Viscosity and Newtonian Fluids

Why viscosity matters

Viscosity is the single fluid property that turns the elegant, frictionless world of ideal flow into the messy, energy-dissipating reality that engineers actually design for. Without it there would be no boundary layers, no drag, no lubrication, no pressure drop in pipes — and no way to pump crude oil, hydroplane a tire, or float a crankshaft on a film of oil thinner than a human hair. It appears in nearly every dimensionless group in fluid mechanics and heat transfer.

• Reynolds number. Re = ρVL/μ decides whether flow is laminar or turbulent, setting drag, mixing, and heat transfer.
• Pipe and duct sizing. Viscosity fixes the friction factor and pumping power for pipelines, HVAC ducts, and hydraulic lines.
• Lubrication. A journal bearing survives only because a viscous oil film builds hydrodynamic pressure to carry the shaft load.
• Boundary layers. The no-slip condition plus viscosity create the thin sheared region that governs aerodynamic drag and heat flux.
• Process industries. Polymers, paints, foods, and drilling muds are engineered around their (often non-Newtonian) viscosity.
• Damping. Shock absorbers and dashpots convert motion to heat by forcing viscous fluid through an orifice.
• Metrology. Engine oils are graded (SAE 10W-40, ISO VG 46) entirely by their viscosity-temperature behavior.

How it works, step by step

Imagine two large parallel plates separated by a gap h filled with fluid. Hold the bottom plate fixed and drag the top plate sideways at velocity U with a force F spread over area A. This is the classic Couette flow thought experiment that defines viscosity.

1. No-slip condition. The fluid molecules touching each plate move with that plate. Fluid at the bottom is stationary; fluid at the top moves at U. This is an experimental fact, not an assumption you can skip.
2. Linear velocity profile. In the steady state the velocity varies linearly across the gap, so the velocity gradient (shear rate) is du/dy = U/h, a constant with units of s⁻¹.
3. Shear stress. The dragging force per unit area is the shear stress τ = F/A, in pascals (Pa = N/m²).
4. Newton's law of viscosity. Experiment shows τ scales linearly with the shear rate: τ = μ (du/dy). The proportionality constant μ is the dynamic viscosity. Rearranged, μ = τ / (du/dy) = (F/A) / (U/h).
5. Momentum diffusion. Physically, viscosity is molecular momentum transport: fast layers hand momentum to slow layers through molecular collisions (gases) or intermolecular forces (liquids), smearing out velocity differences.
6. Kinematic viscosity. Divide by density to get ν = μ/ρ, the momentum diffusivity in m²/s. It plays the same role for momentum that thermal diffusivity plays for heat.

Every symbol, with SI units:

• τ — shear stress, Pa (N/m²)
• μ — dynamic viscosity, Pa·s (kg·m⁻¹·s⁻¹)
• du/dy — velocity gradient / shear rate, s⁻¹
• ν — kinematic viscosity, m²/s, with ν = μ/ρ
• ρ — mass density, kg/m³
• U — top-plate velocity, m/s; h — gap, m

Newtonian vs non-Newtonian behavior

The defining feature of a Newtonian fluid is that μ is a single constant — the plot of τ against du/dy is a straight line through the origin, and its slope is the same whether you shear it gently or violently. Water, air, most gases, glycerine, and light mineral oils are all Newtonian. Non-Newtonian fluids break this linearity, and we describe them with an apparent viscosity μ_app = τ / (du/dy) that itself depends on shear rate. The most common engineering model is the power law (Ostwald–de Waele): τ = K (du/dy)ⁿ, where K is the consistency index (Pa·sⁿ) and n is the flow behavior index.

Fluid type	τ vs du/dy behavior	Power-law index n	Everyday examples
Newtonian	Linear through origin; μ constant	n = 1 (K = μ)	Water, air, glycerine, thin oil
Shear-thinning (pseudoplastic)	Apparent viscosity falls as shear rises	n < 1	Blood, ketchup, paint, polymer melts
Shear-thickening (dilatant)	Apparent viscosity rises as shear rises	n > 1	Cornstarch slurry, dense suspensions
Bingham plastic (yield stress)	No flow until τ > τ₀, then linear	τ = τ₀ + μ_p(du/dy)	Toothpaste, drilling mud, mayonnaise
Thixotropic	Viscosity drops with time under shear	time-dependent	Non-drip paint, yogurt, gels

Shear-thinning is by far the most industrially useful behavior: paint spreads thin under the brush (high shear) but thickens on the wall so it will not sag (near-zero shear); blood thins in narrow capillaries where shear rates are high. Shear-thickening is rarer but dramatic — a cornstarch suspension can be run across but will swallow you if you stand still. Yield-stress fluids let toothpaste sit on the brush without dripping until you squeeze past τ₀.

Temperature and pressure dependence

Viscosity is strongly temperature-dependent, and — crucially — liquids and gases move in opposite directions.

• Liquids fall with temperature. Intermolecular attraction dominates; heating lets molecules slip past neighbors, so μ drops. Water goes from 1.79 mPa·s at 0 °C to 1.00 mPa·s at 20 °C to 0.28 mPa·s at 100 °C. The Andrade/Arrhenius form μ = A·exp(B/T) fits well.
• Gases rise with temperature. Momentum is carried by molecular collisions; heating speeds molecules and increases momentum exchange, so μ grows. Sutherland's law gives μ = μ₀ (T/T₀)^(3/2)·(T₀ + S)/(T + S); for air S ≈ 111 K.
• Pressure. For liquids, viscosity rises modestly with pressure (important in elastohydrodynamic bearing contacts, where GPa pressures can multiply oil viscosity by orders of magnitude). Gas viscosity is nearly pressure-independent at ordinary conditions.

Because engine oils see wide temperature swings, the SAE viscosity grading (e.g. 10W-40) specifies cold-crank behavior (the "W" winter number) and the high-temperature viscosity at 100 °C. The viscosity index (VI) quantifies how flat the viscosity-temperature curve is — a high VI oil changes little between cold start and operating temperature.

Worked example: force to shear an oil film

A flat plate of area A = 0.5 m² slides on a h = 0.5 mm film of SAE 30 oil (μ = 0.29 Pa·s at 40 °C) at U = 3 m/s. Find the shear stress and the dragging force, assuming a linear (Couette) profile.

Shear rate: du/dy = U/h = 3 / 0.0005 = 6000 s⁻¹.

Shear stress: τ = μ (du/dy) = 0.29 × 6000 = 1740 Pa.

Drag force: F = τ·A = 1740 × 0.5 = 870 N.

Viscous power dissipated (all converted to heat in the film): P = F·U = 870 × 3 = 2610 W. This is exactly why bearings and dampers get hot, and why lubricant selection is a thermal problem as much as a mechanical one. Halve the gap and you double both the force and the heat; switch to a lighter oil at operating temperature and you cut both, at the cost of a thinner load-bearing film.

Fluid (typical)	Dynamic viscosity μ	In centipoise (cP)	Relative to water
Air (20 °C)	0.0181 mPa·s	0.018 cP	~1/55
Water (20 °C)	1.00 mPa·s	1 cP	1×
Blood (37 °C, ~high shear)	3–4 mPa·s	3–4 cP	~3–4×
SAE 30 motor oil (40 °C)	~0.29 Pa·s	~290 cP	~290×
Glycerine (20 °C)	~1.41 Pa·s	~1410 cP	~1410×
Honey (20 °C)	~10 Pa·s	~10 000 cP	~10 000×

Common misconceptions and failure modes

• "Thicker means denser." Viscosity is not density. Engine oil is less dense than water but ~300× more viscous. Kinematic viscosity ν = μ/ρ separates the two.
• "Air has almost no viscosity." Its dynamic μ is tiny, but its kinematic ν (~1.5e-5 m²/s) is 15× water's — which is why air boundary layers are relatively thick.
• "Ketchup has one viscosity." Non-Newtonian fluids have no single μ; you must state the shear rate. Quoting "the viscosity" of blood or paint without a shear rate is meaningless.
• "Heating always thins a fluid." True for liquids, false for gases — gas viscosity rises with temperature via Sutherland's law.
• "Poise and Pa·s are interchangeable numbers." They are not: 1 Pa·s = 10 P. Mixing them by a factor of 10 is a classic pipeline and lubrication error.
• Bearing failure. If oil viscosity drops too far (overheating, dilution by fuel), the hydrodynamic film collapses to boundary lubrication and the shaft scores — a viscosity failure, not a load failure.
• Slip / non-continuum limits. Newton's law assumes a continuum with no-slip walls; in microfluidics and rarefied gas flow (high Knudsen number) both assumptions break down.