Long-range hydrodynamic response of particulate liquids and liquid-laden solids

In viscous particulate liquids, such as suspensions and polymer solutions, the large-distance steady-state flow due to a local disturbance is commonly described in terms of hydrodynamic screening -- beyond a correlation length $\xi$ the response drops from that of the pure solvent, characterized by its viscosity $\eta_0$, to that of the macroscopic liquid with viscosity $\eta$. For cases where $\eta>>\eta_0$ we show, based on general conservation arguments, that this screening picture, while being asymptotically correct, should be refined in an essential way. The crossover between the microscopic and macroscopic behaviors occurs gradually over a wide range of distances, $\xi<<r<<\xi\sqrt{eta/\eta_0}$. In liquid-laden solids, such as colloidal glasses, gels and liquid-filled porous media, where $\eta-->\infty$, this intermediate behavior takes over the entire large-distance response. The intermediate flow field, arising from the effect of mass displacement rather than momentum diffusion, has several unique characteristics: (i) It has a dipolar shape with a $1/r^3$ spatial decay, negative transverse components, and vanishing angular average. (ii) Its amplitude depends on the liquid properties through $\eta_0$ and $\xi$ alone; thus, in cases where $\xi$ is fixed by geometry (e.g., for particulate liquids tightly confined in solid matrices), the large-distance response is independent of particle concentration. (iii) The intermediate field builds up non-diffusively, with a distance-independent relaxation rate, making it dominant at large distances before steady state has been reached. We demonstrate these general properties in three model systems.