Hydrodynamic screening and viscous drag at finite concentration

Abstract

The concentration dependence of steady linearized viscous flow in a suspension of spheres of number density ρ0 is studied using a new formulation of the ideas of hydrodynamic screening [cf. K. F. Freed and S. F. Edwards, J. Chem. Phys. 61, 3626 (1974)]. A concentration dependent effective pure fluid Navier–Stokes equation is derived via a mean field treatment similar in spirit to Debye–Hückel theory. The effective Navier–Stokes equation is shown to contain concentration dependence through an inverse screening length κ and a renormalized viscosity η[ρ0]. The quantities κ and η[ρ0] are expressed in terms of the zero frequency and small wavevector limit of the particle current autocorrelation function. The Green’s function of the effective Navier–Stokes equation (screened Oseen tensor) is determined for the case of incompressible flow. The velocity field v(r) for steady, incompressible flow past a sphere of radius R is then calculated for both slip and stick boundary conditions. From v(r), the concentration dependent drag coefficient ζ[ρ0] for a sphere of radius R is determined in terms of κ and η[ρ0]. The result gives a generalization of Stokes’ law to finite concentration suspensions. Finally, two simple approximation treatments of the particle current autocorrelation function are proposed and the forms they yield for η[ρ0] and κ are discussed.