Propagation of hydrodynamic interactions between particles in a compressible fluid

Abstract

Hydrodynamic interactions are transmitted by viscous diffusion and sound propagation, and the temporal evolution of hydrodynamic interactions by both mechanisms is studied using direct numerical simulation in this paper. The hydrodynamic interactions for a system of two particles in a fluid are estimated using the velocity correlation of the particles. In an incompressible fluid, hydrodynamic interactions propagate instantaneously at the infinite speed of sound followed by a temporal evolution due to viscous diffusion. Conversely, sound propagates in a compressible fluid at a finite speed, which affects the temporal evolution of the hydrodynamic interactions through an order-of-magnitude relationship between the time scales of viscous diffusion and sound propagation. The hydrodynamic interactions are characterized by introducing the ratio of these time scales as an interactive compressibility factor.

Retarded hydrodynamic interaction between two spheres immersed in a viscous incompressible fluid

“…The effect is similar to that studied by Zwanzig and Bixon [42] for a single sphere. The effect was seen in the computer simulation of Tatsumi and Yamamoto [32]. As discussed above, the use of the approximation Eq.…”

“…The approximation satisfies reciprocity, but is clearly rather drastic. Ardekani and Rangel [16] in their first approach, and Tatsumi and Yamamoto [32] made the same approximation.…”

“…The retarded interactions are of interest in situations on the timescale of velocity relaxation, as well as in situations where oscillating forces or torques are applied. The theory of velocity relaxation and correlation functions can be applied in the discussion of computer simulations of Brownian motion [32]. Oscillating forces occur in the numerical study of swimming of two unequal spheres [44]- [46].…”

Collinear velocity relaxation of two spheres in a viscous incompressible fluid

“…The reaction is instantaneous due to the assumption of incompressibility. In a compressible fluid the reaction would take some time due to the finite velocity of sound, as seen in the computer simulation of Tatsumi and Yamamoto [9]. From Eq.…”

“…The dynamics of two colloidal spheres immersed in a compressible fluid was studied in computer simulation by Tatsumi and Yamamoto [9]. In their theoretical analysis these authors used a simple Green function approximation to the mobility function [10].…”