Dynamic Viscosity (η): The Pascal-Second in Fluid Mechanics

Definition and Dimensional Form

Newton’s law of viscosity

ISO 80000-4 defines dynamic viscosity, symbol η (Greek letter eta), through Newton’s law of viscosity: τ = η · (du/dy), where τ is the shear stress in pascals and du/dy is the velocity gradient normal to the direction of flow. The SI coherent unit follows directly as Pa·s, equivalent to kg·m⁻¹·s⁻¹ when reduced to base units. Because the pascal itself equals N·m⁻², you can also view Pa·s as N·s·m⁻², an expression often used in instrumentation handbooks. Regardless of the representation, η captures how much tangential force per unit area is required to achieve a unit shear rate.

Symbols, units, and related quantities

In practice engineers often work with submultiples such as millipascal-seconds (mPa·s) or centipoise (cP). One centipoise equals 1 mPa·s, which explains why legacy ASTM tables listing water at 20 °C with a viscosity of 1.002 cP remain compatible with the SI value 1.002 mPa·s. ISO allows the use of the symbol μ in some engineering contexts, but η remains the principal recommendation to avoid confusion with magnetic permeability μ₀. Kinematic viscosity ν, measured in m²·s⁻¹, complements η through the relation ν = η/ρ, tying shear resistance to density. Knowing when to switch between ν and η is crucial when you work with dimensionless groups such as the Reynolds number.

Momentum diffusion perspective

Dynamic viscosity’s dimensional analysis highlights its role in momentum diffusion. The kg·m⁻¹·s⁻¹ signature tells us that viscosity bridges mass (inertia), length scales, and time. By contrast, the thermal diffusivity α carries units m²·s⁻¹, reminding designers that viscous and thermal transport respond differently to geometry. Keeping these relationships clear reduces the risk of data-entry mistakes when you populate computational fluid dynamics (CFD) material libraries.

Key Concepts, Models, and Symbols

Newtonian versus non-Newtonian behaviour

Newtonian fluids exhibit constant η regardless of shear rate, making water, light oils, and dilute gases easy to characterise. Non-Newtonian fluids exhibit shear-thinning, shear-thickening, viscoplastic, or viscoelastic responses. Engineers often describe these behaviours with constitutive equations such as the power-law τ = K(du/dy)ⁿ, Bingham plastic models that include a yield stress τᵧ, or Maxwell and Oldroyd-B models incorporating elastic moduli. Even when deploying these complex equations, ISO notation keeps η reserved for the zero-shear or reference-shear dynamic viscosity.

Temperature dependence and correlations

Temperature exerts a strong influence on viscosity. Liquids usually become less viscous as temperature rises, often following an Arrhenius-type relationship η(T) = η₀·exp(Eₐ/RT). Engineers sometimes linearise the dependence over practical ranges with the Andrade equation or empirically fitted charts. Gases behave differently: their viscosity increases with temperature, captured by Sutherland’s law μ(T) = μ₀·(T/T₀)^1.5·(T₀ + S)/(T + S). These relations are indispensable when you populate process simulators or specify pump sizing at off-design conditions.

Pressure effects and property linkages

Pressure effects are usually minor for gases at low pressure but can be substantial for lubricants and polymer melts. ISO 3448 viscosity grades, for example, classify industrial oils by their kinematic viscosity at 40 °C, implicitly assuming a known density. Accurate reporting therefore requires both η and ρ values, ideally referenced to the same temperature. This interplay explains why the Reynolds number explainer emphasises property evaluation at operating conditions.

Applications Across Industries

Mechanical and automotive systems

In mechanical engineering, dynamic viscosity defines lubrication regimes. The Stribeck curve, which relates friction coefficient to η·U/P (viscosity × speed ÷ load), distinguishes boundary, mixed, and hydrodynamic lubrication. Selecting the correct lubricant grade prevents metal-to-metal contact in bearings, gearboxes, and turbines. Automotive engineers specify oil viscosity classes (such as SAE 0W-20) to balance cold-start flow with high-temperature film strength, while aerospace designers rely on synthetic esters that maintain viscosity under extreme thermal cycling.

Process industries and food technology

Process industries depend on accurate viscosity data for pump sizing, heat exchanger design, and residence time calculations. High-viscosity polymer melts require twin-screw extruders and specially designed dies to maintain throughput without excessive pressure. Food technologists manipulate viscosity to control mouthfeel and stability, using hydrocolloids to keep sauces or yogurts within target shear-thinning ranges. In biotechnology, controlling viscosity ensures homogeneous bioreactor mixing, adequate oxygen transfer, and consistent downstream filtration performance.

Earth and environmental sciences

Earth scientists evaluate mantle convection and lava flow hazards by constraining η across enormous ranges, from 10² Pa·s for basaltic lava to 10²¹ Pa·s for the Earth’s upper mantle. Oceanographers study how viscosity interacts with density stratification to influence turbulence and nutrient transport. Even finance teams encounter viscosity when estimating pumping energy consumption in sustainability audits, linking technical property data back to the calculators catalogued on this site, such as the thermal storage sizing tool that incorporates fluid property effects on charging efficiency.

Why Dynamic Viscosity Matters

Dynamic viscosity stitches together fundamental physics and everyday engineering. It determines how momentum diffuses, sets the energy cost of pumping, and influences thermal management strategies. Without consistent viscosity data, process models diverge, equipment underperforms, and safety margins erode. In metrology, Pa·s measurements tie back to base units through mechanical impedance and time standards, highlighting the interdependence of SI quantities.

For students, mastering viscosity opens the door to understanding boundary layers, turbulence onset, and rheological phenomena. For practitioners, it guides procurement, maintenance scheduling, and regulatory compliance. Keeping η aligned with ISO 80000 notation, documenting measurement traceability, and cross-referencing relevant calculators ensures the concept remains actionable rather than abstract.

As industries pursue energy efficiency and decarbonisation, viscosity monitoring gains even more value. Low-viscosity lubricants can reduce drivetrain losses, while accurate modeling of high-viscosity biofuels prevents pump cavitation. Integrating these insights into the broader Units & Measures library keeps your projects grounded in rigorous, interoperable measurement science.