Answering this comment //Nice story but I am afraid that this won't work for three reasons. The first reason is that the law of Bernoulli is only valid for a tube. The funnel which you have created is open at one side so instead of flowing in the direction of the smaller cross section, the flow will simply bend away sidewards.//You forgot that the air will impact the blade with an inclination so it will not spread as the vector force will not impact the surface blade perpendicular. I do not see it going upwards or sidewards the funnel. It is also important that you try to visualize that another reason why the air will not bend bend away sidewards is because there is air pushing to come inside the funnel and I do not see neither the air inside the funnel trying to fly away in the same direction that the incoming air is directing. A small detail is the air impact in the blade surface inside the funnel, which will do it at higher velocity than the air that impact the blade surface outside the funnels.
Yes, Bernoulli's principle can still be applied to an open funnel using air as the fluid, but with some important considerations.
Bernoulli's principle for air:
The principle assumes that the fluid (in this case, air) is incompressible, steady, and the flow is non-turbulent (laminar). Air is a compressible fluid, meaning its density can change with pressure and temperature, which complicates the application of Bernoulli's equation in certain cases. However, for low-speed flow or when the pressure changes are small (i.e., in situations where the compressibility effects are minimal), Bernoulli's principle can still serve as a good approximation.
Applying Bernoulli's principle to an open funnel with air:
Wide end of the funnel: The air moves more slowly here, and its pressure tends to be relatively higher.
Narrow end of the funnel: The air speeds up as it enters the narrower section. According to Bernoulli’s principle, the increase in velocity results in a decrease in pressure at this point.
In situations where the flow is not highly compressible, Bernoulli’s equation can be simplified and applied to the air flowing through the funnel. However, if the air speeds up significantly (such as in a high-speed flow), the compressibility of air becomes more pronounced, and the velocity and pressure relationship may not follow Bernoulli's principle exactly. In this case, more complex models of fluid dynamics would need to be used.
Practical considerations:
If the air is moving slowly (subsonic flow), Bernoulli’s principle provides a reasonable approximation for the relationship between velocity and pressure.
For faster (transonic or supersonic) flows, the changes in air density due to compressibility need to be considered, and Bernoulli’s principle alone would not be sufficient.
Conclusion:
For everyday situations involving low-speed airflow through an open funnel, Bernoulli's principle can be applied as a good approximation. However, if the flow speeds up significantly, especially to the point of approaching the speed of sound, compressibility effects will become more important, and Bernoulli's principle may not apply as directly.