Velocity Stokesian

From flow visualization and angular velocity measurements, Poe and Acrivos (P12) concluded that the analysis leading to Eqs. (10-37) and (10-38) is valid only for Roq <0.1, while for Re > 6 a sphere rotates unsteadily and the wake is oscillatory. Theoretical or numerical treatments appear to be lacking beyond the near-Stokesian range until much higher Reynolds numbers. 

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. 

The aerodynamic diameter dj, is the diameter of spheres of unit density po, which reach the same velocity as nonspherical particles of density p in the air stream Cd Re) is calculated for calibration particles of diameter dp, and Cd(i e, cp) is calculated for particles with diameter dv and sphericity 9. Sphericity is defined as the ratio of the surface area of a sphere with equivalent volume to the actual surface area of the particle determined, for example, by means of specific surface area measurements (24). The aerodynamic shape factor X is defined as the ratio of the drag force on a particle to the drag force on the particle volume-equivalent sphere at the same velocity. For the Stokesian flow regime and spherical particles (9 = 1, X drag... 

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. 

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. 

Derivation of a modified equation of motion for the particle that accounts approximately for the non-Stokesian motion of the particles is based on the general expression for the drag on a fixed spherical particle in a gas of uniform velocity, U (Brun et al., 1955) ... 

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. 

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) ... 

Consider convection diffusion toward a spherical particle of radius R, which undergoes translational motion with constant velocity U in a binary infinite diluted solution [3], Assume the particle is small enough so that the Reynolds number is Re = UR/v 1. Then the flow in the vicinity of the particle will be Stoke-sean and there will be no viscous boundary layer at the particle surface. The Peclet diffusion number is equal to Peo = Re Sc. Since for infinite diluted solutions, Sc 10 and the flow can be described as Stokesian for the Re up to Re 0.5, it is perfectly safe to assume Pec 1. Thus, a thin diffusion boundary layer exists at the surface. Assume that a fast heterogeneous reaction happens at the particle surface, i.e. the particle is dissolving in the liquid. The equation of convective diffusion in the boundary diffusion layer, in a spherical system of coordinates r, 6, (p, subject to the condition that concentration does not depend on the azimuthal angle [Pg.128]

Rms Minimal radius of drops settling with Stokesian velocity m... 

More rigorous analytical approaches are based on the individual perturbation fields of the primary particles within the aggregate and the corresponding perturbation forces they cause on the other particles. The Kirkwood-Riseman theory uses simplified and averaged expressions for the velocity and force perturbations (Kirkwood and Riseman 1948 Bloomfield et al. 1967 Hess et al. 1986), and provides relatively simple expressions for the hydrodynamic aggregate size (Chen et al. 1984 Lattuada et al. 2004). Stokesian dynamics simulation, which originally aims at the dynamic behaviour of suspensions (Brady and Bossis 1988), provides a... 

Stokesian dynamics is a numerical technique for simulating the dynamic hehaviour of colloidal suspensions (sedimentation, rheology), where the motions of the individual particles is driven hy Brownian and volume forces (including particle interactions) and coupled by hydrodynamic interaction. In a more general approach than in Eq. (4.69), the hydrodynamic forces are traced back to the generalised particle velocities Vp and the velocity gradients E ... 

The multipole expansions of the velocity field (MVE) and hydrodynamic forces (MFE) as employed for Stokesian dynamics allow for a fairly accurate computation of hydrodynamic behaviour of aggregates, yet they impose restriction on shape and overlap of the primary particles. These restrictions can be Ufted, when the flow field around an aggregate is approximated by numerical techniques. 

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. 

Very central to cyclone technology is the dynamically equivalent particle diameter. This is the diameter of an equi-dense sphere that has the same terminal velocity as the actual particle. Calculating this can be difficult in the range of intermediate Reynolds numbers, or when the Cunningham correction is significant. In the region where Stokes drag law applies, we call it the Stokesian diameter. 

The particle sphericity mainly enters the analysis because it influences the particle terminal velocity. We can account for its effect if we use the Stokesian diameter as a measure of particle size x rather than, for instance, a volume or mass equivalent diameter. We recall from Chap. 2 that the Stokesian (or dynamically equivalent ) diameter is the diameter of a sphere having the same terminal settling velocity and density as the particle under consideration. 

Figure 11.3 The pair distribution function determined by Stokesian dynamics simulation for hard spheres at ( ) = 0.45 and Pe = 1000 shown (a) in the full shear plane and (b) in spherical average as a function of pair separation. In (b), a comparison is made to the radial dependence of the isotropic equilibrium structure, which was shown in a planar view in Figure 11.2. Shear flow is as in Figure 11.2, left to right and increasing velocity up the page.

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