# What force accelerates a liquid moving in a narrowing pipe?

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?

• 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. Jun 14, 2020 at 22:01
• Bernoulli’s Principle on Atomic Scale from "Physics Videos by Eugene Khutoryansky". Jun 16, 2020 at 1:40

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) $$\frac{1}{2}\rho v_1^2 + p_1 = \frac{1}{2}\rho v_2^2 + p_2$$

• Nice answer too. Bernoulli's equation helps to understand how pressure changes. Jun 14, 2020 at 9:52
• @Bernhard actually, I think this is the cleanest explanation for me of why the pressure changes with velocity. Jun 14, 2020 at 16:33
• a tapered pipe is sometimes called a bernoulli transformer. Jun 14, 2020 at 16:36
• @JohnDvorak As long as you are aware what the limitations are of applying Bernoulli's equation. Jun 14, 2020 at 17:27
• But why is the pressure in the narrow and wide parts different? Without that, the 'answer' is just wordplay. 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

• Oh c'est chouette. Jun 15, 2020 at 3:37
• No. The narrowing wall of the pipe applies backpressure. Apr 5, 2022 at 11:25

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 $$p_2 A_2$$.