# Can a steady flow have stagnation points?

I am confused by this idea. If a flow is steady, its streamlines are unchanging. Taking the typical example of an airfoil, there is (at least) one streamline which will hit the leading edge of the airfoil and stagnate. Stagnation point is defined as:

a point in a flow field where the local velocity of the fluid is zero.

Now my question is, if the velocity here is zero, and fluid particles that pass through the streamline which results in a stagnation point have nonzero velocity upstream, where do these fluid particles go? Since they are on the streamline, they have to reach the stagnation point? Does this not defy the conservation of mass law?

Further, an online Fluid Mechanics course I have been following has in one lecture shown streamlines going away from the stagnation point. If the velocity at this point is precisely zero (and is unchanging since the flow is steady), then how can the fluid particles go in other directions?

I must be missing something, do enlighten me.

• A common misunderstanding lies in the fact that air particles are not motionless when we say there is zero-velocity wind (or airflow), otherwise pressure would drop to zero. A stagnation point is an infinitely small volume in which the airflow velocity is zero. This is a mathematical definition model, not exactly the physical reality May 16, 2015 at 9:18
• "fluid particles that pass through the streamline " in a steady flow streamlines are trajectories and parcels cannot cross them. A parcel between two streamlines in this case will always be between those two streamlines. Aug 5, 2015 at 14:41
• No mass crosses streamlines. So the fluid particles are immediately above and below the streamline ending in the stagnation point.
– user7241
Aug 5, 2015 at 17:24

The idea of a stagnation point is an idealization. This point is infinitesimally small, and air particles flowing along a streamline which leads into it will slow down on their way. The closer they come to the stagnation point, the slower they flow, and in the end they never arrive at the stagnation point.

In reality, air molecules have a finite size, so they flow either above or below the stagnation point streamline. Even if one particle manages to arrive at the stagnation point and stays in place (which is not possible in theory), a small angle of attack change will wash it away in the next moment.

One streamline goes away from the rear stagnation point which is located in the trailing edge. This point is doubly idealized, because it requires inviscid flow to have a rear stagnation point. Again, molecules flowing along the wall of the airfoil will be slowed down by the stagnation point pressure the closer they are to the trailing edge. Since they arrive either above or below the rear stagnation point, they will accelerate once they have passed the point of highest pressure and move on above or below the streamline emanating from the rear stagnation point.

The concept of a stagnation point is really helpful for understanding flow phenomena, however. There is really a line (in 3D it is a plane) which separates air that will flow over the wing from that which will flow below it. This line changes with angle of attack, and the simple stall warning vane is based on this principle. Once the line ends below the vane, air will press it upwards, closing an electrical contact which activates a buzzer in the cockpit.

Stall warning vane (small metal thingy sticking out from the wing)

• When you say rear stagnation point, do you mean stagnation points at the trailing edge of the airfoil? Dec 1, 2014 at 19:39
• Also, to confirm what I was asking. Then it is theoretically impossible for a steady flow to have a stagnation point right? Dec 1, 2014 at 19:49
• @midnightBlue: Yes, the rear stagnation point is at the trailing edge. Ideally. If your understanding of stagnation points is that they collect air molecules which will never escape them, then yes, those do not exist in practical life. They are practically impossible, but theoretically they are very much alive (but infinitesimally small). Dec 1, 2014 at 20:12
• Perfect! I think I begin to understand it slightly better now. Do you know what is the sensitivity of the warning vane? (a sudden temporary local change in angle of attack could trigger a stall warning?). Also, is the location and set angle of the warning vane an approximately best selection? (i.e. under certain different conditions, would the warning vane be better placed elsewhere with a different preset angle?) Dec 1, 2014 at 20:34
• @midnightBlue: The location and incidence is the result of testing. You want some margin, but the thing should not constantly go off. The sensitivity is quite good: A gust in slow flight can trigger it, but then you can be sure that the wing at this location is just a few degrees away from stalling, so you better speed up. In a good landing the warning will sound a few seconds before the wheels touch the ground. Dec 1, 2014 at 22:38

A stagnation point in a flow does not defy the conservation of mass law. Peter Kämpf already explained that it is an idealization which helps to understand flow phenomena.

The fact that the velocity is zero at a point along a stream line does not mean that mass is collected at that point. You can compare it to a stop sign on a road. Cars arrive at the sign, stop and pull up again. As long as there is sufficient space between successive cars this can be a continuous process representing a steady flow.

Taking this analogy further, the stagnation point at the leading edge of a 2D profile is like a T junction on a one way road. Cars arriving at the T junction stop and continue with either a left or right turn and follow thier paths. Air molecules arrive at the stagnation point and continue from there either along the top or the bottom of the profile.

Velocity is an instantenous phenomenon. It exists at a point.(lets ignore Average velocity here, it isn't necessary that we discuss it for this question.) At the stagnation point, the velocity is 0. However, at a distance dx from the stagnation point, the fluid particle again gains momentum. It sort of is like a pendulum. At the highest point its velocity is zero, but after a differential time dt, it again regains its velocity and starts to move. You see fluid particles like a pendulum need a stimulus to do so, in the case of pendulum its gravity, for a fluid particle, it can be anything from a pressure force to inertia