Lecture 3 Bernoulli’s equation. Airplane wing Rear wing Rain barrel Tornado damage

Work by pressure A1A1 A2A2 v1 v1 v2v2 As an element of fluid moves during a short interval dt, the ends move distances ds 1 and ds 2. Work by pressure during its motion: ρ1 ρ1 ρ2 ρ2 ds 2 ds 1 dV If the fluid is incompressible, the volume should remain constant:

Kinetic and gravitational potential energy Change in kinetic energy: A1A1 A2A2 v1 v1 v2v2 ρ1 ρ1 ρ2 ρ2 ds 2 ds 1 Change in potential energy: y: height of each element relative to some initial level (eg: floor)

Bernoulli’s equation Putting everything together: NB: Bernoulli’s equation is only valid for incompressible, non-viscous fluids with a steady laminar flow!

Static vs flowing fluid Cylindrical container full of water. Pressure at point A (h A below surface): Or gauge pressure: hAhA x A hAhA A x Now we drill a small hole at depth h A. Point A is now open to the atmosphere!

Container with hole Assume the radius of the container is R = 15 cm, the radius of the hole is r = 1 cm and h A = 10 cm. How fast does water come out of the hole? R = 15 cm h A = 10 cm yAyA yByB A x B x Bernoulli at points A and B (on the surface): Continuity at points A and B: (Eqn 1) (Eqn 2)

For once, let’s plug in some numbers before the end: Therefore, This is equivalent to taking v B ~ 0 (the container surface moves very slowly because the hole is small ―compared to the container’s base) DEMO: Container with holes

h ●A●A ●B●B flow Measuring fluid speed: the Venturi meter A horizontal pipe of radius R A carrying water has a narrow throat of radius R B. Two small vertical pipes at points A and B show a difference in level of h. What is the speed of water in the pipe? Venturi effect: High speed, low pressure Low speed, high pressure 2 equations for v A, v B

DEMO: Tube with changing diameter

Partially illegal Bernoulli Gases are NOT incompressible Bernoulli’s equation cannot be used It can be used if the speed of the gas is not too large (compared to the speed of sound in that gas). But… i.e., if the changes in density are small along the streamline

Example: Why do planes fly? High speed, low pressure Low speed, high pressure Net force up (“Lift”) DEMO: Paper sucked by blower. DEMO: Beach ball trapped in air.

ACT: Blowing across a U-tube A U-tube is partially filled with water. A person blows across the top of one arm. The water in that arm: A.Rises slightly B.Drops slightly C.It depends on how hard is the blowing. The air pressure is lower where the air is moving fast. This is how atomizers work!

Aerodynamic grip Tight space under the car ➝ fast moving air ➝ low pressure Race cars use the same effect in opposite direction to increase their grip to the road (important to increase maximum static friction to be able to take curves fast) Lower pressure Higher pressure Net force down

Tornadoes and hurricanes Strong winds ➝ Low pressures v in = 0 v out = 250 mph (112 m/s) Upward force on a 10 m x 10 m roof: Weight of a 10 m x 10 m roof (0.1 m thick and using density of water –wood is lighter than water but all metal parts are denser): The roof is pushed off by the air inside !

The suicide door The high speed wind will also push objects when the wind hits a surface perpendicularly! Air pressure decreases due to air moving along a surface. Modern car doors are never hinged on the rear side anymore. If you open this door while the car is moving fast, the pressure difference between the inside and the outside will push the door wide open in a violent movement. In modern cars, the air hits the open door and closes it again. Delahaye Type 135

Curveballs Speed of air layer close to ball is reduced (relative to ball) Boomerangs are based on the same principle (Magnus effect) Speed of air layer close to ball is increased (relative to ball)

Beyond Bernoulli In the presence of viscosity, pressure may decrease without an increase in speed. Example: Punctured hose (with steady flow). Speed must remain constant along hose due to continuity equation. Ideal fluid (no viscosity) Real fluid (with viscosity) Friction accounts for the decrease in pressure. Lower jet.

The syphon The trick to empty a clogged sink: AxAx x B h Thin hose → v A ~ 0 P A = P B = P atm

ACT: Wooden brick When a uniform wooden brick (1 m x 1 m x 2 m) is placed horizontally on water, it is partially submerged and the height of the brick above the water surface is 0.5 m. If the brick was placed vertically, the height of brick above the water would be: A.0.5 m B.1.0 m C.1.5 m. 0.5 m The displaced volume in both cases needs to be the same: half of the volume of the brick.