motion when forces act on it, but the motion “caused” depends on the nature of the body. Mathematically, the purpose of the constitutive relations is to supply connections between kinematic, mechanical, and thermal fields that are
Lift forces
It is observed (cf. Segre and Silberberg 1961, Segre and Silberberg 1962a, Segre and Silberberg 1962b) that spheres in laminar Poiseuille flow through a pipe (at low Re) accumulate in an annulus some distance from the tube axis. Following these initial observations, a number of investigators verify this ‘tubular pinch’ effect and attempt to explain the lateral (or lift) force acting on the spheres (cf. Denson, Christiansen, & Salt, 1966; Eichorn & Small, 1964; Jeffrey & Pearson, 1965; Oliver,
A general and frame-invariant relationship for fI
In proposing the constitutive relation for the diffusive body force, or the interaction force, in a multicomponent system, many investigators (cf. Anderson & Jackson, 1967; Drew & Segel, 1971; Homsy, El-Kaissy, & Didwania, 1980) generalize the problem of a single spherical particle undergoing slow rectilinear motion in an infinite fluid by introducing a void fraction dependence for certain coefficients. The interaction force is, in general, a function of the fluid pressure gradient; the density
Summary and conclusions
In this paper we have discussed the importance of the Principle of Material Frame-indifference in multiphase flows. This is especially important in formulation of the virtual mass effects and the lift forces. We have provided the basic governing equations for a particulate mixture. We have shown the importance of lift forces and why they cannot be ignored in many applications. A general procedure using Representation Theorems is provided to derive a general expression for a properly frame
Notation
acceleration vector relative acceleration between components a radius of the sphere interaction coefficients, to 5 body force vector Basset force coefficient virtual mass coefficient symmetric part of velocity gradient interaction force vector F volume fraction dependence of drag identity tensor gradient of velocity vector p fluid pressure R radius of the pipe Reynolds number stress tensor reference velocity component velocity vector spin tensor position vector Greek letters second
Acknowledgements
I would like to thank the two anonymous referees for helpful suggestions and bringing to my attention papers of Asmolov and Cox. I am also grateful to Professor K. R. Rajagopal and Dr. Jack Halow for reading an earlier version of this paper, and for their many helpful and encouraging comments in general. This paper is dedicated to the memory of Professor C. Truesdell whose contribution to Continuum Mechanics is well known. He was and continues to be a source of inspiration to me.
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