Using a finite-element program, the drag on an unbounded fluid in an (inelastic) power-law fluid is estimated. Comparison with upper and lower bounds, and with experimental data is given. Some remarks on wall effects are also made, and it is shown that wall effects are negligible forn ⩽ 0.5.

Reply to Chhabra's remarks

The drag on a sphere in a power-law fluid

Hill's extremum principles are not directly applicable to an Ellis model fluid. A method of adapting Hill's principles to the Ellis model was developed and used to calculate upper and lower bounds on the drag coefficient for a sphere moving slowly through such a fluid. Available experimental data were compared with the average of the upper and lower bounds. Agreement was poor with Slattery's data and good with Turian's data. Some reasons to suspect Slattery's data have been pointed out.

Upper and lower bounds on the drag coefficient of a sphere in a power‐model fluid

Previous work on the creeping flow of viscoelastic fluids past a sphere is reviewed Theoretical analyses available in the literature were obtained for weakly elastic fluids and therefore they predict only a small influence of fluid elasticity on the drag In this paper, an approximate theoretical analysis is given for the creeping flow past a rigid sphere in an unbounded medium The analysis uses a variational principle to solve the equations of motion and continuity in conjunction with the Carreau constitutive equation The theoretical results are presented in terms of a correction factor to the Newtonian drag coefficient The correction factor is a function of the power law flow behaviour indexn, the ratio of limiting viscosities (η
0 − η∞)/η0 and a dimensionless timeΛ which reflects the elastic nature of the fluids The results are presented in graphical form covering a realistic range of these dimensionless groups In order to verify the theoretical predictions, the drag coefficient of a number of spheres was measured in a series of shear thinning elastic test fluids The flow properties of the test fluids were independently measured with a Weissenberg Rheogoniometer The power law index of the test fluids varied between 10 and 04 Particle Reynolds number based onη
0 was in the range of 4⋅10−6 to 4⋅10−2 The difference between theoretically predicted values of drag coefficient and the experimentally measured values is less than ±75% In addition, it is found that the Carreau viscosity equation can be used to predict the elastic parameter of primary normal stress difference with moderate to good accuracy for all the polymer solutions used in this work

Creeping motion of spheres through shear-thinning elastic fluids described by the Carreau viscosity equation

Two simple methods are presented for the characterization of inelastic power law fluids from falling sphere data. The methods involve the application of shear rate or shear stress correction factors which have been derived theoretically using Slattery's solution for creeping flow about spheres. Flow curves obtained using these methods are in excellent agreement with those measured on a Weissenberg rheogoniometer for 0.83 > n > 1.0.



The experimentally determined drag coefficients are found to be in good agreement with the predictions of Slattery's creeping flow first approximation solution.



The wall correction factors of Faxen and Francis appear to be valid for inelastic non-Newtonian fluids up to a diameter ratio of at least 0.08.



On presente, a partir de donnees de spheres qui tombent, deux methodes simples pour caracteriser des fluides inelastiques obeissant a la loi de l'energie. Lesdites methodes impliquent l'application des facteurs de correction des taux ou des contraintes de cisaillement qu'on a obtenus theoriquement en employant la solution de Slattery pour un ecoulement de fluage autour de spheres. Les courbes d'ecoulement qu'on a obtenues au moyen des methodes en question concordent tres bien avec celles qu'on a mesurees sur un rheogoniometre lorsque 0.83 > n > 1.0.



On a constate que les coefficients de resistance a la traǐnee determines experimentalement concordaient bien avec les previsions de la premiere solution d'approximation de Slattery relativement a l'ecoulement de fluage.



Les facteurs de correction pour les parois de Faxen et Francis semblent ětre valables dans le cas de fluides inelastiques et non newtoniens jusqu'a un rapport d'au moins 0.08 entre les diametres.

Characterisation of inelastic power‐law fluids using falling sphere data

Drag coefficients of a slowly moving sphere in non-newtonian fluids

Any experimental work on the flow of a polymer solution or any theoretical analysis on the basis of a visoelastic constitutive equation does not always bring out viscoelastic effects but may be showing a non-Newtonian viscosity effect. Therefore, in order to obtain a clear understanding about viscoelastic effects, it is desirable to have a sufficient knowledge of the non-Newtonian viscosity effect. To facilitate this, finite-difference numerical solutions of non-Newtonian flow were carried out using a non-Newtonian viscous model for the Reynolds numbers of 0.1, 1.0, 20 and 60. Drag force measurements and flow visualization experiments were also performed over a wide range of experimental conditions using polymer solutions. The present work appears to support the following idea: When compared with the Newtonian case on the basis of DV∞P/η0, where η0 is the zero shear viscosity, it is on account of the non-Newtonian viscosity that the friction and pressure drags decrease, that the separating vortices behind the sphere become larger, and that no shift occurs in the streamlines. On the other hand, it is due to viscoelasticity that the normal force drag increases, that the separating vortices behind the sphere become smaller, and that an upstream shift occurs in the streamlines.

Reply to Chhabra's remarks

Citations

The drag on a sphere in a power-law fluid

References

Upper and lower bounds on the drag coefficient of a sphere in a power‐model fluid

Creeping motion of spheres through shear-thinning elastic fluids described by the Carreau viscosity equation

Characterisation of inelastic power‐law fluids using falling sphere data

Drag coefficients of a slowly moving sphere in non-newtonian fluids

An investigation of non-newtonian flow past a sphere