Zero velocity at a point in a streamline

Question

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?

D. Halsey · Answer

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

Zero velocity at a point in a streamline

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