THE SEDIMENTATION OF A SPHERE THROUGH AN ELASTIC FLUID .2. TRANSIENT MOTION

Direct comparisons of time-dependent finite-element simulations and ex perimental observations are presented for the transient benchmark prob lem of a sphere accelerating from rest in a cylindrical tube of viscoe lastic fluid. Finite-element calculations of the trajectory of the sph ere using a nonlinear dumbbell model and a nonlinear network model are compared with experimental measurements obtained using a digital vide o-imaging system. The test fluid is a highly elastic polyisobutylene B oger fluid, and comparisons are carried out over a wide range of Debor ah numbers, 0 less than or equal to De less than or equal to 11, and f or a range of sphere/tube radius ratios, 0.12 less than or equal to a/ R less than or equal to 0.63. In the experiments, the sphere shows a v elocity overshoot with a magnitude that is a strong function of the de nsity of the sphere, the radius ratio of the geometry and the Deborah number of the flow. This transient motion is heavily over-damped as a result of the high solvent viscosity of the Boger fluid. The numerical calculations show that significant differences in the transient veloc ity of the sphere are predicted by the network and dumbbell models, pa rtly as a consequence of the variations in the time-dependent viscomet ric functions predicted in the start-up of simple shear flow. At short times, the flow is governed by the linear viscoelastic response of th e fluid; however, at longer times and higher strains, nonlinear fluid rheology becomes increasingly important. A good description of the exp erimentally observed trajectory of the sphere can be obtained with a m ultimode formulation of the nonlinear Phan-Thien-Tanner network model by incorporating both a spectrum of relaxation times and a set of nonl inear model parameters which accurately describe the normal stress res ponse of the fluid in steady shear and in transient uniaxial elongatio n.