Particle Settling Velocity

As soon as grains are lifted above the bed during the entrainment process, they begin to fall back to the bed. ^[1] The distance that grains travel downcurrent before coming to rest on the bed depends upon the drag and lift forces exerted by the current - including turbulence - and the settling velocity of the particles. ^[1] Settling velocity is therefore not a secondary detail but a primary control on how far any grain travels during a transport episode. A grain with a low settling velocity can be carried far downflow before turbulence can no longer support it; a grain with a high settling velocity drops out quickly near the source.

Terminal Fall Velocity

A particle initially accelerates as it falls through a fluid, but acceleration gradually decreases until a steady rate of fall called the terminal fall velocity is achieved. ^[1] For small particles, terminal fall velocity is reached very quickly. ^[1]

The rate at which particles settle after reaching fall velocity is a function of the viscosity of the fluid and the size, shape, and density of the particles. ^[1] The settling rate is determined by the interaction of upwardly directed forces - owing to buoyancy of the fluid and viscous resistance (drag) to fall - and downwardly directed forces arising from gravity. ^[1]

Stokes’s Law

For slow laminar flow at low concentrations of particles and low grain Reynolds numbers (R_eg), the drag coefficient C_D has been determined to equal 24/R_eg. ^[1] Substituting this value for C_D yields Stokes’s Law of settling: ^[1]

V = (ρ_s − ρ_f)gd² / 18μ

This law is often simplified to V = CD² (in cm/sec), where C is the constant (ρ_s − ρ_f)g / 18μ and D is particle diameter expressed in centimetres. ^[1] Values of C have been calculated for a range of common laboratory temperatures; thus, settling velocity V can be determined quickly for any value of particle diameter D. ^[1]

Stokes’s Law is elegant in its implications: settling velocity scales with the square of the diameter. This means that doubling grain size quadruples the settling velocity. A fine sand grain settles roughly 100 times faster than a clay particle of the same density - which is why rivers can carry clay in suspension for hundreds of kilometres while sand drops out within a few kilometres of the source.

Limitations of Stokes’s Law

Experimental determination of particle fall velocity shows that Stokes’s Law accurately predicts settling velocity only for particles less than about 0.1 to 0.2 mm in diameter. ^[1] Larger particles have fall velocities lower than those predicted by Stokes’s Law, apparently owing to inertial (turbulent) effects caused by the increased rates of fall of these larger grains. ^[1] Thus, the Stokes equation cannot be used for determining the settling velocity of sand, a very important component of most sediment. ^[1]

Other Factors That Reduce Fall Velocity

Fall velocity is decreased by decrease in temperature (which increases viscosity), decrease in particle density, and decrease in the sphericity (the degree to which the shape of a particle approaches the shape of a sphere) of the particles. ^[1] Most natural particles are not spheres, and departure from spherical shape decreases fall velocity. ^[1] Fall velocity is also decreased by increasing concentration of suspended sediment in the fluid, which increases the apparent viscosity and density of the fluid, and by turbulence. ^[1]

These multiple dependencies mean that settling velocity in a natural river or ocean current is not a fixed property of a grain - it shifts as temperature, turbulence, and sediment concentration change. Field measurement of particle size distributions in natural suspensions must account for all these effects, which is one reason mechanical grain-size analysis (settling columns, sieve analysis) requires careful temperature control and standardised procedures.

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