Stokes’ Second Flow Problem Revisited for Particle–Fluid Suspensions

Abstract

An alternative analytical model is proposed for hydrodynamics of incompressible Newtonian fluids with suspended solid particles. Unlike existing single-phase models that do not distinguish the velocity field of suspended particles from the velocity field of host fluid, the present model accounts for the relative shift between the two velocity fields and assumes that its effect can be largely captured by substituting the inertia term of Navier–Stokes equations with the acceleration field of the mass center of the representative unit cell. The proposed model enjoys a relatively concise mathematical formulation. The oscillating flow of a particle–fluid suspension between two flat plates is studied with the present model, and detailed results are presented for Stokes’ second flow problem on the oscillating flow of a suspension half-space induced by an oscillating plate with specific examples of dusty gases and nanofluids. Remarkably, leading-order asymptotic expressions derived by the present model, for the effect of suspended particles on the decay index and wavenumber of the velocity field, are shown to be identical to known results derived based on the widely adopted Saffman model for dusty gases. It is hoped that the present work could offer a relatively simplified and yet reasonably accurate model for hydrodynamic problems of particle–fluid suspensions.