Settling of Particles; Threshold of Motion

Stokes Settling

Stokes developed a analysis of the settling of spherical particles involving the balance between the net gravity force acting on the particle and the viscous drag exerted by the fluid.

The viscous drag force acting on a particle will depend on its size, rand velocity, w, and the viscosity, m, of the fluid medium:

Eq 1:

Analyzing dimensionally:

Eq 2:

By inspection a=b=c=1. Without developing the hydrodynamics, the viscous drag force acting on a particle is:

Eq 3:

The net gravitational force (gravity less buoyancy) will be depend on the volume, density, r , and the acceleration of gravity, g:

Eq 4:

At terminal velocity, these two forces are balanced:

Eq 5:

Solving for w:

Eq 6:

Typically is 1500 kg m and for seawater µ is .01 kg ms so:

D	w	1 meter descent	1000 meter descent
1 mm	87 cm s-1	1.15 x 100 s ~ 1 second	1.15 x 103 s ~ 19 minutes
0.1 mm	0.87 cm s-1	1.15 x 102 s ~ 2 minutes	1.15 x 105 s ~ 31 hours
0.01 mm	0.0087 cm s-1	1.15 x 104 s ~ 3 hours	1.15 x 107 s ~ 1/3 year
0.001 mm = 1 m m	0.000087 cm s-1	1.15 x 106 s ~ 12 days	1.15 x 109 s ~ 35 years

The Critical Stress

One the critical stress is just exceeded, particles will advance in the direction of flow due to irregular jumps or less commonly rolls. This mode of tranport is termed the bedload and conceptually can be thought of as being deterministic, that is the behavior of a particle once in motion is dominanted by the gravity force. As the stress is further increased, particles will also begin to be suspended in solution and subject to turbulent forces. This mode of transport is called the suspended load. Due to these two modes of transport there will be a flux of material across a plane perpendicular to the flow. Our ultimate goal is to determine this mass flux by integrating the product of the velocity profile and concentration profile.

The motion of sediment can be parameterized in a number of ways. The oldest of these is due to Hjulstrom who summarized observational data in terms of fluid velocity and grain size:

Figure 1

There is a envelope of values for small particles, contrasting unconsolidated and consolidated/cohesive sediment. This reflects the importance of interparticle forces because of the higher ratio of surface area to volume.

There are a number of variants of the Hjulstrom diagram, using grain diameter as one parameter and some measure of the stress as the other (via the quadratic stress law: u, u₁₀₀ or stress itself: u*):

Figure 2

Shields first suggested a more general scaling, based on an analysis of the forces acting on particles. His analysis suggested an organization of the data based on two factors.

• The first is the entrainment factor, , which scales the boundary stress to the density contrast between the particles and the fluid, -, and the particle size, D and thus compares the applied stress to the gravitational force acting on an individual particle:

  Eq 7:

  g is the gravitational acceleration.

• The other is a Reynolds number, relevant to an individual particle, comparing turbulent and viscous forces:

  Eq 8:

For non-cohesive sediments, under a wide range of experimental conditions, a boundary is found in this parameter space (entrainment factor, Reynolds number) between no motion of the sediment and active erosion of the sediment:

Figure 3

As an example consider the situation for Re=10 where (reading from Figure 3) the critical is 3 x 10:

Eq 9:

Solving each of the two equations for D:

Eq 10:

Equating these expressions for D:

Eq 11:

and solving for u*, we find:

Eq 12:

For grains of density 2700 g/cm in water of viscosity 10^-3 kg ms:

Eq 13:

That is 1.6 cm/s is the scale of the rms velocity fluctuations necessary to just begin to transport a 0.6 mm grain.