How to explain the Venturi effect with Kinetic Theory?

Question

From a macroscopic perspective a fluid flowing through a pipe gets accelerated when the pipe's cross section gets narrower. According to $F= ma$ a force must be present to do this. This force is usually said to be the pressure gradient force between the wider and narrower cross section: the fluid has a higher pressure behind than in front which results in a net force.

But there is a problem with this macroscopic explanation: There is no pressure difference before there is a velocity difference. You cannot explain how the first fluid particle got initially accelerated because there simply is no pressure difference before it got accelerated. Acceleration needs a foregoing pressure difference and the pressure difference needs a velocity difference.

So, there is no macroscopic explanation of the Venturi effect. On a microscopic level you see that fluid particles actually don't accelerate; it's just that their random movement gets directed and internal kinetic energy gets converted to external kinetic energy.

How do you explain the Venturi effect on a microscopic level with Kinetic Theory? How does it come about that the molecules' random movement gets directed and the molecules hit the walls of the pipe less strongly?

Even if you don't have a definitive answer, give me your best one. :-)

Answer 1

I don't know if it will help you, if it doesn't, sorry for wasting your time. But this is the best explanation I have at this moment.

The gas must be flowing faster because of the lower cross sectional area. Assuming no density change, the only way to maintain the same volumetric flow rate is to increase the flow speed. Now consider the perspective of a gas element flowing through this venturi. As it approaches the restriction, it must accelerate, which requires a force. The only way that a force could be accelerating it though is if the pressure is higher behind it and lower ahead. Thus, if a flow accelerates (without a pump or fan or something adding energy to the flow), the pressure must be decreasing, since the flow must be going from a region of higher pressure to a region of lower pressure. Similarly, once it gets past the restriction, it is decelerating, and thus the pressure ahead must be higher and the pressure behind lower.

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This is all I have. Please let me know if it was helpful. Thanks for posting such an important question.

Answer 2

You said it yourself the molecules have direction rather than randomly moving about. Picture a wide spot in a river, slow water, maybe eddy currents, random water flow. River narrows, water is directed through the slot. I agree with you, speed before pressure differential. What do you think about this, molecules entering Venturi creating vacuum which sucks the following molecules. And maybe this vacuum created by high velocity molecules is creating the pressure differential rather than the force of the wider section pushing the molecules through. That's my two cents.