Density & Viscosity of Newtonian Fluids

Core properties that connect “what a fluid is” to how it resists deformation.

foundationsunitsconstitutive-laws
⏱️ About 16 min

Why does honey pour slowly while water splashes and spreads? The answers start with two properties you can measure: density and viscosity.

💡
The big idea: Density links mass to volume, while viscosity links shear stress to velocity gradients. For Newtonian fluids, that link is linear and predictable.
🎯 By the end, you'll be able to
  • Define density and specific gravity and use consistent SI units
  • Apply Newton’s law of viscosity to compute shear stress
  • Differentiate dynamic and kinematic viscosity
  • Describe how temperature typically affects viscosity in liquids vs gases

Density and why it matters

Density is mass per unit volume:

SI unit: kg/m3. In transport problems, density often controls inertia, buoyancy, and hydrostatic pressure.

Many engineering calculations boil down to careful unit handling—especially when converting between kg/m3, g/cm3, and specific gravity.

\[ \rho = \frac{m}{V} \]
Definition of density.

Specific gravity (relative density)

Specific gravity (SG) is a dimensionless ratio comparing a fluid’s density to a reference (typically water at 4°C for liquids):

If SG = 0.85, the fluid is 85% as dense as water and tends to float on water.

\[ \mathrm{SG} = \frac{\rho}{\rho_{\text{ref}}} \]
Specific gravity is a ratio of densities.
🔑 Use consistent reference values

In many problems you can take water density as 998 kg/m3 at 20°C. If a problem specifies a different reference temperature, use that.

Newton’s law of viscosity (Newtonian fluids)

When adjacent layers of a moving fluid slide past each other, the fluid develops a shear stress that resists the motion. For a Newtonian fluid, shear stress is proportional to the velocity gradient.

Examples that are well-approximated as Newtonian over common conditions: water, air, many light oils.

\[ \tau = \mu \frac{du}{dy} \]
Newton’s law of viscosity for simple shear.

Dynamic vs kinematic viscosity

Dynamic viscosity μ (Pa·s) measures resistance to shear. Kinematic viscosity ν (m2/s) is dynamic viscosity divided by density:

Kinematic viscosity appears naturally when momentum diffusion is compared to inertia (for example in the Reynolds number).

\[ \nu = \frac{\mu}{\rho} \]
Relationship between kinematic and dynamic viscosity.
✨ Temperature trends (qualitative)

For liquids, viscosity typically decreases strongly with temperature (molecules slip past each other more easily). For gases, viscosity typically increases with temperature (faster molecules transport momentum more effectively).

📝 Worked example: A Newtonian fluid is confined in a thin film between two large parallel plates. The bottom plate is stationary. The top plate moves at 0.50 m/s. The gap is 2.0 mm and the velocity profile is linear. If the fluid has μ = 0.80 Pa·s, compute the shear stress in the fluid.
  1. For linear shear, the velocity gradient is du/dy = Δu/Δy = 0.50 m/s ÷ 0.0020 m = 250 s⁻¹.
  2. Apply Newton’s law of viscosity: τ = μ(du/dy) = 0.80 Pa·s × 250 s⁻¹ = 200 Pa.
✓ 200 Pa
✏️ Practice: Compute the kinematic viscosity ν of water at 20°C given μ = 1.002×10⁻³ Pa·s and ρ = 998 kg/m³.
m^2/s
Solution
  1. Use ν = μ/ρ.
  2. ν = (1.002×10⁻³ Pa·s) ÷ (998 kg/m³) = 1.004×10⁻⁶ m²/s (since 1 Pa·s = 1 kg/(m·s)).

Check your understanding

1. Which statement is true for a Newtonian fluid in simple shear?
Newton’s law of viscosity states τ = μ(du/dy), a linear relation.
2. Dynamic viscosity μ has units of:
μ is measured in Pa·s (equivalently kg/(m·s)).
3. If a liquid’s temperature increases, its viscosity typically:
Liquids generally become less viscous as temperature rises.
🔎 Related lessons
Next: Manometry & Hydrostatic Pressure
✅ Key takeaways
  • Density relates mass and volume and drives hydrostatics and buoyancy
  • Specific gravity is a dimensionless density ratio to a reference fluid
  • Newtonian fluids obey τ = μ(du/dy) with constant μ for given conditions
  • Kinematic viscosity ν = μ/ρ connects viscosity to momentum diffusion
➡️ Next you’ll use density directly in hydrostatic pressure and manometer relationships.
Want to test yourself on this? Try the Chemical Aptitude test →