CHAPTER01ARTICLE02

FLUID MECHANICS · 01–02 / BASIC

Density, pressure & viscosity

Definitions, SI units, validity conditions and a worked water-property example for density, pressure and viscosity in fluid mechanics.

9 min read2026-08-13Definitions, equations & units checkedJA version

Abstract

Density measures mass per unit volume, pressure is the isotropic normal stress in a fluid at rest, and viscosity quantifies resistance to the rate of deformation. Together they characterize inertia, mechanical state and momentum transport.

Density: connecting mass and volume

For a uniform sample, mean density ρ is mass m divided by volume V. Continuum mechanics uses a local field ρ(x, t), defined over a sampling volume that is small macroscopically but still large relative to molecular scales.

Liquid density is often treated as constant over ordinary pressure ranges. Gas flows, high-speed flows and problems with large temperature differences generally require density to vary with pressure and temperature.

Equation (1)Mean density

\rho=\frac{m}{V}

$\rho$: Density[ $\mathrm{kg/m^3}$ ]
$m$: Mass[ $\mathrm{kg}$ ]
$V$: Volume[ $\mathrm{m^3}$ ]

Physical role: Appears in mass conservation and inertia; it fixes how much matter occupies a given volume.
State dependence: Generally depends on temperature and pressure and may change abruptly across a phase boundary.

Pressure: stress normal to a surface

In a fluid at rest, the stress acting across any plane through a point is normal to that plane and independent of its orientation. Its magnitude is pressure p. When the stress is nearly uniform over a finite area, it can be evaluated as normal force Fₙ divided by area A. The pascal is defined by 1 Pa = 1 N/m² [1].

Absolute pressure uses perfect vacuum as its zero; gauge pressure uses the surrounding atmospheric pressure. Equations of state and comparisons with vapor pressure normally require absolute pressure.

Equation (2)Uniform normal stress

p=\frac{F_{\mathrm n}}{A}

$p$: Pressure[ $\mathrm{Pa}$ ]
$F_{\mathrm n}$: Normal force[ $\mathrm{N}$ ]
$A$: Loaded area[ $\mathrm{m^2}$ ]

Equation (3)Absolute and gauge pressure

p_{\mathrm{abs}}=p_{\mathrm{gauge}}+p_{\mathrm{atm}}

$p_{\mathrm{abs}}$: Absolute pressure[ $\mathrm{Pa}$ ]
$p_{\mathrm{gauge}}$: Gauge pressure[ $\mathrm{Pa}$ ]
$p_{\mathrm{atm}}$: Ambient atmospheric pressure[ $\mathrm{Pa}$ ]

Viscosity: transport of momentum across velocity gradients

Consider simple shear, u = u(y). In a Newtonian fluid, shear stress τ is proportional to shear rate du/dy; the coefficient μ is the dynamic viscosity, not the kinematic viscosity [2, 3].

Kinematic viscosity ν is μ divided by density ρ and is useful when comparing viscous momentum diffusion with inertia. Both μ and ν depend on temperature; for liquids, μ generally decreases as temperature rises.

Equation (4)Newton's law of viscosity

\tau=\mu\frac{\mathrm d u}{\mathrm d y}

Newtonian fluid in simple shear

$\tau$: Shear stress[ $\mathrm{Pa}$ ]
$\mu$: Dynamic viscosity[ $\mathrm{Pa{\cdot}s}$ ]
$\mathrm d u/\mathrm d y$: Shear rate[ $\mathrm{s^{-1}}$ ]

Equation (5)Kinematic viscosity

\nu=\frac{\mu}{\rho}

$\nu$: Kinematic viscosity[ $\mathrm{m^2/s}$ ]
$\mu$: Dynamic viscosity[ $\mathrm{Pa{\cdot}s}$ ]
$\rho$: Density[ $\mathrm{kg/m^3}$ ]

Newtonian fluid: Stress and shear rate are linearly related, so μ can be treated as a state-dependent material property.
Non-Newtonian fluid: Paints, slurries and polymer solutions may have an apparent viscosity that depends on shear rate or flow history.

Worked example: kinematic viscosity of water at 20 °C

For water near 0.1 MPa and 20 °C, use the representative rounded values ρ = 998.2 kg/m³ and μ = 1.002 × 10⁻³ Pa·s. These values are consistent with the IAPWS-based reference correlations described by NIST; precision work should evaluate the standard formulations at a stated temperature and pressure [4, 5].

Substitution in Equation (5) gives ν = (1.002 × 10⁻³) / 998.2 ≈ 1.004 × 10⁻⁶ m²/s. A property-table value should always be reported with temperature, pressure, units and appropriate significant figures.

Illustrative water properties near 20 °C
Quantity	Symbol	Value	SI unit
Density	ρ	998.2	kg/m³
Dynamic viscosity	μ	1.002 × 10⁻³	Pa·s
Kinematic viscosity	ν	1.004 × 10⁻⁶	m²/s

References

Sources accessed 13 August 2026. Independent expert review has not yet been completed.