THE SEDIMENTATION OF A SPHERE THROUGH AN ELASTIC FLUID .1. STEADY MOTION

The first direct comparisons of finite element simulations and detaile d 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 techn ique of laser Doppler velocimetry is used to probe the kinematics of t he fluid over a wide range of Deborah numbers, 0.4 less than or equal to De less than or equal to 9, and dimensionless radius ratios, 0.12 l ess than or equal to a/R less than or equal to 0.64. These observation s are augmented in Part 2 of this work (Rajagopalan et al., 1995) by d igital video-imaging studies and fully-implicit time-dependent numeric al simulations of the initial acceleration of the sphere from rest and the resulting overshoot in the velocity that arises from fluid viscoe lasticity. Numerical simulations are reported for the upper-convected Maxwell (UCM) model, the Chilcott-Rallison model (each with a single r elaxation time constant) and the multimode Phan-Thien-Tanner model. Fo r 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 f luid theology in both viscometric shear flows and transient extensiona l experiments. However, the range of computationally attainable De is limited by the inability to resolve intense stress boundary layers. Bo th experimental measurements and numerical calculations indicate the w all correction factor for the motion of a sphere through a viscoelasti c fluid is a sensitive function of the radius ratio and the Deborah nu mber. They also show that non-Newtonian effects in the strong extensio nal how near the rear stagnation point result in the formation of a pr onounced viscoelastic wake effect extending up to 30 sphere radii behi nd the sphere and corresponding to a downstream shift in the fluid str eamlines. However there is no experimental indication of the formation of a negative wake or a flow instability in the wake, and the flow re mains stable for all radius ratios and Deborah numbers investigated ex perimentally.