Free Stokes' Law Calculator

Understanding Stokes' Law and Its Applications

The Stokes' Law Calculator functions as both a terminal velocity calculator and a fluid viscosity calculator, designed for analyzing spherical particle motion inside a falling ball viscometer. This classic setup consists of a vertical tube filled with a viscous fluid. When a small, dense sphere is released into the tube, it accelerates until the drag force from the fluid balances the net gravitational force, reaching a constant speed known as the terminal velocity. By measuring that velocity, you can determine the fluid's dynamic viscosity using the Stokes law formula.

What Is Dynamic Viscosity?

Dynamic viscosity quantifies a fluid's intrinsic resistance to shearing or flow. A high‑viscosity fluid like honey deforms slowly under stress, whereas a low‑viscosity fluid like water flows readily. The SI unit of dynamic viscosity is the pascal‑second ($\text{Pa} \cdot \text{s}$). Understanding viscosity is essential for predicting the spherical particle drag encountered in many engineering and scientific settings.

The Terminal Velocity Equation

For a small sphere moving slowly (creeping flow) in a viscous fluid, the terminal velocity follows from Stokes' law. The terminal velocity $v$ of a spherical particle is given by:

$$v = \frac{g \, d^{2} (\rho_{p} - \rho_{m})}{18 \, \mu}$$

where:

• $g$ is the gravitational acceleration (on Earth, $g \approx 9.80665\ \text{m/s}^{2}$),
• $d$ is the particle diameter,
• $\rho_{p}$ and $\rho_{m}$ are the densities of the particle and the fluid, respectively,
• $\mu$ is the dynamic viscosity of the fluid.

This stokes drag calculator lets you solve for any of these variables. To find an unknown viscosity, simply enter the remaining quantities, and the tool returns the answer instantly. If you need density values, a separate density calculator can provide them.

Worked Example: Terminal Velocity of an Aluminum Sphere in Oil

Consider a 1 cm diameter aluminum sphere ($\rho_{p} = 2710\ \text{kg/m}^{3}$) falling in an oil with density $\rho_{m} = 850\ \text{kg/m}^{3}$ and dynamic viscosity $\mu = 0.38\ \text{Pa} \cdot \text{s}$.

1. Compute the density difference: $\rho_{p} - \rho_{m} = 1860\ \text{kg/m}^{3}$.
2. Calculate $g \, d^{2}$: $9.81 \times (0.01)^{2} = 9.81 \times 10^{-4}\ \text{m}^{3}/\text{s}^{2}$ (with $g \approx 9.81\ \text{m/s}^{2}$).
3. Apply the formula: $$v = \frac{9.81 \times 10^{-4} \times 1860}{18 \times 0.38} \approx 0.27\ \text{m/s}$$

The result (0.27 m/s) matches the expected terminal velocity for this setup, confirming the practical use of the Stokes law formula.

Extending Your Knowledge

Beyond terminal velocity, you may want to convert between dynamic and kinematic viscosity (the ratio of dynamic viscosity to density). This calculator ecosystem includes converters for poise‑to‑stokes transformations and tools for exploring the viscosity of air at various temperatures. All these resources work together to give you a complete picture of fluid behavior.