^{1,a)}and John W. M. Bush

^{1}

### Abstract

We present the results of a combined experimental and theoretical investigation of sedimentation in a stratified fluid. The theory of sedimentation in a homogeneous ambient is extended to include the influence of a spatially varying particle settling speed. The results of an experimental investigation of latex particles settling in a stably stratified salt water solution are reported. Density variations in the suspending fluid reduce the particle settling speed which increases particle concentrations, thus enhancing the effects of hindered settling. A criterion is developed for the convective instability of an initially uniform suspension settling in a stably stratified ambient. If, as depth increases, the magnitude of the ambient density gradient decreases sufficiently rapidly, an initially uniform particle concentration will give rise to a statically unstable density profile. Experimental observations provide qualitative verification of this new stability criterion.

The authors thank Tom Peacock for his selfless help in setting up the experiments and Jeff Parsons for a number of valuable discussions. F.B. was supported in part by a PSG-B fellowship from NSERC and by the Schlumberger Foundation. J.B. gratefully acknowledges support from an NSF Career Grant No. CTS-0130465.

I. INTRODUCTION

II. THEORETICAL DEVELOPMENTS

A. Homogeneous ambient

B. Stratified ambient

III. CONVECTIVE INSTABILITIES

IV. EXPERIMENTS

A. Time evolution of the top interface

B. Concentration jump

C. Instability

V. CONCLUSION

## Figures

Schematic illustration of the evolution of particle concentration along characteristics in a homogeneous ambient. (a) If , with , characteristics travel faster in the underlying region and an expansion fan forms from an initial discontinuity in particle concentration. (b) If increases with , a shock may develop from a continuous initial distribution. The bottom region shows the accumulation of particles at their packing concentration, .

Schematic illustration of the evolution of particle concentration along characteristics in a homogeneous ambient. (a) If , with , characteristics travel faster in the underlying region and an expansion fan forms from an initial discontinuity in particle concentration. (b) If increases with , a shock may develop from a continuous initial distribution. The bottom region shows the accumulation of particles at their packing concentration, .

The time evolution of an initial particle concentration jump of particles of radius and density settling in a stable linear density gradient, with density at the top and at the bottom, as computed via Eqs. (14) and (15). Here initial particle concentrations are below and above . Characteristics are divergent, leading to the formation of an expansion fan.

The time evolution of an initial particle concentration jump of particles of radius and density settling in a stable linear density gradient, with density at the top and at the bottom, as computed via Eqs. (14) and (15). Here initial particle concentrations are below and above . Characteristics are divergent, leading to the formation of an expansion fan.

Time evolution of a concentration jump of particles of radius and density settling in a stable linear density gradient, with density at the top and at the bottom, computed via Eqs. (14) and (17). Initial concentrations are below and above . Characteristics are convergent, thus the concentration jump remains sharp. The rate of descent of the upper interface and concentration jump are markedly different.

Time evolution of a concentration jump of particles of radius and density settling in a stable linear density gradient, with density at the top and at the bottom, computed via Eqs. (14) and (17). Initial concentrations are below and above . Characteristics are convergent, thus the concentration jump remains sharp. The rate of descent of the upper interface and concentration jump are markedly different.

If particles settle through an ambient density jump, , instabilities may develop if the concentration below the jump is greater than the initial concentration in the bottom region .

If particles settle through an ambient density jump, , instabilities may develop if the concentration below the jump is greater than the initial concentration in the bottom region .

A schematic illustration of the apparatus used in our experimental study. The tank is filled from below via the Oster double-bucket technique to obtain a stably stratified ambient with either a suspension of uniform concentration or a particle concentration jump.

A schematic illustration of the apparatus used in our experimental study. The tank is filled from below via the Oster double-bucket technique to obtain a stably stratified ambient with either a suspension of uniform concentration or a particle concentration jump.

Position of the top interface of a suspension with settling in a stratified ambient with a (a) weak and (b) strong density gradient. The dashed curves correspond to the theoretical predictions deduced by neglecting hindered settling and the solid curves are obtained by using Eq. (16). The stars indicate experimental measurements of the progression of the top interface. A typical error bar is shown. The corresponding density profile is shown on the top right.

Position of the top interface of a suspension with settling in a stratified ambient with a (a) weak and (b) strong density gradient. The dashed curves correspond to the theoretical predictions deduced by neglecting hindered settling and the solid curves are obtained by using Eq. (16). The stars indicate experimental measurements of the progression of the top interface. A typical error bar is shown. The corresponding density profile is shown on the top right.

Progression of a particle concentration jump, with and , settling in a stably stratified ambient with a (a) weak and (b) strong density gradient. The dashed curves correspond to theoretical predictions deduced by neglecting hindered settling and the solid curves are obtained by combining Eqs. (7) and (17). The stars are experimental measurements of the progression of the concentration jump and the dotted line indicates the recorded progression of the top interface. A typical error bar is shown. The corresponding density profile is shown on the top right.

Progression of a particle concentration jump, with and , settling in a stably stratified ambient with a (a) weak and (b) strong density gradient. The dashed curves correspond to theoretical predictions deduced by neglecting hindered settling and the solid curves are obtained by combining Eqs. (7) and (17). The stars are experimental measurements of the progression of the concentration jump and the dotted line indicates the recorded progression of the top interface. A typical error bar is shown. The corresponding density profile is shown on the top right.

A schematic illustration indicating the bulk stability of stably stratified ambients through which an initially uniform particle concentration settles. (a) If the fluid density gradient is constant or decreases with height such that (24) is satisfied, the resulting bulk density profile will remain statically stable. (b) and (c) A uniform initial particle concentration settling in a density profile such that the fluid density gradient vanishes with depth becomes statically unstable. (d) Density jumps also result in the formation of instabilities.

A schematic illustration indicating the bulk stability of stably stratified ambients through which an initially uniform particle concentration settles. (a) If the fluid density gradient is constant or decreases with height such that (24) is satisfied, the resulting bulk density profile will remain statically stable. (b) and (c) A uniform initial particle concentration settling in a density profile such that the fluid density gradient vanishes with depth becomes statically unstable. (d) Density jumps also result in the formation of instabilities.

The convective motion prompted by an initially uniform concentration of particles settling in a constant density gradient overlying a region of constant density. The corresponding initial and final density profiles are shown in (d). The formation of centimetric plumes beneath the critical height is observed. Pictures were taken at 30 s intervals and the region of high concentration is seen to remain stationary while particle plumes are shed continuously. Note that the contrasts were accentuated using MATLAB; particles are also present in the lower region. Scale bars are 1 cm.

The convective motion prompted by an initially uniform concentration of particles settling in a constant density gradient overlying a region of constant density. The corresponding initial and final density profiles are shown in (d). The formation of centimetric plumes beneath the critical height is observed. Pictures were taken at 30 s intervals and the region of high concentration is seen to remain stationary while particle plumes are shed continuously. Note that the contrasts were accentuated using MATLAB; particles are also present in the lower region. Scale bars are 1 cm.

Images of the top interface as it settles through (a) a stably stratified ambient in a stable region and later through (b) a region of constant density where convective motion occurred . The initial and final density profiles are shown in Fig. 9(d). The density remains approximately constant for and decreases linearly for . (a) The top interface is horizontal and relatively sharp when particles settle in a density gradient and do not generate convective overturning. (b) When the top interface reaches a region of constant density in which particles generate convective rolls, the interface becomes diffuse and tilted. Scale bars are 1 cm.

Images of the top interface as it settles through (a) a stably stratified ambient in a stable region and later through (b) a region of constant density where convective motion occurred . The initial and final density profiles are shown in Fig. 9(d). The density remains approximately constant for and decreases linearly for . (a) The top interface is horizontal and relatively sharp when particles settle in a density gradient and do not generate convective overturning. (b) When the top interface reaches a region of constant density in which particles generate convective rolls, the interface becomes diffuse and tilted. Scale bars are 1 cm.

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