# Stokes number Drag coefficient Cd for a sphere as a function of Reynolds number Re, as obtained from laboratory experiments. The dark line is for a sphere with a smooth surface, while the lighter line is for the case of a rough surface. The numbers along the line indicate several flow regimes and associated changes in the drag coefficient:
•2: attached flow (Stokes flow) and steady separated flow,
•3: separated unsteady flow, having a laminar flow boundary layer upstream of the separation, and producing a vortex street,
•4: separated unsteady flow with a laminar boundary layer at the upstream side, before flow separation, with downstream of the sphere a chaotic turbulent wake,
•5: post-critical separated flow, with a turbulent boundary layer.

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). 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

$t_{0}={\frac {\rho _{p}d_{p}^{2}}{18\mu _{g}}}$ where $\rho _{p}$ is the particle density, $d_{p}$ is the particle diameter and $\mu _{g}$ is the gas 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 $\mathrm {Stk} \gg 1$ , 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%.

## 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.

${\text{Stk}}_{\text{e}}={\text{Stk}}{\frac {24}{{\text{Re}}_{o}}}\int _{0}^{{\text{Re}}_{o}}{\frac {d{\text{Re}}^{\prime }}{C_{D}({\text{Re}}^{\prime }){\text{Re}}^{\prime }}}$ Where ${\text{Re}}_{o}$ is the "particle free-stream Reynolds number",

${\text{Re}}_{o}={\frac {\rho _{g}|\mathbf {u} |d_{p}}{\mu _{g}}}$ An additional function $\psi ({\text{Re}}_{o})$ was defined by, this describes the non-Stokesian drag correction factor,

${\text{Stk}}_{e}={\text{Stk}}\cdot \psi ({\text{Re}}_{o})$ It follows that this function is defined by, $\psi$ describes the non-Stokesian drag correction factor for a spherical particle

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

$\psi ({\text{Re}}_{o})={\frac {3({\sqrt {c}}{\text{Re}}_{o}^{1/3}-\arctan({\sqrt {c}}{\text{Re}}_{o}^{1/3}))}{c^{3/2}{\text{Re}}_{o}}}$ Where the constant $c=0.158$ . 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 as:

$c/c_{0}=1+(u_{0}/u-1)\left(1-{\frac {1}{1+\mathrm {Stk} (2+0.617u/u_{0})}}\right)$ where $c$ is particle concentration, $u$ 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,

$\mathrm {Stk} ={\frac {u_{0}V_{s}}{dg}}$ where $V_{s}$ is the particle's settling velocity, $d$ is the sampling tubes inner diameter, and $g$ is the acceleration of gravity.