The slow motion of a rigid particle in a second-order fluid

Abstract

The purpose of the present paper is to reach some general conclusions on the motion of rigid particles in a homogeneous shear flow of a viscoelastic fluid. Under the basic assumption of nearly Newtonian slow flow, the creeping-motion equations for a second-order fluid with characteristic time constants κ0(2) and κ0(11) can be employed. It is shown that the κ0(2) contributions to the hydrodynamic force F and couple G depend upon the hydrodynamic force, couple and stresslet which act upon the particle in a Newtonian fluid (termed F(1), G(1) and S(1), respectively). Since this relation involves time derivatives of F(1) and G(1), a little reflexion is needed to realize that the modification of the classical Stokes law for steady translation in a quiescent fluid can have no κ0(2) term. Since no results of such generality are possible for the κ0(11) contributions we focus attention on transversely isotropic particles. Employing the concept of material tensors, the symmetry of such particles dictates the form these tensors adopt. This alone is sufficient to show that sedimentation in a quiescent fluid is accompanied by a change in orientation until a stable terminal orientation is attained. Depending upon the type of particle only one of the two orientations, axis of symmetry parallel or perpendicular to the external force, is stable. Another result concerns two-dimensional shear flow, for which we show that the symmetry axis has to drift through various Jeffery orbits until an equilibrium orientation is reached. While the orbits C = 0 and C = ∞ are equilibrium orbits for every transversely isotropic particle there may be a third such preferred orbit, which we denote by C*. In order for these orbits to be stable certain restrictions have to hold, showing that the orbits C = 0 and C* cannot both be stable. For the special case of a rigid tridumbbell of axis ratio s the orbit C* does not exist. If s > 1 the drift for this particle is into the orbit C = 0 while for s < 1 it is into the orbit C = ∞. This agrees qualitatively quite well with experimental results obtained for rods and disks. No quantitative comparison is possible; the particle shape influences the result quantitatively owing to its effect on the combination of the fluid parameters κ0(2) and κ0(11).