Although I've attached a homework problem here, I'm only using it for reference for a conceptual question on stagnation pressure.
From my understanding, in the photo attached above, the pressure at point 3 and 4 of the U-tube (I've drawn two red arrows pointing at) should have the same pressure since they're at the same height and the fluids in the U-tube system is in equilibrium (if this is not the case, e.g. pressure at left side is greater than that of the right, then the mercury in the right side would exit the U-tube). I was able to prove that the two pressures are equal (I think) by calculating the stagnation pressure at P3 and P4 by the following calculations:
P3=P1 + 1/2 pair v1
P4=P2 + 1/2 pair v2
Equating P3 with P4 yields
P1 + 1/2 pair v1 = P2 + 1/2 pair v2
This gives the same equation as if I calculate P1 and P2 by applying Bernouli's equation to the long wide-narrow tube.
However, my friend just told me that the pressure at P3 and P4 are not the same and that the concept of stagnation pressure cannot be applied the way I did since stagnation pressure only occurs if the fluid's streamline is perpendicular to the surface it is colliding. This is exemplified by Khan Academy's video on Pitot Tube where in the photo I attached below, the pressure in the lower tube (with the mouth facing air flow) is the stagnation pressure; whereas the pressure in the upper tube (with the mouth not facing air flow) is the same as the surrounding pressure.
Therefore, I'm a bit confused as to when exactly can the concept of stagnation pressure be applied? If the way I applied is wrong to deduce P3 and P4 are of equal pressure, then are P3 and P4 of unequal pressure? If so then it appears to be weird since this would imply mercury would be pushed to the left/right of the U-tube due to the pressure difference.