Streamlines, Streaklines, and Pathlines

Flow Lines: In the Eulerian description, three types of curves are commonly used to describe fluid motion:

Streamline: A streamline is a curve that is instantaneously tangent to the fluid velocity throughout the flow field.
Pathline: A path line is the trajectory of a fluid particle of fixed identity.
Streakline: A streak line is the curve obtained by connecting all the fluid particles that will pass, or have passed through, a fixed point in space.

These are defined and described here assuming that the fluid velocity vector, \(\mathbf{u}\), is known at every point of space and instant of time throughout the region of interest. These curves coincide when the flow is steady and are often valuable for understanding fluid motion.

Streamlines are parallel to the velocity vector. Streamlines are obtained at an instant in time; in an unsteady flow, time \( t \) is held constant in this equation. For 2D flow:

\[ \frac{dy}{dx} \bigg|_{\text{streamline}} = \frac{v_y(x,y)}{v_x(x,y)} \]

For pathlines, describe the position of a specific particle using:

\[ \frac{dx}{dt} \bigg|_{\text{particle}} = v_x(x, y, t), \quad \frac{dy}{dt} \bigg|_{\text{particle}} = v_y(x, y, t) \]

Solving these equations simultaneously gives the trajectory \( x_p(t) \), \( y_p(t) \) of the particle.

Streaklines are determined differently. If a particle is released at \( (x_0, y_0) \) at time \( t_0 \), its pathline is given by:

\[ x_{\text{particle}}(t) = x(t, x_0, y_0, t_0), \quad y_{\text{particle}}(t) = y(t, x_0, y_0, t_0) \]

Instead of interpreting this as the position of a single particle over time, we rewrite this as a streakline:

\[ x_{\text{streakline}}(t) = x(t, x_0, y_0, t_0), \quad y_{\text{streakline}}(t) = y(t, x_0, y_0, t_0) \]

This equation represents the instantaneous positions of all particles released up to time \( t \).