Long-Range Dynamic Correlations in Confined Suspensions

Hydrodynamic interactions between particles confined in a liquid-filled linear channel are known to be screened beyond a distance comparable to the channel width. Using a simple analytical theory and lattice Boltzmann simulations, we show that the hydrodynamic screening is qualitatively modified when the time-dependent response and finite compressibility of the host liquid are taken into account. Diffusive compression modes in the confined liquid cause the particles to have velocity correlations of unbounded range, whose amplitude decays with time only as $t^{-3/2}$.

Figure 1

Coupling diffusion coefficient of a particle pair in a channel as a function of interparticle distance, obtained from lattice Boltzmann simulations. The coefficient is measured from the net displacement (time-integrated velocity) of one particle in response to a unit-momentum impulse imparted to the other. The particle diameter is $\sigma =h/1.5$.

Figure 2

Velocity cross-correlation function of a particle pair in a channel as a function of time. Different curves correspond to different interparticle distances (from left to right): $x/h=1.33$ (black line), 3.33 (red line), and 6.67 (blue line). The particle diameter is $\sigma =h/1.5$. (a) Lattice-Boltzmann simulation results. The correlation is measured from the time-dependent velocity of one particle in response to a unit-momentum impulse imparted to the other. The inset shows the long-time behavior on a log-log scale, the dashed line indicating a slope of $-3/2$. (b) Results from the analytical approximation [inversion of Eq. (3)].

Figure 3

Collective diffusion coefficient of an array of particles as a function of interparticle separation. Lattice-Boltzmann simulation results (solid line) are presented alongside the analytical result [Eq. (5); dashed line]. The inset shows the collective coefficient as expected from the hydrodynamic-screening description. The particle diameter is $\sigma =h/1.5$.