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 longrange 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 new method is applied to problems in the rheology of both structured and random suspensions, and accurate results are obtained with much larger numbers of particles. With access to larger N, the high-frequency dynamic viscosities and short-time self-diffusivities of random suspensions for volume fractions above the freezing point are now studied. 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 the method can readily be extended to other low-Reynolds-number-flow problems. IntroductionNumerical simulations of the behaviour of suspensions of particles provide a valuable tool for understanding many complex rheological phenomena. Through simulations both the macroscopic suspension properties and the suspension microstructure have been studied, and insight into structure-property relationships has been obtained Foss & Brady 2000). Determining the hydrodynamic interactions among particles in Stokes flow (small Reynolds number), however, can be a complicated and computationally expensive task, mainly owing to the long-range nature of the interactions and the presence of strong lubrication effects when particles are in close proximity to each other. The fluid velocity disturbance caused by a particle on which a net external force acts decays as 1/r, where r is the distance from the particle, and therefore the hydrodynamic interactions cannot be truncated and no simple pairwise-additive approximation can be made. In addition, the presence of lubrication effects makes conventional numerical techniques (such as the boundary-integral technique) expensive computationally when two particles approach each other.Durlofsky, Brady & Bossis (1987) developed a method that successfully accounts for both the many-body interactions and the near-field lubrication effects by splitting the hydrodynamic interactions into a far-field mobility calculation and a pairwise additive resistance calculation. The main advantage of the method is that a relatively small number of unknowns per particle is sufficient to solve many dynamic simulation problems adequately. The main disadvantage of the method, however, is that it

No abstract

Self-diffusion in a monodisperse suspension of non-Brownian particles in simple shear flow is studied using accelerated Stokesian dynamics (ASD) simulation. The availability of a much faster computational algorithm allows the study of large systems (typically of 1000 particles) and the extraction of accurate results for the complete shear-induced self-diffusivity tensor. The finite, and often large, autocorrelation time requires the mean-square displacements to be followed for very long times, which is now possible with ASD. The self-diffusivities compare favourably with the available experimental measurements when allowance is made for the finite strains sampled in the experiments. The relationship between the mean-square displacements and the diffusivities appearing in a Fokker-Planck equation when advection couples to diffusion is discussed.

The axisymmetric capillary pinch-off of a viscous fluid thread of viscosity $\lambda\mu$ and surface tension $\gamma$ immersed in a surrounding fluid of viscosity $\mu$ is studied. Similarity variables are introduced (with lengthscales decreasing like $\tau$, the time to pinch-off, in a rapidly translating frame) and the self-similar shape is determined directly by a combination of modified Newton iteration and a standard boundary-integral method. A large range of viscosity ratios is studied ($0.002\,{\leq}\,\lambda\,{\leq}\, 500$) and asymmetric profiles are observed for all $\lambda$, with conical shapes far from the pinching point, in agreement with previous time-dependent studies. The stability of the steady solutions is investigated and oscillatory instability is found for $\lambda \ge 32$. For $\lambda\,{\ll}\, 1$ an asymptotic scaling of $\lambda^{1/2}$ is suggested for the slopes of the far-field conical shapes. These compare well with the quantitative predictions of a one-dimensional theory based on Taylor's (1964) analysis of a slender bubble.

After capillary pinchoff of a fluid thread or drop, the newly created drop tips recoil due to the large local curvature. Similarity solutions for the postpinchoff recoil of an axisymmetric inviscid fluid of density 1 and surface tension ␥ immersed in a surrounding fluid of density 2 are obtained over a range of the density ratio Dϭ 2 / 1 . The far-field shape of the two new drops and the far-field dipole potentials are prescribed from known prepinching solutions ͓D. Leppinen and J. R. Lister, Phys. Fluids 15, 568 ͑2003͔͒ and the positions and self-similar shape of the two recoiling tips are calculated. The momentum of the prepinching flow makes a significant difference to the recoiling shapes. Capillary waves are observed, in agreement with previous two-dimensional studies and analytical calculations, and the wave frequency is found to increase with D. The recoil of a single axisymmetric drop ͑with a conical far-field shape͒ under surface tension is also studied as a function of D and the far-field cone angle 0 . Capillary waves are again observed, and the results for small values of 0 are shown to agree well with previous asymptotic predictions. The related problem of violent jet emission, following the formation of a near-conical structure with very high curvature at its tip, is also discussed and its similarity with the recoiling cone problem investigated.