Say there is a liquid which behaves like an incompressible fluid and is flowing steadily through a pipe which is moving from a cross section of area $A_1$ to the cross section of area $A_2$, where $A_2$ is less than $A_1$. As per the continuity equation, $v_2>v_1$ and so the liquid seems to be accelerating. What force is causing this acceleration?
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1$\begingroup$ I've removed a number of comments that were attempting to answer the question and/or responses to them. Please keep in mind that comments should be used for suggesting improvements and requesting clarification on the question, not for answering. $\endgroup$– David ZJun 14, 2020 at 22:01
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2$\begingroup$ Bernoulli’s Principle on Atomic Scale from "Physics Videos by Eugene Khutoryansky". $\endgroup$– NayukiJun 16, 2020 at 1:40
3 Answers
You are right. From continuity of the incompressible fluid you have $$A_1 v_1 = A_2 v_2.$$ So obviously the velocity is changing. Thus the fluid is accelerated, and therefore there must be a force causing this acceleration. In this case the force comes from the pressure difference between the wide and the narrow part of the pipe.
(image from ResearchGate - Diagram of the Bernoulli principle)
This can be described by Bernoulli's equation ($p$ is pressure, $\rho$ is density) $$\frac{1}{2}\rho v_1^2 + p_1 = \frac{1}{2}\rho v_2^2 + p_2$$
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$\begingroup$ Nice answer too. Bernoulli's equation helps to understand how pressure changes. $\endgroup$– BernhardJun 14, 2020 at 9:52
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3$\begingroup$ @Bernhard actually, I think this is the cleanest explanation for me of why the pressure changes with velocity. $\endgroup$ Jun 14, 2020 at 16:33
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$\begingroup$ a tapered pipe is sometimes called a bernoulli transformer. $\endgroup$ Jun 14, 2020 at 16:36
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$\begingroup$ @JohnDvorak As long as you are aware what the limitations are of applying Bernoulli's equation. $\endgroup$– BernhardJun 14, 2020 at 17:27
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$\begingroup$ But why is the pressure in the narrow and wide parts different? Without that, the 'answer' is just wordplay. $\endgroup$ Nov 25, 2021 at 13:11
There is more mass per area behind than ahead of the constriction so since ppressure is force divided by area there develops a pressure difference
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$\begingroup$ No. The narrowing wall of the pipe applies backpressure. $\endgroup$ Apr 5, 2022 at 11:25
The simple answer is pressure.
As you state, from the continuity equation you can see that the velocity $v_2>v_1$. The next step is a momentum balance (like any balance in fluid dynamics: $\frac{d}{dt}=in-out+production$). The momentum flowing into the system is smaller than the momentum flowing out of the system ($\rho A_1 v_1^2 < \rho A_2 v_2^2 = \rho A_1 v_1^2 \frac{A_1}{A_2}$, $\frac{A_1}{A_2}>1$).
The actual force is not the pressure itself, but the pressure difference, or, actually, the difference in force, because on the left the force is $p_1 A_1$ and on the right $p_2 A_2$.
From another conceptual point. Suppose you have a garden hose. Than you have a fixed pressure drop. Now, if you squeeze it, you create a contraction. Part of the pressure drop is now needed to accelerate the fluid at the exit. The overall flowrate decreases, because you also need pressure to overcome frictional forces.