The Plane Parallel Flow of a Density-Stratified Fluid With and Without a Suspension of Particles

1973 ◽  
Vol 40 (2) ◽  
pp. 326-330 ◽  
Author(s):  
J. Val. Healy
Keyword(s):  
Particle Density ◽  
Infinite Plate ◽  
Axial Flow ◽  
Wall Stress ◽  
Spherical Particles ◽  
Uniform Density ◽  
Asymptotic Results ◽  
Infinite Region ◽  
Dilute Suspension ◽  
Long Cylinder

The response of a viscous incompressible nondiffusive fluid containing a dilute suspension of small spherical particles and occupying a semi-infinite region bounded by an infinite plate, when the latter is slowly moved parallel to itself in a prescribed manner normal to the direction of stratification, is studied. In particular, when the system density decays exponentially with distance from the plate, the problem is equivalent to the axial flow of the same system in a long tube. When the particle density vanishes, the problem reduces to the unsteady diffusion of vorticity in a long cylinder, with the “diffusivity” a function of the stratification length, whose solution is well known. A degeneracy occurs in the solution as the density approaches uniformity and this is examined using asymptotic expansions and some exact and asymptotic results for an impulsively moved plate are given. It is found that, for a given stratification, the ratio of the wall stress with particles to that without, increases first and then, after the non-equilibrium overshoot, it continues to increase with time; this is in contrast to the case of uniform density, when the ratio goes asymptotically to a constant value. When the particle density vanishes and the kinematic viscosity of the fluid goes to infinity at very large distances from the plate, the fluid there acquires, in a finite time, a finite velocity that strongly depends on the stratification length.

scholarly journals The unsteady Kármán problem for a dilute particle suspension

2003 ◽  
Vol 474 ◽  
pp. 379-409 ◽  
Author(s):  
M. R. FOSTER ◽  
P. W. DUCK ◽  
R. E. HEWITT
Keyword(s):  
Three Dimensional ◽  
Axial Flow ◽  
Continuous Phase ◽  
Spherical Particles ◽  
Gravitational Acceleration ◽  
Direction Parallel ◽  
Rotating Surface ◽  
Discrete Phase ◽  
Dilute Suspension ◽  
Rotating Boundary

We consider the unsteady three-dimensional Kármán flow induced by the impulsive rotation of an infinite rotating plane immersed in an incompressible viscous fluid with a dilute suspension of small solid monodisperse spherical particles. The flow is described in terms of a ‘dusty gas’ model, which treats the discrete phase (particles) and the continuous phase (fluid) as two continua occupying the same space and interacting through a Stokes drag mechanism. The model is extended to allow for a local gravitational acceleration in a direction parallel to the axis of rotation, and is valid for cases in which gravity acts either in the same direction as or in the opposite direction to the Ekman axial flow induced by the rotation of the plane.Analysis based on the theory of characteristics shows that the role of gravity is crucial to the treatment of the discrete-phase equations, particularly in regard to the appropriate boundary conditions to be applied at the solid surface. Other notable features include the presence of an essential singularity in the solution when gravity is absent; indeed this phenomenon may help to explain some of the difficulties encountered in previous studies of this type. If the gravitational force is directed away from the rotating surface, a number of other interesting features arise, including the development of discontinuities in the particle distribution profiles, with corresponding particle-free regions contained between the interface and the rotating boundary. These ‘shock’ features can be associated with a critical axial location in the boundary layer at which a balance is achieved between Ekman suction induced by the rotating boundary and the influence of gravitational effects acting to move particles away from the boundary.

Inertial effects in dusty Rayleigh–Taylor turbulence

2021 ◽  
Vol 926 ◽  
Author(s):  
Marta Magnani ◽  
Stefano Musacchio ◽  
Guido Boffetta
Keyword(s):  
Turbulent Mixing ◽  
Particle Density ◽  
Mixing Layer ◽  
Uniform Density ◽  
Heavy Particles ◽  
Turbulent Mixing Layer ◽  
Gravity Forces ◽  
Dilute Suspension ◽  
Two Phases ◽  
Set Up

We investigate the dynamics of a dilute suspension of small, heavy particles superposed on a reservoir of still, pure fluid. The study is performed by means of numerical simulations of the Saffman model for a dilute particle suspension (Saffman, J. Fluid Mech., vol. 13, issue 1, 1962, pp. 120–128). In the presence of gravity forces, the interface between the two phases is unstable and evolves in a turbulent mixing layer which broadens in time. In the case of negligible particle inertia, the particle-laden phase behaves as a denser fluid, and the dynamics of the system recovers to that of the incompressible Rayleigh–Taylor set-up. Conversely, particles with large inertia affect the evolution of turbulent flow, delaying the development of turbulent mixing and breaking the up–down symmetry within the mixing layer. The inertial dynamics also leads to particle clustering, characterised by regions with higher particle density than the initial uniform density, and by the increase of the local Atwood number.

Settling Behavior of Particles in Bubble Containing Newtonian Fluids: Experimental Study and Model Development

2021 ◽  
Author(s):  
Silin Jing ◽  
Xianzhi Song ◽  
Zhaopeng Zhu ◽  
Buwen Yu ◽  
Shiming Duan
Keyword(s):  
Reynolds Number ◽  
Drag Coefficient ◽  
Volume Concentration ◽  
Settling Velocity ◽  
Particle Density ◽  
Particle Diameter ◽  
Newtonian Fluids ◽  
Spherical Particles ◽  
Bubble Volume ◽  
Particle Reynolds Number

Abstract Accurate description of cuttings slippage in the gas-liquid phase is of great significance for wellbore cleaning and the control accuracy of bottom hole pressure during MPD. In this study, the wellbore bubble flow environment was simulated by a constant pressure air pump and the transparent wellbore, and the settling characteristics of spherical particles under different gas volume concentrations were recorded and analyzed by highspeed photography. A total of 225 tests were conducted to analyze the influence of particle diameter (1–12mm), particle density (2700–7860kg/m^3), liquid viscosity and bubble volume concentration on particle settling velocity. Gas drag force is defined to quantitatively evaluate the bubble’s resistance to particle slippage. The relationship between bubble drag coefficient and particle Reynolds number is obtained by fitting the experimental results. An explicit settling velocity equation is established by introducing Archimedes number. This explicit equation with an average relative error of only 8.09% can directly predict the terminal settling velocity of the sphere in bubble containing Newtonian fluids. The models for predicting bubble drag coefficient and the terminal settling velocity are valid with particle Reynolds number ranging from 0.05 to 167 and bubble volume concentration ranging from 3.0% to 20.0%. Besides, a trial-and-error procedure and an illustrative example are presented to show how to calculate bubble drag coefficient and settling velocity in bubble containing fluids. The results of this study will provide the theoretical basis for wellbore cleaning and accurate downhole pressure to further improve the performance of MPD in treating gas influx.

The instability of a dispersion of sedimenting spheroids

1989 ◽  
Vol 209 ◽  
pp. 521-542 ◽  
Author(s):  
Donald L. Koch ◽  
Eric S. G. Shaqfeh
Keyword(s):  
Particle Number ◽  
Particle Density ◽  
Particle Number Density ◽  
Density Fluctuations ◽  
Hydrodynamic Interactions ◽  
Spherical Particles ◽  
Inertial Effects ◽  
Number Density ◽  
Characteristic Particle ◽  
Particle Length

It is shown that hydrodynamic interactions between non-Brownian, non-spherical, sedimenting particles give rise to an increase in the number of neighbouring particles in the vicinity of any given particle. This result suggests that the suspension is unstable to particle density fluctuations even in the absence of inertia; a linear stability analysis confirms this inference. It is argued that the instability will lead to convection on a lengthscale (nl)−½, where l is a characteristic particle length and n is the particle number density. Sedimenting suspensions of spherical particles are shown to be stable in the absence of inertial effects.

scholarly journals Can terminal settling velocity and drag of natural particles in water ever be predicted accurately?

2020 ◽  
Author(s):  
Onno J. I. Kramer ◽  
Peter J. de Moel ◽  
Shravan K. R. Raaghav ◽  
Eric T. Baars ◽  
Wim H. van Vugt ◽  
...  
Keyword(s):  
Drinking Water ◽  
Measurement Errors ◽  
Settling Velocity ◽  
Particle Density ◽  
Fixed Bed ◽  
Drinking Water Treatment ◽  
Liquid System ◽  
Solid Particles ◽  
Spherical Particles ◽  
Fluidised Bed

Abstract. Natural particles are frequently applied in drinking water treatment processes in fixed bed reactors, in fluidised bed reactors, and in sedimentation processes to clarify water and to concentrate solids. When particles settle, it has been found that in terms of hydraulics, natural particles behave differently when compared to perfectly round spheres. To estimate the terminal settling velocity of single solid particles in a liquid system, a comprehensive collection of equations is available. For perfectly round spheres, the settling velocity can be calculated quite accurately. However, for naturally polydisperse non-spherical particles, experimentally measured settling velocities of individual particles show considerable spread from the calculated average values. This work aimed to analyse and explain the different causes of this spread. To this end, terminal settling experiments were conducted in a quiescent fluid with particles varying in density, size and shape. For the settling experiments, opaque and transparent spherical polydisperse and monodisperse glass beads were selected. In this study, we also examined drinking water related particles, like calcite pellets and crushed calcite seeding material grains, both applied in drinking water softening. Polydisperse calcite pellets were sieved and separated to acquire more uniformly dispersed samples. In addition, a wide variety of grains with different densities, sizes and shapes were investigated for their terminal settling velocity and behaviour. The derived drag coefficient was compared with well-known models such as Brown–Lawler. A sensitivity analysis showed that the spread is caused to a lesser extent by variations in fluid properties, measurement errors and wall effects. Natural variations in specific particle density, path trajectory instabilities and distinctive multi-particle settling behaviour caused a slightly larger degree of spread. In contrast, greater spread is caused by variations in particle size, shape and orientation.

scholarly journals Secondary Instabilities in Incompressible Axisymmetric Boundary Layers: Effect of Transverse Curvature

2012 ◽  
Vol 134 (2) ◽  
Author(s):  
N. Vinod ◽  
Rama Govindarajan
Keyword(s):  
Boundary Layer ◽  
Laminar Flow ◽  
Axial Flow ◽  
Base Flow ◽  
Transition To Turbulence ◽  
Flow Past A Cylinder ◽  
Incompressible Boundary ◽  
Layer Flow ◽  
Secondary Instabilities ◽  
Long Cylinder

The secondary instability of the incompressible boundary layer in the axial flow past a cylinder is studied. The laminar flow is shown to be always stable at high transverse curvatures to secondary disturbances. Because the primary mode is stable as well, (Tutty et al., 2002, “Boundary Layer Flow on a Long Thin Cylinder,”. Phys. Fluids, 14(2), pp. 628–637), this implies that the boundary layer on a thin long cylinder may undergo transition to turbulence by means very different from that on a flat plate. The azimuthal wavenumber of the least stable secondary modes (m±) are related to that of the primary (n) by m+ = 2n and m− = −n. The base flow is shown to be inviscidly stable at any curvature.

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