Streamlines, pathlines & streaklines

Definitions, equations, experimental interpretations and the steady-flow condition under which streamlines, pathlines and streaklines coincide.

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

Abstract

The three curves used to describe a flow are distinguished by what is held fixed during observation. They coincide in steady flow but generally differ in unsteady flow.

Streamline

A streamline is a curve tangent to the velocity field at a fixed instant. At every point, its tangent has the direction of the local velocity vector.

A streamline is constructed with time held fixed. In an unsteady flow its shape therefore changes with the observation time. Parallelism of the curve element and velocity vector gives the differential equation below.

EQ. 01Differential equation of a streamline

\frac{\mathrm{d}x}{u}=\frac{\mathrm{d}y}{v}=\frac{\mathrm{d}z}{w},\qquad (t=t^{\ast})

(1)

$x,\,y,\,z$: Spatial coordinates[ $\mathrm{m}$ ]
$u,\,v,\,w$: Velocity components[ $\mathrm{m/s}$ ]
$t^{\ast}$: Observation time defining the streamline[ $\mathrm{s}$ ]

Pathline

A pathline is the trajectory traced by one specified fluid particle as time advances. It is obtained by selecting and continuously tracking that particle.

A pathline is a Lagrangian description focused on one particle moving through the velocity field. Writing particle position as a function of time gives the initial-value problem below.

EQ. 02Particle kinematics

\frac{\mathrm{d}\boldsymbol{x}_{\mathrm p}}{\mathrm{d}t}=\boldsymbol{u}\!\left(\boldsymbol{x}_{\mathrm p}(t),t\right),\qquad \boldsymbol{x}_{\mathrm p}(t_0)=\boldsymbol{x}_0

(2)

$\boldsymbol{x}_{\mathrm p}(t)$: Position of the marked particle[ $\mathrm{m}$ ]
$\boldsymbol{u}(\boldsymbol{x},t)$: Eulerian velocity field[ $\mathrm{m/s}$ ]
$\boldsymbol{x}_0$: Initial position at time t₀[ $\mathrm{m}$ ]

Streakline

A streakline is the locus of particles that have passed through the same fixed point by the observation time. It is seen when dye or smoke is released continuously from a fixed source.

The release position is fixed while particle-release time varies. The positions, at observation time, of particles released from that point at different times form the streakline.

EQ. 03Set representation of a streakline

\mathcal{S}(t)=\left\{\boldsymbol{X}(t;\boldsymbol{x}_{\mathrm s},\tau)\mid \tau\le t\right\}

(3)

$\mathcal{S}(t)$: Streakline at time t
$\boldsymbol{X}(t;\boldsymbol{x}_{\mathrm s},\tau)$: Position at t of the particle that passed xₛ at time τ[ $\mathrm{m}$ ]
$\tau$: Time at which the particle passed the fixed point[ $\mathrm{s}$ ]

Comparison of the three lines

Comparison of the three lines
Aspect	01Streamline	02Pathline	03Streakline
Held fixed	Observation time	Tracked particle	Particle-release position
Definition	A curve tangent to the local velocity at a fixed time	The trajectory of one fluid particle over time	The locus of particles that passed through one fixed point
Required information	Velocity field at one instant	Position history of one particle	Positions of particles released at different times
Experimental example	Curve integrated from an instantaneous velocity field	Trajectory of a marked particle	Dye or smoke continuously released from a fixed source

In steady flow the velocity field is time-independent, so streamlines, pathlines and streaklines coincide.

References

Sources accessed 11 August 2026. Independent expert review has not yet been completed.