Stokesian particles

Here u is the particle velocity, U/ i.s the local fluid velocity, and / is the Stokes friction coefficient. We call particles that obey this equation of motion Stokesian particles. The use of (4.2S) is equivalent to employing (4.19), neglecting the acceleration terms containing the gas density. Because (4.19) was derived for rectilinear motion, the extension to flows with velocity gradients and curved streamlines adds further uncertainty to this approximate method. 

For Stokesian particles, rvi o impaction regimes are similar when the Stokes, interception, and Reynolds numbers are the same. The impaction efficiency. as in the case of diffusion, is defined as the ratio of the volume of gas cleared of particles by the collecting element to the total volume swept out by the collector. (Refer to Fig. 4.5 for the case of the cylinder.) If all panicles coming within one radius of the collector adhere, then we obtain... 

This analysis provides a lower anchor point for curves of impaction efficiency as a function of Stokes number. It applies also to non-Stokesian particles, discussed in the next section, because the point of vanishing efficiency corresponds to zero relative velocity between particle and gas. Hence Stokes law can be used to approximate the particle motion near the stagnation point. This is one of the few impaction problems for which an analytical solution is possible. 

An approximate analysis of the particle motion and cyclone performance can be carried out by setting up a force bulance for Stokesian particles in the radial direction ... 

Small particles in a turbulent gas dilfuse from one point to another as a result of the eddy motion. The eddy diffusion coefficient of the particles will in general differ from that of the carrier gas. An expression for the particle eddy diffusivity can be derived for a Stokesian particle, neglecting the Brownian motion. In carrying out the analysis, it is assumed that the turbulence is homogeneous and that there is no mean gas velocity. The statistical properties of the system do not change with time. Essentially what we have is a stationary, uniform turbulence in a large box. This is an approximate representation of the core of a turbulent pipe flow, if we move with the mean velocity of the flow. 

The analysis is similar to that used in Chapter 2 to derive the Stokes-Einstein relation for the diffusion coefficient. Again we consider only the one-dimensional problem. Particles originally present in the differential thickness around, v = 0 (Chapter 2) spread through the fluid a a result of the turbulent eddies. If the particles are much smaller than the size of the eddies, the equation of particle motion for Stokesian particles, based on (4.24) (see associated discussion), is... 

The drag coefficient for non-Stokesian particles can be represented by the expression... 

Turbulence may also lead to coagulation as a re.sull of inertial effects. When particles of different sizes (masses) are present in the same accelerating eddy, a relative motion is induced between the particles that may lead to collision. Again the scale of the particle motion is confined to distances < X. The mean. square relative velocity between the particles can be approximated using the force balance for Stokesian particles (Chapter 4) ... 

The particles in the wall region have a gravitationally induced vertically downward velocity corresponding to the terminal velocity Upy, of a spherical Stokesian particle (see expression (6.3.1)) of radius Cp. 

Sometimes the theoretical or computational approach to description of molecular structure, properties, and reactivity cannot be based on deterministic equations that can be solved by analytical or computational methods. The properties of a molecule or assembly of molecules may be known or describable only in a statistical sense. Molecules and assemblies of molecules exist in distributions of configuration, composition, momentum, and energy. Sometimes, this statistical character is best captured and studied by computer experiments molecular dynamics, Brownian dynamics, Stokesian dynamics, and Monte Carlo methods. Interaction potentials based on quantum mechanics, classical particle mechanics, continuum mechanics, or empiricism are specified and the evolution of the system is then followed in time by simulation of motions resulting from these direct... 

For a spherical particle with radius a moving at low Reynolds number, the drag force is Stokesian,... 

Finally, some distortion of the particle size distribution is caused by deviations from Stokes law during particle sedimentation. These deviations are a function of the particle size such that the retardation of Stokesian settling becomes larger with increasing particle size. This effect can be evaluated theoretically, however, and corrections can be made to determine size distributions experimentally. 

Gravitational sedimentation causes a change in the particle size distribution anywhere in and below the cloud compared with the size distribution at stabilization time. Thus, to reconstruct the size distribution at stabilization time, corrections must be applied to the size distributions measured in the samples. These corrections were calculated by assuming Stokesian settling modified by a drag slip correction. It was assumed further that at stabilization time the cloud was axially symmetric and consisted of spherical particles. Wind and diffusion effects were neglected. 

In Stokesian dynamics (Section VIII), a direct simulation is made starting with a randomly chosen initial particle configuration. The structure is allowed to evolve as part of the detailed fluid-mechanical solution, and the hydro-dynamic particle interactions are determined at least to the extent of assuming pairwise additivity of these interactions. The momentum tracer method (Section VIII) is characterized by the interesting feature that the particulate phase of the suspension is at rest. The static configuration of this suspension is... 

A second prominent feature here is the ergodic character (or lack thereof) of the process, depending on the rationality or irrationality of <. This leads inevitably to the fascinating question, Does a real system choose between these values of , and if so, how The boundaries themselves remain neutral with respect to the choice of whenever they are compatible with the flow. Thus, for a slide flow, the walls must be parallel to the slides, whereas for a tube flow, they must be parallel to the tube. In both cases there remains an additional degree of freedom, which is precisely the choice of f. Other examples of indeterminancy arise from the neglect of fluid and particle inertia, as already discussed in Section I (see also the review in Leal, 1980). Whether or not inclusion of nonlinear inertial effects can remove the above indeterminacy, as it often does for the purely hydrodynamic portion of the problem, is a question that lies beyond the scope of the present (linear) Stokesian context. 

For dense suspensions of spherieal particles, an espeeially aecurate method ealled Stokesian dynamics has been developed by Bossis and Brady (1989). In Stokesian dynamics, one solves a generalized form of Eq. (1-40), in which the simple Stokes law for the drag force on sphere i, = — (x/ — v ), is replaced by a more accurate tensor expression that accounts for the hydrodynamic interactions—that is, the disturbances to the solvent velocity field produced by the relative motions of the other spheres. The Stokesian dynamics method accounts for hydrodynamic interactions among widely separated spheres by a multipole expansion, as well as for closely spaced ones by a lubrication approximation. Results from this method appear in Figs. 6-8 and 8-8. 

Figure 8.8 Relative viscosity t]/t s versus Mason number for an ER fluid consisting of hydrated lithium poly(methacrylate) particles in a chlorinated hydrocarbon studied by Marshall et al. (1989) with = 0.23 at various field strengths, compared to predictions of two-dimensional Stokesian dynamics simulations (closed symbols) with and without near-field (NF) interactions at areal fraction = 0.4. Since p in the above was taken from the polarization model with Eq. (8-2), while the experiments were carried out under dc fields for which the effective polarization should be controlled by conduc-tivities [Eq. (8-2a)], the quantitative agreement between simulations and experiment is presumably...

Other samples of alkaline slurry were subjected to particle size analysis by sedimentation. With the —43 xm + 1.2 [im fraction this analysis was done in a 50-mm-diameter settling column of dilute slurry with a tared pan at the base to record continuously the mass of sedimented solid. The data were analyzed by the method of Oden (8), and the particle size distribution (Stokesian diameter), expressed on a mass percent basis, was calculated. 

A typical output from the sedimentation balance for —43 xm + 1.2 xm material is shown in Figure 1. The occurrence of distinct peaks indicates that groups of closely sized particles are present, the smallest being about 6 xm in effective (Stokesian) diameter. The frequent occur-... 

The particle size distribution for the humic acid fraction is depicted in Figure 4. No material sedimented out until the most extreme conditions were applied (40,000 rpm for 24 hr), when some lightening of color at the top of the solution was observed. The sedimented particles had a Stokesian diameter of around 2 nm, which means that a particle size gap of three orders of magnitude exists between these and the next largest particles detected (5 xm). From the experimentally determined coal particle density of 1.43 g/cm, it was calculated that a solid sphere of diameter 2 nm would have a molecular mass of 4000. If the molecules were rod-shaped, even smaller molecular masses would be predicted. Literature values of the molecular mass of regenerated humic acids range between 800 and 20,000, with the values clustering around 1,000 and 10,000 (i5, 16, 17). 

No particles exist in alkali-digested coal solutions between 6 xm and 2 nm Stokesian diameter. 

The fluid-particle interaction closures applied in the modern single particle momentum balances originate from the classical work on the Newton s second law as applied to a small rigid sphere in an unsteady, non-uniform flow limited to Stokesian flow conditions Rep [Pg.554]

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