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 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
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%.
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.
Where is the "particle free-stream Reynolds number",
It follows that this function is defined by,
Considering the limiting particle free-stream Reynolds numbers, as then and therefore . Thus as expected there correction factor is unity in the Stokesian drag regime. Wessel & Righi evaluated for from the empirical correlation for drag on a sphere from Schiller & Naumann.
Where the constant . 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
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.
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