Stokes velocity

In most colloidal suspensions tire particles have a tendency to sediment. At infinite dilution, spherical particles with a density difference Ap with tire solvent will move at tire Stokes velocity... 

For example, for equal volumes of gas and liquid ( =0.5), Eq. (266) predicts that the Stokes velocity (which is already very small for relatively fine dispersions) should be reduced further by a factor of 38 due to hindering effects of its neighbor bubbles in the ensemble. Hence in the domain of high values and relatively fine dispersions, one can assume that the particles are completely entrained by the continuous-phase eddies, resulting in a negligible convective transfer, although this does not preclude the existence of finite relative velocities between the eddies themselves. 

The foregoing expressions give the suspension velocity (Fs) relative to the single particle free settling velocity, V0, i.e., the Stokes velocity. However, it is not necessary that the particle settling conditions correspond to the Stokes regime to use these equations. As shown in Chapter 11, the Dallavalle equation can be used to calculate the single particle terminal velocity V0... 

Consider the case of a solid sphere falling through a stagnant fluid, in which the sphere is soluble. This is a problem that could be solved on the computer, taking into consideration the change of shape of the sphere due to the differences of mass transfer at different locations. Even if that were the objective, we should do a modeling study before the more detailed analysis and simulation. We assume that the sphere falls under gravity and attains the Stokes velocity at all times we shall return to examine this assumption later. Thus its downward velocity is... 

The droplet simultaneously experiences a frictional force due to the dynamics of the surrounding fluid that opposes its movement. Under laminar flow conditions, the frictional force is given by Ff = 67tr orv, where ri0 is the shear viscosity of the medium and vthe velocity with which the droplet moves. Under steady-state conditions, the so-called Stokes velocity (v) emerges from the force balance ... 

The nature of the lower vesicular zone is not particularly dependent on flow thickness beyond size compression due to lava overburden. As bubbles rise to escape the rising lower crystallization front, the size of the largest bubble caught depends on the velocity of the front, and once the velocity (slowing with the square-root of time like a cooling half space) is reduced below the Stokes velocity of the smallest bubbles in the distribution, all can escape and the lower boundary of the massive zone (Sahagian et al. 1989) is defined at that point. This is true of any flow thickness, so that the only factor that controls the nature of the lower vesicular zone (relative to that of the upper vesicular zone, which is much more complex) is the overlying pressure of the lava. A thicker flow would result in proportionally smaller size mode, which is the basis of the entire analysis for paleoelevation. 

As the front descends and slows further, bubbles that started lower in the flow have time to reach the front. These bubbles may have escaped the lower crystallization front and had time on their way upward to interact and coalesce the larger bubbles catching and joining with smaller ones with lower terminal Stokes velocities. These coalesced bubbles reach the upper crystallization front and lead to the formation of a highly vesicular zone that includes the largest bubbles in the flow. Below that zone, only the smaller... 

Inserting equation (6.7) in equation (6.2) gives the relationship between the particle diameter and its Stokes velocity ... 

As t approaches infinity, u approaches the Stokes velocity = pg/X as given in equation (6.8). 

At high pressures, for particles much larger than X, the discontinuity effect gives rise to slip between the particle and the gas leading to the following modification to the Stokes velocity ... 

For moderately concentrated suspensions, 0.2 xp> 0.01, the sedimentation is reduced as a result of hydrodynamic interaction between the particles, which no longer sediment independently of each other. The sedimentation velocity, v, can be related to the Stokes velocity by the following equation. 

In this case, account must be taken of the hydrodynamic interaction between the droplets, which reduces the Stokes velocity to a value v given, by the following expression [23] ... 

In the case of a small heavy sphere falling through a suspension of large particles (hxed in space), we have A. > 1 the respective expansions, corresponding to Equation 5.265, were obtained by Fuentes et al. ° hi the opposite case, when A. 1, the suspension of small background spheres will reduce the mean velocity of a large heavy particle (as compared with its Stokes velocity ) because the suspension behaves as an effective fluid of larger viscosity as predicted by the Einstein viscosity formula. ... 

Similarly, the leading-order approximation in the vicinity of B is the Stokes velocity field that satisfies the boundary condition... 

Brenner (1980) has explored the subject of solute dispersion in spatially periodic porous media in considerable detail. Brenner s analysis makes use of the method of moments developed by Aris (1956) and later extended by Horn (1971). Carbonell and Whitaker (1983) and Koch et al. (1989) have addressed the same problem using the method of volume averaging, whereby mesoscopic transport coefficients are derived by averaging the basic conservation equations over a single unit cell. Numerical simulations of solute dispersion, based on lattice scale calculations of the Navier-Stokes velocity fields in spatially periodic structures, have also been performed (Eidsath et al., 1983 Edwards et al., 1991 Salles et al., 1993). These simulations are discussed in detail in the Emerging Areas section. 

Polydispersity. In practice, we are often interested in the amount of material arriving in the sediment (or cream) layer per unit time. An instrumental relation results if v is divided by the maximum sedimentation distance (height of the liquid) H by using the Stokes velocity we obtain... 

FIGURE 13.12 Creaming rate relative to the Stokes velocity v/vs of O-W emulsions of various volume fraction (p and of average droplet size under gravity ( ) or in a centrifuge (O)- (Results from the author s laboratory.)... 

This relation coincides with the boundary condition for a viscous flow around solid spheres. In this approximation the velocity distribution at Re l is expressed by Stokes formula. From Stokes velocity distribution v(z,0) it is easy to calculate the viscous stresses acting on the surface of the sphere and the equilibrating surface tension gradient... 

In the formulation of the convective diffusion equation a Stokes velocity field can be taken into account and the weak motion of the bubble surface is neglected. This can be justified only at a sufficiently high degree of retardation. This can be estimated from the results as follows. 

The main distinction of the theory of a dynamic adsorption layer formed under weak and strong retardation arises when formulating the convective diffusion equation. At weak retardation the Hadamard-Rybczynski hydrodynamic velocity field is used while at strong retardation the Stokes velocity field. Different formulas for the dependence of the diffusion layer thickness on Peclet numbers are obtained. The problem of convective diffusion in the neighbourhood of a spherical particle with an immobile surface at small Reynolds numbers and condition (8.74) is solved, so that the well-known expression for the density distribution of the diffusion flow along the surface can be used. As a result, Eq. (8.10) takes the form (Dukhin, 1982),... 

The same has to be done with condition (8.92). Clearly, the effect of a surfactant on surface motion is small, if the established surface tension gradient is small as compared with the viscous stress gradient calculated on the basis of Stokes velocity distribution... 

Substituting v for a potential velocity field, Sutherland s formula is obtained. When we substitute Stokes velocity field into Eq. (10.25) and use Eq. (10.22), we obtain... 

Johnson et al. (B12) followed Friedlander s (F2) solution based on the Stokes velocity profiles around solid particles, and numerically calculated the external coefficients for Ar,. < 1. Only a slight difference in the Nusselt number was observed when the velocity profiles of Stokes and Hadamard were postulated. These calculations showed that the transfer coefficient ratio of drops and solids increases from 1 for (Ape), = 1 to 3 for (Ape) = 10 . In the absence of oscillation, similar results may be expected at moderately higher Reynolds numbers (Cl, H3). 

X terms require detailed knowledge of the Stokes velocity fields themselves. In dimensional form the and terms are independent of viscosity, and are directly proportional to the fluid density. 

Here, 8 denotes the Dirac delta function, and I the idemfactor [cf. Eqs. (193)-(194)]. Physically, the Cartesian tensor equivalent of V, namely, represents the rth component of the Stokes velocity field at r. due to a unit point force exerted on the fluid at r in the yth direction. 

O Brien (Ola) applied the Acrivos-Taylor analysis to the case where the Reynolds number, though small, is not identically zero as in the Stokes flow case. The analysis is vastly more complicated because, to any order in the Reynolds number, the velocity field v appearing in Eq. (298) is now expressed in terms of two, locally valid expansions [the inner and outer expansions of v given by the Proudman-Pearson analysis (PI 1)], rather than the single Stokes velocity field. For a solid spherical particle she obtains for small Pe and Re... 

This relation is self-obvious, since at the initial moment, particles in an infinite diluted suspension precipitate with Stokes velocity. 

The drop sinks in the liquid with Stokes velocity under the action of gravity without getting deformed. The expression for the drop s velocity was derived in [2] and is written as... 

Denote the minimum radius of drops by R s, provided that they descend with Stokes velocity (see the formula (18.16)) ... 

Here t is the residence time of mixture in the separator is the time it takes for the drop with radius Ra to leave the layer of height Du if this drop moves with the Stokes velocity u o = 2gApR /9p( -... 

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