Streamlines, Streakline, Pathline

Streamline

A streamline is a line that is tangent to the velocity vectors throughout the flow field.

For steady flows (nothing changes with time at a given location in the flow field), streamlines are trajectories of fluid particles.

Note that we assume inviscid fluids/flows, which have zero viscosity:

$$\tau =0\mu =0\text{or}\frac{du}{dy}=0$$

The angle can be found with:

$$\frac{dy}{dx}=\tan \theta =\frac{v}{u}$$

or

$$y=\int \frac{v}{u}dx$$

Example

Let’s say we have a velocity field $\mathbf{V}=\frac{V_0}{l}(x\mathbf{i}-y\mathbf{j})$.

• Get the location where $V=V_0$.
• Note that the two components of $V$ are $u=\frac{V_{0}x}{l},v=-\frac{V_{0}y}{l}$
• Then:

$$\begin{array}{rl}\frac{dy}{dx}& =-\frac{y}{x}\\ \frac{dy}{y}& =-\frac{dx}{x}\\ \int \frac{dy}{y}& =-\int \frac{dx}{x}\\ \ln y& =-\ln x+c\\ xy& =c\end{array}$$

Streakline

A streakline consists of all particles in a flow that have previously passed through a common point.

• Time dependent?

Pathline

A pathline is the trajectory of a given fluid particle

Differences between lines

• For steady flows, the three lines are the same
• For unsteady flows, they can be very different