What analytical solutions exist for sphere drag?

For slow, viscosity-dominated flow, Stokes's solution Cd = 24/Re is exact. Oseen's solution, adding a little inertia, gives Cd = (24/Re)(1 + 3Re/16). At higher Reynolds numbers there is no analytical solution, and empirical correlations such as Schiller–Naumann serve as reference values.

How far can the Stokes solution be used?

Against the empirical correlation, it is 3% low at Re 0.1, 8.5% at 0.5 and 13% at 1. Oseen agrees well to about 0.5 but is 11% high at 2 and 66% high at 10. For CFD verification, use Stokes below Re 0.1 and Oseen for roughly 0.1–1.

What to watch in low-Reynolds CFD verification?

Domain size. In Stokes flow the sphere's influence decays only inversely with distance, so nearby outer boundaries inflate drag. For a sphere in a cylinder, a domain ten diameters across gave +26%, fifty diameters +4.4%, and even two hundred +1.1%. Make the domain large enough or apply a wall-effect correction.

Sphere Drag (Stokes/Oseen Solution)

Category: V&V – Analytical Solutions | Revised 2026-10-01

CAE visualization for sphere drag theory - technical simulation diagram

Drag on a Sphere (Stokes/Oseen Solutions)

Theory: a sphere in slow flow

Overview

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I'm simulating fine-particle settling with CFD and want to verify on sphere drag first. Should I compare with Stokes's formula?

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A good choice, with two cautions. First, Stokes's solution is exact only at very small Reynolds numbers; approaching 1, inertia shifts it by over 10%. Choose a reference that fits your Reynolds number, or you'll wrongly judge a correct CFD as mismatched. Second, domain size: in slow flow the sphere disturbs a very wide region, so nearby outer boundaries inflate drag. Handle these two and sphere drag is an excellent verification case.

Drag coefficient formulas

$$ C_D^{Stokes} = \frac{24}{Re}, \qquad C_D^{Oseen} = \frac{24}{Re}\left(1 + \frac{3}{16}Re\right), \qquad C_D^{SN} = \frac{24}{Re}\left(1 + 0.15\,Re^{0.687}\right) $$

$Re = \rho U d/\mu$ is based on sphere diameter. Stokes neglects inertia exactly; Oseen approximately includes far-field inertia; Schiller–Naumann (SN) is an empirical fit up to Re about 800.

$$ v_t = \frac{(\rho_p - \rho_f)\,g\,d^2}{18\mu} \ \ (\text{Stokes}), \qquad v_t = \sqrt{\frac{4(\rho_p - \rho_f)\,g\,d}{3\rho_f C_D}} $$

Terminal velocity of a settling particle. Stokes's form holds only at small Reynolds numbers; for larger particles, iterate drag coefficient and Reynolds number together.

Coffee Break Trivia

Stokes, Oseen and a paradox

In 1851 Stokes, studying air resistance on pendulums, derived the drag 3πμdU on a sphere in slow flow. Solving a cylinder with the same approximation, however, yields no solution (Stokes's paradox), and it was noted that even for the sphere inertia can't be neglected far away. In 1910 Oseen showed how to include far-field inertia approximately, resolving the inconsistency. Later Millikan used Stokes's law on falling oil drops to measure the electron's charge, work that led to a Nobel Prize.

Worked examples

Example 1: validity of analytical solutions

Re	Stokes	Oseen	Empirical (SN)	Stokes deviation	Oseen deviation
0.1	240	244	247	−3.0%	−1.2%
0.5	48.0	52.5	52.5	−8.5%	+0.1%
1	24.0	28.5	27.6	−13.0%	+3.3%
2	12.0	16.5	14.9	−19.5%	+10.8%
10	2.40	6.90	4.15	−42.2%	+66.2%
100	0.24	4.74	1.09	−78.0%	+334%

Example 2: domain size

In slow flow, how much drag on a sphere on a cylinder's axis increases with the ratio of cylinder diameter D to sphere diameter d (Haberman–Sayre wall correction):

Domain D/d	5	10	20	50	100	200
Drag increase	+68%	+26%	+11.7%	+4.4%	+2.2%	+1.1%

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Even a domain ten diameters across overstates drag by 26%?

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In slow flow, the velocity disturbance around the sphere decays only inversely with distance. So even distant boundaries, where velocity is fixed, confine the flow and raise drag. Building the domain with the 10–20-diameter habit of high-Reynolds work goes badly wrong at low Reynolds numbers; for 1% verification you need around 200 diameters. Alternatively, correct the reference value with the wall-effect formula rather than enlarging the domain.

Example 3: sand-grain settling

Diameter	Stokes	Iterated empirical	Re	Stokes overestimate
50 µm	2.25 mm/s	2.18 mm/s	0.11	3%
100 µm	9.00 mm/s	7.98 mm/s	0.80	13%
200 µm	36.0 mm/s	24.8 mm/s	4.96	45%
500 µm	225 mm/s	78.6 mm/s	39.2	187%

Verification workflow

Choose the Reynolds number and a matching reference (Stokes, Oseen or empirical).
Make the domain large (over 100 diameters at low Re) or apply wall correction.
Refine near the sphere surface and check grid dependence on three or more levels.
Split drag into pressure and friction parts and compare with the Stokes ratio (1:2).
Vary Reynolds number and check the trend against the empirical correlation.

Coffee Break Trivia

“Always 20% high”

Verifying particle CFD, a team computed sphere drag at Re 0.05 and got about 20% above Stokes no matter how fine the grid. The problem wasn't the grid: the domain was only 12 diameters across. In slow flow even distant boundaries matter; enlarging the domain to 200 diameters brought the difference to about 1%. Building domains with high-Reynolds instincts produces unexpected error at low Reynolds numbers.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Comparing with Stokes at Re 1	Correct CFD judged wrong	Use Oseen or empirical
Small domain	Drag overestimated	100–200 diameters
Stokes settling for large particles	Velocity grossly overestimated	Iterate with Cd
Coarse near-surface grid	Friction drag wrong	Resolve the boundary layer
Comparing total drag only	Cancelling errors missed	Split pressure and friction

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I'd like to try related calculations.

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Try the sphere drag calculator, settling velocity calculator and drag coefficient calculator. Other flow verification cases include laminar pipe flow, Couette flow and lid-driven cavity.

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