Calculate Stokes Law Drag Force and Terminal Velocity Online

What is Stokes' Law?

Stokes' Law describes the drag force experienced by a small sphere moving slowly through a viscous fluid. It states that the drag force $F_d$ is directly proportional to the fluid viscosity $\mu$, the sphere radius $r$, and the velocity $v$: $F_d = 6\pi \mu r v$. This relationship holds when the flow is laminar, meaning viscous forces dominate over inertial forces (Reynolds number $Re < 1$).

The law also gives the terminal settling velocity $V_t$ a sphere reaches when the drag force exactly balances the net gravitational force (weight minus buoyancy): $V_t = g d^2 (\rho_p - \rho_m) / (18\mu)$. This terminal velocity equation is fundamental to sedimentation analysis, particle size determination, and viscometry across many scientific and engineering fields.

Stokes' Law is named after the Irish mathematician and physicist George Gabriel Stokes, who first derived it in 1851. It remains one of the most widely used relationships in fluid mechanics for creeping flow regimes and forms the basis for instruments such as the falling-ball viscometer and the hydrometer.

How to Use the Stokes' Law Calculator

This calculator supports two equation modes. Select Drag Force to work with $F_d = 6\pi \mu r v$, or Terminal Velocity to work with $V_t = g d^2 (\rho_p - \rho_m) / (18\mu)$. Then choose which variable to solve for. Enter the known values and the calculator will compute the result instantly with a step-by-step breakdown.

Drag Force Mode: Solve for drag force $F_d$, fluid viscosity $\mu$, sphere radius $r$, or velocity $v$. Supports multiple units for each variable.
Terminal Velocity Mode: Solve for terminal velocity $V_t$, gravitational acceleration $g$, particle diameter $d$, particle density $\rho_p$, fluid density $\rho_m$, or fluid viscosity $\mu$.

Stokes' Law Formulas

The two primary equations are:

$$F_d = 6\pi \mu r v$$

$$V_t = \frac{g d^2 (\rho_p - \rho_m)}{18\mu}$$

Where:

$F_d$ = viscous drag force (N)
$\mu$ = fluid dynamic viscosity (Pa·s or kg/(m·s))
$r$ = sphere radius (m)
$v$ = sphere velocity relative to fluid (m/s)
$V_t$ = terminal settling velocity (m/s)
$g$ = gravitational acceleration (9.81 m/s² on Earth)
$d$ = particle diameter (m); $d = 2r$
$\rho_p$ = particle density (kg/m³)
$\rho_m$ = fluid density (kg/m³)

Applications of Stokes' Law

Stokes' Law has numerous practical applications across science and engineering. In environmental engineering, it is used to design sedimentation basins and clarifiers for water and wastewater treatment by predicting how quickly particles of different sizes settle. In soil science, the hydrometer method uses Stokes' Law to determine particle size distribution by measuring the rate at which soil particles settle in water.

In the pharmaceutical industry, Stokes' Law helps formulate suspension drugs by predicting particle settling behavior to ensure consistent dosing. The petroleum industry uses it for drilling fluid rheology and reservoir engineering. In hematology, the erythrocyte sedimentation rate (ESR) test -- a common blood test for inflammation -- is based on Stokes' Law applied to red blood cells settling in plasma.

Viscometry is another major application. Falling-ball and rising-bubble viscometers directly apply Stokes' Law to measure fluid viscosity by timing the descent or ascent of a sphere or bubble through the test fluid.

Limitations and Common Mistakes

Stokes' Law is strictly valid only for rigid spheres in unbounded Newtonian fluids at Reynolds numbers below about 0.1, with acceptable accuracy up to $Re \approx 1$. At higher Reynolds numbers, inertial effects become significant and the drag follows a quadratic relationship ($F_d \propto v^2$) rather than the linear Stokes drag.

Using diameter instead of radius: The drag force formula uses radius $r$, not diameter $d$. Remember $F_d = 6\pi \mu r v$, not $6\pi \mu d v$.
Ignoring buoyancy: The terminal velocity equation accounts for the density difference $\rho_p - \rho_m$. Omitting the fluid density overestimates settling speed.
Applying to non-spherical particles: Stokes' Law assumes perfect spheres. Irregular particles experience different drag and require shape correction factors.
Forgetting the Reynolds number check: Always verify $Re < 1$ before applying Stokes' Law to ensure the creeping flow assumption holds.

Frequently Asked Questions

What is the difference between Stokes drag and Newton drag?

Stokes drag applies at low Reynolds numbers ($Re < 1$) where viscous forces dominate, giving a linear relationship $F_d \propto v$. Newton drag applies at high Reynolds numbers ($Re > 1000$) where inertial forces dominate, giving a quadratic relationship $F_d \propto v^2$. In the transition region between these regimes, neither simple formula is accurate and empirical drag coefficient curves must be used.

How does particle size affect settling velocity?

Terminal velocity scales with the square of particle diameter: $V_t \propto d^2$. This means halving the particle size reduces terminal velocity to one quarter. A sand particle of 200 $\mu$m settles in seconds in still water, while a clay particle of 2 $\mu$m takes hours to settle the same distance. This dramatic size dependence is why sedimentation is an effective method for particle size classification.

When does Stokes' Law break down?

Stokes' Law breaks down when the Reynolds number exceeds about 1. At this point, inertial effects start to become significant and the drag force is no longer strictly proportional to velocity. For water at room temperature, this typically occurs for particles larger than about 100 $\mu$m settling at their terminal velocity. For air, the limit is around 50 $\mu$m. Above $Re \approx 1000$, the drag follows the Newtonian quadratic law instead.

How is Stokes' Law used in falling-ball viscometry?

In a falling-ball viscometer, a sphere of known density and diameter is dropped through the test fluid, and its terminal velocity is measured. Using Stokes' Law rearranged for viscosity: $\mu = g d^2 (\rho_p - \rho_m) / (18 V_t)$, the fluid viscosity can be calculated directly. The ASTM D1343 standard describes this method for measuring viscosity of Newtonian liquids in the laminar regime.

What is the erythrocyte sedimentation rate (ESR) test?

The ESR test measures how quickly red blood cells settle in a tube of blood over one hour. It is based on Stokes' Law: red blood cells have an effective diameter of about 7 $\mu$m and density of 1100 kg/m³ in plasma of density 1025 kg/m³ and viscosity 0.0014 Pa·s. The normal ESR range is 0-20 mm/hr. Elevated ESR values suggest inflammation in the body, making it a widely used non-specific screening test.

Can Stokes' Law be applied to gases?

Yes, Stokes' Law applies to any Newtonian fluid, including gases. It is used in aerosol science to model the settling of fog droplets, dust particles, and airborne pollutants. However, for very small particles (below about 1 $\mu$m in air), the no-slip boundary condition at the particle surface breaks down and a Cunningham slip correction factor must be applied to the Stokes drag formula.

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Stokes Law Calculator