Textbook Example #1: Halliday and Resnick

A hollow tube has a disk DD attached to its end. When air is blown through the tube, the disk attracts the card CC. Let the area of the card be A and let v be the average airspeed between the card and the disk. Calculate the resultant upward force on CC, Neglect the card's weight; assume that v0 << v, where v0 is the airspeed in the hollow tube.

Problem #HR-74-P

This problem is from chapter 16, page 390 of Fundamentals of Physics, Third Edition by Halliday and Resnick, John Wiley. Professor Halliday's approach is to apply Bernoulli's Principle to calculate pressure drop as a function of velocity.

Solution

(from Professor Halliday, by email)

Apply Bernoulli to the top (1) and bottom (2) of the card, using the average velocity of the fluid and set the energy equal.

p₁ + dV₁²/2 + dgy₁ = p₂ + dV₂²/2 + dgy₂

(1)

Since y₁ ~ y₂and v₂= 0,

p₁ + dV₁²/2 = p₂

(2)

Rearrange

p₂ - p₁ = dV₁²/2

(3)

Force = pressure * area

F = (p₂ - p₁) = A/2 * dV₁²

so force equals one half of the product of the area, the density and the square of the average velocity.

(4)

Discussion

In general, the flows above and below the card are not in the same flow stream. To the extent the flow expands radially, the the physics might violate the assumptions of constant average velocity and constant density.

Equation (1):

First, the fluid velocity above the card is not constant, and there is no particular reason to take an average velocity.

Second, the fluid above the card does not flow, in a closed loop, around the bottom of the card, and back over the top of the card again. There is simply no closed-loop to link these two regions, so the claim that these two values must be equal, may have no basis.

Third, the fluid in the cylinder, at the center of the card has higher density than that at the open edge of the card, so the constant density provision may not apply.

Equation (4):

The equation has force proportional to the square of the velocity. As such, there is no limit to the force [F], and it can be arbitrarily large. This, in turn, implies the pressure above the card can be arbitrarily small, even negative.