Anomalous Rolling of Spheres down an Inclined Plane

Y. J. Liu, J. Nelson, J. Feng and D. D. Joseph

*J. Non-Newtonian Fluid Mech.* 50, 305-329 (1993)

**Abstract **- A sphere in air will roll down a plane that is tilted
away from the vertical. The only couple acting about the point of contact
between the sphere and the plane is due to the component of the weight of
the sphere along the plane, provided that air friction is negligible. If
on the other hand the sphere is immersed in a liquid, hydrodynamic forces
will enter into the couples that turn the sphere, and the rotation of the
sphere can be anomalous, i.e., as if rolling up the plane while it falls.
In this paper we shall show that anomalous rolling is a characteristic phenomenon
that can be observed in every viscoelastic liquid tested so far. Anomalous
rolling is normal for hydrodynamically levitated spheres, both in Newtonian
and viscoelastic liquids. Normal and anomalous rolling are different names
for dry and hydrodynamic rolling. Spheres dropped at a vertical wall in Newtonian
liquids are forced into anomalous rotation and are pushed away from the wall
while in viscoelastic liquids, they are forced into anomalous rotaion, but
are pushed toward the wall. If the wall is inclined and the fluid is Newtonian,
the spheres will rotates normally as for dry rolling, but the same sphere
rotates anomalously in viscoelastic liquids when the angle of inclination
from the vertical is less than some critical value. The hydrodynamic mechanisms
underway in the settling of circular particles in a Newtonian fluid at a
vertical wall are revealed by an exact numerical simulation based on a
finite-element solution of the Navier-Stokes equation and Newton's equations
of motion for a rigid body.