The lift on a small sphere in a slow shear flow

doi:10.1017/S0022112065000824

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The lift on a small sphere in a slow shear flow

Journal Article•DOI•

The lift on a small sphere in a slow shear flow

Philip Geoffrey Saffman¹•Institutions (1)

California Institute of Technology¹

01 Jun 1965-Journal of Fluid Mechanics (Cambridge University Press)-Vol. 22, Iss: 02, pp 385-400

TL;DR: In this article, it was shown that a sphere moving through a very viscous liquid with velocity V relative to a uniform simple shear, the translation velocity being parallel to the streamlines and measured relative to streamline through the centre, experiences a lift force 81·2μVa2k½/v½ + smaller terms perpendicular to the flow direction, which acts to deflect the particle towards the streamline moving in the direction opposite to V.

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Abstract: It is shown that a sphere moving through a very viscous liquid with velocity V relative to a uniform simple shear, the translation velocity being parallel to the streamlines and measured relative to the streamline through the centre, experiences a lift force 81·2μVa2k½/v½ + smaller terms perpendicular to the flow direction, which acts to deflect the particle towards the streamlines moving in the direction opposite to V. Here, a denotes the radius of the sphere, κ the magnitude of the velocity gradient, and μ and v the viscosity and kinematic viscosity, respectively. The relevance of the result to the observations by Segree & Silberberg (1962) of small spheres in Poiseuille flow is discussed briefly. Comments are also made about the problem of a sphere in a parabolic velocity profile and the functional dependence of the lift upon the parameters is obtained.

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Journal Article•DOI•

An investigation of particle trajectories in two-phase flow systems

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S. A. Morsi¹, A. J. Alexander²•Institutions (2)

University of Surrey¹, Loughborough University²

26 Sep 1972-Journal of Fluid Mechanics

TL;DR: In this article, the authors describe a theoretical investigation into the response of a spherical particle to a one-dimensional fluid flow, and the motion of the spherical particle in a uniform 2D fluid flow about a circular cylinder.

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Abstract: This paper describes a theoretical investigation into (i) the response of a spherical particle to a one-dimensional fluid flow, (ii) the motion of a spherical particle in a uniform two-dimensional fluid flow about a circular cylinder and (iii) the motion of a particle about a lifting aerofoil section. In all three cases the drag of the particle is allowed to vary with (instantaneous) Reynolds number by using an analytical approximation to the standard experimental drag-Reynolds-number relationship for spherical particles.

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2,401 citations

Journal Article•DOI•

Sediment transport; Part I, Bed load transport

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van Rijn, C Leo

01 Oct 1984-Journal of Hydraulic Engineering

TL;DR: In this article, a method is presented which enables the computation of the bed-load transport as the product of the saltation height, the particle velocity and the bed load concentration.

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Abstract: A method is presented which enables the computation of the bed-load transport as the product of the saltation height, the particle velocity and the bed-load concentration. The equations of motions for a solitary particle are solved numerically to determine the saltation height and particle velocity. Experiments with gravel particles (transported as bed load) are selected to calibrate the mathematical model using the lift coefficient as a free parameter. The model is used to compute the saltation heights and lengths for a range of flow conditions. The computational results are used to determine simple relationships for the saltation characteristics. Measured transport rates of the bed load are used to compute the sediment concentration in the bed-load layer. A simple expression specifying the bed-load concentration as a function of the flow and sediment conditions is proposed. A verification analysis using about 600 (alternative) data shows that about 77% of the predicted bed-load-transport rates are within 0.5 and 2 times the observed values.

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1,653 citations

Cites background or methods from "The lift on a small sphere in a slo..."

...Further, it was assumed that the lift coefficient (aL) computed by Saffman (37) for viscous flow (aL =1....
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...Saffman (37) showed theoretically that for viscous flow the lift force due to the particle rotation is less by an order of magnitude than that due to the shear effect and may therefore be neglected....
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...For a sphere moving in a viscous flow, Saffman (37) derived the following expression:...
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Journal Article•DOI•

Discrete particle simulation of particulate systems: Theoretical developments

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Haiping Zhu¹, Zongyan Zhou¹, Runyu Yang¹, Aibing Yu¹•Institutions (1)

University of New South Wales¹

01 Jul 2007-Chemical Engineering Science

TL;DR: This paper reviews the work in this area with special reference to the discrete element method and associated theoretical developments, and covers three important aspects: models for the calculation of the particle–particle and particle–fluid interaction forces, coupling of discrete elements method with computational fluid dynamics to describe particle-fluid flow, and the theories for linking discrete to continuum modelling.

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1,563 citations

Journal Article•DOI•

On predicting particle-laden turbulent flows

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Said Elghobashi¹•Institutions (1)

University of California, Irvine¹

01 Jun 1994-Flow Turbulence and Combustion

TL;DR: An overview of the challenges and progress associated with the task of numerically predicting particle-laden turbulent flows is provided and suggestions are made for improving closure modelling of some important correlations.

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Abstract: The paper provides an overview of the challenges and progress associated with the task of numerically predicting particle-laden turbulent flows The review covers the mathematical methods based on turbulence closure models as well as direct numerical simulation (DNS) In addition, the statistical (pdf) approach in deriving the dispersed-phase transport equations is discussed The review is restricted to incompressible, isothermal flows without phase change or particle-particle collision Suggestions are made for improving closure modelling of some important correlations

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1,328 citations

Cites background from "The lift on a small sphere in a slo..."

...The terms on the right side are respectively the forces due to viscous and pressure drag, fluid pressure gradient and viscous stresses, inertia of virtual mass, viscous drag due to unsteady relative acceleration (Basset), buoyancy, and Saffman's lift force due to shear in the carrier flow [31, 30]....
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Journal Article•DOI•

The physics of wind-blown sand and dust

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Jasper F. Kok¹, Eric J. R. Parteli, Timothy I. Michaels, Diana Bou Karam•Institutions (1)

Cornell University¹

20 Jan 2012-arXiv: Earth and Planetary Astrophysics

TL;DR: The physics of aeolian saltation, the formation and development of sand dunes and ripples, the physics of dust aerosol emission, the weather phenomena that trigger dust storms, and the lifting of dust by dust devils and other small-scale vortices are reviewed.

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Abstract: The transport of sand and dust by wind is a potent erosional force, creates sand dunes and ripples, and loads the atmosphere with suspended dust aerosols This article presents an extensive review of the physics of wind-blown sand and dust on Earth and Mars Specifically, we review the physics of aeolian saltation, the formation and development of sand dunes and ripples, the physics of dust aerosol emission, the weather phenomena that trigger dust storms, and the lifting of dust by dust devils and other small-scale vortices We also discuss the physics of wind-blown sand and dune formation on Venus and Titan

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1,175 citations

Cites background from "The lift on a small sphere in a slo..."

...The Saffman lift force is caused by the sharp gradient in the fluid velocity above the particle bed which, due to Bernoulli’s principle, creates a lower pressure above the particle than below it (Saffman 1965)....
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...However, at lower roughness Reynolds numbers of ∼1–10, characteristic for saltation in air, the theoretical treatments of Saffman (1965) and McLaughlin (1991) predict a lift force that is substantially smaller than the drag force....
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References

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Journal Article•DOI•

The transverse force on a spinning sphere moving in a viscous fluid

[...]

S. I. Rubinow¹, Joseph B. Keller²•Institutions (2)

Stevens Institute of Technology¹, New York University²

01 Nov 1961-Journal of Fluid Mechanics

TL;DR: In this article, the Stokes and Oseen expansions of the Oseen equation are used to calculate the transverse force of a spinning sphere in a viscous fluid, which is in such a direction as to account for the curving of a pitched baseball, the long range of a flying golf ball, etc.

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Abstract: The flow about a spinning sphere moving in a viscous fluid is calculated for small values of the Reynolds number. With this solution the force and torque on the sphere are computed. It is found that in addition to the drag force determined by Stokes, the sphere experiences a force FL orthogonal to its direction of motion. This force is given by .Here a is the radius of the sphere, Ω is its angular velocity, V is its velocity, ρ is the fluid density and R is the Reynolds number, . For small values of R, the transverse force is independent of the viscosity μ. This force is in such a direction as to account for the curving of a pitched baseball, the long range of a spinning golf ball, etc. It is used as a basis for the discussion of the flow of a suspension of spheres through a tube.The calculation involves the Stokes and Oseen expansions. A representation of solutions of the Oseen equations in terms of two scalar functions is also presented.

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929 citations

"The lift on a small sphere in a slo..." refers methods in this paper

...The result (4.15) was given by Rubinow & Keller (1961) and applied by them to the Segr6 & Saberberg phenomenon, but the present work shows that unless the rotation speed is very much greater than the rate of shear, and for a freely rotating particle R = QK, the lift force due to particle rotation…...
[...]

Journal Article•DOI•

The motion of rigid particles in a shear flow at low Reynolds number

[...]

Francis P. Bretherton¹•Institutions (1)

University of Cambridge¹

01 Oct 1962-Journal of Fluid Mechanics

TL;DR: In this paper, the authors investigated how far the orbit of a particle of more general shape in a non-uniform shear in the presence of rigid boundaries may be expected to be qualitatively similar.

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Abstract: According to Jeffery (1923) the axis of an isolated rigid neutrally buoyant ellipsoid of revolution in a uniform simple shear at low Reynolds number moves in one of a family of closed periodic orbits, the centre of the particle moving with the velocity of the undisturbed fluid at that point. The present work is a theoretical investigation of how far the orbit of a particle of more general shape in a non-uniform shear in the presence of rigid boundaries may be expected to be qualitatively similar. Inertial and non-Newtonian effects are entirely neglected.The orientation of the axis of almost any body of revolution is a periodic function of time in any unidirectional flow, and also in a Couette viscometer. This is also true if there is a gravitational force on the particle in the direction of the streamlines. There is no lateral drift. On the other hand, certain extreme shapes, including some bodies of revolution, will assume one of two orientations and migrate to the bounding surfaces or to the centre of the flow. In any constant slightly three-dimensional uniform shear any body of revolution will ultimately assume a preferred orientation.

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813 citations

"The lift on a small sphere in a slo..." refers background in this paper

...…1932, p. 596; the notation here is slightly different) 13, Po = ;r;r2n+l’ n t For the two-dimensional problem of a cylinder moving in a simple shear (Bretherton 1962a), the bmic equation for the outer expansion can be reduced to a second-order equation that describes diffusion from a line source…...
[...]
...For motion at small Reynolds number, it was pointed out by the author (1956b), and more fully by Bretherton (1962b) that no sideways force on a single rigid spherical particle can be derived on the basis of the creeping flow equations whatever the velocity profile and relative size of particle and…...
[...]
...…Goldsmith & Mason (1962) have pointed out that the deformation of non-rigid particles will produce a lateral migration across streamlines, and Bretherton (1962b) has shown that rigid particles of an extreme shape may likewise migrate, the remarkable observations of Segr6 & Silberberg (1962),…...
[...]

Journal Article•DOI•

Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder

[...]

Ian Proudman¹, J. R. A. Pearson¹•Institutions (1)

University of Cambridge¹

01 May 1957-Journal of Fluid Mechanics

TL;DR: In this paper, the Navier-Stokes equation is replaced by a set of differential equations for the coefficients ψn and Ψn, but only one set of physical boundary conditions is applicable to each expansion (the no-slip conditions for the Stokes expansion, and the uniform-stream condition for the Oseen expansion).

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Abstract: This paper is concerned with the problem of obtaining higher approximations to the flow past a sphere and a circular cylinder than those represented by the well-known solutions of Stokes and Oseen. Since the perturbation theory arising from the consideration of small non-zero Reynolds numbers is a singular one, the problem is largely that of devising suitable techniques for taking this singularity into account when expanding the solution for small Reynolds numbers.The technique adopted is as follows. Separate, locally valid (in general), expansions of the stream function are developed for the regions close to, and far from, the obstacle. Reasons are presented for believing that these ‘Stokes’ and ‘Oseen’ expansions are, respectively, of the forms where (r, θ) are spherical or cylindrical polar coordinates made dimensionless with the radius of the obstacle, R is the Reynolds number, and vanish with R. Substitution of these expansions in the Navier-Stokes equation then yields a set of differential equations for the coefficients ψn and Ψn, but only one set of physical boundary conditions is applicable to each expansion (the no-slip conditions for the Stokes expansion, and the uniform-stream condition for the Oseen expansion) so that unique solutions cannot be derived immediately. However, the fact that the two expansions are (in principle) both derived from the same exact solution leads to a ‘matching’ procedure which yields further boundary conditions for each expansion. It is thus possible to determine alternately successive terms in each expansion.The leading terms of the expansions are shown to be closely related to the original solutions of Stokes and Oseen, and detailed results for some further terms are obtained.

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777 citations

"The lift on a small sphere in a slo..." refers methods in this paper

...Although the technique of matching inner and outer expansions is available to overcome this problem (Kaplun & Lagerstrom 1957; Proudman & Pearson 1957), the case of bodies in shear flow is rendered difficult by.the fact that fundamental solutions of the Oseen-like equation for the outer expansion…...
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Journal Article•DOI•

Behaviour of macroscopic rigid spheres in Poiseuille flow Part 1. Determination of local concentration by statistical analysis of particle passages through crossed light beams

[...]

G. Segré¹, A. Silberberg¹•Institutions (1)

Weizmann Institute of Science¹

01 Sep 1962-Journal of Fluid Mechanics

TL;DR: In this paper, an apparatus is described for determining particle passages through any selected point on a tube cross-section, based on the simultaneous blocking out of two mutually perpendicular light beams by a particle passing through their common region.

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Abstract: An apparatus is described for determining particle passages through any selected point on a tube cross-section. The method depends on the simultaneous blocking out of two mutually perpendicular light beams by a particle passing through their common region. The coincidence is registered and counted electronically. At higher particle concentrations coincidences are also registered arising from a pair-wise occupation of the light beams by two particles. An analysis is presented showing that these pair coincidences can be allowed for exactly in terms of experimentally measurable quantities.The reliability and reproducibility of the method is discussed and illustrated by examples from sphere suspensions in Poiseuille flow.

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559 citations

Journal Article•DOI•

The flow of suspensions through tubes. I. Single spheres, rods, and discs

[...]

H.L Goldsmith¹, S. G. Mason¹•Institutions (1)

McGill University¹

01 Jun 1962-Journal of Colloid Science

TL;DR: In this article, the motions of single rigid spheres, rods, and discs, and of fluid drops suspended in liquids undergoing Poiseuille flow have been studied, with the exception of effects due to interaction with the wall and neglect of particle size, provided that the equivalent axis ratio re was used instead of the particle axis ratio rp.

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285 citations

"The lift on a small sphere in a slo..." refers background in this paper

...Although Goldsmith & Mason (1962) have pointed out that the deformation of non-rigid particles will produce a lateral migration across streamlines, and Bretherton (1962b) has shown that rigid particles of an extreme shape may likewise migrate, the remarkable observations of Segr6 & Silberberg…...
[...]