Numerical simulations of eccentricity and end effects in falling ball rheometry

Abstract

Finite element calculations, theoretical predictions, and experimental observations are combined to analyze wall and end effects for a wide range of ball sizes in falling ball rheometry. We find that reflection solutions, finite element calculations, and experimental data are indistinguishable for small balls approaching the cylinder bottom. For larger falling balls, finite element modeling predicts a significant increase in drag in comparison with the reflection solution. Our experimental measurements are in very good agreement with the predictions of the finite element calculations and confirm the relative drag increase. For balls falling off the centerline of the cylinder, analytical solutions are restricted to the limiting cases of very small balls and large balls only slightly smaller than the containing cylinder. In this report, intermediate ball sizes are studied with three‐dimensional finite element codes. The finite element predictions for the intermediate ball sizes are qualitatively similar to the results of reflection solutions for very small balls, with the minimum drag found when the center of the falling ball is located approximately 40% of a cylinder radius off the centerline. This qualitative similarity suggests an ad hoc modification of the centerline wall correction. This modified expression is shown to predict to within 1% the terminal velocity during eccentric settling of balls of intermediate sizes.