Stokesian dynamics

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... 

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... 

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. 

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...

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 principles of Stokesian dynamics were developed by Durlofsky, Brady, and Bossis in the 1980s (Durlofsky et al. 1987 Brady and Bossis 1988). The computational costs increase with the third power of particle number (N ). In 2001, Sierou and Bossis introduced a modified numerical procedure for solving the matrix equations, called the accelerated Stokesian dynamics, which considerably reduced the computational effort (A logAO-... 

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. 

Besides these investigations first attempts to study shear flow have been made, using BD simulations. We should also finally mention that the BD approach can be applied to suspensions as well. For these systems, there exist highly developed simulation schemes, which employ rather accurate diffusion tensors taking into account both the fmiteness of the Brownian particles as well as the hydrodynamic interactions with the periodic images via Ewald sums. For more information on these so-called Stokesian Dynamics , see, e.g.. Ref. 94. Due to the computational complexity, only moderate system sizes can be studied. 

Saffman PG (1965) The lift on a small sphere in a slow shear flow. J Ruid Mech 22 385-400 Schlauch E, Ernst M, Seto R, Briesen H, Sommerfeld M, Behr M (2013) Comparison of three simulation methods for colloidal aggregates in Stokes flow finite elements, lattice Boltzmann and Stokesian dynamics. Comput Huids 86 199-209... 

Seto R, Botet R, Briesen H (2011) Hydrodynamic stress on small colloidal aggregates in shear flow using Stokesian dynamics. Phys Rev E 84 041405... 

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