Viscosity

Newton's law of viscosity

Imagine a thin layer of fluid between two parallel plates separated by a gap h. The bottom plate is stationary; the top plate moves at speed U, dragging the fluid. In steady state the fluid develops a linear velocity profile from 0 at the bottom plate to U at the top — couette flow. The shear stress τ on the moving plate is what you have to apply per unit area to keep it moving at U. Newton in 1687 conjectured that for "ordinary fluids" τ is proportional to the velocity gradient:

τ = μ (du/dy)

τ      shear stress (Pa = N/m²)
du/dy  shear rate or velocity gradient (1/s)
μ      dynamic viscosity (Pa·s)

The constant μ is the dynamic viscosity. It is a material property that depends on temperature and pressure but, for a Newtonian fluid, not on the shear rate or its history. Newton's relation defines viscosity; its experimental verification took until the late 19th century when Couette flow rheometers became reliable.

Geometrically, viscosity is the rate at which momentum is transported across a velocity gradient. Faster fluid layers drag slower ones via the same molecular collisions and entanglements that hold the bulk fluid together. The dimensional analysis is satisfying: Pa·s = (N/m²)(s) = (kg·m/s²/m²)(s) = kg/(m·s) — momentum density divided by length, which is exactly what diffusivity of momentum should look like.

Dynamic vs kinematic viscosity

Two quantities deserve the name "viscosity" and you have to know which one a textbook means.

Dynamic viscosity μ (also called absolute viscosity). The constant in Newton's law. SI units Pa·s. The CGS unit is the poise (P), with 1 P = 0.1 Pa·s; the centipoise cP = 0.001 Pa·s is a common practical unit, conveniently set so that water at 20°C is 1 cP exactly. Dynamic viscosity is what enters force balances directly — the drag on a sphere in Stokes flow is F = 6πμRv.

Kinematic viscosity ν = μ/ρ. Dynamic viscosity divided by density. SI units m²/s. The CGS unit is the stokes (St), with 1 St = 10⁻⁴ m²/s; the centistokes cSt = 10⁻⁶ m²/s is conveniently set so that water at 20°C is 1 cSt. Kinematic viscosity is the diffusivity of momentum and is what enters dimensionless groups like the Reynolds number Re = vL/ν and the Prandtl number Pr = ν/α.

The two can pull in opposite directions. Air has very small dynamic viscosity (μ_air ≈ 0.018 cP versus μ_water = 1 cP) but, because air is a thousand times less dense, kinematic viscosity is actually 15× larger than water (ν_air ≈ 1.5×10⁻⁵ m²/s vs ν_water ≈ 1.0×10⁻⁶ m²/s). Air is "thinner" in the everyday sense (lower μ) but momentum diffuses through it faster (higher ν) — both statements are true and neither is a contradiction.

Viscosities of common fluids

Viscosity spans an enormous range. Liquid helium and gases sit near 10⁻⁵ Pa·s; bitumen is closer to 10⁸ Pa·s. Twelve orders of magnitude separate the two, and most engineering fluids occupy the middle.

Fluid	Temperature	μ (cP)	μ (Pa·s)
Air	20°C	0.018	1.8×10⁻⁵
Water	20°C	1.00	1.0×10⁻³
Water	100°C	0.28	2.8×10⁻⁴
Blood	37°C	3–4	3–4×10⁻³
Olive oil	20°C	84	8.4×10⁻²
Glycerin	20°C	1410	1.41
Honey	20°C	~10000	~10
Peanut butter	20°C	~250000	~250
Bitumen	20°C	~10¹¹	~10⁸

The University of Queensland's pitch-drop experiment, started in 1927, illustrates how thick bitumen really is: pitch was poured into a sealed funnel, allowed to settle, and the funnel was unsealed. Drops fall about once per decade. Inferred viscosity: 2.3×10⁸ Pa·s — about 200 billion times more viscous than water.

Worked example: pressure drop in a hospital IV line

Consider a saline IV running through tubing of inner diameter 2 mm and length 1.2 m at flow rate 100 mL/h. What is the required hydrostatic pressure (the height the bag must hang above the patient)? This is a Hagen–Poiseuille calculation:

ΔP = 8 μ L Q / (π R⁴)

μ = 1.0×10⁻³ Pa·s   (saline at 20°C ≈ water)
L = 1.2 m
Q = 100 mL/h = 100 × 10⁻⁶ / 3600 = 2.78×10⁻⁸ m³/s
R = 1.0 mm = 1.0×10⁻³ m

ΔP = 8 × 0.001 × 1.2 × 2.78×10⁻⁸ / (π × (10⁻³)⁴)
   = 2.67×10⁻¹⁰ / 3.14×10⁻¹²
   = 85 Pa

That converts to a column of water of about 85 / (1000 × 9.81) ≈ 8.7 mm. So an IV bag hung even 1 cm above a peripheral cannula will drive flow at the prescribed rate — and in practice the bag is mounted half a metre or so above the patient, providing a generous excess to overcome cannula resistance and venous back-pressure. The R⁴ dependence is the key: halving the cannula diameter raises the required ΔP sixteenfold, which is why nurses prefer the largest cannula a patient's vein will tolerate.

Now repeat for blood instead of saline, μ = 3.5 cP at body temperature, and the required pressure rises to 300 Pa or about 3 cm H₂O. That is the pressure your aortic root must develop to push blood through a single capillary; in practice the heart works against the integrated resistance of the entire systemic circulation, ~13 kPa = 100 mmHg.

Where viscosity comes from at the molecular level

The microscopic origin of viscosity differs between gases and liquids — and gets you opposite temperature dependences.

In a gas, viscosity comes from collisional momentum transfer. A faster molecule wandering across a velocity gradient eventually collides with slower molecules, transferring some of its forward momentum to them. Maxwell's kinetic theory gives μ_gas ≈ ⅓ ρ v̄ λ, where v̄ is mean thermal speed and λ is mean free path. Combined with v̄ ∝ √T and λ ∝ T/P (at fixed density), this predicts μ_gas ∝ √T at fixed pressure — independent of pressure to first order, and increasing with temperature. Air viscosity rises 30 % between 0°C and 100°C.

In a liquid, molecules are too crowded for the gas picture to apply. Instead, viscosity reflects the energy cost for a molecule to "jump" from one cage of neighbours to another, and that cost is exponentially sensitive to T. Eyring's transition-state theory gives μ_liquid ≈ A exp(E_a/RT), with E_a the activation energy for molecular flow. Liquid viscosity therefore decreases exponentially with temperature: water drops 4× between 0°C and 100°C, motor oil drops 10×, glycerin drops 30×.

The crossover from liquid-like to gas-like behaviour as a fluid passes through its critical point reveals both mechanisms operating simultaneously. Supercritical fluids are exploited industrially (supercritical CO₂ extraction of caffeine, supercritical water oxidation of waste) precisely because their viscosity is gas-like (low) while their density is liquid-like (high), giving very high mass transport.

Non-Newtonian fluids

A Newtonian fluid is one whose viscosity is independent of shear rate. Many real fluids — most of them, by population — are not Newtonian. Their effective viscosity depends on how fast you shear them, sometimes on the history of shear, and sometimes on whether shear stress exceeds a threshold at all.

Power-law (Ostwald–de Waele) fluids: τ = K(du/dy)ⁿ. With n = 1, Newtonian. With n < 1, shear-thinning (pseudoplastic) — viscosity decreases with shear rate. Examples: blood (n ≈ 0.8 at moderate shear), polymer solutions, paint, ketchup. With n > 1, shear-thickening (dilatant) — viscosity increases with shear rate. Examples: concentrated cornstarch suspensions, sand-water mixtures.

Bingham plastics: τ = τ_y + μ_p(du/dy). No flow occurs until shear stress exceeds a yield stress τ_y. Toothpaste, mayonnaise, drilling muds, lava and snow obey this approximately. Toothpaste sits in a tube without flowing because its yield stress (~200 Pa) exceeds gravitational stress on the column.

Herschel–Bulkley: τ = τ_y + K(du/dy)ⁿ. Combines yield stress with power-law shear rate dependence. Most semi-solid foods and personal-care products fall in this category, which is why food rheologists make a living with two-parameter fits.

Viscoelastic fluids: the stress depends not only on instantaneous shear rate but on the deformation history. Polymer melts, biological tissue and silly putty exemplify this. They can show "die swell" (extruded jet expands to greater than orifice diameter), Weissenberg climbing (rod stirred in a polymer solution causes the solution to climb the rod against gravity), and elastic recoil after sudden stress release.

Where viscosity shows up

• Engine oil grading. The SAE multigrade system (10W-40, 5W-30) specifies kinematic viscosity at two reference temperatures: cold start (−10 to −35°C, low shear) and operating temperature (100°C, high shear). Modern oils use polymer "viscosity index improvers" that uncoil at high T to keep ν roughly constant, the rare commercial application of designed viscoelasticity.
• Blood viscosity in the microcirculation. Whole blood at 37°C is about 3–4 cP at high shear and rises to 50+ cP at very low shear, when red blood cells aggregate into rouleaux. The Fåhræus–Lindqvist effect adds another twist: in capillaries narrower than 300 μm, the apparent viscosity drops below bulk values because RBCs travel single-file along the centreline.
• Polymer extrusion. Plastics processing depends entirely on the temperature- and shear-rate-dependent viscosity of polymer melts, typically 100–10000 Pa·s at processing temperatures. Designing a film-blowing or injection-moulding line is rheology engineering: you have to predict ν over a 10⁵-fold shear-rate range.
• Geological flow. Earth's mantle has effective viscosity ~10²¹ Pa·s — a million billion billion times more viscous than water — yet flows on geological timescales by solid-state creep mechanisms. Plate tectonics is mantle convection: Rayleigh number Ra ~10⁷ in the mantle, just above the Rayleigh–Bénard convection threshold.
• Lubrication and tribology. A 1 μm oil film between two steel surfaces sliding at 1 m/s sustains shear rates of 10⁶ s⁻¹. Engine bearings, hydraulic systems and machine tools depend on hydrodynamic lubrication, where viscosity generates a load-bearing pressure film without solid-solid contact.

Measuring viscosity

Practical rheometry uses geometries chosen so that shear rate is uniform across the sample and shear stress is easy to compute from torque or pressure.

The capillary viscometer times how long a known volume of fluid takes to flow through a calibrated glass tube under gravity; the Hagen–Poiseuille formula converts time to ν. Ostwald and Ubbelohde tubes with different tube diameters cover ν from 0.5 to 100000 cSt and are the industry standard for petroleum and oils.

The rotational viscometer places fluid between two coaxial cylinders, one rotating at controlled angular velocity and the other measuring torque. Cone-and-plate and parallel-plate variants allow tiny sample volumes and well-defined shear rate. Modern stress-controlled rheometers can apply oscillatory shear to extract storage modulus G' and loss modulus G''(ω) for full viscoelastic characterisation.

The falling-ball viscometer measures the terminal velocity of a sphere falling through fluid; Stokes's law gives μ from observed velocity. Used for very viscous fluids where flow times become impractical.

Variants and extensions

• Bulk (volume) viscosity κ. Newton's law extended: total stress includes a term κ(∇·u) that resists volumetric expansion or compression. For incompressible flow κ does not appear. For sound attenuation, shock waves and gas mixtures undergoing relaxation, bulk viscosity is essential. CO₂ has κ ≈ 1000 μ — sound absorbs strongly through it.
• Reynolds-averaged (eddy) viscosity ν_t. In turbulence modelling, the Reynolds stresses are absorbed into an effective turbulent viscosity that augments the molecular value. Eddy viscosity in the atmospheric boundary layer can be 10⁵× the molecular ν, dominated entirely by turbulent rather than molecular momentum transfer.
• Apparent viscosity. For non-Newtonian fluids, the effective μ_eff = τ/(du/dy) defined at a particular shear rate. Always quote the shear rate when reporting apparent viscosity for blood, polymer or food samples.
• Trouton viscosity (extensional). Resistance to stretching rather than shear. For Newtonian fluids it is exactly 3μ; for polymer solutions and viscoelastic fluids it can be 1000× the shear viscosity, which is why polymer additives can dramatically extend the lifetime of liquid jets.
• Reynolds-averaged Navier–Stokes. The standard engineering CFD treatment splits velocity into mean and fluctuating components and time-averages the Navier–Stokes equation. The unknown Reynolds stresses are then closed by k-ε, k-ω or other turbulence models that recast the closure as an eddy viscosity ν_t depending on local turbulent kinetic energy.

Common pitfalls

• Mixing μ and ν unit systems. Pa·s, P, cP, kg/m·s all measure dynamic viscosity but with factor-of-10 conversions that bite. Always carry units explicitly. Water = 1 cP = 10⁻³ Pa·s = 0.01 P.
• Assuming a fluid is Newtonian when it isn't. Blood, paints, slurries and most foods are non-Newtonian — applying μ values measured at one shear rate at a different rate gives nonsense. Specify shear rate when quoting apparent viscosity, and check that your application range matches the measurement range.
• Forgetting temperature dependence. A 10°C change can move μ by 25 % for water, 200 % for honey. Engineering specs must give μ at the operating temperature, not 20°C.
• Confusing kinematic viscosity with diffusivity. ν is momentum diffusivity, with units m²/s — same as thermal diffusivity α and mass diffusivity D. The dimensionless ratios Pr = ν/α and Sc = ν/D are essential in heat and mass transfer; treating ν as dimensionless or as energy-related quantity is wrong.
• Using bulk viscosity values in incompressible problems. The bulk viscosity κ does not appear in incompressible Navier–Stokes; quoting it where it cannot act gives a wrong sense of how strongly a fluid dissipates. Use bulk viscosity only for sound, shocks, and compressible flows.