Eulerian vs Lagrangian descriptions, streamlines, pathlines, and streaklines
Put an arrow at every point in space showing how fast and which way the fluid is moving there. That field of arrows is all you need to describe a flow.
Switch between field types and change the strength. Longer arrows mean faster flow, and color runs from blue (slow) to red (fast). Hover over any point to read its velocity.
A velocity field assigns a velocity vector to every point in space (and time). Rather than tracking individual molecules, we describe the fluid as a continuum—at each location, we know how fast and in which direction the fluid is moving. This is the foundation of all fluid dynamics analysis.
Each arrow represents the velocity vector at that grid point.
Click anywhere on the canvas to seed a streamline from that point. Small arrowheads along each curve show which way the fluid goes.
Streamlines are curves that are everywhere tangent to the velocity field. The tracer integrates through the velocity field using a 4th-order Runge-Kutta method—the same numerical technique used in professional CFD software.
The defining property of a streamline is that at every point along the curve, the local velocity vector is tangent to the curve. Mathematically, if the streamline is parameterized as (x(s), y(s)), then:
For steady flows (where the velocity field does not change with time), streamlines, pathlines, and streaklines all coincide. For unsteady flows, they differ—which is why the Eulerian and Lagrangian views can look quite different even though they describe the same physics.
Both views show the same vortex drifting slowly to the right. Compare the fixed measuring points on one side with the parcels riding the flow on the other.
There are two fundamental ways to describe fluid motion. The Eulerian approach fixes measurement points in space and records how velocity changes at those fixed locations over time. The Lagrangian approach follows individual fluid parcels as they move through the flow, tracking each parcel's trajectory.
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