Sedimentation of a sphere near a plane wall: weak non-Newtonian and inertial effects

Abstract

A reciprocal theorem is used to predict the steady three-dimensional creeping motion of a sphere sedimenting near a single vertical plane wall. Weak non-Newtonian effects up to second order in Deborah number are calculated as are inertial effects up to first order in Reynolds number. For the Newtonian case, a sphere translates parallel to the wall and simultaneously rotates as if rolling down the wall. At low Deborah numbers, the theoretical calculations indicate that fluid elasticity results in a migration of the sphere away from the wall and a drag decrease greater than that predicted in the unbounded case. A second normal stress difference enhances the drift velocity and shear thinning enhances the drag reduction. Shear thinning in the fluid viscosity tends to cause the sphere to rotate more slowly, which may lead to “anomalous” rotation at higher Deborah numbers. Intertial effects, on the other hand, cause no modification to the rotation speed, but do produce a drift velocity away from the wall. In addition, illustrative experiments are performed with spheres sedimenting through a polyacrylamide solution in a sufficiently large tank to focus on the interaction with a single vertical boundary. A digital imaging system is used to quantify both the sedimentation and migration speeds and the rotation rate. The observations show that the low Deborah number expansions fail to capture, even qualitatively, all of the effects observed at finite Deborah numbers.