The constitutive relation of suspensions of noncolloidal particles in viscous fluids

Abstract

The motion of noncolloidal particles convected by a nonhomogeneous and nonstationary viscous fluid flow is investigated, assuming that inertial effects can be neglected. It appears that the particle volumetric flux is the sum of a convective part, VEφ, and a diffusive term, −Dself⋅∇φ, where VE is the Eulerian mean velocity of a test particle, φ is the particle volume fraction, and Dself is the coefficient of self-diffusion. The latter measures the local temporal growth of the mean square displacement of a tracer particle from its average position and can be written as Dself(r)=D(r,r), where the cross-diffusion tensor D(r1,r2) is the time integral of the velocity cross-correlation function. On the other hand, the Eulerian mean velocity VE is the sum of the coarse-grained average particle velocity, V̄, and a drift velocity, Vd(r)=−[(∂/∂r2)⋅DT(r,r2)]r2=r. This last term, which is identically zero when the suspended particles are passive tracers, indicates that the suspended particles tend to move toward regions with smaller diffusivities. This result demonstrates that the motion of each suspended particle is a random process satisfying a generalized nonlinear Langevin equation, where the fluctuating term is described through the cross diffusivity D.