What is the Stokes Number (Stk)?
In the fields of fluid mechanics and multiphase particle dynamics, understanding how solid particles or liquid droplets behave when moving within a fluid is of paramount importance. The Stokes number ($Stk$) is a fundamental dimensionless number that characterizes the behavior of particles suspended in a fluid flow. Specifically, it represents the ratio of the characteristic time of a particle (its relaxation time) to the characteristic time of the flow field.
Put simply, the Stokes number indicates how well a particle can follow the streamlines of the fluid carrying it. If the value is small, the particle closely follows the fluid flow; if the value is large, the particle's own inertia dominates, causing it to detach from the fluid streamlines and continue along its original trajectory.
You can use our Stokes Number Calculator to quickly evaluate the particle relaxation time and flow timescale based on your specific particle properties and flow conditions.
The Stokes Number Formula and Components
For spherical particles operating within the Stokes drag regime (where the particle Reynolds number is low), the Stokes number is typically calculated using the following formula:
$$ Stk = \frac{\tau_p}{\tau_f} = \frac{\rho_p \cdot d_p^2 \cdot U \cdot C_c}{18 \cdot \mu \cdot L} $$
The variables in this equation are:
- $\tau_p$ (Particle Relaxation Time): The time required for a particle to adjust to changes in the fluid velocity (seconds).
- $\tau_f$ (Flow Timescale): The time it takes for the fluid to pass a characteristic obstacle or distance (seconds).
- $\rho_p$ (Particle Density): The mass per unit volume of the particle ($kg/m^3$).
- $d_p$ (Particle Diameter): The physical size of the particle (meters).
- $U$ (Fluid Velocity): The characteristic velocity of the carrier fluid ($m/s$).
- $C_c$ (Cunningham Slip Correction): A factor that corrects for non-continuum effects when analyzing very small (nano/micro) aerosols.
- $\mu$ (Dynamic Viscosity): The internal resistance of the fluid to flow ($Pa\cdot s$).
- $L$ (Characteristic Length): The dimension of the flow domain or an obstacle within the flow (e.g., the diameter of a filter fiber) (meters).
Particle Relaxation Time ($\tau_p$)
This parameter depends on the mass (inertia) of the particle and the drag force exerted on it by the fluid. Because it is proportional to the square of the diameter ($d_p^2$), larger particles have significantly longer relaxation times, meaning they are much slower to react to changes in the fluid's direction.
Flow Timescale ($\tau_f = \frac{L}{U}$)
This scale is determined by the fluid's velocity and the physical dimensions of the system. At high speeds (large $U$) or in miniaturized systems (small $L$), the flow timescale becomes very short. This rapid change leaves less time for the particle to adapt, thereby increasing the Stokes number and making inertial effects more pronounced.
Interpreting the Stokes Number
The magnitude of the Stokes number provides engineers with critical insights into system behavior:
- Low Stokes Number ($Stk \ll 1$, typically $< 0.1$): The particle's response time is much shorter than the flow's characteristic time. The particle acts almost as a tracer, perfectly following the fluid's streamlines. Inertial effects are entirely negligible. A common example is light smoke drifting seamlessly with air currents in a room.
- High Stokes Number ($Stk \gg 1$, typically $> 10$): The particle's response time is too long. Even if the fluid turns a sharp corner, the particle's momentum carries it forward in a straight line. Raindrops striking a car windshield or dust particles colliding with filter fibers (inertial impaction) are classic examples.
- Intermediate Stokes Number ($Stk \approx 1$): The particle partially follows the streamlines but deviates noticeably when the fluid turns, due to centrifugal forces. This is the most complex regime to model and is critical for optimizing particle separation and filtration equipment.
Realistic Example and Practical Scenario
Scenario: We want to analyze whether dust particles suspended in an HVAC system will strike a fine mesh filter wire or flow around it.
Data:
- Air velocity ($U$): $2 \ m/s$
- Air dynamic viscosity ($\mu$): $1.8 \times 10^{-5} \ Pa\cdot s$
- Filter wire diameter (Characteristic length, $L$): $0.005 \ m \ (5 \ mm)$
- Particle density ($\rho_p$): $1200 \ kg/m^3$
- Particle diameter ($d_p$): $20 \ \mu m \ (20 \times 10^{-6} \ m)$
- Cunningham correction ($C_c$): $\approx 1$ (negligible for this size)
Calculation Steps:
- Particle Relaxation Time ($\tau_p$):
$\tau_p = \frac{1200 \cdot (20 \times 10^{-6})^2 \cdot 1}{18 \cdot 1.8 \times 10^{-5}} \approx 0.00148 \ s$ - Flow Timescale ($\tau_f$):
$\tau_f = \frac{0.005}{2} = 0.0025 \ s$ - Stokes Number ($Stk$):
$Stk = \frac{0.00148}{0.0025} \approx 0.592$
Interpretation: An $Stk$ of $0.592$ indicates that the particle's inertia is significant enough that it cannot perfectly adjust to the air's sudden curve around the wire. Consequently, a large fraction of these $20 \ \mu m$ particles will deviate from the streamlines and impact the wire, suggesting the filter will have good capture efficiency for this size range.
Difference Between Stokes and Reynolds Numbers
Students often confuse the Stokes number with the Reynolds number, but they measure entirely different physical phenomena:
- Reynolds Number ($Re$): Represents the ratio of inertial forces to viscous (friction) forces within a fluid. It dictates the flow regime (laminar vs. turbulent).
- Stokes Number ($Stk$): Represents the ratio of a particle's relaxation time to the flow's timescale. It dictates the particle's tendency to deviate from the flow path.
While high Reynolds numbers imply turbulent, chaotic fluid motion, high Stokes numbers imply that the particles ignore the fluid motion altogether, moving independently due to their own momentum.
Important Considerations and Edge Cases
When calculating the Stokes number, several assumptions must be kept in mind:
- Stokes Drag Regime Assumption: The standard formula is derived assuming a low particle Reynolds number ($Re_p < 1$), where Stokes' law of linear drag applies. For particles moving at very high relative velocities, non-linear drag corrections are required.
- Cunningham Effect in Nano-Aerosols: For very small particles (e.g., below $0.1 \ \mu m$), the fluid can no longer be treated as a continuous medium. The particles start to "slip" between the air molecules. In such cases, the Cunningham slip correction factor ($C_c$) must be greater than 1. Ignoring it will result in underestimating the particle's relaxation time.
- Selecting the Characteristic Length: The choice of $L$ is highly dependent on the problem. It could be the diameter of a pipe, the width of a bluff body, or the Kolmogorov length scale in turbulent flows. Selecting the wrong characteristic length will yield a Stokes number that does not reflect physical reality.
For a deeper analysis of complex multiphase systems, our Stokes Number Calculator provides an efficient way to model particle dynamics reliably and accurately.
