Stokes' Law and Milk Fat Separation

Stokes' Law — The Equation

Stokes' Law gives the terminal velocity of a small sphere falling (or rising) through a viscous fluid under gravity:

Key implications:

• Velocity scales with radius squared — doubling globule size quadruples rise rate
• Velocity depends on the density difference — if particle and fluid have equal density, no separation
• Higher fluid viscosity slows separation — cold milk separates more slowly than warm milk
• Negative density difference (fat lighter than water) means particles rise, not fall

Application to Milk Fat Globules

Native milk fat globules range from 0.1 to 10 µm diameter, with an average around 3–4 µm. Milk fat density at 40°C is approximately 915 kg/m³; skim milk density is approximately 1,033 kg/m³. So fat globules are about 12% less dense than the surrounding fluid and rise.

Calculation for a typical milk fat globule

For a 4 µm diameter fat globule at 40°C in skim milk:

• r = 2 µm = 2 × 10⁻⁶ m
• ρ_p = 915 kg/m³
• ρ_f = 1,033 kg/m³
• μ (skim milk at 40°C) = ~1.0 × 10⁻³ Pa·s

Substituting:

v = 2 × 9.81 × (2×10⁻⁶)² × (915−1033) ÷ (9 × 1.0×10⁻³)
v = −1.0 × 10⁻⁶ m/s = approximately 3.6 mm/hour rise (negative sign confirms rise, not fall)

That's why a glass of warm whole milk left to stand will show a visible cream line within an hour or two. Cold milk (4°C) has higher viscosity (~2.0 × 10⁻³ Pa·s), halving the rise rate to ~1.8 mm/hour.

Centrifugal Separation — Extending Stokes' Law

Gravity alone is too slow for commercial dairy separation. A centrifuge replaces gravitational acceleration g with centripetal acceleration ω²r_c, where ω is the angular velocity (radians/second) and r_c is the distance from the axis. The modified Stokes equation:

A typical milk separator operating at 6,000 rpm with r_c = 0.3 m gives:

• ω = 2π × 100 = 628 rad/s
• ω²r_c = 628² × 0.3 ≈ 118,000 m/s²
• G-force ≈ 12,000 g

That's 12,000 times faster separation than under gravity alone. The 4 µm fat globule that rises at 3.6 mm/hr under gravity moves at ~12 m/s under centrifugal force — small wonder the separator achieves complete fat separation in seconds.

What This Means for Plant Design

Why warm milk separates better

At 40°C, milk viscosity is about half that at 4°C. Per Stokes' Law, separation velocity doubles. This is why milk is heated to 40°C before separation — not for hygiene but for fluid mechanics.

Why homogenisation prevents creaming

Homogenisation reduces fat globule size from ~4 µm to ~0.5 µm. Stokes velocity scales with r², so separation rate drops by a factor of (4/0.5)² = 64. The cream-line rise rate becomes 0.056 mm/hour — over the shelf life of pasteurised milk, less than a millimetre of separation, invisible to consumers.

Why centrifuges have a lower size limit

Even at very high G-forces, separation requires time for the particles to travel from the bulk fluid to the separator wall or disk surface. The smaller the particle, the longer the residence time required. Most modern dairy separators are sized to separate fat globules down to ~0.5 µm at design throughput — smaller globules pass through with the skim.

Why bactofugation works

Bacterial spores have density ~1,200 kg/m³, significantly denser than milk (1,033 kg/m³). Although small (1–2 µm), the high density difference allows centrifugal removal at high G-forces. Bactofuges typically operate at higher G than standard cream separators for this reason.

Limitations of Stokes' Law

Stokes' Law assumes:

• Spherical particles — valid for milk fat globules (which are spherical) but not for irregular sediment particles
• Low Reynolds number (Re < 0.1) — laminar flow around the particle. True for micron-sized particles in milk; breaks down for larger / faster particles
• Newtonian fluid — constant viscosity. Reasonably true for low-solids milk; breaks down for high-solids concentrate (40%+ solids)
• Dilute suspension — particles don't interact. Breaks down at high fat content (cream > 10% fat) where hindered settling occurs
• Rigid particles — fat globules are actually deformable; this becomes significant at high shear

For dense suspensions (cream, concentrate), modified equations (Richardson-Zaki, Krieger-Dougherty) better predict behaviour. In practice, separator sizing relies more on supplier empirical data than pure theory.

Related Physics in Dairy

• Sedimentation — sediment settling in milk tanks; same physics, reversed direction
• Membrane fouling — particles settling under flux towards a membrane surface; the basis for back-pulse cleaning
• Spray drying droplet trajectories — Stokes drag plus gravity, in a hot air stream
• Ice cream freezing — air bubble rise during freezing affects overrun retention
• Whipping — partial coalescence of fat globules around air bubbles; rising velocity of bubbles affected by viscosity

Frequently Asked Questions

What is Stokes' Law?

Stokes' Law gives the terminal velocity of a small spherical particle falling or rising through a viscous fluid: v = 2gr²(ρ_p−ρ_f)/9μ. In dairy, it predicts how fast milk fat globules rise (creaming) and is the foundation for centrifugal separator design.

How fast does milk cream rise?

For a 4 µm fat globule in warm milk (40°C), Stokes' Law predicts about 3.6 mm/hour rise. In cold milk (4°C) viscosity is twice as high, halving the rise rate to ~1.8 mm/hour. That's why a glass of warm milk shows visible cream-line within an hour or two.

Why is milk heated before separation?

At 40°C, milk viscosity is about half that at 4°C. Per Stokes' Law, halving viscosity doubles separation velocity. So separators operate at 40–55°C to maximise throughput and separation efficiency — not for hygiene reasons.

Why doesn't homogenised milk separate?

Homogenisation reduces fat globule size from ~4 µm to ~0.5 µm. Stokes velocity scales with radius squared, so separation rate drops by a factor of about 64. Over the shelf life of pasteurised milk, the separation is less than a millimetre — invisible.

What G-force does a milk separator operate at?

Typical dairy separators operate at 5,000–10,000 g, generated by spinning at 5,000–9,000 rpm depending on the design. Bactofuges run higher (up to 15,000+ g) to separate small dense particles like bacterial spores.

Can Stokes' Law predict separation of bacteria?

Yes, to an extent. Bacterial cells and spores have densities of about 1,100–1,200 kg/m³ (denser than milk at 1,033) so they settle / centrifuge towards the outer wall. Small size (1–2 µm) means very slow settling under gravity, but at separator G-forces (5,000–15,000 g) removal becomes practical. This is how bactofuges work.

What are the limits of Stokes' Law in dairy?

Stokes' Law assumes spherical particles, dilute suspension, Newtonian fluid, low Reynolds number and rigid particles. It works well for predicting milk fat globule rise in dilute systems. It breaks down for dense suspensions (cream above ~10% fat), high-viscosity concentrate (above 40% solids), and irregular particles. In practice, separator suppliers use empirical correlations alongside Stokes theory.

Further reading: John Watson publishes articles on dairy industry topics on LinkedIn. Browse all articles by John Watson on LinkedIn →

See related: Milk separator, Milk standardisation (cream removal), Pearson's Square, Cream production, Homogenisation, Butter making, Membrane filtration, all dairy science information, consultancy services.