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Airflow is motion of air, described in the continuum approximation by fields of velocity, pressure, mass density and temperature. A fan can sustain a pressure difference, buoyancy can drive a warm plume, and external motion can carry air past a body. The driving mechanism and boundary geometry determine the resulting flow.

For a characteristic speed $U$ much smaller than the speed of sound, an incompressible flow model can often describe mechanical motion when thermal density changes are also small. The Mach number tests the acoustic comparison, while the Reynolds number compares inertial and viscous effects. These are different approximations: low Mach number does not require laminar flow. A fixed probe measures an Eulerian coordinate reading; smoke markers can visualize paths and mixing, with the limitation that their motion must follow the air closely enough for the question being studied.

The current configuration is the spatial arrangement of a material body at the time being studied. A deformation map $\mathbf{x}=\mathbf{\chi }(\mathbf{X},t)$ gives the current position $\mathbf{x}$ of the particle labelled by reference position $\mathbf{X}$. Fields expressed as functions of $\mathbf{x}$ use Eulerian coordinates.

A marked rubber strip illustrates the distinction: its original marks identify material particles, while their observed positions after stretching form the current configuration. A current length or area can differ from the corresponding quantity in the reference configuration. The Cauchy stress tensor measures force per current area, and mass density per current volume changes under compression even though each marked material region retains its mass. Specifying the configuration is therefore part of specifying a measurement, rather than a change in the underlying physical body.

An Eulerian coordinate labels a fixed spatial location at which a continuum field is evaluated. A probe fixed in a pipe records $\mathbf{v}({\mathbf{x}}_0,t)$ as different fluid particles pass it. This description is convenient when the apparatus or boundary is fixed and the identities of individual particles are unimportant.

It differs from a Lagrangian coordinate, which labels one material particle. If the velocity field varies with position, that particle experiences changes both because the field evolves and because it moves into a different region. Its material derivative is therefore $D/Dt={\partial }_t+\mathbf{v}\cdot \nabla$. In a steady narrowing pipe, ${\partial }_t\mathbf{v}=0$ at every Eulerian coordinate, yet a travelling particle can accelerate through the convective term. Steady observations do not imply zero particle acceleration.

The Eulerian time derivative measures a continuum field's rate of change at a fixed Eulerian coordinate. For a velocity field $\mathbf{u}$, the material derivative instead follows a fluid element and satisfies $Df/Dt={\partial }_{t}f+\mathbf{u}\cdot \nabla f$. A nonuniform steady flow has zero Eulerian time derivative of velocity but can have nonzero material acceleration.

A Lagrangian coordinate labels a material particle and follows its identity as the body moves. Taking the particle's position $\mathbf{X}$ in a reference configuration as its label gives the deformation map $\mathbf{x}=\mathbf{\chi }(\mathbf{X},t)$. Differentiation at fixed $\mathbf{X}$ gives that particle's velocity and acceleration.

A floating marker in water approximates a Lagrangian trajectory, while a fixed current meter records an Eulerian coordinate description. The distinction is useful because a particle can cross regions with different velocity even when every fixed-location reading is steady. For a conserved material element, mass conservation becomes $\rho J={\rho }_0$, relating its current mass density to its changing volume. Material labels are especially natural for elasticity, where the same region's initial and final lengths are compared to define strain.

For a scalar fluid property $f(\mathbf{x},t)$, the Eulerian time derivative ${\partial }_{t}f$ measures change at a fixed Eulerian coordinate. The material derivative, also called the Lagrangian time derivative, measures change while following one fluid element. If its Lagrangian trajectory is $\mathbf{X}(\mathbf{a},t)$, where $\mathbf{a}$ is its fixed Lagrangian coordinate, then ${\partial }_t\mathbf{X}=\mathbf{u}(\mathbf{X},t)$.

The chain rule therefore givesThe last term accounts for entering a region where the property differs. For example, a spatially varying but time-independent velocity field has zero Eulerian time derivative, while a moving fluid element can have nonzero material acceleration. The same formula applies componentwise to a vector field.

The velocity of a fluid element is the time derivative of its position with its Lagrangian coordinates $X,Y$ held fixed. Direct differentiation givesReplacing the Lagrangian coordinates by the current Eulerian coordinates therefore gives the velocity fieldThe distinction between following a fluid element and differentiating at fixed Eulerian coordinates is what connects the given deformation map to this velocity field.

At a fixed point beside a river, the Eulerian coordinate description measures $\mathbf{u}(\mathbf{x},t)$. Its local acceleration ${\partial }_t\mathbf{u}$ measures change at that fixed point. Following a floating parcel gives a Lagrangian trajectory $\mathbf{X}(t)$ satisfying $\dot{\mathbf{X}}=\mathbf{u}(\mathbf{X},t)$. The chain rule gives its physical material acceleration:The extra term is the convective acceleration. For a river that narrows steadily, an observer at one fixed position measures a constant velocity, yet a parcel speeds up as it reaches the narrowing. A parcel following a steady bend also experiences a centripetal acceleration even when its speed is constant. An Eulerian calculation of the physical acceleration must include the same convective acceleration; the two descriptions do not predict different forces.

Follow a conserved mass element from its Lagrangian coordinate $\mathbf{q}$ to its Eulerian coordinate $\mathbf{x}=\mathbf{q}+b\mathbf{p}$. The integrated continuity equation giveswhere $\bar{\rho }={\rho }_0/a^3$ and the Lagrangian labels are chosen with uniform reference mass per $d^{3}q$. These equations hold before shell crossing, when the oriented determinant is positive. If ${\mu }_i$ are the eigenvalues of $\partial p_i/\partial q_j$, thenFor an irrotational displacement the deformation matrix is symmetric, so these eigenvalues are real. Using the conventional positive collapse eigenvalues ${\lambda }_i=-{\mu }_i$, equivalently the eigenvalues of $-\partial p_i/\partial q_j$, givesThere is a sign mismatch in the printed pairing of this last expression with eigenvalues of $+\partial p_i/\partial q_j$: for the displayed plus-sign displacement map, those eigenvalues give plus signs in the determinant. For example, $p_1=c{q}_1$, $p_2=p_3=0$ makes $x_1=(1+bc)q_1$, so mass conservation gives $1+\delta =(1+bc{)}^{-1}$, not $(1-bc{)}^{-1}$. The corrected eigenvalue convention restores the intended formula without changing the map.

Now take $\mathbf{p}=(p(q_1),0,0)$ and $J=1+b{p}^{\prime }$. The physical trajectory is $\mathbf{r}=a\mathbf{x}$. Differentiating at fixed $\mathbf{q}$ givesThe physical derivative in the perturbed direction is $\partial /\partial r_1=[a(1+b{p}^{\prime }){]}^{-1}\partial /\partial q_1$. The two transverse directions still have the background acceleration. ThereforeUse the pressureless, $\Lambda =0$ background acceleration equation $3\ddot{a}/a=-4\pi G\bar{\rho }$ to obtainThis proves the planar exactness of the Zeldovich approximation for the requested full force-divergence equation. The displacement depends on only one coordinate, but the divergence includes all three physical directions; discarding the transverse background terms would give the wrong result. The proof is valid until shell crossing. With the planar gravitational boundary condition fixing the possible spatially uniform force, it is the exact single-stream dust evolution; it does not describe the multistream density by a single invertible map after crossing.

Stokes drift is the difference between the mean velocity of moving fluid particles and the mean velocity measured at fixed spatial positions. An oscillating particle samples a slightly different part of the velocity field during each part of its cycle. Correlation between its displacement and the spatial variation of the field can therefore produce a net drift even when the first-order Eulerian coordinate velocity average is zero.

For a small-amplitude deep-water gravity wave with surface elevation $\eta =a\cos (kx-\omega t)$, no imposed mean current and mean particle height $z\le 0$, the leading drift isIt follows the propagation direction and decreases with depth. Its being quadratic in wave amplitude explains why the linear particle orbits nearly close but do not close exactly.^[ref]

A marked fluid particle and a fixed velocity probe measure different averages, so a wave-tank comparison must also account for any compensating return current. Surface float motion can include additional effects from wind or the float's own response. Stokes drift is consequently a precise difference of averages, rather than the name for every observed motion of floating debris.