The sedimentation of a sphere through an elastic fluid. Part 1. Steady motion

Abstract

The first direct comparisons of finite element simulations and detailed point-wise experimental velocity measurements are presented for the international benchmark problem of a sphere sedimenting axially under gravity through a cylindrical tube of viscoelastic fluid. In addition to measurements and calculations of the viscoelastic correction to the drag force exerted by the fluid on the sphere, the non-invasive technique of laser Doppler velocimetry is used to probe the kinematics of the fluid over a wide range of Deborah numbers, 0.4≤De≤9, and dimensionless radius ratios, 0.12≤ a R ≤0.64 . These observations are augmented in Part 2 of this work (Rajagopalan et al., 1995) by digital video-imaging studies and fully-implicit time-dependent numerical simulations of the initial acceleration of the sphere from rest and the resulting overshoot in the velocity that arises from fluid viscoelasticity. Numerical simulations are reported for the upper-convected Maxwell (UCM) model, the Chilcott-Rallison model (each with a single relaxation time constant) and the multimode Phan-Thien-Tanner model. For the radius ratio of a R = 0.5 , flow simulations using the UCM model have been extended well above the commonly reported limit point of De = 1.6 through careful mesh refinement. The multimode simulations with a spectrum of time constants allow a quantitative description of the fluid rheology in both viscometric shear flows and transient extensional experiments. However, the range of computationally attainable De is limited by the inability to resolve intense stress boundary layers. Both experimental measurements and numerical calculations indicate the wall correction factor for the motion of a sphere through a viscoelastic fluid is a sensitive function of the radius ratio and the Deborah number. They also show that non-Newtonian effects in the strong extensional flow near the rear stagnation point result in the formation of a pronounced viscoelastic wake effect extending up to 30 sphere radii behind the sphere and corresponding to a downstream shift in the fluid streamlines. However there is no experimental indication of the formation of a negative wake or a flow instability in the wake, and the flow remains stable for all radius ratios and Deborah numbers investigated experimentally.