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

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

where $t_{0}$ is the relaxation time of the particle (the time constant in the exponential decay of the particle velocity due to drag), $u_{0}$ is the fluid velocity of the flow well away from the obstacle, and $l_{0}$ is the characteristic dimension of the obstacle (typically its diameter) or a characteristic length scale in the flow (like boundary layer thickness).^[1] 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 about one, 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

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

where $\rho _{p}$ is the particle density, $d_{p}$ is the particle diameter and $\mu _{g}$ is the fluid dynamic viscosity.^[2]

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 $\mathrm {Stk} \gg 1$ , particles will detach from a flow especially where the flow decelerates abruptly. For $\mathrm {Stk} \ll 1$ , particles follow fluid streamlines closely. If $\mathrm {Stk} <0.1$ , tracing accuracy errors are below 1%.^[3]

^ Raffel, M.; Willert, C. E.; Scarano, F.; Kahler, C. J.; Wereley, S. T.; Kompenhans, J. (2018). Particle Image Velocimetry (3rd ed.). Switzerland [u.a.]: Springer International Publishing. ISBN 978-3-319-68851-0.
^ 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. (2007-10-09). Springer Handbook of Experimental Fluid Mechanics. Springer. ISBN 978-3-540-25141-5.

[1] Raffel, M.; Willert, C. E.; Scarano, F.; Kahler, C. J.; Wereley, S. T.; Kompenhans, J. (2018). Particle Image Velocimetry (3rd ed.). Switzerland [u.a.]: Springer International Publishing. ISBN 978-3-319-68851-0.

[2] Brennen, Christopher E. (2005). Fundamentals of multiphase flow (Reprint. ed.). Cambridge [u.a.]: Cambridge Univ. Press. ISBN 9780521848046.

[3] Cameron Tropea; Alexander Yarin; John Foss, eds. (2007-10-09). Springer Handbook of Experimental Fluid Mechanics. Springer. ISBN 978-3-540-25141-5.

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