pathline

A pathline is the trajectory that a single fluid particle traces through space as it moves with the flow. If the velocity field of the fluid is v(x, t), then the pathline x(t) is defined by the ordinary differential equation dx/dt = v(x(t), t) with a specified initial position x(t0) = x0. The solution x(t) gives the particle’s position at time t.

In steady flow, where the velocity field is time-independent, pathlines coincide with streamlines, which are curves tangent to the velocity field at a given instant. In unsteady flows, pathlines generally differ from streamlines; a streakline (the locus of fluid that has passed through a fixed point) is also distinct, and all three concepts can diverge in general.

Pathlines are often visualized by seeding the flow with tracer particles or dyes and tracking their motion, or by integrating particle trajectories in numerical simulations (such as computational fluid dynamics) using the velocity field. When only discrete velocity data are available, interpolation and numerical integration methods (for example, Runge-Kutta) are used to construct pathlines. The choice of initial time t0 and initial position x0 determines the computed pathline; in unsteady flows different starting times yield different pathlines for the same region.

Pathlines are used to study transport and mixing in environmental flows, atmospheric and oceanic currents, pollutant dispersion, blood flow in hemodynamics, and various industrial processes. They provide a Lagrangian perspective complementary to the Eulerian view provided by velocity fields and streamlines.