# Why is a piece of paper lifted upward even when blown below?

According to Wikipedia,

A correct explanation of why the paper rises would observe that the plume follows the curve of the paper and that a curved streamline will develop a pressure gradient perpendicular to the direction of flow, with the lower pressure on the inside of the curve. Bernoulli's principle predicts that the decrease in pressure is associated with an increase in speed, i.e. that as the air passes over the paper it speeds up and moves faster than it was moving when it left the demonstrator's mouth. But this is not apparent from the demonstration.

1. Why does the air speed up while passing over the paper?

the plume follows the curve of the paper

1. Does that mean, the paper cannot be planer and has to be curved for this effect to occur?

2. How does this explain how the paper will move upward when one blows along the bottom of the paper?

Before I start the answer, I'd like to point out that Bernoulli's principle is not applicable directly when you're comparing air flow from two different sources ( or two different flow fields, according to Wikipedia ). It only relates the speed and pressure of air within a single flow field.

Now let's consider the first case where you blow over the sheet of paper. The paper would not rise if it were flat, even though you are blowing air across the top of it at a furious rate. Bernoulli's principle does not apply directly in this case. This is because the air on the two sides of the paper did not start out from the same source. The air on the bottom is air from the room, but the air on the top came from your mouth where you actually increased its speed without decreasing its pressure by forcing it out of your mouth. As a result the air on both sides of the flat paper actually has the same pressure, even though the air on the top is moving faster. The reason that a curved piece of paper does rise is that the air from your mouth speeds up even more as it follows the curve of the paper, which in turn lowers the pressure according to Bernoulli's principle.

If Bernoulli's principle were to hold true in the first case, it would then imply that the paper would droop downward in the second case, when air is blown below the paper. But this is clearly not the case. The upward pressure gradient in this downward-curving flow adds to the atmospheric pressure at the lower surface of the paper. This resulting pressure gradient is the source of lift in the second case. I hope this answers your question.

Reference: Wikipedia

EDIT - A formal derivation of the mathematical relation between curved streamlines and pressure gradients can be found here: https://ocw.mit.edu/courses/aeronautics-and-astronautics/16-01-unified-engineering-i-ii-iii-iv-fall-2005-spring-2006/fluid-mechanics/f20_fall.pdf

EDIT 2 - Lift is also generated by the Coanda effect. Here a couple of links:

How is lift generated due to Coanda effect?

http://en.wikipedia.org/wiki/Coand%C4%83_effect

• Why would the paper even try to droop downward? If it droops downward for similar reasons as in the first case, why will it move upward again? What is the upward pressure gradient of downward curving flow? Also, when the air is blown over the paper, by "the air from your mouth speeds up even more as it follows the curve of the paper", do you mean concave upward or concave downward curve? – Archisman Panigrahi May 15 '17 at 6:33
• I'm sorry about the drooping thing. It's edited now. Now, about the pressure gradients, whenever the air follows a curved path, a pressure gradient is created perpendicular to the flow with higher pressure on the outside and lower pressure on the inside. The mathematical relation between the curvature of the streamlines and the pressure difference can be derived from Newton's second law. – Anish Bhattacharya May 15 '17 at 7:02
• Also, when air is blown over the paper, the curve is concave upward. – Anish Bhattacharya May 15 '17 at 7:02
• I think I still have not understood the case when air is blown below. Why does not the paper droop downwards by similar arguments when air is blown up? It seems to me that if the curve is concave upward in both cases, then only the paper will move upwards in both cases. Is it so? If it is so, why does the paper become concave upward in both cases? Is it due to Gravity or something related to flow of air? – Archisman Panigrahi May 15 '17 at 7:08
• If the paper is concave downward, then the curvature of the streamlines become such that the pressure is higher on the top than the bottom. This would then lead to the paper moving downward. But that doesn't happen. – Anish Bhattacharya May 15 '17 at 7:14