1.4 Path Lines, Streamlines, and Stream Functions

A path line is a line along which a fluid particle actually travels. Since it is the time history of the position of a fluid particle, it is best described using the Lagrangian description. Since the particle incrementally moves in the direction of the velocity vector, the equation of a path line is given by

the integration being performed with X ₀, Y ₀, and Z ₀ held fixed.

A streamline is defined as a line drawn in the flow at a given instant of time such that the fluid velocity vector at any point on the streamline is tangent to the line at that point. The requirement of tangency means that the streamlines are given by the equation

While in principle the streamlines can be found from equation (1.4.2), it is usually easier to pursue a method utilizing the continuity equation and stream functions described in the following.

A stream surface (or stream sheet) is a collection of adjacent streamlines, providing a surface through which there is no flow. A stream tube is a tube made up of adjoining streamlines.

For steady flows (time-independent), path lines and streamlines coincide. For unsteady flows (time-dependent), path lines and streamlines may differ. Generally path lines are more difficult to find analytically than are streamlines, and they are of less use in practical applications.

The continuity equation imposes a restriction on the velocity components. It is...