A new computational method is presented to resolve hydrodynamic interactions acting on solid particles immersed in incompressible host fluids. In this method, boundaries between solid particles and host fluids are replaced with a continuous interface by assuming a smoothed profile.This enabled us to calculate hydrodynamic interactions both efficiently and accurately, without neglecting many-body interactions. The validity of the method was tested by calculating the drag force acting on a single cylindrical rod moving in an incompressible Newtonian fluid. This method was then applied in order to simulate sedimentation process of colloidal dispersions.

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We study the electric double layer by combining the effects of ion finite size and dielectric decrement. At high surface potential, both mechanisms can cause saturation of the counter-ion concentration near a charged surface. The modified Grahame equation and differential capacitance are derived analytically for a general expression of a permittivity ε(n) that depends on the local ion concentration, n, and under the assumption that the co-ions are fully depleted from the surface. The concentration at counter-ion saturation is found for any ε(n), and a criterion predicting which of the two mechanisms (steric vs. dielectric decrement) is the dominant one is obtained. At low salinity, the differential capacitance as function of surface potential has two peaks (so-called camel-shape). Each of these two peaks is connected to a saturation of counter-ion concentration caused either by dielectric decrement or by their finite size. Because these effects depend mainly on the counter-ion concentration at the surface proximity, for opposite surface-potential polarity either the cations or anions play the role of counter-ions, resulting in an asymmetric camel-shape. At high salinity, we obtain and analyze the crossover in the differential capacitance from a double-peak shape to a uni-modal one. Finally, several nonlinear models of the permittivity decrement are considered, and we predict that the concentration at dielectrophoretic saturation shifts to higher concentration than those obtained by the linear decrement model.

Abstract. Previously, we have proposed a direct simulation scheme for colloidal dispersions in a Newtonian solvent [Phys.Rev.E 71,036707 (2005)]. An improved formulation called the "Smoothed Profile (SP) method" is presented here in which simultaneous time-marching is used for the host fluid and colloids. The SP method is a direct numerical simulation of particulate flows and provides a coupling scheme between the continuum fluid dynamics and rigid-body dynamics through utilization of a smoothed profile for the colloidal particles. Moreover, the improved formulation includes an extension to incorporate multi-component fluids, allowing systems such as charged colloids in electrolyte solutions to be studied. The dynamics of the colloidal dispersions are solved with the same computational cost as required for solving non-particulate flows. Numerical results which assess the hydrodynamic interactions of colloidal dispersions are presented to validate the SP method. The SP method is not restricted to particular constitutive models of the host fluids and can hence be applied to colloidal dispersions in complex fluids.

The hydrodynamic interactions of a suspension of self-propelled particles are studied using a direct numerical simulation method which simultaneously solves for the host fluid and the swimming particles. A modified version of the "Smoothed Profile" method (SPM) is developed to simulate microswimmers as squirmers, which are spherical particles with a specified surface-tangential slip velocity between the particles and the fluid. This simplified swimming model allows one to represent different types of propulsion (pullers and pushers) and is thus ideal to study the hydrodynamic interactions among swimmers. We use the SPM to study the diffusive behavior which arises due to the swimming motion of the particles, and show that there are two basic mechanisms responsible for this phenomena: the hydrodynamic interactions caused by the squirming motion of the particles, and the particle-particle collisions. This dual nature gives rise to two distinct time-and length-scales, and thus to two diffusion coefficients, which we obtain by a suitable analysis of the swimming motion. We show that the collisions between swimmers can be interpreted in terms of binary collisions, in which the effective collision radius is reduced due to the collision dynamics of swimming particles in viscous fluids. At short time-scales, the dynamics of the swimmer is analogous to that of an inert tracer particle in a swimming suspension, in which the diffusive motion is caused by fluid-particle collisions. Our results, along with the simulation method we have introduced, will allow us to gain a better understanding of the complex hydrodynamic interactions of self-propelled swimmers.