Properties of Time-Dependent Stokes Flow and the Regularization ofVelocity Flucutations in Particle Suspensions
Solid particles, suspended in a fluid and subject to an external force, such as gravity, settle if the density of the particles is higher than the density of the fluid. This process, which separates fluid and particles, is called sedimentation. It is important for a variety of processes in nature an...
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|Summary:||Solid particles, suspended in a fluid and subject to an external force, such as gravity, settle if the density of the particles is higher than the density of the fluid. This process, which separates fluid and particles, is called sedimentation. It is important for a variety of processes in nature and industry.
While naturally occuring particles come in a variety of shapes with smooth and rough surfaces, many of the fundamental issues of the sedimentation process can be studied with spherical particles. Experimentally, this situation can be realized, for instance, for glass beads which settle in a vessel. For these systems, theoretical predictions and experiments agree about the average settling rate of the particles, which depends on the concentration of the particles in the fluid.
In contrast, there is much controversy about the fluctuations of the particle velocities. Available theories assume that the flow past a particle obeys the steady Stokes equation, i.e. the flow is proportional to the instantaneous particle velocity and decays as $1/$distance from the particle. Caflisch and Luke (Phys. Fluids, 28:759, 1985) pointed out that due to these long-range hydrodynamic interactions, the fluctuations diverge for a homogeneous distribution of particles: At a distance $r$ from a particle, the fluctuations of the flow produced by that particle decay as the velocity squared, i.e. as $1/r^2$.
Since a spherical shell at distance $r$ and of width $dr$ contains $r^2 dr$ particles, all shells contribute equally to the fluctuations and the integral over all shells diverges with the volume of the vessel. While numerical simulations confirm the divergence, it is in marked contrast to experiments, which show that the flow is correlated on a finite length and that correlation length and fluctuations are finite and independent of the vessel size. To uncover the discrepancy between simulations and experiments, a number of theoretical studies have looked at effects of the wall, vertical stratification of the sediment or particle concentration fluctuations. However, no conclusive explanation of the discrepancies has emerged.
All theories and simulations for sedimentation above assume that the flow past the particles is steady Stokes flow. This assumption is believed to be fairly robust. In this work, the assumption is dropped and the flow is assumed to be unsteady Stokes flow, i.e. the temporal evolution of the velocity field is taken into account. How does unsteady Stokes flow differ from steady Stokes flow? The differences between steady and unsteady flow are known since the equations were formulated.
In his seminal 1851 article, which is most of all famous for the equations which describe the steady flow around a steadily moving sphere, Stokes also provided the equations for the time-dependent flow past a sphere whose center of mass is in oscillatory motion. However, the consequences of the differences between the two cases and in particular their implications for sediment fluctuations have not been studied. As we show here, taking the time-dependence into account goes a long way towards resolving many of the puzzles.
In order to see the effects of a time-dependent solution rather more clearly and not clouded by technicalities, we consider the situation of diffusional spreading from localized, sat fluctuating sources. For a homogeneous distribution of sources
it is shown that the fluctuations of the concentration field are finite if the full time-dependent equation is used for the concentration field. In contrast, if the concentration field is approximated by the quasisteady concentration field,
the fluctuations diverge. The concentration field model suggests new experiments to check time-dependence effects.
Moreover, we show that the velocity fluctuations of a particle suspension are finite if the flow around the particles is described by the time-dependent Stokes equation. The fluctuations are regularized essentially due to a cut-off of the steady Stokes flow velocity field. Taking this cut-off into account, the velocity fluctuations for a particle suspension is found in agreement with experimental measurements.
Time-dependence of Stokes flow is shown to be important for the dynamics of a particle suspension. This thesis initiates further investigations on particle suspensions and related problems, which take into account time-dependent Stokes flow.|