Long range memory effects in flows involving abrupt changes in geometry: Part 2: the expansion/contraction/expansion problem

We give further theoretical consideration to the solution of flow problems involving abrupt changes in geometry using implicit rheological equations of state, so that the stress components are dependent variables along with the velocity components and the pressure.Attention is confined to the expansion/contraction/expansion problems and main interest centres on the effect of long range memory on flow characteristics, particularly the recirculating vortices arising from the presence of the abrupt changes in geometry.     

Variational principles and entropy production in creeping flow of non-newtonian fluids

The extension of Helmholtz—Korteweg's theorem to purely viscous non-Newtonian fluids yields a variational principle for steady creeping flow for a rather wide class of fluids. Only for a restricted subclass (Newtonian and power-law fluids) does the quantity to be minimized coincide with the total rate of entropy production. It is argued that the Helmholtz—Korteweg theorem is a purely mechanical result which has no significant thermodynamic content.     

A critical re-appraisal of the jet-thrust technique for normal stresses,with particular reference to axial velocity and stress rearrangement at the exit plane

It is shown that the thrust, T, exerted by a jet on the tube from which it flows, and the corresponding die-swell ratio, D, are closely related and dependent on the axial velocity and stress profiles at the exit plane. Velocity-profile data, calculated by Tanner using a finite element method, have been used to demonstrate that for a Newtonian liquid the reduction in measured thrust from the expected value arises from a re-arranged, non-parabolic axial velocity profile and the related re-arranged non-zero axial stress profile at the exit plane. The axial stress re-arrangement is the major effect.Using the correction-curve thus derived to determine the normal stresses, ν₁ + $\text{1}\text{2}$ν₂ aqueous and non-aqueous polymer solutions gives values that are higher than the “correct” results by a significant, substantial amount. The difference is not due to neglect of the second normal stress difference, ν₂, nor to the neglect of the wall pressure at the exit plane, which is shown experimentally to be very small. It is suggested that the difference, which is a function only of the shear stress (or rate of shear) at the wall, may arise from a difference in the stress profile associated with the velocity re-arrangement at the exit between Newtonian liquids and elasticoviscous liquids for which the extensional viscosity may be high.