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A stagnation streamline is a streamline that ends on the surface of an object, resulting in a stagnation point. Consider now the stagnation streamline for a symmetric object, which is along the centerline as shown below. Fluid will keep on entering from the left, so I assume that at some point the fluid has to escape to either side of the object (up or down). But at the centerline we know that $v=0$. So how is this streamline physical if it does not satisfy continuity?

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My initial guess is that the streamline is infinitely thin and therefore nothing is building up, however I would appreciate any additional explanation or links to resources :)

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A fluid is approximated as a continuous medium. That works on a macroscopic scale, but it is composed of molecules. So you can't push the idea of continuity to a mathematical limit. You break the approximation.

So there is a small region at the stagnation point where molecules pile up and bounce up or down.

Never the less, people do use the mathematical limit to describe a perfectly continuous fluid. This behaves very much like what a real fluid behaves like. In that approximation, you are right. The streamline is infinitely thin and the volume of the stagnation point is $0$.

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To add to mmesser314's answer: the stagnation streamline ends at a stagnation point in the flow. at that point, a parcel of air in the free stream has been brought to a halt, and its kinetic energy of motion is converted into potential energy of compression and the pressure at that point rises to the stagnation pressure.

Mathematically, the stagnation point occupies zero volume, but this is an idealization. In reality, there will be a small volume where the flow velocity falls to a low value, rather than there being a zero volume at zero velocity.

That increased pressure urges the air parcel in the stagnation zone to find a way to leak out of the zone and escape, rather than being carried for example all the way across the Atlantic Ocean by the leading edge of the wing of a Boeing Something-or-other series 800.

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Do you know the antique paradoxon of the race between Achilles and the turtle? The turtle gets a head start. When the race starts, Achilles quickly catches up to the position where the turtle started, but by that time the turtle has moved a bit ahead. Repeat, and again the turtle is ahead. Ad infinitum, so Achilles will never win against the turtle.

In reality, the time slices get infinitely small and the time in that paradoxon only approaches the point where Achilles overtakes the turtle. In reality, time does not stop but moves at a constant pace.

The stagnation streamline is similar. In theory, the fluid in that streamline will slow down util it reaches a complete stop at the stagnation point.

In reality, no single molecule of the fluid will stay exactly in the stagnation streamline, but will jiggle around, bouncing into other molecules and be bounced around by them. So it should be no surprise that no molecule will stay forever in the stagnation point.

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