Weglinien

Weglinien refer to the pathlines of fluid motion: the trajectories followed by individual fluid particles as time advances within a velocity field. In mathematical terms, a Weglinie is the curve x(t) that satisfies the differential equation dx/dt = v(x(t), t) with an initial position x(t0) = x0. This concept is part of the Lagrangian description of fluid flow, contrasting with the Eulerian viewpoint that describes the velocity field v(x,t) at fixed points in space.

In steady (time-independent) flows, Weglinien coincide with streamlines, which are lines tangent to the velocity vector at a given instant. In unsteady flows, Weglinien, streamlines, and streaklines generally differ. A streakline is the locus of all particles that have passed through a specific point, whereas a Weglinie follows a single particle through time.

Weglinien are commonly visualized by introducing tracer particles or dyes into the flow or by reconstructing particle trajectories from velocity field data. Techniques such as particle image velocimetry (PIV) provide velocity fields that can be used to compute approximate Weglinien, helping to illustrate how fluid parcels move.

Applications of Weglinien span engineering and geophysical contexts, including the analysis of mixing and transport in pipes and channels, pollutant dispersion in air or water, and nutrient transport in oceans. Understanding Weglinien aids in predicting transport phenomena, assessing safety margins in systems involving fluid flow, and informing design choices that influence mixing and separation processes.