why do airplanes fly?
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Then what makes an airplane fly? 
The wing pushes air down; Newton's third law pushes the wing up. Bernoulli's law has little or nothing to do with it.
 Warren 
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AM 
Indeed Andrew, Newton's third law is not the only important part here. Bernouilli's law tells us which structure the wings has to be
marlon 
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What you have is air being deflected up by the leading edge. The air that would be sitting on top of the wing as the wing moves is, therefore, moving upward above the wing, leaving a partial vacuum above the trailing edge of the wing. The result of all this, is downward movement of air because the air underneath is pushing up on the wing, the air has to move down. But the mechanism is a little more subtle than the wing just pushing the air down (although that is part of it  angle of attack). AM 
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So, Bernoulli's does explain part of it, is just not the whole story. Also, Newton's 3rd is more an effect than a cause or an explanation. In a high angle of attack situation, its easy to see why air gets directed down, but that doesn't explain how you can get lift at zero geometric aoa. 
I will add a few comments here, but first to PBRMEASAP:
You made a few important remarks concerning irrotational flow in the other thread, I'll hope to get back to those later on. I will focus here on the (in 2D) TWO integrated expressions we can make of Newton's 2. law related to streamlines, in the stationary case: 1. The quantity which is conserved ALONG the streamline (i.e, what is given in Bernoulli's equation. 2. The integral of Newton's 2. law ACROSS the streamlines (Crocco's theorem) Since the "stationary" case is only possible in the wing's rest frame, my comments will use this as the frame of reference henceforth (note that in the ground frame, in which the fluid is at rest in infinity, the timedependent position of the wing will mean that the equivalent velocity field is timedependent, according to the coordinate transformation given by Galilean relativity.) But first, a few comments on chroot's post: chroot gives an absolutely correct description of a flight situation, in that if the net effect on the air from the wing is a downwards deflection of the air, then by Newton's 3.law the air must impart an upwards force on the wing, i.e, lift. However, I tend to regard this analysis as a GLOBAL analysis, in that it looks at a control volume of air surrounding the wing and calculates the net momentum flux out of that control volume. This is, of course, both a permissible and intelligent way of viewing the problem, but what I would like to proceed with here, is what I call a LOCAL analysis, i.e, directly relating the air's acceleration in the vicinity of the wing and the forces acting upon it. That is, Newton's 2.law locally applied on the wing. I'll post more a bit later. 
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Thanks everyone for your posts. Keep 'em coming :). 
I always marveled at how massive an airplane is and yet still get off the ground gracefully.

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Regards, Nenad 
I will proceed with a local analysis of the (inviscid) flow over the wing, and its relation to lift.
For the present purposes, I assume that the fluid leaves the trailing edge in a smooth, tangential manner (apart from the formation of a thin wake region, this is what happens in reality, and in inviscid theory is known as the Kutta hypothesis). Let us glance at the result from chroot's global analysis: This relates the net downwards deflection of the fluid with the lift force. But, if the fluid velocity upstream was strictly horizontal, that means that the fluid necessarily have experienced CENTRIPETAL acceleration, i.e, the streamlines must become CURVED when passing about the wing. Locally speaking, the necessity of the presence of centripetal acceleration is a "trivial" insight, since the wing itself is curved.. But, those forces causing a particle's trajectory to curve, rather than accelerate the particle along a straight line, are the forces ortogonal to the trajectory, rather than the forces tangential to the trajectory. In the case of the inviscid fluid where we neglect gravity, the force directly related to the curvation of the streamlines is given by the component of the pressure gradient normal to the streamlines, rather than the tangential component of the pressure gradient. Furthermore, since by global analysis we may conclude that streamlines MUST curve in order for us to have any lift at all, it follows that the component of the pressure gradient most directly relevant for flight is the normal component, rather than the tangential component. But, Bernoulli's equation essentially relates pressure values as given by the tangential component of the gradient (i.e, through the formation of the dot product between the pressure gradient and the streamline tangent, and then integrating). From the above, it should seem more natural to fix our attention first upon the insights from Crocco's theorem, rather than upon Bernoulli's equation. 
Now, let's see how the presence of lift is plausible when considering Crocco's theorem, and typical airfoil shapes.
I'll get back to symmetrical wing shapes with an effective angle of attack later. Now, let our first airfoil consist of a horizontal underside, and a curved form on the upper side, and let gravity be negligible: We also assume that if we either go infinitely far from the wing horizontally or vertically, we end up in the uniform freestream with constant pressure. 1. Vertical pressure distribution beneath the wing: Since the underside is basically horizontal, we may assume that the streamlines underneath are practically straight horizontal lines (as they are in infinity), that is, particles passing beneath the wing don't experience any centripetal acceleration to speak of. But that means, that the normal component of the pressure gradient on the underside is zero, i.e, a measure of the pressure directly beneath the wing is the freestream pressure to be found at (vertical) infinity. 2. Vertical pressure distribution above the wing: By assuming the typical negative curvature of the top foil, the pressure must increase upwards from the wing in order for the fluid to traverse the curve as determined by the wing. Extending that increase up to infinity in the vertical direction, we may conclude that the typical pressure at the upper foil must be LOWER than the freestream pressure. But, combining 1+2 indicates the presence of lift.. Now, we may invoke Bernoulli: Knowing that the pressure is typically lower on the upper side than the lower side, the measure of the velocity at the top of the foil must be greater than the measure of the velocity at the downside, i.e, we have a net CIRCULATION about the wing. The relation between lift and circulation is known as KuttaJakowski's theorem. Note that the "increase" of velocity at the top foil relative to the underside is consistent with the presence of a stronger centripetal acceleration on the upper side. 
Not to pick on you, warren, but...
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While Newton's laws can be used to calculate the net quantity of lift, they don't describe the airflow over the wing itself. 
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Of course, you arrived at this result from a different angle that is very enlightening. It just seems to me that Jukowski's theorem and Bernoulli's theorem are related in a way that makes it hard to say that one completely solves the problem, while the other is less important or practically irrelevant. Could you shed some more light on the distinction? Also, what is Crocco's theorem? I am not familiar with it. And please continue with the explanation, if you don't mind. I'm getting a lot out of it. 
You are completely right that one cannot dismiss Bernoulli's equation, i.e, basically the tangential component of Newton's 2.law; but neither must one dismiss that component of Newton's 2.law which is normal to the streamlines.
This is, however, what is ordinarily done when people try to argue from Newton's 2.law, and solely use the tangential integral (Bernoulli's equation). We need the full vector equations here (i.e, what happens in "both" directions), otherwise we simplify our "explanation" to the point of misconstruction. EDIT: The basic weakness by trying to use only Bernoulli's equation (i.e, the tangential integral relation), is that you don't have any solid arguments for why the velocity should be higher on the upper side (and hence, lower pressure). By connecting pressure differences to (effective) curvatures (or, rather, centripal accelerations), you DO get a rather powerful argument. But that requires an analysis of the dynamics normal to the streamlines.. I'll get back tomorrow. 
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:smile: 
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Here is how I would calculate the downward force: [tex]F_{down} = v_ydm/dt = v_y\rho dV/dt = v_y\rho A_{le}ds/dt = \rho A_{le}v^2sin\theta[/tex] where [itex]A_le[/itex] is the vertical crosssection area of the leading edge, [itex]v_y[/itex] is the vertical component of the upwardly deflected air, v is the speed of the wing relative to the air and [itex]\theta[/itex] is the upward angle of the deflected air. The upward lift is the pressure differential x wing area  F_down. So: [tex](P_{bottom}  P_{top}) A_w  \rho A_{le}v^2sin\theta = F_{up}[/tex] where A_w is the area of the whole wing. So if: [tex]\Delta PA_w > \rho A_{le}v^2sin\theta[/tex] you should get lift. I am not sure how to determine the pressure difference between the top and bottom surfaces! I'll have to think about it. But I don't see that the pressure difference is strongly related to the vertical speed of the deflected air. But as I say, I may be missing something. My sense is that you should get just as much pressure difference if the deflection is at a small upward angle. What is important is that the wing cross section has to encounter a sufficient volume of air as it moves so that enough air is moving upward above the wing as the wing passes under it. AM 
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