Dynamic simulation of hydrodynamically interacting particles

Abstract

A general method for computing the hydrodynamic interactions among N suspended particles, under the condition of vanishingly small particle Reynolds number, is presented. The method accounts for both near-field lubrication effects and the dominant many-body interactions. The many-body hydrodynamic interactions reproduce the screening characteristic of porous media and the 'effective viscosity ' of free suspensions. The method is accurate and computationally efficient, permitting the dynamic simulation of arbitrarily configured many-particle systems. The hydrodynamic interactions calculated are shown to agree well with available exact calculations for small numbers of particles and to reproduce slender-body theory for linear chains of particles. The method can be used to determine static (i.e. configuration specific) and dynamic properties of suspended particles that interact through both hydrodynamic and non-hydrodynamic forces, where the latter may be any type of Brownian, colloidal, interparticle or external force. The method is also readily extended to dynamically simulate both unbounded and bounded suspensions.

Suspensions of prolate spheroids in Stokes flow. Part 1. Dynamics of a finite number of particles in an unbounded fluid

“…At larger separations, the viscous interactions are weaker, and the bodies simply drift apart (figure 4a). Similar effects had been observed for highly symmetric arrangements of spheres (Durlofsky et al 1987); four spheres placed at the corners of a square in the vertical plane, for example, fall in a viscous fluid following a pattern in which the top spheres first move inward and faster than the ones on the bottom, eventually overtaking these to form a new square which is the mirror image of the original configuration. This scenario is repeated ad injinitum in the absence of external perturbations.…”

“…In contrast, its inverse vanishes, and its effect will be swamped by that of all other particles, leading to particle overlap in any scheme using pairwise additivity of velocities (Bossis & Brady 1984).) These observations lie at the heart of the method called Stokesian dynamics (Durlofsky, Brady & Bossis 1987). The N-body mobility tensor is first approximated using the more accurate pairwise additivity of velocities.…”

“…The mobility tensor M is constructed by generalizing the procedure originally applied to dispersions of spheres by Durlofsky et al (1987) (Claeys 1991). The velocity disturbance due to each particle is first decomposed into the contributions of the irreducible moments of the stress density on its surface: at the zeroth order, the effect of the force exerted by the particle on the fluid is quantified; at the next level, the perturbation due to the first moment is evaluated.…”

Rheology of sheared suspensions of rough frictional particles

“…Since they are very short-range in nature, they can usually not be fully resolved with the typical grids used in the numerical model and they consequently require an ad hoc model. Our lubrication model rests on the ideas already used in SD (Brady & Bossis 1988;Durlofsky et al 1987) and FCM (Yeo & Maxey 2010a). Let us consider a system of N p spherical particles suspended in a linear Stokes flow and let U be the 6N p vector of translational/rotational velocities U =( U , Ω) T and F =(F , T ) T the 6N p vector of hydrodynamic forces/torques exerted by the fluid on the particles.…”

Numerical Determination of Truncation Orders in the Correction Method for Stokes Equations

“…Unfortunately, we do not have a nice explicit expression for DN such as (4 A large amount of research has been studied to develop numerical tools to approximate the friction operator, such as [5], [6], [8], [9], [13]. Recently, in [15], the authors developed a method which is called the correction method for computing very accurate numerical solutions.…”