Pathlines

Pathlines are the trajectories traced by individual fluid particles as they move through a flow over time. If the fluid velocity field is denoted by u(x, t), a particle that is at position x0 at time t0 follows a path x(t) that satisfies the differential equation dx/dt = u(x(t), t) with the initial condition x(t0) = x0. The curve x(t) in space, as time advances, is the pathline of that particle.

Pathlines are related but distinct from other flow visualization constructs. A streamline is a curve that is everywhere tangent to the velocity field at a single instant in time; a streakline is the locus of all particles that have passed through a fixed point. In steady flows, where u does not depend on time, pathlines, streamlines, and streaklines coincide. In unsteady flows, they generally differ, reflecting the history-dependent nature of particle motion.

From a modeling perspective, pathlines are a Lagrangian description of the flow, focusing on individual particle histories. They are obtained by numerical integration of the particle equation of motion, or by tracking tracer particles in simulations. In experiments, dye or other tracers can visualize pathlines, revealing transport and mixing patterns in real fluids.

Pathlines require a time-dependent velocity field to be defined, distinguishing them from instantaneous field lines. They provide insight into material transport, dispersion, and the temporal evolution of flows, and are a fundamental tool in fluid dynamics for understanding how substances move within a fluid.