Long-Range Dynamic Correlations in Confined Suspensions Derek Frydel and Haim Diamant ∗ Raymond & Beverly Sackler School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel (Dated: June 17, 2010) 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 / $^{2}$. PACS numbers: 83.50.Ha 47.56.+r 82.70.Dd 47.60.Dx As particles move through a liquid, the flows that they create correlate their motions. These hydrodynamic in- teractions play a central role in the dynamics of partic- ulate liquids [1]. In unbounded liquids the correlation is dominantly mediated by diffusive shear modes (i.e., diffusion of transverse momentum) as accounted for by the Navier-Stokes equation for an incompressible liquid. Consequently, the perturbation caused by the forced mo- tion of a particle in the liquid can be represented at large distances as emanating from a momentum monopole. The resulting flow velocity field decays with distance as 1 /r , leading to long-range hydrodynamic interaction be- tween particles and the well studied phenomenon of long time tails in their velocity correlation functions [2]. further established analytically [10, 11], confinement by rigid boundaries strongly affects the sound modes in a compressible liquid. These modes change from under- damped propagation with velocity c$_{s}$ to overdamped dif- fusion with diffusivity D$_{s}$ ∼ c 2 $_{s}$h$^{2}$/ν , where ν is the kine- matic viscosity of the liquid. The resulting sound dif- fusion is fast, D$_{s}$ ∼ 1 m $^{2}$/s for water in micron-scale confinement, compared to ν ∼ 10 − 6 m $^{2}$/s for the diffu- sion coefficient of shear modes, yet the latter are sup- pressed by the boundaries, leaving the diffusive compres- sion modes as the sole mechanism for hydrodynamic in- teraction at interparticle distances r > h . Furthermore, similar to the appearance of long time tails due to shear- stress diffusion in unbounded suspensions [2], the diffu- sive nature of compression modes in confined liquids leads to a (negative) long time tail in the velocity autocorrela- tion function of a suspended particle [9]. In this Letter we show that those diffusive compression modes have far reaching implications for the dynamic correlations be- tween distant particles in a channel. When the suspension is confined by rigid boundaries, the hydrodynamic interaction is qualitatively different [3]. Because of momentum transport into the boundaries the liquid no longer conserves momentum, and the ampli- tude of shear modes, developing in the liquid in response to particle motion, decays exponentially with distance. The correlations are now dominantly mediated by com- pression modes, and the perturbation due to particle mo- tion can be represented at large distances as arising from an effective mass dipole. When the suspension is confined between two plates (quasi-two-dimensional suspension), the resulting flow field remains long-ranged, decaying as 1 /r 2 (as for a mass dipole in two dimensions) [4]. How- ever, since in one dimension (1D) a steady dipole has a vanishing effect, the steady flow due to local particle motion in a linear channel has no far field and decays exponentially with r/h , h being the channel width [5, 6]. This hydrodynamic screening in pores and channels is well documented (e.g., [6, 7]) and widely employed (for instance, in studying the dynamics of confined polymers [8]). We employ a simple phenomenological approach, in which only the unconfined dimensions (in the current case the single dimension along the channel) are retained, whereas all the details in the confined dimensions, such as the cross-section of the channel and the precise boundary conditions at its walls, are averaged over and reduced to an effective friction term. Such a lubrication approach [12] leads to tremendous simplification of the analysis while yielding the correct physics at distances larger than the confinement width. It was successfully applied in the past to various problems of confined hydrodynamics (e.g., [13]) and, in particular, was shown to correctly reproduce the diffusive sound modes in confinement [9, 11]. We support the calculations by three-dimensional lattice-Boltzmann simulations [14]. In the simulations spherical particles of diameter σ move along the axis of a channel of length L and a square cross-section of side h . The channel is filled with an ambient fluid of mass density ρ$_{0}$ , shear viscosity η , bulk viscosity ζ , and sound veloc- ity c$_{s}$ . The channel and particles are rigid, and a no-slip boundary condition is imposed at their surfaces using the second-order bounce-back scheme [14]. Periodic bound- The analysis above assumes that the host liquid is at steady state and is incompressible, thus omitting lon- gitudinal sound modes. Since the Reynolds number of relevant colloidal systems is typically smaller than 10 − $^{6}$, and the sound velocity in the liquid, c$_{s}$ , is of order 10 3 m/s, both assumptions seem safely valid. Yet, as was first pointed out in Ref. [9] based on simulations, and arXiv:1001.1898v2 [cond-mat.soft] 3 Jul 2010