# Stokes number

The **Stokes number** (**Stk**), named after George Gabriel Stokes, is a dimensionless number characterising the behavior of particles suspended in a fluid flow. The Stokes number is defined as the ratio of the characteristic time of a particle (or droplet) to a characteristic time of the flow or of an obstacle, or

where is the relaxation time of the particle (the time constant in the exponential decay of the particle velocity due to drag), is the fluid velocity of the flow well away from the obstacle and is the characteristic dimension of the obstacle (typically its diameter). A particle with a low Stokes number follows fluid streamlines (perfect advection), while a particle with a large Stokes number is dominated by its inertia and continues along its initial trajectory.

In the case of Stokes flow, which is when the particle (or droplet) Reynolds number is low enough that the particle drag coefficient is inversely proportional to the Reynolds number itself, the characteristic time of the particle can be defined as

where is the particle density, is the particle diameter and is the gas dynamic viscosity.^{[1]}

In experimental fluid dynamics, the Stokes number is a measure of flow tracer fidelity in particle image velocimetry (PIV) experiments where very small particles are entrained in turbulent flows and optically observed to determine the speed and direction of fluid movement (also known as the velocity field of the fluid). For acceptable tracing accuracy, the particle response time should be faster than the smallest time scale of the flow. Smaller Stokes numbers represent better tracing accuracy; for , particles will detach from a flow especially where the flow decelerates abruptly. For , particles follow fluid streamlines closely. If , tracing accuracy errors are below 1%.^{[2]}

## Application to anisokinetic sampling of particles

For example, the selective capture of particles by an aligned, thin-walled circular nozzle is given by Belyaev and Levin^{[3]} as:

where is particle concentration, is speed, and the subscript 0 indicates conditions far upstream of the nozzle. The characteristic distance is the diameter of the nozzle. Here the Stokes number is calculated,

where is the particle's settling velocity, is the sampling tubes inner diameter, and is the acceleration of gravity.

## References

- ↑ Brennen, Christopher E. (2005).
*Fundamentals of multiphase flow*(Reprint. ed.). Cambridge [u.a.]: Cambridge Univ. Press. ISBN 9780521848046. - ↑ Cameron Tropea; Alexander Yarin; John Foss (eds.).
*Springer Handbook of Experimental Fluid Mechanics*. Springer. ISBN 978-3-540-25141-5. - ↑ Belyaev, SP; Levin, LM (1974). "Techniques for collection of representative aerosol samples".
*Aerosol Science*. Pergammon Press.**5**: 325–338. doi:10.1016/0021-8502(74)90130-X.

## Further reading

- Fuchs, N. A. (1989).
*The mechanics of aerosols*. New York: Dover Publications. ISBN 0-486-66055-9. - Hinds, William C. (1999).
*Aerosol technology: properties, behavior, and measurement of airborne particles*. New York: Wiley. ISBN 0-471-19410-7. - Snyder, WH; Lumley, JL (1971). "Some Measurements of Particle Velocity Autocorrelation Functions in a Turbulent Flow".
*Journal of Fluid Mechanics*. Cambridge Univ Press.**48**: 41–71. Bibcode:1971JFM....48...41S. doi:10.1017/S0022112071001460. - Collins, LR; Keswani, A (2004). "Reynolds number scaling of particle clustering in turbulent aerosols".
*New Journal of Physics*.**6**(119). Bibcode:2004NJPh....6..119C. doi:10.1088/1367-2630/6/1/119.