Stokes Drag and Terminal Settling
Balance viscous drag against weight and buoyancy for a tiny settling sphere.
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Stokes Drag and Terminal Settling
An isolated rigid sphere settling in an unbounded Newtonian fluid under gravity experiences viscous drag that opposes its motion. When creeping-flow assumptions apply (very low Reynolds numbers), this drag is described by Stokes' law. The balance between gravitational forces (weight and buoyancy) and viscous drag determines the sphere's terminal velocity.
Worked Example
Consider a tiny rigid sphere with a radius of 5 μm and density 2000 kg/m3, falling through a Newtonian fluid with density 1000 kg/m3 and viscosity1 mPa·s under gravity (g = 9.81 m/s2).
- Convert inputs to base SI units:
r = 5 × 10-6 m
μ = 1 × 10-3 Pa·s - Calculate the net gravitational force (weight minus buoyancy):
Fg = (ρparticle - ρfluid) · (4/3 π r3) · g
Fg = (2000 - 1000) · (4/3 · π · (5 × 10-6)3) · 9.81 ≈ 5.1365 × 10-12 N - At terminal velocity, the net gravitational force equals the magnitude of Stokes drag (Fd = 6 π μ r |v|). Solving for terminal velocity v:
v = 2 · (ρparticle - ρfluid) · g · r2 / (9 μ)
v = 2 · (1000) · 9.81 · (5 × 10-6)2 / (9 × 10-3) ≈ 5.45 × 10-5 m/s - Check the creeping-flow assumption by computing the Reynolds number (Re):
Re = ρfluid · |v| · 2 r / μ
Re = 1000 · 5.45 × 10-5 · 2(5 × 10-6) / (1 × 10-3) ≈ 0.000545
Since Re << 1, the creeping-flow assumption holds, and Stokes' law is valid.
Interactive Experiment
Use the Stokes Drag and Terminal Settling module to observe the balance of forces.
- Set the Radius (um) to 5,Particle density (kg/m^3) to 2000, and Viscosity (mPa*s) to 1. Observe that the particle settles downward with a terminal velocity of 5.45e-05 m/s.
- Change the Particle density (kg/m^3) to 1000. Notice that the particle becomes neutrally buoyant: the net gravitational force is zero, the terminal velocity is zero, and the motion is Neutral.
- Change the Particle density (kg/m^3) to 500. The net gravitational force becomes negative (buoyancy dominates weight), and the particle rises with a negative terminal velocity.
- Increase the Radius (um) to 10. Observe how the terminal velocity magnitude increases by a factor of 4 due to ther2 scaling relationship.
Limitations and Assumptions
This model assumes creeping flow, explicitly requiring the Reynolds number (Re) to be less than 0.005, so inertial forces can be neglected. It applies only to an isolated, perfectly rigid sphere in an unbounded, continuous Newtonian fluid. This idealized terminal state neglects transient inertial acceleration, boundary effects from vessel walls, hydrodynamic interactions with other particles, non-Newtonian fluid behaviors, and random thermal fluctuations (Brownian motion) which become significant for sub-micrometer particles.
References
The principles of viscous drag and terminal velocity under creeping flow are standard topics in university physics and fluid mechanics.
- OpenStax, University Physics Volume 1. Provides foundational reading on drag forces and the resulting terminal speed for objects falling through fluids. Available at: https://openstax.org/books/university-physics-volume-1/pages/6-4-drag-force-and-terminal-speed.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Stokes Drag and Terminal Settling module.
Try Stokes Drag and Terminal Settling on Riano →