Stokes' Law Drag Calculator
Enter the sphere radius, velocity and fluid viscosity to compute the viscous drag from F=6πηrv, plus the terminal velocity v_t=2r²Δρg/(9η).
Input Data
Results
At a glance:Stokes' law gives the viscous drag force on a sphere moving slowly through a fluid: F_d = 6πηrv, where η is the dynamic viscosity, r the sphere radius and v the velocity. It is valid at low Reynolds number (Re<1), where viscous forces dominate inertia. Balancing this drag against the net gravitational force (weight minus buoyancy) gives the terminal velocity v_t = 2r²(ρ_s−ρ_f)g/(9η), and the corresponding drag coefficient is C_d = 24/Re. The law is a cornerstone of low-Reynolds-number hydrodynamics and underlies sedimentation, aerosol deposition and many microfluidic phenomena.
Formula
Viscous drag: F_d = 6πηrv
Terminal velocity: v_t = 2r²(ρ_s − ρ_f)g / (9η)
Drag coefficient: C_d = 24 / Re
$$F_d = 6\pi\eta r v, \quad v_t = \frac{2r^{2}(\rho_s-\rho_f)g}{9\eta}, \quad C_d = \frac{24}{Re}$$How to Use
- Enter the sphere radius r, velocity v and fluid viscosity η.
- The calculator returns the viscous drag F_d = 6πηrv.
- It also derives the terminal velocity v_t and the drag coefficient C_d=24/Re.
Case Studies
Sedimentation of a sand grain
A quartz grain (ρ_s=2650 kg/m³, r=0.5 mm) settling in water (ρ_f=1000, η=0.001) reaches a small terminal velocity.
The drag F_d = 6πηrv balances the net weight, giving v_t = 2r²(ρ_s−ρ_f)g/(9η).
This is the basis of particle-size analysis by sedimentation.
When Stokes' law breaks down
The law holds only for Re<1 (slow motion, small spheres, high viscosity).
At Re>1 an Oseen correction or full Navier-Stokes solution is needed.
A raindrop falling in air has Re≫1, so Stokes' law does not apply.
FAQ
When does Stokes' law apply?
When Re<1 (low speed, small sphere or high viscosity). For Re>1 use the Oseen correction or a Navier-Stokes numerical solution; a raindrop in air has Re≫1 and is outside the Stokes regime.
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.