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 formula_2 is the relaxation time of the particle (the time constant in the exponential decay of the particle velocity due to drag), formula_3 is the fluid velocity of the flow well away from the obstacle and formula_4 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 
where formula_6 is the particle density, formula_7 is the particle diameter and formula_8 is the fluid dynamic viscosity.
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 formula_9, particles will detach from a flow especially where the flow decelerates abruptly. For formula_10, particles follow fluid streamlines closely. If formula_11, tracing accuracy errors are below 1%. 
Relaxation time and tracking error in Particle Image Velocimetry.
The Stokes number provides a means of estimating the quality of Particle Image Velocimetry data sets, as previously discussed. However, a definition of a characteristic velocity or length scale may not be evident in all applications. Thus, a deeper insight of how a tracking delay arises could be drawn by simply defining the differential equations of a particle in the Stokes regime. A particle moving with the fluid at some velocity formula_12 will encounter a variable fluid velocity field as it advects. Let's assume the velocity of the fluid, in the Lagrangian frame of reference of the particle, is formula_13. It is the difference between these velocities that will generate the drag force necessary to correct the particle path:
The stokes drag force is then:
The particle mass is:
Thus, the particle acceleration can be found through Newton's second law:
The cut-off frequency and the particle transfer function, plotted on the side panel, allows for the assessment of PIV error in unsteady flow applications and its effect on turbulence spectral quantities and kinetic energy.
Particles through a shock wave.
The bias error in particle tracking discussed in the previous section is evident in the frequency domain, but it can be difficult to appreciate in cases where the particle motion is being tracked to perform flow field measurements (like in particle image velocimetry). A simple but insightful solution to the above-mentioned differential equation is possible when the forcing function formula_18 is a Heaviside step function; representing particles going through a shockwave. In this case, formula_19 is the flow velocity upstream of the shock; whereas formula_20 is the velocity drop across the shock.
The step response for a particle is a simple exponential:
To convert the velocity as a function of time to a particle velocity distribution as a function of distance, let's assume a 1-dimensional velocity jump in the formula_22 direction. Let's assume formula_23 is positioned where the shock wave is, and then integrate the previous equation to get:
Considering a relaxation time of formula_26 (time to 95% velocity change), we have:
This means the particle velocity would be settled to within 5% of the downstream velocity at formula_29 from the shock. In practice, this means a shock wave would look, to a PIV system, blurred by approximately this formula_29 distance. 
For example, consider a normal shock wave of Mach number formula_31 at a stagnation temperature of 298 K. A propyleneglycol particle of formula_32 would blur the flow by formula_33; whereas a formula_34 would blur the flow by formula_35 (which would, in most cases, yield unacceptable PIV results).
Although a shock wave is the worst-case scenario of abrupt deceleration of a flow, it illustrates the effect of particle tracking error in PIV, which results in a blurring of the velocity fields acquired at the length scales of order formula_29.
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 &amp; Rosner.
formula_37
Where formula_38is the "particle free-stream Reynolds number",
formula_39
An additional function formula_40 was defined by; this describes the non-Stokesian drag correction factor,
formula_41
It follows that this function is defined by,
formula_42
Considering the limiting particle free-stream Reynolds numbers, as formula_43 then formula_44 and therefore formula_45. Thus as expected there correction factor is unity in the Stokesian drag regime. Wessel &amp; Righi evaluated formula_46 for formula_47 from the empirical correlation for drag on a sphere from Schiller &amp; Naumann.
formula_48
Where the constant formula_49. 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 as:
where formula_51 is particle concentration, formula_52 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 formula_54 is the particle's settling velocity, formula_55 is the sampling tubes inner diameter, and formula_56 is the acceleration of gravity.