streamlines and velocity field
Lay a grid over flowing water and draw, at each point, an arrow for which way the water goes and how fast. That is a velocity field. Instead of chasing one drop to the end, you stick a camera at each place in space and read the speed of the water passing there. That view is the Eulerian view. Drag the probe and you see the direction and the length of the arrow change from place to place. Keep following those arrows and joining them and one smooth curve is drawn. That is a streamline. A streamline shows the path the water is taking at that instant, and it is the first step in reading a flow as a picture. The length of an arrow is the speed and the direction is the path.
First feel the velocity field itself. Each arrow of the grid is the velocity of the water passing that place. The direction is the way the water goes and the length is the speed. Drag the probe sideways and in some places the arrow leans up and in some places it leans down. You are not following the fate of one drop. You are seeing the whole distribution of velocity that fills the space, and a weather map that draws the wind as arrows is the picture to hold. That map is the velocity field, and fluid mechanics starts by reading this field. Arrows that differ from place to place make one map.
Now follow those arrows. Drop one small seed into the water and the seed goes a little the way the arrow under its feet points, and at the new place it follows that place's arrow again, and it repeats this without end. That trace is a streamline. Drag the height of the seed to change the starting point and each starting point draws a different streamline. If the arrows are a dotted guide, the streamline is the smooth path drawn by joining those dots, and a leaf floating on a river draws a line that is exactly a streamline. A different start makes a different path, and the water follows that path.
A flow is one of two kinds, steady or unsteady. A steady flow is one whose velocity at each point does not change as time passes. Look at one place in a river and the water keeps flowing, but the speed and the direction at that place stay the same, and the streamline picture stays as time passes. An unsteady flow has a velocity at a point that changes with time, and a wavy sea is like that. Switch the two with a button and in a steady flow the streamlines stay fixed in one shape, while in an unsteady flow the streamlines a little later differ from now and slip. The formulas ahead mostly assume a steady flow, which is easier to handle. A fixed picture is a steady flow and a shifting picture is an unsteady flow.
A streamline has one decisive property. At every point on a streamline the velocity points exactly along the tangent of the streamline, which is natural because a streamline was drawn by following the arrows in the first place. Drag a point along the streamline and you see the velocity arrow always lying against the curve, touching it. One important conclusion comes out of this. No water flows across a streamline. The velocity only follows the streamline, so there is no part that crosses it, and a streamline works like an invisible wall. This property is decisive when you count the flow rate through a cross section. An arrow that touches the curve keeps the water inside the path.
Last, look at a river. Where the river is wide the streamlines spread loosely, and at a narrow stretch they press close to each other. Drag the throat width to squeeze the river and the narrower it gets the denser the streamlines become. Dense streamlines mean the water flows fast there, because the same amount of water has to pass a narrow path and so it has no choice but to go faster. This is why the current grows strong in a narrow rapid, and the spacing of the streamlines is a map of speed. This intuition becomes an exact formula in the continuity equation. A dense line is fast water and a loose line is slow water.