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I get that with Bernoulli's Principle an increase in area = a decrease in velocity = an increase in pressure. I also understand the concept of pressure being force per unit area.

My difficulty is with understanding what is meant by pressure in this context. Is it the pressure exerted on the fluid from the pipe's walls (with an equal pressure exerted by the fluid on the walls)? Is it something else?

If it is the pressure exerted on the fluid by the walls of the pipe, why does it decrease when cross-sectional area decreases? I can see why from a mathematical perspective, that it occurs from how the formulae can be rearranged, but to me that just doesn't intuitively make sense. I just can't viscerally understand it, and a lot of how I understand things comes from visualising it, which YouTube videos unfortunately have not helped me with (as they explain how the formulae work, and I can see the decreasing area which in my head means pressure should increase [even though I can recognise it does not]).

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  • $\begingroup$ I just wanted to say that I think this is a GREAT question, and I pretty much feel exactly the same. All this "blah blah energy conservation blah blah potential energy converted to kinetic energy" just isn't doing anything for me. I guess we don't understand less than most other people do, we just need a deeper level of understanding to be satisfied. $\endgroup$ – MaxD Feb 14 at 1:04
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The static pressure is the compressive longitudinal stress exerted by the pipe walls on the fluid and vice versa, but perhaps more importantly in this context, it is also the compressive longitudinal stress exerted on every little parcel of fluid by the parcels of fluid next to it, in every direction. If the pressure didn't decrease along the length of the pipe, then a parcel of fluid (call it parcel $\mathsf{A}$) would be subjected to the same pressure by the parcel of fluid in front of it (resulting in a backward force on parcel $\mathsf{A}$) as by the parcel of fluid behind it (resulting in a forward force on parcel $\mathsf{A}$); hence, there would be no net force on parcel $\mathsf{A}$, and parcel $\mathsf{A}$ would not be able to accelerate, as it must do to achieve the increasing velocity needed to carry the (constant, by conservation of volume) flow rate through a decreasing cross-sectional area. The actual situation, of pressure decreasing in the forward direction along the pipe, means that the forward force on parcel $\mathsf{A}$ is smaller than the backward force on parcel $\mathsf{A}$, there is a net force, and the necessary acceleration can happen. Bernoulli's equation is nothing more or less than Newton's second law relating the pressure variation to the acceleration for a steady, incompressible, inviscid flow.

When the flow first starts up from stationary (an unsteady flow that is outside the scope of Bernoulli's equation), there is a brief period when the flow rate isn't constant along the pipe, so that fluid builds up in some cross-sections and is evacuated from others, which has to be accommodated by the development of a pattern of (small) circumferential strains in the material of the pipe walls, with that pattern becoming steady when the flow rate is constant along the pipe, i.e. when those circumferential strains are just right to sustain the pressure variation implied by Bernoulli's equation and conservation of volume.

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  • $\begingroup$ Ok from your answer I get that Static Pressure is both the pressure exerted on the fluid by the pipe walls (with corresponding force exerted on the walls by the fluid), and that it is the pressure exerted by every particle of fluid on every adjacent particle of fluid. I also understand that the downstream pressure must decrease in order for the required velocity increase. I accept the forward pressure drop is required for the velocity increase, but I cannot tell what causes this pressure drop, as intuitively a smaller cross-sectional area seems like it should increase pressure? $\endgroup$ – Dale117 Feb 13 at 14:34
  • $\begingroup$ @Dale117 What experiences do you think led you to develop that intuition? (Not an idle question: I can think of three main possibilities, each requiring a different explanation). $\endgroup$ – Daniel Hatton Feb 13 at 14:58
  • $\begingroup$ Just based off P = F/A, with A decreasing leading to a higher pressure, and I cannot see any decreasing force. $\endgroup$ – Dale117 Feb 13 at 15:09
  • $\begingroup$ @Dale117 Make that four main possibilities, then. There is indeed a decreasing force, since each transverse "slice" of fluid must accelerate (increase its velocity) to maintain conservation of volume, so there must be a net force in the forward direction on each slice of fluid, i.e. there must be a smaller compressive longitudinal force on the front of the slice than on the back. $\endgroup$ – Daniel Hatton Feb 13 at 15:36
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It might help to consider the special case of Hatton's excellent answer, where the velocity is very small. A small piston at one end of a tube, then an expansion to a large piston at the other end, is an idealised hydraulic jack. When not moving it is considered to have the same pressure at each end, so can support a car at the large end with a small force on the small piston. A slow motion of the small piston then creates enough deceleration of the fluid in the expansion to increase the pressure at the large end and accelerate the car.

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Let's start with a few clarifying points:

  • for INCOMPRESSIBLE flow an increase in area = a decrease in velocity = an increase in (static) pressure (this is simply the continuity equation mass flow rate=density * velocity * area in which density is assumed constant)
  • This pressure here is the pressure exerted by the fluid on a cross-section normal to the flow. In the case of a circular pipe, this is a circular cross sectional area (the same area used in the continuity equation).
  • The pressure distribution on the walls is more complicated due to boundary layer effects and the definition of pressure. Of course, there is a force exerted on the pipe wall that is met by an opposite force from the fluid to maintain a force balance. There are also forces acting along the flow direction (shear) that complicate matters further. This question is related: Why is the direction of pressure always perpendicular to surface area for fluids? Does this help any?
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