A 3-DIMENSIONAL COMPUTATION OF THE FORCE AND TORQUE ON AN ELLIPSOID SETTLING SLOWLY THROUGH A VISCOELASTIC FLUID

The orientation of an ellipsoid falling in a viscoelastic fluid is stu died by methods of perturbation theory. For small fall velocity, the f luid's theology is described by a second-order fluid model. The soluti on of the problem can be expressed by a dual expansion in two small pa rameters: the Reynolds number representing the inertial effect and the Weissenberg number representing the effect of the non-Newtonian stres s. Then the original problem is split into three canonical problems: t he zeroth-order Stokes problem for a translating ellipsoid and two fir st-order problems, one for inertia and one for second-order theology. A Stokes operator is inverted in each of the three cases. The problems are solved numerically on a three-dimensional domain by a finite elem ent method with fictitious domains, and the force and torque on the bo dy are evaluated. The results show that the signs of the perturbation pressure and velocity around the particle for inertia are reversed by viscoelasticity. The torques are also of opposite sign: inertia turns the major axis of the ellipsoid perpendicular to the fall direction; n ormal stresses turn the major axis parallel to the fall. The competiti on of these two effects gives rise to an equilibrium tilt angle betwee n 0 degrees and 90 degrees which the settling ellipsoid would eventual ly assume. The equilibrium tilt angle is a function of the elasticity number, which is the ratio of the Weissenberg number and the Reynolds number. Since this ratio is independent of the fall velocity, the pert urbation results do not explain the sudden turning of a long body whic h occurs when a critical fall velocity is exceeded. This is not surpri sing because the theory is valid only for slow sedimentation. However, the results do seem to agree qualitatively with 'shape tilting' obser ved at low fall velocities.