Flow Is a Velocity Field; Streamlines Follow It
Lay a grid over flowing water and draw, at each point, an arrow for which way and how fast the water moves there. This is the velocity field. Instead of chasing one droplet all the way downstream, you pin a camera at each point in space and read the speed of whatever water passes through it. This viewpoint is called the Eulerian view. Drag the probe around and you see the arrow's direction and length change from place to place. And if you keep following these arrows and join them up, you trace a single smooth curve: a streamline. A streamline shows the path the water is taking at that instant. It is the first step in reading a flow as a picture.
First, get a feel for the velocity field itself. Each arrow on the grid is the velocity of the water passing through that spot: the direction is where the water goes, the length is how fast. Drag the probe sideways. At some spots the arrow tilts up, at others down. You are not following the fate of one droplet, but seeing the whole distribution of velocity that fills the space at once. Think of a weather map with little wind arrows: that map is a velocity field, and fluid mechanics begins with reading one.
Now let us follow those arrows. Imagine dropping a tiny seed into the water. The seed moves a little in the direction the arrow under it points, then at its new spot follows that spot's arrow, and repeats this endlessly. Its trail is a streamline. Drag the seed's height to change the starting point, and each start draws a different streamline. If the arrows are dotted guidance, a streamline is the smooth road drawn by joining the dots. Float a leaf on a river and the line it traces is exactly a streamline.
There are two kinds of flow: steady and unsteady. In steady flow, the velocity at each point does not change over time. Watch one spot in a river: the water keeps flowing, but the speed and direction at that spot stay the same. So the streamline picture stays put as time passes. In unsteady flow, the velocity at a point changes with time, like a choppy sea. Toggle between the two. In steady flow the streamlines hold one fixed shape; in unsteady flow the streamlines a moment later no longer match the ones now, and the two drift apart. Most of the formulas ahead assume the easier, steady case.
A streamline has one decisive property: at any point on it, the velocity points exactly along the streamline's tangent. Of course it does, since the streamline was drawn by following the arrows in the first place. Drag the point along the streamline and you see the velocity arrow always hugging the curve, tangent to it. Here is the key consequence: no water ever flows across a streamline. Since the velocity only runs along it, there is no component crossing it. So a streamline acts like an invisible wall. This property is decisive in the next lesson, when we count the flow passing through a cross-section.
Finally, look at a river. Where the river is wide the streamlines spread out loosely, but at a narrows where the channel pinches, the streamlines crowd tightly together. Drag the throat width to squeeze the river. The narrower it gets, the denser the streamlines become. Densely packed streamlines mean the water there flows fast: the same amount of water forced through a narrow gap has no choice but to speed up. This is why the current quickens at a narrow rapid. The spacing of streamlines is a map of speed, and this intuition becomes an exact formula in the next lesson, the continuity equation.