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I need help in understanding Bernoulli's principle, namely, the idea of total pressure. Imagine a situation where there is water flow in a divergent pipe. Now imagine two scenarios. The first, we suddenly put a wall in the smaller section of pipe. All the fluid stagnates such that the total pressure is exerted over the wall. Now imagine the second scenario, where we suddenly put a wall in the wider section, and again the flow completely stagnates. Given that total pressure remains constant through the pipe, the pressure on both the small wall and larger wall is the same. So this is where my understanding breaks. Given the smaller wall has smaller area, the force on the smaller wall must be smaller (by P=F/A), and the force on the larger wall must be larger, but where does the extra force that is exerted on the larger wall come from?

My guess is that there isn't actually any extra force, and that it's because the constant total pressure idea works only for streamlines. In reality, won't the pressure exerted on the larger wall be less if all the fluid would stagnate?

See image below for diverging-pipe.

enter image description here

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  • $\begingroup$ divergent pipe?? a figure may help $\endgroup$
    – lineage
    Commented Jun 19, 2021 at 4:17
  • $\begingroup$ @lineage thanks for the suggestion. I've added an image of the pipe in question. $\endgroup$ Commented Jun 19, 2021 at 22:43
  • $\begingroup$ In reality, the pressure exerted on the wall would be less.. which wall larger or smaller? $\endgroup$
    – lineage
    Commented Jun 19, 2021 at 22:55
  • $\begingroup$ @lineage The pressure exerted on the larger wall. $\endgroup$ Commented Jun 19, 2021 at 23:31

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Bernoulli principle can be used to treat stagnant bodies of water as it is based on conservation of energy applied to fluids. Even if the fluid is unmoving wrt. lab., energy conservation still applies - conservation between work done by external force supplying the pressure and the potential energy in the field the fluid is in. This is the reason we have the familiar $P=P_{atm}+\rho g h$ for gauge pressure at fluid depth.

In your case of divergent pipe, I assume your are ignoring potential energy changes along the pipe. For the stagnant case this reduces Bernoulli's eqn. to $$ P=const. $$

So the forces on the two walls must be unequal. The origin of the excess force is the uniformity of fluid pressure through out its bulk, aka. Pascal's law. This stems from the fact that at equilibrium, the net force on a fluid element must be zero. In a way you could say that the agent responsible for pressurizing the pipe thereby providing uniform pressure throughout the stagnant fluid body is also responsible for the excess force. In this regard, your question is, in theory, similar to the working principle of hydraulics. Remember, there is more fluid pushing on a larger wall.

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  • $\begingroup$ @SaxonMilton yours is the stagnant case right? $\endgroup$
    – lineage
    Commented Jun 19, 2021 at 23:41
  • $\begingroup$ yes that's right $\endgroup$ Commented Jun 19, 2021 at 23:44
  • $\begingroup$ @SaxonMilton so there'll be no mass flow rate $\endgroup$
    – lineage
    Commented Jun 19, 2021 at 23:46
  • $\begingroup$ Yes, I understand this, but if we consider the case immediately after insertion of the hypothetical wall, given the immediately prior constant mass flow rate, we can assume at this moment mass would be distributed across this wall as it would have been had it not been there, i.e. less kg/m^2 than in the smaller section of pipe. $\endgroup$ Commented Jun 19, 2021 at 23:49
  • $\begingroup$ @SaxonMilton wait a second - so first of all you are asking what happens to the immediate pressure on the larger wall when its interrupted abruptly during flow? And secondly, even though prior to insertion, this region was at larger pressure than the region of narrower cross-section, should the immediate pressure not drop? And further still drop more than the narrower regions uninterrupted flow pressure? $\endgroup$
    – lineage
    Commented Jun 19, 2021 at 23:55

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