Über den Einfluß von Wänden auf die Bewegung einer Kugel in einer reibenden Flüssigkeit

Ladenburg, R.

doi:10.1002/andp.19073280806

Cited by 132 publications

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“…All analytical equations mentioned in this section are valid for unbounded flows only. Ladenburg [14] as well as Faxén [13] determined theoretically and e.g. Fidleris and Whitmore [15] and Sutterby [16] found experimentally that the influence of walls is to increase the drag.…”

Section: Introductionmentioning

confidence: 96%

Lattice Boltzmann simulations to determine drag, lift and torque acting on non-spherical particles

2009

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Section: Introductionmentioning

confidence: 96%

Lattice Boltzmann simulations to determine drag, lift and torque acting on non-spherical particles

2009

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“…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.…”

Section: And Pearson Ismentioning

confidence: 97%

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

Turian

1967

AIChE Journal

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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...

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“…No entanto, SCS examinaram o que eles chamam de "enganosa simplicidade" desse método, coletando dados de velocidade terminal por meio de filmagem de esferas de aço em queda dentro de tubos cilíndricos de diversos diâmetros, contendo glicerina, e concluíram que o métodó e válido apenas para números de Reynolds R d <0,5, concordando com Liao [3], ainda assim desde que se faça uma correção na velocidade devido ao efeito do diâmetro finito do tubo. A proximidade da esfera com a parede do tubo aumenta o gradiente de velocidade no fluido em torno da esfera, aumentando, consequentemente, a força viscosa e tornando necessário corrigir a sua expressão, dada pela equação (2), por um fator adimensional λ 1 , chamado fator de Ladenburg [16], que depende da relação entre os diâmetros da esfera e do tubo, ficando…”

Section: Número De Reynolds Força E Coeficiente De Arrastounclassified

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

Vertchenko

2017

Rev. Bras. Ensino Fís.

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Propomos a correção, por um fator adimensional, do termo quadrático na velocidade da força de arrasto de Oseen para que ela seja usada na determinação do coeficiente de viscosidade da glicerina através da medição da velocidade terminal de esferas em queda dentro do fluido. Esse fator incorpora o efeito da parede do recipiente sobre o movimento das esferas, analogamenteà correção da força de Stokes pelo fator de Ladenburg, e permite compatibilizar a força de Oseen com os dados experimentais do coeficiente de arrasto até números de Reynolds próximos a 30. Admitindo uma relação de proporcionalidade entre esses fatores, o coeficiente de viscosidade passa a ser o coeficiente linear na relação entre a viscosidade que seria obtida se fosse considerada apenas a força de Stokes e o produto da velocidade terminal pelo diâmetro das esferas. A experiência com esferas de aço em queda dentro de tubos de diferentes diâmetros forneceu valores para o coeficiente de viscosidade compatíveis com a temperatura em que se trabalhou. Esse método pôde ser aplicado até números de Reynolds da ordem da dezena, relaxando a restrição de se trabalhar com números de Reynolds inferiores a 0,5 na determinação da viscosidade através do método de Stokes. Palavras-chave: Força de arrasto, viscosidade, número de Reynolds.A correction to the term with quadratic dependency of the velocity in the Oseen s drag force by a dimensionless factor is proposed in order to determine the viscosity of glycerin through the measurement of the terminal velocity of spheres falling inside the fluid. This factor incorporates the effect of the container s wall over the movement of the spheres, analogously to the correction of the Stoke s force by the Landenburg factor, and permits to make the Oseen s force compatible with the experimental data for the drag coefficient up to Reynolds number near 30. Admitting a proportionality relation between these factors, the viscosity coefficient turns into a linear coefficient in the relation between the parameter that corresponds to the viscosity that would be determined if only the Stokes' force was considered and the product of the terminal velocity by the diameter of the spheres. The experiment with steel spheres falling inside tubes of different diameters showed values for the viscosity coefficient of glycerin compatible with the expected, for the temperature range worked. This method might be applied to Reynolds number of order of ten of the sphere's motion, relaxing the restriction of working with Reynolds numbers inferior to 0.5 for the determination of the viscosity through the Stokes' method. Keywords: Drag force, viscosity, Reynolds number. IntroduçãoA viscosidade está associada a uma espécie de atrito interno que causa fricção entre as camadas do fluido que se movimentam com velocidades diferentes eé explicada, ao nível microscópico, como devendo-seà transferência de momento linear entre as partículas que compõem o fluido. Tecnicamente elaé caracterizada pelo coeficiente de viscosidade dinâmico, η, definido como a razão ...

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Über den Einfluß von Wänden auf die Bewegung einer Kugel in einer reibenden Flüssigkeit

Cited by 132 publications

References 0 publications

Lattice Boltzmann simulations to determine drag, lift and torque acting on non-spherical particles

Lattice Boltzmann simulations to determine drag, lift and torque acting on non-spherical particles

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

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

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