# 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

${\displaystyle \mathrm {Stk} ={\frac {t_{0}\,u_{0}}{l_{0}}}}$

where ${\displaystyle t_{0}}$ is the relaxation time of the particle (the time constant in the exponential decay of the particle velocity due to drag), ${\displaystyle u_{0}}$ is the fluid velocity of the flow well away from the obstacle and ${\displaystyle l_{0}}$ 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 less than unity, the particle drag coefficient is inversely proportional to the Reynolds number itself. In that case, the characteristic time of the particle can be written as

${\displaystyle t_{0}={\frac {\rho _{p}d_{p}^{2}}{18\mu _{g}}}}$

where ${\displaystyle \rho _{p}}$ is the particle density, ${\displaystyle d_{p}}$ is the particle diameter and ${\displaystyle \mu _{g}}$ 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 ${\displaystyle \mathrm {Stk} \gg 1}$ , particles will detach from a flow especially where the flow decelerates abruptly. For ${\displaystyle \mathrm {Stk} \ll 1}$ , particles follow fluid streamlines closely. If ${\displaystyle \mathrm {Stk} <0.1}$ , tracing accuracy errors are below 1%.[2]

## Non-Stokesian drag regime

The preceding analysis will not be accurate in the ultra-Stokesian regime. i.e. if the particle Reynolds number is much greater than unity. Assuming a Mach number much less than unity. A generalized form of the Stokes number was demonstrated by Israel & Rosner.[3]

${\displaystyle {\text{Stk}}_{\text{e}}={\text{Stk}}{\frac {24}{{\text{Re}}_{o}}}\int _{0}^{{\text{Re}}_{o}}{\frac {d{\text{Re}}^{\prime }}{C_{D}({\text{Re}}^{\prime }){\text{Re}}^{\prime }}}}$

Where ${\displaystyle {\text{Re}}_{o}}$ is the "particle free-stream Reynolds number",

${\displaystyle {\text{Re}}_{o}={\frac {\rho _{g}|\mathbf {u} |d_{p}}{\mu _{g}}}}$

An additional function ${\displaystyle \psi ({\text{Re}}_{o})}$ was defined by,[3] this describes the non-Stokesian drag correction factor,

${\displaystyle {\text{Stk}}_{e}={\text{Stk}}\cdot \psi ({\text{Re}}_{o})}$

It follows that this function is defined by,

${\displaystyle \psi ({\text{Re}}_{o})={\frac {24}{{\text{Re}}_{o}}}\int _{0}^{{\text{Re}}_{o}}{\frac {d{\text{Re}}^{\prime }}{C_{D}({\text{Re}}^{\prime }){\text{Re}}^{\prime }}}}$

Considering the limiting particle free-stream Reynolds numbers, as ${\displaystyle {\text{Re}}_{o}\rightarrow 0}$ then ${\displaystyle C_{D}({\text{Re}}_{o})\rightarrow 24/{\text{Re}}_{o}}$ and therefore ${\displaystyle \psi \rightarrow 1}$ . Thus as expected there correction factor is unity in the Stokesian drag regime. Wessel & Righi [4] evaluated ${\displaystyle \psi }$ for ${\displaystyle C_{D}({\text{Re}})}$ from the empirical correlation for drag on a sphere from Schiller & Naumann.[5]

${\displaystyle \psi ({\text{Re}}_{o})={\frac {3({\sqrt {c}}{\text{Re}}_{o}^{1/3}-\arctan({\sqrt {c}}{\text{Re}}_{o}^{1/3}))}{c^{3/2}{\text{Re}}_{o}}}}$

Where the constant ${\displaystyle c=0.158}$ . The conventional Stokes number will significantly underestimate the drag force for large particle free-stream Reynolds numbers. Thus overestimating the tendency for particles to depart from the fluid flow direction. This will lead to errors in subsequent calculations or experimental comparisons.

## 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[6] as:

${\displaystyle c/c_{0}=1+(u_{0}/u-1)\left(1-{\frac {1}{1+\mathrm {Stk} (2+0.617u/u_{0})}}\right)}$

where ${\displaystyle c}$ is particle concentration, ${\displaystyle u}$ 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,

${\displaystyle \mathrm {Stk} ={\frac {u_{0}V_{s}}{dg}}}$

where ${\displaystyle V_{s}}$ is the particle's settling velocity, ${\displaystyle d}$ is the sampling tubes inner diameter, and ${\displaystyle g}$ is the acceleration of gravity.

## References

1. Brennen, Christopher E. (2005). Fundamentals of multiphase flow (Reprint. ed.). Cambridge [u.a.]: Cambridge Univ. Press. ISBN 9780521848046.
2. Cameron Tropea; Alexander Yarin; John Foss, eds. (2007-10-09). Springer Handbook of Experimental Fluid Mechanics. Springer. ISBN 978-3-540-25141-5.
3. Israel, R.; Rosner, D. E. (1982-09-20). "Use of a Generalized Stokes Number to Determine the Aerodynamic Capture Efficiency of Non-Stokesian Particles from a Compressible Gas Flow". Aerosol Science and Technology. 2 (1): 45–51. Bibcode:1982AerST...2...45I. doi:10.1080/02786828308958612. ISSN 0278-6826.
4. Wessel, R. A.; Righi, J. (1988-01-01). "Generalized Correlations for Inertial Impaction of Particles on a Circular Cylinder". Aerosol Science and Technology. 9 (1): 29–60. Bibcode:1988AerST...9...29W. doi:10.1080/02786828808959193. ISSN 0278-6826.
5. L, Schiller & Z. Naumann (1935). "Uber die grundlegenden Berechnung bei der Schwerkraftaufbereitung". Zeitschrift des Vereines Deutscher Ingenieure. 77: 318–320.
6. Belyaev, SP; Levin, LM (1974). "Techniques for collection of representative aerosol samples". Aerosol Science. 5 (4): 325–338. Bibcode:1974JAerS...5..325B. doi:10.1016/0021-8502(74)90130-X.
• Fuchs, N. A. (1989). The mechanics of aerosols. New York: Dover Publications. ISBN 978-0-486-66055-4.
• Hinds, William C. (1999). Aerosol technology: properties, behavior, and measurement of airborne particles. New York: Wiley. ISBN 978-0-471-19410-1.
• Snyder, WH; Lumley, JL (1971). "Some Measurements of Particle Velocity Autocorrelation Functions in a Turbulent Flow". Journal of Fluid Mechanics. 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): 119. Bibcode:2004NJPh....6..119C. doi:10.1088/1367-2630/6/1/119.