Stokesian

In an inelastic, time-independent (Stokesian) fluid the extra stre.ss is considered to be a function of the in.stantaneous rate of defomiation (rate of strain). Therefore in this case the fluid does not retain any memory of the history of the deformation which it has experienced at previous stages of the flow. 

In the simplest case of Newtonian fluids (linear Stokesian fluids) the extra stress tensor is expressed, using a constant fluid viscosity p, as... 

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... 

Lieb WR, Stein WD (1986) Non-stokesian nature of transverse diffusion within human red blood cells. J Membrane Biol 92 111-119. 

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

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. 

Bielski, W., Telega, JJ. and Wojnar, R. (1999) Macroscopic Equations for Nonstationary Flow of Stokesian Fluid through Porous Elastic Medium. Arch. Mech. 51, 243-274... 

Telega, J.J. and Bielski, W. (2002) Nonstationary Flow of Stokesian Fluid through Random Porous Medium with Elastic Skeleton. Poromechanics II, 569-574, Lisse Abbington Exton (PA) Tokyo... 

As a reference to something more familiar, consider the case of a fluid where incompressibility is enforced via a Lagrange multiplier. For a Stokesian fluid, it is assumed that the constitutive variables (stress, energy, heat flux) are a function of density, p, temperature, T, rate of deformation tensor, d, and possibly other variables (such as the gradients of density and temperature). Exploiting the entropy inequality in this framework produces the following constitutive restriction for the Cauchy stress tensor [10]... 

Molecular-dynamics-like simulation of suspensions ( Stokesian dynamics )... 

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. 

Four novel approaches to contemporary studies of suspensions are briefly reviewed in this final section. Addressed first is Stokesian dynamics, a newly developed simulation technique. Surveyed next is a recent application of generalized Taylor dispersion theory (Brenner, 1980a, 1982) to the study of momentum transport in suspensions. Third, a synopsis is provided of recent studies in the general area of fractal suspensions. Finally, some novel properties (e.g., the existence of antisymmetric stresses) of dipolar suspensions are reviewed in relation to their applications to magnetic and electrorheolog-ical fluid properties. 

For simplicity, a set of constitutive equations for a Stokesian fluid without memory is... 

Computational methods are increasingly valuable supplements to experiments and theories in the quest to understand complex liquids. Simulations and computations can be aimed at either molecular or microstructural length scales. The most widely used molecular-scale simulation methods are molecular dynamics. Brownian dynamics, and Monte Carlo sampling. Computations can also be performed at the continuum level by numerical solutions of field equations or by Stokesian dynamics methods, described briefly below. 

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. 

I I I I I (j) = 0.45, N = 27 Total Viscosity n,-A Hydrodynamic Viscosity ° Brownian Viscosity yy Figure 6.8 Stokesian dynamics simulation results for the steady shear viscosity at 0 = 0.45 also shown are, the separate contributions of the Brownian and hydrodynamic stresses. (From... 

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. 

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