Spatially periodic suspensions

When N is infinite, so that suspended particles are dispersed throughout the entire fluid domain, condition (2.5) is replaced by an equivalent one that prescribes the average fields U0 and G. These are explicitly given for spatially periodic suspensions in Section VII. [See also the paragraph following Eq. (2.28).]... 

The foregoing results may be discussed in terms of spatially periodic suspensions, which represent the only exactly analyzable suspension models currently available for concentrated systems. Since spatially periodic models are discussed in the next section, the remainder of this section may be omitted at first reading. 

Second, the spatially periodic model suggests further interpretations and experiments. That no kink exists in the viscosity vs. concentration curve may be related to the fact that the average dissipation rate remains finite at the maximum kinematic concentration limit, ma>. Infinite strings of particles are formed at this limit. It may thus be said that although the geometry percolates, the resulting fields themselves do not, at least not within the context of the spatially periodic suspension model. 

This section begins with an account of spatially periodic suspension models embodying a single particle (a solid sphere in most cases) per unit cell. Rigidity... 

In the present rheological context, lattice deformation may be regarded as arising from the transport of neutrally buoyant lattice points suspended within a macroscopically homogeneous linear shear flow. The local vector velocity field v at a general (interstitial or particle interior) point R of such a spatially periodic suspension can be shown to be of the form... 

The body 6 is illustrated in Fig. 2 for the simple case of a spatially periodic suspension of circular disks subjected to the flow given by Eq. (7.8). It is evident from the figure that any point lying within the body will later be found within the collision disk, contradicting the specified impenetrability condition imposed on these rigid bodies. 

This subsection describes general dynamical properties of the interstitial velocity and pressure fields, from which the instantaneous rheological properties of the spatially periodic suspension are deduced, followed by the requisite time averaging for self-reproducing structures (Adler et al., 1985). As a consequence of Eq. (7.4), the velocity field may be decomposed into respective linear (R G) and spatially periodic (v(R)) contributions. With account taken of the interstitial velocity vector v, the velocity field may be written as... 

The previous analysis may be extended to spatially periodic suspensions whose basic unit cell contains not one, but many particles. Such models would parallel those employed in liquid-state theories, which are widely used in computer simulations of molecular behavior (Hansen and McDonald, 1976). This subsection briefly addresses this extension, showing how the trajectories of each of the particles (modulo the unit cell) can be calculated and time-average particle stresses derived subsequently therefrom. This provides a natural entree into recent dynamic simulations of suspensions, which are reviewed later in Section VIII. 

Consider a spatially periodic suspension whose basic unit cell contains N particles, all of whose sizes, shapes, and orientations may be different. The particulate phase of the system is completely specified geometrically by the values of the 3N spatial coordinates rN, and 3N orientational coordinates eN of the (generally nonspherical) particles, as in Section II,A. A particle is identified by the pair of scalar and vector indices i (i = 1,..., N) and n,... 

Specifics of this approach are outlined in the next few paragraphs. Consider a spatially periodic suspension, one whose density p is everywhere constant and whose kinematic viscosity v(r) is everywhere a spatially periodic function of position, albeit perhaps discontinuous. (Generalization to the case of... 

An overbar denotes coarse-grained quantities. The Newtonian constitutive Eq. (8.20c), in effect, defines the configuration-specific, anisotropic, kinematic viscosity-tetradic T (=vijkl) of the spatially periodic suspension. Subject to the attenuation conditions,... 

Accompanying the impeded particle rotation is the (kinematical) existence of an internal spin field 12 within the suspension, which is different from one-half the vorticity to = ( )V x v of the suspension. The disparity to — 2 between the latter two fields serves as a reference-frame invariant pseudovector in the constitutive relation T = ((to — 12), which defines the so-called vortex viscosity ( of the suspension. Expressions for (( ) as a function of the volume of suspended spheres are available (Brenner, 1984) over the entire particle concentration range and are derived from the prior calculations of Zuzovsky et ai (1983) for cubic, spatially-periodic suspension models. 

It has been shown (Adler et al. 1990) from a rigorous asymptotic, lubrication-theory analysis that lubrication concepts cannot lead to a singular behavior of the viscosity of a spatially periodic suspension in which layers of particles slide past one another. This means that the use of Eq. (9.3.8), for example, which employs lubrication concepts to characterize suspension viscosity is limited to suspensions where particle layering does not take place, for example, where the microstructure is random. 

FIG. 11 Spatially periodic suspensions of fractal aggregates. The aggregate in (a) contains 1024 cubic particles of size a- it was built with the hierarchical model with linear trajectories. The deterministic self-similar flake in (b) is at the third-generation stage with b = 5. 

First, the observed critical concentration l 0.65 may be compared with the maximum kinematic concentration (0max = n/4 a 0.785) possible for a two-dimensional suspension of circular disks undergoing simple shear. That the actual theoretically predicted one may be rationalized in terms of spatially periodic packings allowing the existence of more concentrated systems than disordered packings. According to Berryman... 

Spatially periodic models of suspensions (Adler and Brenner, 1985a,b Adler et al., 1985 Zuzovsky et al, 1983 Adler, 1984 Nunan and Keller, 1984) constitute an attractive subject for theoretical treatment since their geometrical simplicity permits rigorous analysis, even in highly concentrated systems. In particular, when a unit cell of the spatially periodic arrangement contains but a single particle, the underlying kinematical problems can be... 

Consider a suspension composed of an ordered, repetitive, three-dimensional array of identical, rigid spheres immersed in an otherwise homogeneous fluid continuum and extending indefinitely in every direction. From a formal point of view, the lattice A, representing the group of translational self-coincidence symmetry operations of this spatially periodic medium, consists of the set of points... 

Consider the centers of the identical spherical particles of radii a to be instantaneously located at the lattice points R . As such, the simplest geometric state exists, in which only one particle is contained within each unit cell. When the latter suspension is sheared, the three basic lattice vectors 1( (1 = 1,2, 3) (or, equivalently, the dyadic L) become functions of time t. Under a homogeneous deformation, the lattice composed of the sphere centers remains spatially periodic, although its instantaneous spatially periodic configuration necessarily changes with time. 

To conclude this subsection, we expose an interesting paradox arising from the time dependence of the particle configuration. As discussed in Section III, Frankel and Acrivos (1967) developed a time-independent lubrication model for treating concentrated suspensions. Their result, given by Eq. (3.7), predicts singular behavior of the shear viscosity in the maximum concentration limit where the spheres touch. Within the spatially periodic framework, the instantaneous macroscopic stress tensor may be calculated for the lubrication limit, e - 0. The symmetric portion of its deviatoric component takes the form (Zuzovsky et al, 1983)... 

In a companion pair of contributions, Mauri and Brenner (1991a,b) introduce a novel scheme for determining the rheological properties of suspensions. Their approach extends generalized Taylor-Aris dispersion-theory moment techniques (Brenner, 1980a, 1982)—particularly as earlier addressed to the study of tracer dispersion in immobile, spatially periodic media (Brenner, 1980b Brenner and Adler, 1982)—from the realm of material... 

Equation (8.22) constitutes the means whereby the configuration-specific kinematic viscosity of the suspension may be computed from the prescribed spatially periodic, microscale, kinematic viscosity data v(r) by first solving an appropriate microscale unit-cell problem. Its Lagrangian derivation differs significantly from volume-average Eulerian approaches (Zuzovsky et al, 1983 Nunan and Keller, 1984) usually employed in deriving such suspension-scale properties. 

There is, of course, an analogy between the study of a fixed and regular suspension and that of a spatially periodic medium. Dilute suspensions of particles have been intensively investigated by O Brien and coworkers (cf., for instance. Ref. 10 for a recent review of these works). Only a few contributions deal with nondilute suspensions an alternative and efficient approach is to use a cell model as Levine and Neale [11] did in order to take into account the effect of the finite solid volume void fraction. 

T 0 compute the transport properties of these aggregates, a spatially periodic pattern of such particles was created. For the sake of convenience, one can say that the center of gravity of each aggregate is located at the center of each unit cell of size N. a however, because of the periodicity condition, the precise position of the cluster inside the unit cell does not matter. was chosen equal to 32, 48, 64, and 96 usually, is large with respect to the gyration radius, and the resulting solid concentration cj) is smaller than 0.02. The structure of the suspension is represented in Fig. 1 la. 

T 0 have a complete overview of conductivity, it was found useful to determine the macroscopic conductivity Gq of a suspension of uncharged aggregates, which requires solving a simple Laplace equation in the spatially periodic geometry described in Sec. II. 

An illustrative example is a snapshot in Fig. 19 from a periodic-box DNS of a Hquid—solid suspension with monodisperse spherical particles coUiding as a result of the turbulence level imposed via a random forcing technique it was performed by Ten Cate et al (2004) by means of an LB technique. The snapshot shows both the instantaneous Hquid velocity field (denoted by the arrows) and the spatial distribution of the rate of energy dissipation in the liquid (denoted by the color code) along with the instantaneous position of the particles in some cross-sectional area through the periodic box. In addition, Derksen and Van den Akker (2007) demonstrated the usefiilness of the periodic-box DNS implementation for investigating the effect of... 

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