Eulerian vs Lagrangian descriptions, streamlines, pathlines, and streaklines
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.
Below, explore several canonical velocity fields. Notice how arrow length encodes speed and arrow direction encodes the local flow direction.
Each arrow represents the velocity vector at that grid point. Arrow color maps from blue (slow) to red (fast). Try switching between field types and adjusting the strength to see how the flow patterns change. Hover over any point to inspect the exact velocity components.
Streamlines are curves that are everywhere tangent to the velocity field. Click anywhere on the canvas to seed a streamline from that point. The tracer integrates through the velocity field using a 4th-order Runge-Kutta method—the same numerical technique used in professional CFD software. Small arrowheads along each curve indicate the flow direction.
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. Both views below show the same time-varying flow: a vortex that drifts slowly to the right.
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.