An airplane is moving through the air at a velocity v=200m/s. Streamlines just over the top of the wing are compressed to 80% of their original area, and those under the wing are not compressed at all.

Question

An airplane is moving through the air at a velocity v=200m/s. Streamlines just over the top of the wing are compressed to 80% of their original area, and those under the wing are not compressed at all.
a) Determine the velocity of the air just over the wing.
b) Find the difference in the pressure between the air just over the wing,P and that under the wing, P'.
c) Find the net upward force on both wings due to the pressure difference, if the area of the wig is 40m^2 and the density of the air is 1.3kg/m^3.

Bot · Answer

a) To determine the velocity of the air just over the wing, we can use the principle of continuity, which states that the product of the cross-sectional area A, the fluid (air) density ρ, and the fluid velocity v must be constant in the streamline:

A * ρ * v = constant

Let's assume the original area (before the compression) is A₀, so the compressed area is A₁ = 0.8 * A₀. The original velocity is v₀ = 200 m/s.

A₀ * ρ * v₀ = A₁ * ρ * v₁

We can cancel ρ because it's constant in the streamline:

A₀ * v₀ = A₁ * v₁

Now we can solve for v₁:

v₁ = (A₀ / A₁) * v₀ = (1 / 0.8) * 200 = 200 / 0.8 = 250 m/s

The velocity of the air just over the wing is 250 m/s.

b) To find the difference in pressure between the air just over the wing (P) and under the wing (P'), we can use Bernoulli's equation, which relates the pressure, density, and velocity of a fluid in a streamline:

P + (1/2) * ρ * v^2 = constant

Applying this equation to the air over the wing (1) and under the wing (0):

P₀ + (1/2) * ρ * v₀^2 = P₁ + (1/2) * ρ * v₁^2

Solving for the pressure difference P - P':

∆P = P₁ - P₀ = (1/2) * ρ * (v₀^2 - v₁^2)

Using the given density of the air, ρ = 1.3 kg/m³, and the calculated velocities:

∆P = (1/2) * 1.3 * (200^2 - 250^2) = 0.65 * (-7500) = -4875 Pa

The pressure difference between the air just over the wing and under the wing is -4875 Pa (meaning the pressure under the wing is higher).

c) To find the net upward force on both wings due to the pressure difference, we can use the formula:

F = ∆P * A

Using the given area of each wing (40 m²), and since there are two wings, we multiply the force by 2:

F = 2 * (-4875) * 40 = -390000 N

The net upward force on both wings due to the pressure difference is -390000 N (which is a downward force, as expected from a negative pressure difference).