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Terminal settling velocity of a single particle

The tool calculates the terminal settling velocity of a single in an infinite domain.
Grain settling is very important in sedimentology as it governs the transport and deposition of sediment in river or oceanic flows.
The equation of Ferguson and Church, see Ferguson and Church 2004, is used in the tool for calculating the terminal settling velocity. The fall velocity of a particle is a function of the particle diameter, the fluid viscosity and density and the particle density. The equation of Ferguson and Church is of simple explicit form. Therefore, it is easy to use in a computer codes or other applications in e.g. geomorphology. The equation is based on dimensional analysis. Furthermore, the equation is valid for the entire range of viscous to turbulent conditions.

A nice exploration of grain settling, including python code, can be found at
hinderedsettling.com

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Short mathematical description

The terminal settling velocity of a single particle in an infinite domain surrounded by a stagnant fluid reads:

\begin{equation}
w = \frac{R g d_p^2}{C_1\nu + \sqrt{\left(0.75 C_2 R g d_p^3\right)} }
\end{equation}

in which $w$ is the terminal settling velocity, $g$ the acceleration due to gravity and $d_p$ the particle size diameter. The kinematic viscosity is expressed as $\nu$ and $R$ is the submerged specific gravity. $R$ is defined as follows:

\begin{equation}
R = \frac{\rho_p-\rho_f}{\rho_f}
\end{equation}

where $\rho_p$ is density of the particle and $\rho_f$ is density of the surrounding fluid. The parameters $C_1$ and $C_2$ take the value of $18$ and $0.4$ for smooth spheres respectively. For natural grains these values are slightly higher. For natural grains Ferguson and Church suggest values of $18$ and $1.0$ for $C_1$ and $C_2$ when sieve diameters are used and $20$ and $1.1$ where nominal diameters are used.

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References

Ferguson, R. and Church, M.
*
A Simple Universal Equation for Grain Settling Velocity
*
Journal of Sedimentary Research,
**
2004
**
, Vol. 74(6), pp. 933-937