Interest in the interaction of acoustic waves with particulate mixtures has a long history—dating back to the work of Rayleigh in the 19th century. This interest has intensified over the last fifteen years as advances in electronics and instrumentation science have brought the possibility of using ultrasound to characterize colloidal mixtures both in the laboratory and in-process, and in both of these contexts a small number of instruments are currently in use. The characterization of colloidal mixtures by ultrasound requires a formal theoretical basis which relates the properties of the mixture, particularly the dispersed phase particle size distribution (PSD), to the complex wavenumber governing propagation. The number of theoretical treatments is vast, having evolved over more than a century. This paper is intended to provide a review of these developments in a form which will enable new researchers in the field to climb a very steep learning curve in a relatively short time. We discuss definitions and production techniques for colloidal mixtures and the basic physical phenomena underlying wave propagation through them. We identify two approaches to the propagation problem—scattering and coupled-phase; these are treated both separately and comparatively, particularly in relation to limitations that arise when the concentration of particles is high and the basic theories break down. We introduce the basic method for the measurement of PSD and show how dynamic effects such as flocculation and crystallization can be observed and modelled. The core of all ultrasonic characterization procedures is the physical measurement of the ultrasonic wave attenuation coefficient and phase velocity as functions of frequency; here we discuss these techniques on the basis that what is observable or measurable about a colloid depends on both its physical properties and the frequency bandwidth available for measurement. This paper concludes with our view on future developments of measurement technique and theoretical treatments.

https://iopscience.iop.org/article/10.1088/0034-4885/68/7/R01/pdf

Ultrasound techniques for characterizing colloidal dispersions

Recent work on the drag experienced by an entrained particle in relative motion through a fluid is reviewed. It is shown that fluid turbulence, acceleration, particle shape and orientation, and particle-fluid mass transfer can all have a significant effect on the value of the drag force, particularly when the non-idealities cause a change in the flow pattern around the particle. Conditions under which the drag force can be predicted with certainty are still very limited, and some potentially valuable directions for future research are suggested.



On fait une revue des travaux recents sur la trainee a la quelle est sujette une particule entrainee dans un mouvement relatif dans un fluide. On voit que la turbulence du fluide, l'acceleration, la forme et l'orientation des particules et le transfert de masse entre les particules et le fluide sont des facteurs qui peuvent tous avoir un effet appreciable sur la valeur de la force de trainee, surtout lorsque l'absence de conditions ideales produit un changement dans l'allure de l'ecoulement autour de la particule. Les conditions qui permettent de predire avec certitude la force de trainee sont encore tres limitees et l'on propose des directives potentiellement valables pour les travaux futurs de recherche.

Motion of entrained particles in gas streams

The theory of the scattering of compression waves in viscous fluids is examined. The effects of fluid viscosity, of differences in density and in elastic modulus between the particles and the fluid, of heat transfer and of concentration are considered. Ultrasonic phase velocity and attenuation are derived. Results for the phase velocity are compared with several other formulations. The feasibility of using ultrasound to characterise suspensions is discussed.

http://iopscience.iop.org/article/10.1088/0022-3727/21/11/006/pdf

Velocity and attenuation of ultrasound in suspensions of particles in fluids

We present closures for the drag and virtual mass force terms appearing in a two-fluid model for flow of a mixture consisting of uniformly sized gas bubbles dispersed in a liquid. These closures were deduced through computational experiments performed using an implicit formulation of the lattice Boltzmann method with a BGK collision model. Unlike the explicit schemes described in the literature, this implicit implementation requires iterative calculations, which, however, are local in nature. While the computational cost per time step is modestly increased, the implicit scheme dramatically expands the parameter space in multiphase flow calculations which can be simulated economically. The closure relations obtained in our study are limited to a regular array of uniformly sized bubbles and were obtained by simulating the rise behaviour of a single bubble in a periodic box. The effect of volume fraction on the rise characteristics was probed by changing the size of the box relative to that of the bubble. While spherical bubbles exhibited the expected hindered rise behaviour, highly distorted bubbles tended to rise cooperatively. The closure for the drag force, obtained in our study through computational experiments, captured both hindered and cooperative rise. A simple model for the virtual mass coefficient, applicable to both spherical and distorted bubbles, was also obtained by fitting simulation results. The virtual mass coefficient for isolated bubbles could be correlated with the aspect ratio of the bubbles.

Analysis of drag and virtual mass forces in bubbly suspensions using an implicit formulation of the lattice Boltzmann method

Unsteady flow over a stationary sphere with small fluctuations in the free-stream velocity is considered at finite Reynolds number using a finite-difference method. The dependence of the unsteady drag on the frequency of the fluctuations is examined at various Reynolds numbers. It is found that the classical Stokes solution of the unsteady Stokes equation does not correctly describe the behaviour of the unsteady drag at low frequency. Numerical results indicate that the force increases linearly with frequency when the frequency is very small instead of increasing linearly with the square root of the frequency as the classical Stokes solution predicts. This implies that the force has a much shorter memory in the time domain. The incorrect behaviour of the Basset force at large times may explain the unphysical results found by Reeks & Mckee (1984) wherein for a particle introduced to a turbulent flow the initial velocity difference between the particle and fluid has a finite contribution to the long-time particle diffusivity. The added mass component of the force at finite Reynolds number is found to be the same as predicted by creeping flow and potential theories. Effects of Reynolds number on the unsteady drag due to the fluctuating free-stream velocity are presented. The implications for particle motion in turbulence are discussed.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112091000629

Unsteady drag on a sphere at finite reynolds number with small fluctuations in the free-stream velocity

Stokes's linearized equations of motion are used to calculate the flow field generated by a spheroid executing axial translatory oscillations in an infinite, otherwise still, incompressible, viscous fluid. The flow field, expressed in terms of spheroidal wave functions of order one, is used to develop general expressions for the drag on oscillating prolate and oblate spheroids. Formulae for the approximate drag, useful in making calculations, are obtained for small values of an oscillation parameter. These formulae reduce to the Stokes result in the limit when the spheroid becomes a sphere and the steady-state drag for a spheroid as the frequency of oscillation becomes zero. The fluid forces on spheroids of various shapes are compared graphically. The approximate formulae for the drag are integrated over all frequencies to obtain formulae for the drag on spheroids executing general axial translatory accelerations. The fluid resistance on the spheroid is expressed as the sum of an added mass effect, a steady-state drag and an effect due to the history of the motion. A table of added mass, viscous and history coefficients is given.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112072002939

The Stokes-flow drag on prolate and oblate spheroids during axial translatory accelerations

Measured displacement-time relations for spheres gravitationally accelerating from rest through viscous fluids are compared with the displacement-time relations predicted by the Stokes flow solution. The Stokes flow solution, consisting of a viscous drag, an added mass effect, and a history-of-the-motion integral, agrees with experimental values over a range that is larger than expected. The total dimensionless distance for which the theoretical and measured values agree depends on the ratio of sphere density to fluid density. The instantaneous Reynolds number at the point of departure between measurements and theory depends on the Stokes theory terminal Reynolds number. For the experiments reported, the theory is valid to points for which the instantaneous Reynolds number is as high as 10,000.

Robert Y. S. Lai

Papers

The Stokes-flow drag on prolate and oblate spheroids during axial translatory accelerations

Validity of Stokes Theory for Accelerating Spheres