The sedimentation of a suspension of uniform spheres under conditions of viscous flow

Richardson, John F.; Zaki, W.N.

doi:10.1016/0009-2509(54)85015-9

Cited by 577 publications

(360 citation statements)

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“…2.2 , the ODE solutions prescribe the distribution of the particle concentration and ultimately the velocity profile of the slurry in the normal direction as a result of the balanced settling and resuspension forces. Using the kinematic boundary condition (6), and by moving on to the next order in the particle transport equation, we obtain a system of conservation laws in terms of the dependent, conserved variables, U = (h, hφ 0 ):…”

Section: Hyperbolic System Of Pdesmentioning

confidence: 99%

“…Some of the earliest work on the subject investigated gravitational settling/sedimentation of a uniform suspension of particles in a quiescent bulk fluid [5][6][7]. Richardson and Zaki [6] carried out a series of experiments to suggest an expression for the drag force; this expression showed good agreement with theory. In the current study, the interest lies in the gravitational transport of particles within a thin film of viscous oil.…”

Section: Introductionmentioning

confidence: 99%

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Hyperbolic systems of conservation laws in gravity-driven, particle-laden thin-film flows

Mavromoustaki¹,

Bertozzi²

2014

J Eng Math

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Our study focuses on gravity-driven, particle-laden flows which are pertinent to a wide range of industrial and geophysical settings in which transport of suspensions occur. In the present study we employ a previously derived model by Murisic et al. (Physica D: Nonlinear Phenomena, 240, 2011) which uses the lubrication approximation to describe particle-laden films on an incline. The model consists of a coupled system of hyperbolic conservation laws for the interface position and the particle concentration. While it has been shown that the model compares well quantitatively with experiments, it lacks analysis. The objectives of this paper focus on the study of the Riemann problem for this system of conservation laws and how the results relate to experiments. We investigate the governing system analytically and numerically; the equations exhibit rich mathematical structures including double-shock wave solutions, rarefaction waves and singular shocks.

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Section: Hyperbolic System Of Pdesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Hyperbolic systems of conservation laws in gravity-driven, particle-laden thin-film flows

Mavromoustaki¹,

Bertozzi²

2014

J Eng Math

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“…The oscillatory dynamics of such particles can vary a lot depending on the particle's size, shape and its inertia, and the surrounding flow properties. This can be important in a variety of fields ranging from sediment transport and fluidization to multiphase particleand bubble-column reactors [4,5,6]. …”

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confidence: 99%

Mass and Moment of Inertia Govern the Transition in the Dynamics and Wakes of Freely Rising and Falling Cylinders

et al. 2017

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In this Letter, we study the motion and wake-patterns of freely rising and falling cylinders in quiescent fluid. We show that the amplitude of oscillation and the overall system-dynamics are intricately linked to two parameters: the particle's mass-density relative to the fluid m * ≡ ρp/ρ f and its relative moment-of-inertia I * ≡ Ip/I f . This supersedes the current understanding that a critical mass density (m * ≈ 0.54) alone triggers the sudden onset of vigorous vibrations. Using over 144 combinations of m * and I * , we comprehensively map out the parameter space covering very heavy (m * > 10) to very buoyant (m * < 0.1) particles. The entire data collapses into two scaling regimes demarcated by a transitional Strouhal number, Stt ≈ 0.17. Stt separates a mass-dominated regime from a regime dominated by the particle's moment of inertia. A shift from one regime to the other also marks a gradual transition in the wake-shedding pattern: from the classical 2S (2-Single) vortex mode to a 2P (2-Pairs) vortex mode. Thus, auto-rotation can have a significant influence on the trajectories and wakes of freely rising isotropic bodies.Path-instabilities are a common observation in the dynamics of buoyant and heavy particles. Common examples are the fluttering of falling leaves and disks, and the cork-screw and spiral trajectories of air-bubbles rising in water [1,2,3]. The oscillatory dynamics of such particles can vary a lot depending on the particle's size, shape and its inertia, and the surrounding flow properties. This can be important in a variety of fields ranging from sediment transport and fluidization to multiphase particleand bubble-column reactors [4,5,6].Examples of organisms that exploit path-instabilities are plants and aquatic animals. These often make use of passive appendages attached to their bodies (plumed seeds, barbs, tails, and protrusions) to generate locomotion [7,8,9]. Recently, Lācis et al. [7] demonstrated that the interaction between the wake of a falling bluff body and a protrusion clamped to its rear end can generate a sidewards drift by means of a symmetry-breaking instability similar to that of an inverted pendulum. Such kinds of passive interactions are advantageous to locomotion, since no energy needs to be spent by the animal. Instead, the energy can be extracted through fluid-structure interaction.

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“…About settling and deposition, one might see Kynch (1952), Richardson and Zaki (1954), Krone (1962), Mehta (1986), and Winterwerp (2002). On the consolidation process, further readings are in Toorman (1996) and Roberts et al (1998).…”

Section: On the Dynamics Of Cohesive Sediments Transportmentioning

confidence: 98%

A morphodynamic model for cohesive sediments transport

Peixoto

Rosman

Vinzón

2017

RBRH

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The dynamics of cohesive sediments in natural water bodies is of great importance for coastal engineering. Quite often, it is necessary to assess sedimentological processes in regions where cohesive sediments are prevalent, such as in many harbors. Such assessments poses challenges, especially in quantitative evaluations, as the physics involved in the transport, sedimentation and erosion of cohesive sediments is rather complex. For that, by all means, it is important to have analysis tools capable of computing cohesive sediments dynamics. This paper presents the development of a vertically averaged eulerian transport model for suspended cohesive sediments. Such model is currently available on SisBaHiA  , which is a computational modeling system developed and maintained in COPPE/UFRJ. After describing the model implementation, we present a discussion on consistency and validation tests, for each implemented mechanism within the sedimentological process, with a focus in the erosion and sedimentation fluxes. In sequence, we compare model results with the theoretical results and, for some tests, with the results of the simulations performed with the Delft3D  and Telemac  modeling systems. These models are similar to the one presented in this paper, and are well accepted and widely used on professional projects and studies. The tests showed that the developed model results were consistent for all the evaluated mechanism, and hence it's positively verified.Keywords: Erosion and sedimentation; Environmental modeling; Sediments transport; Cohesive sediments. RESUMOA dinâmica de sedimentos coesivos em corpos d'água naturais é um tema de grande importância na prática da engenharia costeira, sendo recorrente a necessidade de serem feitos prognósticos sobre esses fenômenos. Devido ao fato da física que envolve o transporte desses sedimentos ser relativamente complexa, o desenvolvimento de estudos e avaliações em projetos de engenharia se mostram um desafio. Assim, é importante que existam ferramentas de análise capazes de descrever os processos sedimentológicos para sedimentos coesivos. Nesse sentido, este artigo apresenta o desenvolvimento de um modelo euleriano 2DH de transporte advectivo-difusivo de sedimentos coesivos em suspensão. Tal modelo está disponível no SisBaHiA  , um software computacional de modelagem, desenvolvido e mantido na COPPE/UFRJ. Após a implantação, procedeu-se a etapa de verificação dos mecanismos implementados. Os resultados do modelo foram confrontados com resultados teóricos e, para alguns testes, com os resultados das mesmas simulações feitas com o Delft3D  e o Telemac  , que são modelos similares ao desenvolvido neste trabalho. Os testes confirmaram que o modelo desenvolvido apresenta resultados consistentes para todos os mecanismos avaliados, sendo assim positivamente verificado. Palavras-chave:Erosão e sedimentação; Modelagem ambiental; Transporte de sedimentos; Sedimentos coesivos.

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The sedimentation of a suspension of uniform spheres under conditions of viscous flow

Cited by 577 publications

References 0 publications

Hyperbolic systems of conservation laws in gravity-driven, particle-laden thin-film flows

Hyperbolic systems of conservation laws in gravity-driven, particle-laden thin-film flows

Mass and Moment of Inertia Govern the Transition in the Dynamics and Wakes of Freely Rising and Falling Cylinders

A morphodynamic model for cohesive sediments transport

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