Formula & Calculator
Newton's Law of Viscosity
Relates the shear stress in a Newtonian fluid to its viscosity and the local velocity gradient.
Interpretation
Newton's law of viscosity: τ = μ·(dv/dy), relates shear stress to velocity gradient. Defines dynamic viscosity μ. Example: μ=0.001 Pa·s, gradient=100 s⁻¹ → τ=0.1 Pa.
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
Frequently Asked Questions
Newton's law of viscosity states that the shear stress in a fluid is proportional to the velocity gradient: τ = μ · (dv/dy), where τ is the shear stress (Pa), μ is the dynamic viscosity (Pa·s), and dv/dy is the velocity gradient (s⁻¹). It defines a Newtonian fluid.
In SI, dynamic viscosity μ has units of Pa·s = kg/(m·s). The cgs unit is the poise (1 poise = 0.1 Pa·s). Kinematic viscosity ν = μ/ρ has units of m²/s (Stokes).
- Applying it to non‑Newtonian fluids (e.g., slurries, polymers) where μ is not constant.
- Using the wrong velocity gradient – it must be the derivative of velocity with respect to the perpendicular direction.
- Ignoring the temperature dependence of μ.
In a Newtonian fluid, the viscosity is constant at a given temperature. In non‑Newtonian fluids, the apparent viscosity depends on shear rate (e.g., shear‑thinning, shear‑thickening, Bingham plastics).
For liquids, viscosity decreases with increasing temperature. For gases, viscosity increases with temperature. This is captured by empirical correlations (e.g., Sutherland's law for gases).
- Water at 20°C: μ ≈ 0.001 Pa·s (1 cP).
- Air at 20°C: μ ≈ 1.8×10⁻⁵ Pa·s.
- Motor oil: μ ≈ 0.1‑1 Pa·s.
Viscosity is a measure of a fluid's resistance to deformation. It arises from intermolecular forces and molecular momentum transfer.
Common viscometers: capillary (Ubbelohde), rotational (Brookfield), and falling ball. They measure the shear stress vs. shear rate relationship.