What is a fluid?

This technical note defines a fluid by its mechanical response to shear stress and summarizes the constitutive relation for a Newtonian fluid and the range of validity of the continuum approximation.

8 min readUpdated 2026-08-11Terminology, equations & units checkedJA version

Abstract

A fluid is a substance that cannot sustain shear stress in a state of rest and therefore continues to deform while shear stress is applied. This mechanical definition includes both liquids and gases.

Mechanical response to shear stress

In an elastic solid, shear stress corresponds to a finite shear strain, and static equilibrium may be established after deformation. In a fluid, no static equilibrium can balance a sustained shear stress; shear deformation therefore proceeds with time.

Elastic solid

Static equilibrium at finite strain

Fluid

Shear deformation proceeds in time

Constitutive relation for a Newtonian fluid

In a Newtonian fluid, the shear components of the stress tensor are linear in the spatial derivatives of velocity [1, 2]. For unidirectional simple shear, u = u(y), the relation reduces to the following form.

Equation (1)　Newton's law of viscosity

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

Representation for simple shear, u = u(y)

$\tau$: Shear stress[ $\mathrm{Pa}$ ]
$\mu$: Shear viscosity (distinct from kinematic viscosity ν)[ $\mathrm{Pa{\cdot}s}$ ]
$\mathrm d u/\mathrm d y$: Velocity gradient (shear rate)[ $\mathrm{s^{-1}}$ ]

InterpretationFor a fixed velocity difference, reducing the characteristic gap increases the velocity gradient. At constant viscosity, the required shear stress therefore increases in direct proportion.

Continuum hypothesis

Although fluids consist of molecules, macroscopic fluid mechanics represents density ρ, pressure p, velocity u and temperature T as continuous functions of space and time. The approximation is valid when the characteristic length is sufficiently large relative to the molecular mean free path [3].

Field variables: Density ρ, pressure p, velocity u and temperature T
Applicability: Characteristic length L is sufficiently large relative to molecular mean free path λ
Non-continuum effects: High vacuum, rarefied gases, microchannels and internal shock layers require separate assessment

Fluid-mechanical distinction between liquid and gas phases

Liquid phase

Relatively small volume change over ordinary pressure ranges
May form a free surface in a gravitational field
Examples: water, oil and molten metal

Gas phase

Occupies the available container volume
Density change with pressure may be significant
Examples: air, steam and combustion gas [4]

Consequences for fluid-mechanical formulation

Fluid statics

Stress in a fluid at rest is represented by pressure normal to a surface

Transport phenomena

Viscosity relates velocity gradients to shear stress and governs frictional loss

Conservation equations

Local balances of mass, momentum and energy can be formulated for a continuum

NEXT SECTION · 01—02Density, pressure & viscosity

Define the basic properties used to describe fluid state and transport behavior.

References

Terminology, constitutive relations and property classifications were checked against international terminology and public research institutions. Accessed 11 August 2026. Independent expert review has not yet been completed.