advanced34 min studyLesson 171 in Physics

Watching a flow and following its material

Translate between a velocity field and particle paths, and distinguish streamlines from the histories recorded by tracers.

Before you begin

On this page

The question for today

What you’ll understand

Construct a path from a prescribed Eulerian velocity field.
Distinguish pathlines, streamlines, and streaklines in unsteady flow.
Identify the scale assumptions behind a material fluid element.

A camera fixed beside a river and a floating marker describe the same water in different ways. The camera records whatever water passes a location; the marker follows a particular piece of water. Fluid mechanics needs both descriptions because forces are naturally assigned to moving material while instruments and boundaries often occupy fixed positions. Start by stating which observer and which mathematical label a statement uses.

A field and a material label carry different information

An Eulerian velocity assigns a vector to every position and time in a chosen laboratory frame. A Lagrangian position assigns the current location to a material label, often its initial location. That label travels with the fluid element and is not reset when the element enters another region. A fixed thermometer reads a field at one location; a drifting sensor samples that field along its own path.

A fluid element is a continuum averaging volume, containing many molecules while remaining small compared with the scale of variation being studied. Its velocity need not equal any molecule’s instantaneous velocity. This separation can fail in a very dilute gas or across distances comparable with a molecular mean free path. Within a smooth velocity field, integrating the particle equation requires an initial position as well as the field.

\frac{d\mathbf X(\mathbf a,t)}{dt}=\mathbf u(\mathbf X(\mathbf a,t),t),\qquad \mathbf X(\mathbf a,0)=\mathbf a

The initial coordinate a is a material label; X is its later position. The right-hand side evaluates the laboratory velocity at the moving element’s current location.↔ Scroll wide equations horizontally.

An instantaneous direction is not a recorded history

A pathline is the history of one tagged element. A streamline is a curve tangent to the entire velocity field at one specified instant. A streakline joins elements that passed through one release point at different times, as in continuous dye injection. In a steady field these constructions trace the same family of curves, although their parametrizations and which portions are occupied may differ. In unsteady flow they generally differ.

Consider a spatially uniform field whose horizontal component is a positive constant and whose vertical component increases with time. Every streamline at a fixed instant is straight because the velocity direction is the same everywhere then. A tracer curves upward because it meets a new direction at each later instant. A long-exposure image can therefore show curved tracks even though an instantaneous vector plot consists of parallel arrows.

\frac{d\mathbf r}{ds}=\alpha(s)\mathbf u(\mathbf r(s),t_0),\qquad t_0\ \text{fixed}

A streamline uses one frozen time. The scalar α changes the parametrization without changing the tangent direction where the velocity is nonzero; stagnation points require separate treatment.↔ Scroll wide equations horizontally.

Compare descriptions without changing the physical experiment

To convert a particle calculation into a field statement, find which material label currently occupies a location and evaluate that element’s velocity there. This requires an invertible material map in the smooth region under study. A shock or a discontinuity requires additional care; writing a smooth inverse through it does not resolve the physics. Changing from a material description to a spatial description is also distinct from changing the laboratory reference frame.

For measured flows, first decide what was actually recorded: vectors from image pairs, complete tracer paths, or a continuously released dye pattern. Then compare the same observable with a model. A steady streamline computation cannot automatically reproduce a time-dependent streakline photograph. This is a practical model check: agreement of a visually attractive curve with one frame of a video does not establish that its time evolution or transport predictions are correct.

Work it through

Straight streamlines, curved particle path

Use $\mathbf u=(1,t)$ in metres per second with t measured in seconds and its vertical coefficient equal to $1\,\mathrm{m\,s^{-2}}$ . A marker starts at the origin at time zero. Find its position at two seconds and the instantaneous streamline slope then.

Integrate the horizontal equation $dX/dt=1$ with $X(0)=0$ , obtaining $X=t$ metres. The element travels two metres horizontally by the requested time; spatial uniformity does not imply that both velocity components are time independent.
Integrate $dY/dt=t$ with $Y(0)=0$ , obtaining $Y=t^2/2$ metres in the stated units. At two seconds, Y is two metres, so the recorded path is a parabola rather than a line.
Freeze the velocity field at two seconds: every velocity vector is then $(1,2)\,\mathrm{m\,s^{-1}}$ . The streamline slope is two, including at the marker, while the already recorded path contains earlier, shallower directions.

The result

The marker is at (2 m, 2 m). The instantaneous streamline through it has slope two; this tangent agrees locally with the path without making the entire historical path a straight line.

The living notebook

Explore the model

Interactive lab

Interactive lab · fields

Freeze a field before interpreting an arrow

Set source strength q to zero, then vary rotation Ω. Predict that reversing Ω reverses every nonzero arrow while preserving its magnitude at each location. Pause and inspect the arrows as an instantaneous velocity-field illustration; playback does not compute tracer histories.

Local field directionSource strength

Fx at (1, 0): 0.833
Fy at (1, 0): 0.5
Field magnitude there: 0.972

Source strength q1Rotation Ω0.5

t = 0 s

The softened illustration is q·r/(r² + 0.2) + Ω(−y, x). Moving markers indicate local direction, not integrated particle paths or a Maxwell solution. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThis prescribed softened source-plus-rotation field illustrates a spatial description. It does not integrate pathlines or model the time-dependent uniform flow in the worked example.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

At a frozen time, spatial uniformity gives the same direction everywhere, producing straight streamlines when the velocity is nonzero. Time variation can curve particle paths and create material acceleration even without spatial gradients.

Your turn

Try it yourself

A uniform flow has $u_x=2\,\mathrm{m\,s^{-1}}$ and $u_y=3t\,\mathrm{m\,s^{-2}}$ . A marker starts at $(1,0)$ metres at t = 0. Find its location at t = 2 seconds and explain why a streamline drawn only at the final time misses its history.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Integrate each velocity component using its initial position; freeze time only when constructing the final streamline.

Compare with the solution

The coordinates are $X=1+2t$ and $Y=3t^2/2$ , giving $(5,6)$ metres at two seconds. The final velocity is $(2,6)\,\mathrm{m\,s^{-1}}$ , so the final instantaneous streamline has slope three. Earlier streamline slopes were smaller, so that straight line does not represent the marker’s complete path. No force law was needed because the velocity field was prescribed.

Keep the idea

Take it with you

Specify whether a coordinate is a fixed spatial point or a moving material label.
Streamlines are instantaneous geometry; pathlines record a tagged element’s history.
Continuum and smooth-map assumptions delimit the translation between descriptions.

Read further

Original sources and deeper explanations to follow at your own pace.

Optional progress, saved in this browser when available. Revisit any lesson at any time.

Lesson not marked complete.