

Daniel Bernoulli and the making of the fluid equation - Mz
http://plus.maths.org/content/daniel-bernoulli-and-making-fluid-equation/

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marktangotango
The Bernoulli family did a lot of stuff, very interesting how they seemed to
be blessed with mathematical talent:

>>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](http://en.wikipedia.org/wiki/Bernoulli_family)

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lordnacho
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.

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crxpandion
This article, while interesting, unfortunately describes the Equal Transit-
Time Fallacy [1].
[http://en.wikipedia.org/wiki/Lift_(force)#False_explanation_...](http://en.wikipedia.org/wiki/Lift_\(force\)#False_explanation_based_on_equal_transit-
time)

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bewuethr
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.

~~~
shaldengeki
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).

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drostie
It's easier to understand airplane flight with Newton's laws, rather than
Bernoulli's principle:

[http://www.allstar.fiu.edu/aero/airflylvl3.htm](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
    

This equation tells a story: "there is a box moving downstream (v·∇) which
contains some stuff, with a certain concentration (c). The time-rate-of-change
(∂/∂t) of stuff in the box is equal to the stuff moving into the box, minus
the stuff moving out of the box. Some stuff is moving into adjacent boxes
proportional to the concentration difference, in a process called diffusion
(D); everything else is somehow going out of the scope of the fluid flow
(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.

~~~
cma
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.

