An experimental investigation of the flow of aqueous non‐Newtonian high polymer solutions past a sphere

Abstract

The terminal velocities of spheres falling in aqueous solutions of hydraxyethyl cellulose and polyethylene oxide were determined with ruby and steel spheres. For each sphere the terminal velocity was obtained in seven cylinder sizes covering a range of sphere-to-cylinder diameter ratio from 0.0067 to 0.18. The data were used in an extrapolation to correct for the cylindrical wall effect. Several alternative extrapolation procedures for estimating the zero shear viscosity were attempted and the results were compared. An empirical correlation for the drag coefficient was developed in terms of the Ellis parameters of the fluids, and the Faxen wall correction for Newtonian flow. The correlation gives the drag coefficient within 4% for all cases in which the quantity qovt/Dtl/a is less thon 0.3. N E W T O N I A N F L O W PAST A SPHEREThe drag force exerted on a sphere of radius R resulting from the flow past it of an unbounded incompressible Newtonian fluid of viscosity To and approach velocity v, is given by Stokes' law ( 1 2 ) :Equation (1) Oseen's approximation is represented by taking the terms up to and including N R e in Equation (2), whereas Goldstein's solution, purportedly an extension of Oseen's approximation, does not include the logarithmic term. These solutions are classified as the low Reynolds number flow approximations to the Navier-Stokes equations, and they possess certain interesting features, some of which have been discussed by Chester ( 4 ) (with regard to the Oseen approximation) and also by Proudman and Pearson (1 7). to (D/D,) = 0.32. Ladenburg (11) has also presented a correction for the effect of the bottom of the cylinder. This correction, based on the solution for a sphere falling in an infinite fluid bounded at the bottom by an infinite flat plate, appears to overestimate the bottom correction. More recently, Tanner (24) carried out a numerical calculation in which the effects of both the cylindrical wall and the bottom of the container were taken into account. He found that the top and bottom corrections are negligible, provided the sphere is at least one cylinder radius away from either end. These conclusions are corroborated by experimental evidence and, consequently, the need for a top and bottom correction can be eliminated by appropriate design of the falling-sphere system.Many other studies on the flow of Newtonian fluids past a sphere have been reported. These have been reviewed (8, 13) and will not be discussed inasmuch as the results are not employed in the present study. N O N -N E W T O N t A N F L O W PAST A SPHEREReported studies on non-Newtonian flow past a sphere include solutions based on perturbation procedures, solutions based on variational schemes, and empirical correlations and associated experimental studies. Solutions Bored o n Perturbation ProceduresA perturbation approximation for the flow of a ReinerRivlin-Prager fluid ast a sphere was obtained by Rathna ity are constant. The rheological model subject to these restrictions is inadequate in describing non-Newton...

Drag force on a rigid spheroidal particle in a cylinder filled with Carreau fluid

“…Tripathi and Chhabra [16] calculated the drag force on a spheroid moving in a dilatant fluid in the convection-flow regime. Turian [17] and Uhlherr et al [18] conducted the experiment of the sedimentation of a rigid sphere in a circular tube filling with an elastic shear-thinning fluid. Missirlis et al [19] estimated the drag coefficient of a rigid sphere moving near the wall of a circular tube filled with a powerlaw shear-thinning fluid under conditions of creeping flow.…”

Sedimentation of a cylindrical particle in a Carreau fluid

“…It was shown that the influence of the shear-thinning nature of a Carreau fluid on a rigid sphere is greater than that on a bubble. Turian [17] and Uhlherr et al [18] investigated experimentally the sedimentation of a rigid sphere in a cylindrical tube filled with an inelastic power-law fluid. Navez and Walters [19] analyzed the settling of a sphere in shear-thinning polymer solutions.…”

Determinação da viscosidade por meio da velocidade terminal: uso da força de arrasto com termo quadrático na velocidade

“…Espera-se, então, que essas expressões possam ser aplicadas ao movimento de uma esfera dentro de um fluido não [24] fez no estudo de esferas em queda dentro de fluidos não-newtonianos, para os quais a viscosidade depende da tensão de cisalhamentoà qual são submetidos. Esse autor acrescentouà força de Oseen um termo de força com dependência cúbica na velocidade, dado por c3πv 3 /d, sendo cé um parâmetro empírico, constante para cada líquido, para obter Verificamos no Apêndice que a correção da velocidade medida nos tubos pelo fator de Ladenburg, descrito pela equação (7), para obter a velocidade que se teria em um meio infinito, mostrou-se adequada apenas para d/D <0,2.É conveniente esclarecer que os fatores λ 1 e λ 2 são usados no nosso trabalho simplesmente com o propósito de uma descrição empírica para a força de arrasto, e não com o objetivo de relacionar as velocidades das esferas medidas dentro dos tubos com as velocidades que se teria em um meio infinito.…”