Daniel Bernoulli and the making of the fluid equation(plus.maths.org)
plus.maths.org
Daniel Bernoulli and the making of the fluid equation
http://plus.maths.org/content/daniel-bernoulli-and-making-fluid-equation/
4 comments
Great article.
I find it quite helpful to have historical context when learning about math and science. Somehow, learning the who/when/how helps immensely in learning the what. For instance, the book A Brief History of Everything by Bill Bryson is very good at presenting things with a context.
I find it quite helpful to have historical context when learning about math and science. Somehow, learning the who/when/how helps immensely in learning the what. For instance, the book A Brief History of Everything by Bill Bryson is very good at presenting things with a context.
This article, while interesting, unfortunately describes the Equal Transit-Time Fallacy [1].
http://en.wikipedia.org/wiki/Lift_(force)#False_explanation_...
Does it? I only see talk of accelerated flow on top of the wing leading to reduced pressure, nothing about arriving at the trailing edge at a certain time.
Right, and for what reason would the flow atop the wing be faster than below?
The equal-time assumption is what leads to the idea that flow on top is faster (and therefore, pressure is lower).
The equal-time assumption is what leads to the idea that flow on top is faster (and therefore, pressure is lower).
It's easier to understand airplane flight with Newton's laws, rather than Bernoulli's principle:
http://www.allstar.fiu.edu/aero/airflylvl3.htm
In particular, the normal hand-wavey arguments that say that air particles must "meet up" after they are separated are total bunk, so that it's not clear that the upper edge actually has a faster airflow. In fact I had a class at Cornell with someone who was studying how dragonfly wings fly, and in addition to momentum-transfer due to steady streams (like airplane wings) you can also have momentum transfer into vortices which are created by those wings.
In general I feel like this article has a lot of social detail and not very much physics. So, for example, someone who doesn't really understand what pressure is will probably be totally unenlightened on why a moving fluid will have a lower pressure. In particular there is a huge caveat to saying that "pressure + kinetic energy = constant", which is that this is a statement valid on a streamline of the fluid. (A streamline is a curve traced by a small group of particles in the fluid as they go downstream -- averaging out their individual jiggles so that we just see how the fluid overall is flowing. If you inject a steady stream of dye then it often shows a streamline, for example.)
If you're not on a streamline then it might not hold. So for example, cut two holes in a bag, stick some tubes on either side, and have a fan blow into one of those tubes: now you've got air moving at a certain velocity inside the bag; yet it is inflated (higher pressure than the stationary air outside the bag) even though it is moving (which you would think would lower the pressure). Turn the fan around and the bag will be inclined to deflate. The air outside the bag is not on a streamline with the air inside the bag.
The transport equation looks like this:
The interesting thing about fluid mechanics, that leads to the Navier-Stokes equations, is that in classical mechanics we have conservation of momentum, which means that the fluid's own momentum in each direction is a stuff subject to this same story. This causes the term (v·∇) c, in particular, to have two instances of v in it (it becomes "nonlinear") and so everything becomes difficult to analyze -- which is why there's a million-dollar prize to make some substantial progress in understanding the resulting equation.
http://www.allstar.fiu.edu/aero/airflylvl3.htm
In particular, the normal hand-wavey arguments that say that air particles must "meet up" after they are separated are total bunk, so that it's not clear that the upper edge actually has a faster airflow. In fact I had a class at Cornell with someone who was studying how dragonfly wings fly, and in addition to momentum-transfer due to steady streams (like airplane wings) you can also have momentum transfer into vortices which are created by those wings.
In general I feel like this article has a lot of social detail and not very much physics. So, for example, someone who doesn't really understand what pressure is will probably be totally unenlightened on why a moving fluid will have a lower pressure. In particular there is a huge caveat to saying that "pressure + kinetic energy = constant", which is that this is a statement valid on a streamline of the fluid. (A streamline is a curve traced by a small group of particles in the fluid as they go downstream -- averaging out their individual jiggles so that we just see how the fluid overall is flowing. If you inject a steady stream of dye then it often shows a streamline, for example.)
If you're not on a streamline then it might not hold. So for example, cut two holes in a bag, stick some tubes on either side, and have a fan blow into one of those tubes: now you've got air moving at a certain velocity inside the bag; yet it is inflated (higher pressure than the stationary air outside the bag) even though it is moving (which you would think would lower the pressure). Turn the fan around and the bag will be inclined to deflate. The air outside the bag is not on a streamline with the air inside the bag.
The transport equation looks like this:
∂c/∂t + (v·∇) c = D ∇² c + Source − Sink
The interesting thing about fluid mechanics, that leads to the Navier-Stokes equations, is that in classical mechanics we have conservation of momentum, which means that the fluid's own momentum in each direction is a stuff subject to this same story. This causes the term (v·∇) c, in particular, to have two instances of v in it (it becomes "nonlinear") and so everything becomes difficult to analyze -- which is why there's a million-dollar prize to make some substantial progress in understanding the resulting equation.
Also, planes can fly upside down. Flaps probably have to be used to maintain flight, but flaps don't seem to significantly change the surface area of either side of the wing.
>>Over the course of three generations, the Bernoullis produced eight mathematically gifted academics who between them contributed to the foundations of applied mathematics and physics[1]
[1] http://en.wikipedia.org/wiki/Bernoulli_family