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Given a streamline, is it possible that the velocity at some point on it is non-zero but then at some other point(for example, stagnation point) it is zero? I mean does that make sense?

My existing knowledge tells me that a streamline is basically defined by the path traced by a fluid particle(maybe kind of a really very small drop of liquid but consisting of considerably large no. of molecules).Now a point in the streamline cannot be occupied by a single fluid particle forever. It must be replaced by a particle coming from behind. Now if the velocity at a point in the streamline is zero, but still the particle coming from behind must replace the already existing particle, then where does this particle(already existing) go?

Someone please clear my doubt? Exactly where am I doing it wrong?

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    $\begingroup$ Kind of seems to me that you're assuming that the velocity being $0$ means the particle won't move. If I throw a ball in the air, and you consider the velocity at each point of the trajectory, it will have $0$ velocity at the highest point, but it won't stay there, right? It's instantaneous velocity, doesn't have such strong implications by itself. Did I understand your doubt correctly? $\endgroup$
    – Paulo Mourão
    Jul 28, 2020 at 20:20
  • $\begingroup$ Yes you did... and yes your argument is correct, but here I am trying to think where does this fluid particle go unlike the ball, because the trajectory(streamline) is shown to have ended at that particular point(the stagnation point), there's no furthur continuation. So what does that really mean? $\endgroup$
    – user266637
    Jul 28, 2020 at 20:25
  • $\begingroup$ There is continuation though, right, stopping at that point doesnt mean it doesnt continue. Otherwise, you'd have perpetuous accumulation of fluid at that point. I must say I've never gotten much into fluid dynamics but a quick search for stagnation point just tells me that it's where the "local velocity of the fluid is $0$". Wouldn't it be exactly like the ball? $\endgroup$
    – Paulo Mourão
    Jul 28, 2020 at 20:31
  • $\begingroup$ Imagine shooting water in the air. Wouldnt the highest point be an example of a stagnation point? $\endgroup$
    – Paulo Mourão
    Jul 28, 2020 at 20:33
  • $\begingroup$ You are right, absolutely right, but where does it exactly go...? Can you tell me the direction of continuation? I have no idea... And yes, I must tell you that this is just the beginning of my study of fluid mechanics. Cause I am a high school student, so you can understand that I too don't know too much. 😅 $\endgroup$
    – user266637
    Jul 28, 2020 at 20:39

1 Answer 1

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The short answer is that the flow coming into a stagnation point gets pushed to the side, rather than piling up against the fluid ahead of it.

The simplest way to see this mathematically is via the Cauchy-Riemann equations,which were derived from complex-analysis and apply for 2-D functions satisfying Laplace's equation (including inviscid flows in fluid dynamics).

\begin{aligned} &{\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}\\[6pt]&{\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}\end{aligned}

In these equations, x and y are the coordinate directions, and u and v are the corresponding components of the velocity vector in those directions.

The first equation, in particular, shows that as the x component reduces, the y component also changes (i.e. - the flow is diverted to the side.)

For a more detailed explanation, see https://en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations

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  • $\begingroup$ Thanks Sir... 😊 $\endgroup$
    – user266637
    Jul 30, 2020 at 14:56

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