The sedimentation of a sphere through an elastic fluid Part 2. Transient motion

Abstract

Direct comparisons of time-dependent finite-element simulations and experimental observations are presented for the transient benchmark problem of a sphere accelerating from rest in a cylindrical tube of viscoelastic fluid. Finite-element calculations of the trajectory of the sphere using a nonlinear dumbbell model and a nonlinear network model are compared with experimental measurements obtained using a digital video-imaging system. The test fluid is a highly elastic polyisobutylene Boger fluid, and comparisons are carried out over a wide range of Deborah numbers, 0≤ De≤11, and for a range of sphere/tube radius ratios, 0.12≤ a R ≤0.63 . In the experiments, the sphere shows a velocity overshoot with a magnitude that is a strong function of the density 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 velocity of the sphere are predicted by the network and dumbbell models, partly as a consequence of the variations in the time-dependent viscometric functions predicted in the start-up of simple shear flow. At short times, the flow is governed by the linear viscoelastic response of the fluid; however, at longer times and higher strains, nonlinear fluid rheology becomes increasingly important. A good description of the experimentally observed trajectory of the sphere can be obtained with a multimode formulation of the nonlinear Phan-Thien-Tanner network model by incorporating both a spectrum of relaxation times and a set of nonlinear model parameters which accurately describe the normal stress response of the fluid in steady shear and in transient uniaxial elongation.