A solid sphere falling through a Bingham plastic moves in a small envelope of fluid with shape that depends on the yield stress. A finite-element/Newton method is presented for solving the free-boundary problem composed of the velocity and pressure fields and the yield surfaces for creeping flow. Besides the outer surface, solid occurs as caps at the front and back of the sphere because of the stagnation points in the flow. The accuracy of solutions is ascertained by mesh refinement and by calculation of the integrals corresponding to the maximum and minimum variational principles for the problem. Large differences from the Newtonian values in the flow pattern around the sphere and in the drag coefficient are predicted, depending on the dimensionless value of the critical yield stress Yg below which the material acts as a solid. The computed flow fields differ appreciably from Stokes’ solution. The sphere will fall only when Yg is below 0.143 For yield stresses near this value, a plastic boundary layer forms next to the sphere. Boundary-layer scalings give the correct forms of the dependence of the drag coefficient and mass-transfer coefficient on yield stress for values near the critical one. The Stokes limit of zero yield stress is singular in the sense that for any small value of Yg there is a region of the flow away from the sphere where the plastic portion of the viscosity is at least as important as the Newtonian part. Calculations For the approach of the flow field to the Stokes result are in good agreement with the scalings derived from the matched asymptotic expansion valid in this limit.

No abstract

No abstract

No abstract

Laser Doppler velocimetry and flow-induced birefringence are used to measure the rate of deformation and the principal components of the refractive index tensor in a 5% polyisobutylene (PIB) solution in tetradecane (C14) flowing along the centerplane of an abrupt 3.97:1 planar contraction. The stress optical law is used to interpret the birefringence data in terms of the normal stress difference, which is used to calculate a transient elongational viscosity defined along the centerplane. These measurements are compared directly to predictions of six multimode, differential constitutive models (Oldroyd-B, White–Metzner, Acierno et al., Giesekus, Bird–DeAguiar, and Phan-Thien–Tanner) that are fit to steady and small amplitude oscillatory shear flow data for the PIB/C14 solution. The fluid exhibits slight elongational thickening followed by apparent extensional thinning at higher elongation rates. We believe that this ‘‘thinning’’ behavior is due to the decreased residence time of the polymer molecules in the high-strain-rate region as the flow rate (and maximum elongation rate) is increased. The nonlinear constitutive equations, except for the White–Metzner model, are virtually indistinguishable in their description of the dynamical response of the fluid in this experiment; however, the Phan-Thien–Tanner model gives the best quantitative fit to the data. These results point to the need for experiments in which the fluid flowing along the centerline is subjected to a greater total elongational strain.