Accelerated Stokesian dynamics : development and application to sheared non-Brownian suspensions

Abstract

A new implementation of the conventional Stokesian Dynamics (SD) algorithm, called Accelerated Stokesian Dynamics (ASD), is presented. The equations governing the motion of N particles suspended in a viscous fluid at low particle Reynolds number are solved accurately and efficiently, including all hydrodynamic interactions, but with a significantly lower computational cost of O(N ln N). The main differences from the conventional SD method lie in the calculation of the many-body long-range interactions, where the Ewald-summed wave-space contribution is calculated as a Fourier Transform sum, and in the iterative inversion of the now sparse resistance matrix. The ASD method opens up an entire new class of suspension problems that can be investigated, including particles of non-spherical shape and a distribution of sizes, and can be readily extended to other low-Reynolds-number flow problems. The new method is applied to the study of sheared non-Brownian suspensions. The rheological behavior of a monodisperse suspension of non-Brownian particles in simple shear flow in the presence of a weak interparticle force is studied first. The availability of a faster numerical algorithm permits the investigation of larger systems (typically of N = 512 — 1000 particles), and accurate results for the suspension viscosity, first and second normal stress differences and the particle pressure are determined as a function of the volume fraction. The system microstructure, expressed through the pair-distribution function, is also studied and it is demonstrated how the resulting anisotropy in the microstructure is correlated with the suspension non-Newtonian behavior. The ratio of the normal to excess shear stress is found to be an increasing function of the volume fraction, suggesting different volume fraction scalings for different elements of the stress tensor. The relative strength and range of the interparticle force is varied and its effect on the shear and normal stresses is analyzed. Volume fractions above the equilibrium freezing volume fraction (o ≈ 0.494) are also studied, and it is found that the system exhibits a strong tendency to order under flow for volume fractions below the hard-sphere glass transition; limited results for o = 0.60, however, show that the system is again disordered under shear. Self-diffusion is subsequently studied and accurate results for the complete tensor of the shear-induced self-diffusivities are determined. The finite, and oftentimes large, auto-correlation time requires the mean-square displacement curves to be followed for longer times than was previously thought necessary. Results determined from either the mean-square displacement or the velocity autocorrelation function are in excellent agreement. The longitudinal (in the flow direction) self-diffusion coefficient is also determined, and it is shown that the finite autocorrelation time introduces an additional coupled term to the longitudinal self-diffusivity, a term which previous theoretical and numerical results omitted. The longitudinal self-diffusivities for a system of non-Brownian particles are calculated for the first time as a function of the volume fraction.