Suppose I shoot a jet of water at you from a hose. You feel a force, and that force is due to the change in momentum of the water. e.g. if the water emerges from the hose at a speed v, then stops when it hits you, its momentum changes from mv to zero. Yes?
And lets take m to the mass of water hitting you per second, where m is equal to the volume flow rate (volume per second) times the fluid density. Then the change of momentum per second is mv, so the rate of change of momentum is mv.
And rate of change of momentum is just force. So you feel a force mv.
That's known as the inertial force. Its the force caused whenever the momentum of the fluid changes. In Satwik's question the momentum is not changing so there is no inertial force and therefore no contribution to the pressure from inertial forces.
Then there is no pressure on the walls of the pipe.
With a real fluid there is resistance to flow because you have to push on the fluid to get it to flow through the pipe. That means ina real pipe there is a pressure on the walls of the pipe because you are exerting a pressure on the fluid to make it flow.
So because a fluid like water has a mass, any time it is accelerating that means there has to be a pressure.
When you are doing calculations of fluid flow then in theory you should consider both types of force, but luckily in real life often only one type of force dominates and you can neglect the toher.
As a general rule viscous forces only dominate at very low flow rates and small pipes, so mostly the inertial forces dominates.
The Bernoulli equation is derived using the assumption that inertial force dominate. You'll often see it decribed as being applied to inviscid fluids i.e. fluids where the viscosity can be ignored.
Now, the velocity profile we get in a pipe is determined by the viscosity. It's because if there is a velocity profile then there must be shear present in the pipe i.e. adjacent layers of water move relative to each other so we have a non-zero shear rate $\dot\gamma$.
And the shear rate is related to the shear stress $\sigma$ by $\sigma/\dot\gamma = \eta$, where $\eta$ is the viscosity.
The shear rate is dv/dx where x is the distance measured from the wall of the pipe.
So obviously if the average velocity of the fluid is v there must be a non-zero shear rate because the speed has to change from zero to v.
And this cannot happen instantly because then $\dot\gamma = dv/dx = \infty$ and for any non-zero viscosity that would require an infinite stress.
But for inviscid fluids where the viscosity is small $\dot\gamma$ can be high enough that we can ignore the small amount of water flowing at reduced speeds near the edge of the pipe.
We just assume there are no viscous forces so we can use conservation of energy and therefore we get Bernoulli's equation.
So you are quite correct that strictly speaking this is an approximation for the sort of system you showed, but in many cases it is a very good approximation.
Again this is an approximation. We assume that the vertical dimension of the pipe is small compared to the height of the water surface, so Pa is approximately equal to Pb.
