Understanding what a stagnation point is in compressible flow

[boneh3ad]

That wasn't even the original question. The original question was whether the flow velocity had to be zero in all directions for it to be a stagnation point. It does.

 

[Ken G]

The real flow has the appearance of a coordinated stream that moves with uniform velocity until it encounters the plate, whereon it curves round and spreads out with radial symmetry about a centreline and flows at sensibly right angles to its original direction. The solution also contains straight lines AD and BC along the axes, meeting at right angles at S in a sharp corner. These are continuous everywhere and have a defined derivative nearly everywhere. But as S the derivative is indeterminate or undefined.

No, that is not correct. The derivative is well defined there, because the entire flow (in the fluid description we are using) is determined by a velocity field. At the stagnation point (in compressible or incompressible flows), the velocity field is zero (in all directions, as boneh3ad and others stressed), and also has zero divergence. Not undefined divergence, zero divergence. This is also why it takes an infinite time to reach the stagnation point, and why there is no "train pile up" there, and no mathematical difficulties for the idealized fluid model.

But there is a further twist, introduced by my_wan. And this is Physics rather than Maths.
What is the mechanism by which momentum in the AD direction can be turned into momentum in the SB and SC direction?

Momentum in one direction is never "turned into" momentum in another direction. Momentum is a vector, and its components obey the laws of physics entirely independently of each other. To stop a momentum toward the wall, you need a force away from the wall. To start a momentum along the wall, you need a force along the wall. Pressure forces do all this just fine, even at stagnation points. There are no mathematical difficulties presented by stagnation points in the fluid description, and Newton's laws encounter no singularities there. I do not see why you are framing any of this as "twists" or "wrinkles", it is all basic fluid mechanics. But the diagrams and the scenario you are describing may indeed be useful to the OPer, if they simply read past all the intimations of dire singularities here!