Bernoulli and Flow Continuity

 U-Tube Manometer  Used to measure pressure of a fluid  Principles involved: ◦ The pressure is the same in equal elevations of a fluid ◦ The pressure at the bottom of a column of fluid equals the pressure at the top + ρgh

Laminar – When fluid particles move along the same smooth path. The path is called a streamline. Turbulent – When fluid particles flow irregularly, causing changes in velocity.

Source: Wikimedia Commons A B

 When a fluid flows, mass is conserved.  If there are no outlets or inlets, the same mass per unit time will flow everywhere in the stream

 The volume per unit time of a liquid flowing in a pipe is constant throughout the pipe.  We can say this because liquids are not compressible, so mass conservation is identical to volume conservation for a liquid.

 V = Avt  V: volume of fluid (m 3 )  A: cross-sectional area at a point in the pipe  (m 2 )  v: speed of fluid flow at a point in the pipe (m/s)  t: time (s)

 A 1 v 1 = A 2 v 2  A 1, A 2 : cross sectional areas at points 1 and 2  v 1, v 2 : speed of fluid flow at points 1 and 2  ** Av = rate of volume flow (unit is m 3 /s)

 A pipe of diameter 6.0 cm has water flowing through it at 1.6 m/s. How fast is the water flowing in an area of the pipe in which the diameter is 3.0 cm? How much water per second flows through the pipe?

 The water in a canal flows 0.10 m/s where the canal is 12 meters deep and 10 meters across. If the depth of the canal is reduced to 6.5 meters at an area where the canal narrows to 5.0 meters, how fast will the water be moving through this narrower region?

Continuity equation: A 1 v 1 = A 2 v 2 Bernoulli’s principle: “The pressure in a fluid decreases as the fluid’s velocity increases.” Bernoulli’s equation: P + ½ ρv 2 + ρgh = constant

Bernoulli’s equation at different points in a horizontal pipe: P 1 + ½ρv 1 2 = P 2 + ½ρv 2 2 Image source: 2013 Emily Sappington, University of Houston Point 1Point 2Point 3

 P + ρgh + ½ ρv 2 = Constant  P : pressure (must be in Pa)  ρ : density of fluid (kg/m 3 )  g: acceleration due to gravity (9.8 m/s 2 )  h: height above lowest point (m)  v: speed of fluid flow at a point in the pipe (m/s)

Bernoulli’s equation at two different points of varying height P 1 + ½ρv ρgh 1 = P 2 + ½ρv ρgh 2 Source:

 An above-ground swimming pool has a hole of radius 0.10 m in the side 2.0 meters below the surface of the water. How fast is the water flowing out of the hole? How much water flows out each second?

 Water travels through a 9.6 cm diameter fire hose with a speed of 1.3 m/s. At the end of the hose, the water flows out of a nozzle whose diameter is 2.5 cm. What is the speed of the water coming out of the nozzle? If the pressure in the hose is 350 kPa, what is the pressure in the nozzle?

 If the wind is fast enough, the pressure outside your house is much lower than the pressure inside:  kTsg kTsg