Newton’s Law of Viscosity
τ = μ · dv/dy — Solve for shear stress, dynamic viscosity, or velocity gradient.
Basic Mode
Solve for one unknownAdvanced Mode
Finite-difference & presetsResults
Basic Mode- Ready to calculate.
Fluid-Layer Visualization
Velocity profile between two parallel layers. Arrows indicate flow direction.
Velocity & Rheology Charts
Newtonian vs. Non-Newtonian
This calculator applies directly to Newtonian fluids at fixed conditions.
| Behavior | Relationship | Apparent Viscosity | Examples |
|---|---|---|---|
| Newtonian | τ ∝ γ̇ | Constant at fixed T, P | Water, air, light oils |
| Shear-thinning | Nonlinear | Decreases with γ̇ | Polymer solutions, paints |
| Shear-thickening | Nonlinear | Increases with γ̇ | Concentrated suspensions |
| Yield-stress | Threshold τ0 | Model-dependent | Toothpaste, ketchup |
Dynamic vs. Kinematic Viscosity
Dynamic viscosity μ (Pa·s) measures resistance to shear.
Kinematic viscosity ν = μ / ρ (m²/s), where ρ is density.
— m²/s
Reference Tables
A — Symbols
| Symbol | Quantity | SI Unit | Description |
|---|---|---|---|
| τ | Shear stress | Pa | Force per unit area |
| μ | Dynamic viscosity | Pa·s | Resistance to shear |
| v | Velocity | m/s | Fluid layer speed |
| y | Position | m | Distance normal to flow |
| dv/dy | Velocity gradient | s⁻¹ | Local derivative |
| γ̇ | Shear rate | s⁻¹ | Synonym for dv/dy |
| ν | Kinematic viscosity | m²/s | μ / ρ |
| ρ | Density | kg/m³ | Mass per volume |
B — Viscosity Unit Conversions
| Unit | Equivalent |
|---|---|
| 1 Pa·s | 1000 cP • 10 P • 1 N·s/m² • 1 kg/(m·s) |
| 1 mPa·s | 1 cP • 0.001 Pa·s |
| 1 P | 0.1 Pa·s • 100 cP |
| 1 cP | 1 mPa·s • 0.001 Pa·s |
C — Example Fluid Viscosities
| Fluid | Temp (°C) | μ (mPa·s) | Notes |
|---|---|---|---|
| Water | 20 | 1.002 | Reference |
| Water | 40 | 0.653 | Reference |
| Air | 20 | 0.018 | At 1 atm |
| Ethanol | 20 | 1.20 | Reference |
| Methanol | 20 | 0.59 | Reference |
| Glycerol | 20 | 1410 | Highly viscous |
| Glycerol | 40 | 310 | Temperature-dependent |
| Mercury | 20 | 1.55 | Liquid metal |
Values are for reference; always use measured data for design.
What This Tool Offers
How to Use This Calculator
- Select Basic or Advanced mode.
- Choose the quantity to calculate (τ, μ, or dv/dy).
- Enter the known values with appropriate units.
- Optionally pick a fluid preset (Advanced mode).
- Click Calculate.
- Review the primary result, unit conversions, and equation substitution.
- Expand Calculation Breakdown for step-by-step details.
- Inspect the fluid-layer visualization and charts.
- Change parameters to explore their effects.
Formula Explanation
Newton’s Law of Viscosity:
Viscosity describes a fluid's resistance to relative motion between neighboring layers. For a Newtonian fluid at fixed thermodynamic conditions, τ ∝ dv/dy, and μ is the constant of proportionality.
The finite-difference approximation dv/dy ≈ Δv/Δy is reasonable when the velocity profile is linear or nearly so between two points.
Engineering Applications
References & Further Reading
- White, F. M. Fluid Mechanics. McGraw-Hill.
- Munson, B. R., et al. Fundamentals of Fluid Mechanics. Wiley.
- Çengel, Y. A. & Cimbala, J. M. Fluid Mechanics: Fundamentals and Applications. McGraw-Hill.
- Property data should be checked against authoritative thermophysical-property sources for the stated temperature and pressure before design use.
All numerical values are for educational reference; verify with measured data for engineering design.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| tau | Shear stress | Pa |
| mu | Dynamic viscosity | Pa.s |
| dv/dy | Velocity gradient (shear rate) | 1/s |
What it means
Newton’s law of viscosity is the constitutive equation for a Newtonian fluid, which states that the shear stress τ is proportional to the velocity gradient (shear rate) perpendicular to the direction of flow. The proportionality constant is the dynamic viscosity μ. Mathematically, τ = μ (dv/dy). This law applies to fluids like water, air, and most simple liquids. For non‑Newtonian fluids, the relationship is more complex (shear‑thinning, shear‑thickening, etc.). Viscosity is a measure of a fluid’s resistance to deformation. The law is fundamental in fluid mechanics for deriving the Navier‑Stokes equations and for solving problems involving laminar flow in pipes, lubrication, and coating. In engineering, it is used to predict pressure drops and to characterise fluid behaviour. Understanding viscosity is crucial for pump selection, pipeline design, and mixing operations. It also plays a role in food processing, cosmetics, and biomedical applications.
Worked example
Newton's Law of Viscosity – Two Examples
Real‑World| Parameter | Value |
|---|---|
| μ | 0.001 Pa·s |
| dv/dy | 500 s⁻¹ |
| Parameter | Value |
|---|---|
| μ | 0.05 |
| dv/dy | 100 |
Common mistakes
- Viscosity μ: Dynamic viscosity, not kinematic.
- Velocity gradient dv/dy: Shear rate – in s⁻¹.
- Shear stress τ: In Pa (N/m²) when SI units are used.
- Newtonian fluid: This law applies only to Newtonian fluids; non‑Newtonian fluids have a different relationship.
- Sign convention: Shear stress is positive in the direction of flow.
Applications
Newton's law of viscosity, τ = μ·(dv/dy), defines dynamic viscosity as the ratio of shear stress to velocity gradient. It is the basis for understanding fluid flow behaviour, particularly for Newtonian fluids (where viscosity is constant). Engineers use this relationship to calculate shear stresses in pipes, bearings, and viscometers. It is applied in the design of lubrication systems, coatings, and food processing equipment. The law also underpins the development of non‑Newtonian models (e.g., power‑law, Bingham). By understanding viscosity, professionals can select appropriate pumps, predict pressure drops, and design mixing equipment. It is a fundamental concept in rheology and fluid mechanics.
- Design of pipeline and channel flow systems
- Lubrication and tribology analysis in bearings and gears
- Selection of pumps and mixers based on fluid viscosity
- Rheological characterisation of fluids (Newtonian vs. non‑Newtonian)
- Quality control in food, cosmetic, and pharmaceutical industries
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