The field lines of the velocity vector field at a particular instant of time are called streamlines. The streamlines are visualised by taking photographs of the fluid. If the velocity field changes with time, then the streamlines will change also, but remains the same for steady flows.
Streamlines have the following properties:
At any instant, no field crosses a streamline. This follows because at any point on a streamline, the velocity vector is perpendicular to
the normal vector to the streamline. Hence
a measure of the flow rate of the fluid across a streamline, is zero
At any instant of time, distinct streamlines cannot cross. Suppose two streamlines cross at a point A. At each point of a streamline the velocity vector is parallel to the streamline tangent, so the velocity vector at A has two different directions, so the velocity field is not uniquely defined, obviously a contradiction.
If the streamline has equation
and velocity
then
or
We can then find the streamlines by integration.
Example: Find the streamlines for the vector field
The streamlines are parabolae. If
and
then
Example: Find the streamline for the velocity vector field
which integrates to give