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.

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

An experimental study of the drag on hollow cylinders and conglomerates falling in a viscous fluid under Stokes flow conditions is described. The experiments were carried out in a tank of square cross section using silicone oil as the Newtonian fluid. Settling velocities of the free falling objects were measured and corrected to conditions of zero Reynolds number flow in an unbounded fluid. The results reveal that all objects tested have Stokes settling velocities smaller than that of a sphere of equal mass and volume. Measurements are reported in terms of the settling speed ratio defined as the ratio of the Stokes settling speed to that of a sphere of equal mass and volume. For the hollow cylinders two parameters are varied: the aspect ratio (length to outside diameter) and the radius ratio (inner to outer radius). Measurements show that the settling speed ratio decreases markedly as the hollowness of the cylinder increases. Each fixed radius ratio data set exhibits a maximum settling speed ratio near an aspect ratio of 1.65. For conglomerates composed of n spheres two trends appear: one for planar configurations and the other for globular clusters. Experimental data for two spheres and three spheres in point contact are in good agreement with recent theoretical results.

Stokes drag on hollow cylinders and conglomerates

Characteristics of laser-doppler signals from bubbles

A study of wall effects on the motion of a sphere in viscoelastic fluids

New experimental results on the wall effect for sphere motion in cylindrical tubes are presented and discussed for the conditions d/D ≤ 0.9 and Rem ≤ 20000. Extensive comparisons with previous studies have been carried out to evaluate their predictability and to demonstrate the utility of the present results.



The wall factor, defined as the ratio of settling velocity in an unbounded medium to that measured in a cylindrical tube, is found to depend on sphere-to-tube diameter ratio and on sphere Reynolds number. However, for small values of the Reynolds number (Re ≤ 0.5), as well for large values (Re ≥ 1000), the Reynolds number dependence of the wall factor disappears; in these regions, only the dependence on diameter ratio remains.



De nouveaux resultats experimentaux sur l'effet de paroi pour le deplacement des spheres dans des tubes cyclindriques sont presentes et analyses pour les conditions d/D ≤ 0,9 Rem ≤ 2000. On a effectue des comparaisons poussees avec les etudes anterieures afin d'evaluer la caractere predictif du modele et de demontrer l'utilite des presents resultats.



On a trouve que le facteur de paroi, defini comme le rapport entre la vitesse de decantation dans un milieu infini et la vitesse dans un tube cylindrique, dependait du rapport de diametre spheres-tube et du nombre de Reynolds des spheres. Cependant, pour le petites valeurs du nombre de Reynolds (Re ≤ 0.5) comme pour les grands (Re ≤ 1000), la dependance du facteur de paroi par rapport au nombre: de Reynolds disparait; dans ces regions, seule subsiste la dependance par rapport au rapport des diametres.

Wall effect for the fall of spheres in cylindrical tubes at high Reynolds number

A theoretical treatment is presented for the determination of the drag upon a sphere settling along the axis of a long square duct under the condition that the creeping motion equations are applicable. In order to obtain the second reflection velocity field, it was necessary to develop a new general solution in cartesian coordinates to the creeping motion equation, applicable within the domain of a long square duct. Using the second reflection velocity field solution an Faxen's law, a third reflection (first correction to the Stokes' value) drag correction is obtained. The results show that the drag correction for a square container is quite close to (but smaller than) the drag correction produced by a cylinder whose diameter is the same as the duct width.

J. Happel

Papers

The settling of a sphere along the axis of a long square duct at low Reynolds' number