# What is an example of a quantitative analysis for the lift using Newtonian mechanics?

There are several different ways of explaining how an airplane generates lift. The simpler qualitative explanation uses Newtonian mechanics (the third law is invoked to explain how the wing pushes air down and is in turn pushed up by the air) whereas the more quantitative explanation uses aerodynamics and fluid mechanics.

I have been looking to find a simple model of an airplane wing that uses Newtonian mechanics to come to an approximate quantitative description of airplane flight. Everything I have found so far (such as this article) is merely qualitative.

• There was an article in the EAA Sport Aviation magazine many years ago where the author was explaining lift pretty much like your article in your link, and he calculated the tonnage of air that has to be displaced at x feet per second per second to create enough force to hold a 747 up, and it was a lot, basically a package extending a chord or more below and several chords above that had to be displaced. Trying to find it. – John K Sep 7 at 15:22
• If you understand french, I recommend this article/video. In short, math shows all valid explanations are equivalent. – Manu H Sep 8 at 8:55
• This article on the subject, one of the first discussing the issue, may be of help: booksc.xyz/book/45382205/a4710b – xxavier Sep 8 at 20:15
• @xxavier Good reference. The result from eqn (1), however, is from Lifting Line Theory. Using that, it was able to deduce a relationship between flow turning angle and AOA. – JZYL Sep 9 at 14:30

The governing equations of fluid mechanics, Navier-Stokes equations, are derived based on Newtonian mechanics (plus thermodynamics). The circulation theory of lift is derived using a simplified form of N-S equations. It's powerful because it's predictive for important things like lift and induced drag. The Newtonian explanation is not that useful because it's not predictive. Good for party talks, but poor for engineering.

That being said, if you couple it with the concept of control volume and mass conservation, you can derive something. Take a look at NASA Memo 1-16-59L. To summarize, as air moves past the wing, it gets deflected, so the lift in the vertical direction (normal to the incoming flow direction) is:

$$L=\dot{m}\Delta V$$

where $$\Delta V$$ is the portion of air that was induced across the wing interface in the vertical direction, $$\dot{m}$$ is the total mass flow rate of air through the wing. That's the Newtonian part of the story.

Now from the conservation of mass, what comes in = what goes out. So we can express mass flow rate as (assuming the wing is like a giant circle with diameter equal to its span):

$$\dot{m}=\rho A_iV_{i}=\rho\pi(\frac{b}{2})^2V_{\infty}$$

where $$b$$ is the wing span, $$V_{\infty}$$ is the freestream airspeed, $$\rho$$ is air density.

As for the change in vertical airspeed, we can parametrize it by assuming the whole stream tube gets deflected by the wing through an angle $$\epsilon$$. Therefore, $$\Delta V=V_{\infty}\sin \epsilon$$, and:

$$L=\frac{\pi}{4}b^2\rho V_{\infty}^2 \sin \epsilon$$

That's pretty close to the lift equation that we know. In fact, if you express drag as the loss of horizontal airspeed, apply small angle approximation to the second order, then normalize the lift and drag by $$\frac{1}{2}\rho V^2 S_{ref}$$, you get (with $$S_{ref}$$ being the reference wing area, $$A$$ being the aspect ratio):

$$C_{D}=\frac{C_L^2}{\pi A}$$

That's very close to the induced drag expression that we get from the circulation theory of lift. But that's pretty much as far as you can go. You have no way of estimating $$\epsilon$$ from first principle, so the theory isn't all that predictive.

• Thank you for the answer and resource recommendation. You have said that there is no way of estimating $\epsilon$ from first principles, but could we not assume it is the same as the angle of attack of the wing, in effect assuming that the downwash is parallel to the slope of the wing? – Pancake_Senpai Sep 9 at 9:26
• @Pancake_Senpai We cannot. Assuming that gives you incorrect results. – JZYL Sep 9 at 14:38