Stokesian fluid

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... 

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]... 

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

Given the apparent arbitrariness of the assumptions in a purely continuum-mechanics-based theory and the desire to obtain results that apply to at least some real fluids, there has been a historical tendency to either relax the Newtonian fluid assumptions one at a time (for example, to seek a constitutive equation that allows quadratic as well as linear dependence on strain rate, but to retain the other assumptions) or to make assumptions of such generality that they must apply to some real materials (for example, we might suppose that stress is a functional over past times of the strain rate, but without specifying any particular form). The former approach tends to produce very specific and reasonable-appearing constitutive models that, unfortunately, do not appear to correspond to any real fluids. The best-known example is the so-called Stokesian fluid. If it is assumed that the stress is a nonlinear function of the strain rate E, but otherwise satisfies the Newtonian fluid assumptions of isotropy and dependence on E only at the same point and at the same moment in time, it can be shown (see, e.g., Leigh29) that the most general form allowed for the constitutive model is... 

Before leaving this brief discussion of Stokes contribution to the development of the Navier-Stokes equations, we need to point out that his ideas about fluid behavior were more general than the result indicated by Eq. 1-61 would Indicate. In fact, he defined what is now known as a Stokesian fluid (Arts, Sec. 5.21, 1962) in terms of the following four postulates ... 

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. 

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...

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]

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. 

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... 

Here, a, ft, and 5 are material coefficients that can depend on the thermodynamic state, as well as the invariants of E, namely tr E, det E, and (tr E - tr E2). The Stokesian model appears to be a perfectly obvious generalization of the Newtonian fluid model. However, no real fluid has been found for which the model with 5 f 0 is an adequate approximation. We should perhaps, not be surprised by this result as the examples in the all seem to suggest that the assumptions of isotropy, plus instantaneous and linear dependence on E, all seem to break down at the same time in real, complex fluids. 

A. Sierou, J.F. Brady, Accelerated Stokesian d5mamics simulations. J. Fluid Mech. 448, 115-146 (2001). doi 10.1017/S0022112001005912... 

Schlauch E, Ernst M, Seto R, Briesen H, Sommerfeld M, Behr M (2013) Comparison of three simulation methods for coUoidal aggregates in Stokes flow Einite elements, lattice Boltzmann and Stokesian dynamics. Comp Fluids 86 199... 

Brady JF, Bossis G (1988) Stokesian dynamics. Ann Rev Fluid Mech 20 111... 

Sierou A, Brady JF (2001) Accelerated Stokesian Dynamics simulations. J Fluid Mech 448 115... 

Ichiki K (2002) Improvement of the Stokesian dynamics method for systems with finite number of particles. J Fluid Mech 452 231... 

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