A Three-Dimensional Computation of the Force and Torque on an Ellipsoid Settling Slowly through a Viscoelastic Fluid

J. Feng, D. D. Joseph, R. Glowinski and T. W. Pan

*J. Fluid Mech.* **283**, 1-16
(1995)

**Abstract** The orientation of an ellipsoid falling in a
viscoelastic fluid is studied by methods of perturbation theory. For small
fall velocity, the fluid's rheology is described by a second-order fluid
model. The solution of the problem can be expressed by a dual expansion in
two small parameters: the Reynolds number representing the inertial effect
and the Weissenberg number representing the effect of the non-Newtonian stress.
Then the original problem is broken into three canonical problems: the zeroth
order Stokes problem for a translating ellipsoid and two first order problems,
one for inertia and one for second-order rheology. A Stokes operator is inverted
in each of the three cases. The problems are solved numerically on a
three-dimensional domain by a finite element method with fictitious domains,
and the force and torque on the body 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; normal stresses turn the major axis parallel to fall. The competition
of these two effects gives rise to an equilibrium tilt angle between 0^{o
}and 90^{o} which the settling ellipsoid would eventually 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 perturbation results do not
explain the sudden turning of a long body which occurs when a critical fall
velocity is exceeded. This is not surprising because the theory is valid
only for slow sedimentation. However, the results do seem to agree qualitatively
with "shape tilting" observed at low fall velocities.