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
S
t
k
=
t
0
u
0
l
0
{\displaystyle \mathrm {Stk} ={\frac {t_{0}\,u_{0}}{l_{0}}}}
where
t
0
{\displaystyle 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
{\displaystyle u_{0}}
is the fluid velocity of the flow well away from the obstacle, and
l
0
{\displaystyle l_{0}}
is the characteristic dimension of the obstacle (typically its diameter) or a characteristic length scale in the flow (like boundary layer thickness). 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
=
ρ
p
d
p
2
18
μ
g
{\displaystyle t_{0}={\frac {\rho _{p}d_{p}^{2}}{18\mu _{g}}}}
where
ρ
p
{\displaystyle \rho _{p}}
is the particle density,
d
p
{\displaystyle d_{p}}
is the particle diameter and
μ
g
{\displaystyle \mu _{g}}
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
S
t
k
≫
1
{\displaystyle \mathrm {Stk} \gg 1}
, particles will detach from a flow especially where the flow decelerates abruptly. For
S
t
k
≪
1
{\displaystyle \mathrm {Stk} \ll 1}
, particles follow fluid streamlines closely. If
S
t
k
<
0.1
{\displaystyle \mathrm {Stk} <0.1}
, tracing accuracy errors are below 1%.
Contents
Relaxation time and tracking error in particle image velocimetry (PIV)
The Stokes number provides a means of estimating the quality of PIV 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
v
p
(
t
)
{\displaystyle v_{p}(t)}
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
v
f
(
t
)
{\displaystyle v_{f}(t)}
. It is the difference between these velocities that will generate the drag force necessary to correct the particle path:
Δ
v
(
t
)
=
v
f
(
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
v
f
(
t
)
=
V
u
−
Δ
V
H
(
t
)
{\displaystyle v_{f}(t)=V_{u}-\Delta VH(t)}
is a Heaviside step function; representing particles going through a shockwave. In this case,
V
u
{\displaystyle V_{u}}
is the flow velocity upstream of the shock; whereas
Δ
V
{\displaystyle \Delta V}
is the velocity drop across the shock.
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.
Stk
e
=
Stk
24
Re
o
∫
0
Re
o
d
Re
′
C
D
(
Re
′
)
Re
′
{\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
Re
o
{\displaystyle {\text{Re}}_{o}}
is the "particle free-stream Reynolds number",
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
)
(
1
−
1
1
+
S
t
k
(
2
+
0.617
u
/
u
0
)
)