Sphere motion

Atapattu, D. D., Chhabra, R. P. and Uhlherr, P. H. T. J. Non-Newt. Fluid Mech. 59 (1995) 245. Creeping sphere motion in Herschel - Bulkley fluids Flow field and drag. 

Torobin, L. B. and Gauvin, W. H. Can. J. Chem. Eng. 38 (1959) 129, 167, 224. Fundamental aspects of solids-gas flow. Part I Introductory concepts and idealized sphere-motion in viscous regime. Part II The sphere wake in steady laminar fluids. Part III Accelerated motion of a particle in a fluid. 

On Laplace inversion and then inserting the rate kernel into the Noyes expression for the rate coefficient [eqn. (191)], the rate coefficient is seen to be exactly that of the Collins and Kimball [4] analysis [eqn. (25)]. It is a considerable achievement. What is apparent is the relative ease of incorporating the dynamics of the hard sphere motion. The competitive effect comes through naturally and only the detailed static structure of the solvent is more difficult to incorporate. Using the more sophisticated Gaussian approximation to the reactant propagators, eqn. (304), Pagistas and Kapral calculated the rate kernel for the reversible reaction [37]. These have already been shown in Fig. 40 (p. 219) and are discussed in the next section. 

In evaluating the particle resistance due to relative motion to the fluid, it is convenient to begin with the constant velocity case and work to the time-dependent case. Consider the case of a sphere moving in a gravitational field and assume that the gravitational acceleration is in the same direction as the sphere motion. Denote Fz as the resistance. Since the total resistance is due to the surface shear and surface pressure, and the flow is axisymmetric, Fz can be postulated as... 

Reciprocal sphere motion is provided by a linear motor driven by a triangular wave form signal generator. 

Fig. 4 Current recorded during wear experiments for anodic aptplied potential (+2V) in 0.5 M H2SO4 for a sphere motion cycle of 0.2 s...

In the case of creeping sphere motion in Newtonian fluids, the skin and form friction contributions are in the proportion of 2 to 1. This ratio continually decreases with increasing degree of pseudoplasticity, the two components becoming nearly equal at 0.4 these relative proportions, on the other... 

Tuinier R, Dhont JKG, Fan T-H (2006) How depletion affects sphere motion through solutions containing macromolecules. Europhys Lett 75 929-935... 

Ever since the pioneering work of Stokes in 1851, considerable research effort has been expended in studying the hydrodynamic aspects of sphere motion in Newtonian fluids consequently, a voluminous body of information has accrued over the last 150 years or so, dealing with different aspects of sphere motion in incompressible Newtonian fluids. Excellent treatises summarizing the rich literature on this subject are available [e.g., see Happel and Brenner (1965), Clift et al. (1978), O Neill (1981), and Kim and Karrila (1991)]. 

In contrast to the vast literature available on the importance of wall effects on sphere motion in Newtonian and non-Newtonian fluids (Clift et al., 1978 Chhabra, 1993a), no analytical treatments are available which shed any light on the magnitude of the extra retardation effect on nonspherical particles. Therefore, most of the progress in this area has been made through experimental results coupled with dimensional considerations. One can quantify the extent of wall effects in a variety of ways the simplest of all being through a wall factor /, defined as... 

Zhou and Brown measured Dp of stearic-acid-coated silica spheres in polyisobutylene (PIB) chloroform using QELSS(20), as seen in Figure 9.11. Sphere motion was diffusive, with a -dependent linewidth. Inverse Laplace transforms... 

The significance of probe flexibility was examined by Cao, et al, who used QELSS to measure Dp of 32 and 54 nm radius polystyrene spheres, phospho-lipid/cholesterol vesicles, and multilamellar vesicles in aqueous 65 and 1000 kDa polyacrylamides(21). The Dp arose from arelaxation rate that was accurately linear in q -, its concentration dependence was a stretched exponential in c. As seen in Figure 9.12, polyacrylamides are more effective at increasing t] than at reducing Dp, and are more effective at slowing sphere motion than at slowing unilamellar... 

Furthermore, the dynamic equations differ in one substantial respect, namely that colloidal spheres are free to move with respect to each other so long as they do not interpenetrate, but the segments ( beads ) of a single polymer chain are obliged to remain attached to each other for all time. Sphere motion at very large concentrations encounters jamming, in which many spheres all get in each other s way, but polymer chains at far smaller concentrations are said to encounter topological obstacles, similar to those found with a poorly-wound ball of yam. 

Quadrupolar nuclei symmetrically coordinated in ions, such as [ Al(OH2)6] or [ 104] , have no permanent efg to relax them. However, because quadrupolar relaxation is so efficient, they can be markedly affected even by the fluctuating efg s that continually arise from outer-sphere motions. One simplified equation that has been proposed for T iQ) or T2 Q) in the limit of short x is... 

Torobin LB, Gauvin WH Fundamental aspects ofsoHds-gas flow— part 1 introductory concepts and idealized sphere motion in viscous regime, CanJ Chem Eng 3S A29—141,1959a. 

The physical situation of interest m a scattering problem is pictured in figure A3.11.3. We assume that the initial particle velocity v is comcident with the z axis and that the particle starts at z = -co, witli x = b = impact parameter, andy = 0. In this case, L = pvh. Subsequently, the particle moves in the v, z plane in a trajectory that might be as pictured in figure A3.11.4 (liere shown for a hard sphere potential). There is a point of closest approach, i.e., r = (iimer turning point for r motions) where... 

We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

The first of these problems involves relative motion between a rigid sphere and a liquid as analyzed by Stokes in 1850. The results apply equally to liquid flowing past a stationary sphere with a steady-state (subscript s) velocity v or to a sphere moving through a stationary liquid with a velocity -v the relative motion is the same in both cases. If the relative motion is in the vertical direction, we may visualize the slices of liquid described above as consisting of... 

From Eq. (9.1) we see that the viscous force associated with this motion equals [i7(dv/dr)] (area), where the pertinent area is proportional to the surface of the sphere and varies as. This qualitative argument suggests that the viscous force opposing the relative motion of the liquid and the sphere is propor tional to [t7(v /R)] (R ). The complete solution to this problem reveals that both pressure and shear forces arising from the motion are proportional tc 77Rvj., and the total force of viscous resistance is given by... 

This concludes our discussion of the viscosity of polymer solutions per se, although various aspects of the viscous resistance to particle motion continue to appear in the remainder of the chapter. We began this chapter by discussing the intrinsic viscosity and the friction factor for rigid spheres. Now that we have developed the intrinsic viscosity well beyond that first introduction, we shall do the same (more or less) for the friction factor. We turn to this in the next section, considering the relationship between the friction factor and diffusion. 

Eig. 5. Target efficiency of spheres, cylinders, and ribbons. The curves apply for conditions where Stokes law holds for the motion of the particle (see also N j ia Table 5). Langmuir and Blodgett have presented similar relationships for cases where Stokes law is not vaUd (149,150). Intercepts for ribbon or... 

FIG. 6-57 Drag coefficients for spheres, disks, and cylinders =area of particle projected on a plane normal to direction of motion C = over-... 

Hindered Settling When particle concentration increases, particle settling velocities decrease oecause of hydrodynamic interaction between particles and the upward motion of displaced liquid. The suspension viscosity increases. Hindered setthng is normally encountered in sedimentation and transport of concentrated slurries. Below 0.1 percent volumetric particle concentration, there is less than a 1 percent reduction in settling velocity. Several expressions have been given to estimate the effect of particle volume fraction on settling velocity. Maude and Whitmore Br. J. Appl. Fhys., 9, 477—482 [1958]) give, for uniformly sized spheres,... 

Stokes diameter is defined as the diameter of a sphere having the same density and the same velocity as the particle in a fluid of the same density and viscosity settling under laminar flow conditions. Correction for deviation from Stokes law may be necessary at the large end of the size range. Sedimentation methods are limited to sizes above a [Lm due to the onset of thermal diffusion (Brownian motion) at smaller sizes. 

Figure 6 Apparent elastic incoherent structure factor A q(Q) for ( ) denatured and ( ) native phosphoglycerate kinase. The solid line represents the fit of a theoretical model in which a fraction of the hydrogens of the protein execute only vihrational motion (this fraction is given by the dotted line) and the rest undergo diffusion in a sphere. For more details see Ref. 25.

Two physically reasonable but quite different models have been used to describe the internal motions of lipid molecules observed by neutron scattering. In the first the protons are assumed to undergo diffusion in a sphere [63]. The radius of the sphere is allowed to be different for different protons. Although the results do not seem to be sensitive to the details of the variation in the sphere radii, it is necessary to have a range of sphere volumes, with the largest volume for methylene groups near the ends of the hydrocarbon chains in the middle of the bilayer and the smallest for the methylenes at the tops of the chains, closest to the bilayer surface. This is consistent with the behavior of the carbon-deuterium order parameters,. S cd, measured by deuterium NMR ... 

Figure 7.6. A filled. skutterudite antimonide crystal structure. A transition niclal atom (Fc or Co) at the centre of each octahedron is bonded to antimony atoms at each corner. The rare earth atoms (small spheres) are located in cages made by eight octahedra. The large thermal motion of rattling of the rare earth atoms in their cages is believed be responsible for the strikingly low thermal conductivity of these materials (Sales 1997).

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