fluid dynamics two parts 1.fluid flow 2.bernoulli’s equation and applications
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Fluid Dynamics
Two Parts1. Fluid Flow2. Bernoulli’s Equation and Applications
Assumptions for Fluid Flow:
Streamline flow Turbulent flow
1. Non-viscous. (isn’t “sticky”)
2. Incompressible (constant ρ)
3. All particles in cross section travel at the same speed (flow rate)
4. Flow is laminar (no turbulence)
Laminar FlowLaminar flow, type of fluid (gas or liquid) flow in which the fluid travels smoothly or in regular pathsLaminar flow over a horizontal surface may be thought of as consisting of thin layers, or laminae, all parallel to each other.
Laminar Flow• Video:
Flow RateFlow Rate (ƒ): Volume of fluid that passes a particular point in a given timeUnits used to measure Flow Rate = m³/secEquation for: Flow Rate
ƒ = Aν = (m2)(m/s)
(A = cross sectional area)(ν = velocity of fluid)
Rate of Flow
V Avt
AvtR vA
t Rate of flow = velocity x area
vt
Volume = A(vt)
A
Since A1 > A2…
1 1 2 2R v A v A
For an incompressible, frictionless fluid, the velocity increases when the cross-section decreases:
Continuity EquationFlow rates are the same at all points along a closed pipeContinuity Equation:
ƒ₁ = ƒ₂A₁ν₁ = A₂ν₂
Reminder: the equation for Area of a circle: A = πr²
PHet• Fluid Flow
Question: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 through a nozzle whose diameter is 2.5 cm. What is the speed of the water coming out of the nozzle?
Venturi Meter
The higher the velocity in the constriction at Region-2, the lower the pressure... Wait what?
Venturi Effect
Venturi Effect
Airplane Wings
Airplane Wings
How do Plane’s Fly
Video
The Physics of Sailing
Videohttp://science.kqed.org/quest/video/the-physics-of-sailing
/
QuestionA small ranger vehicle has a soft, ragtop roof. When the car is at rest the roof is flat. When the car is cruising at highway speeds with its windows rolled up, does the roof a. bow upward b. remain flatc. bow downward?
QuestionA small ranger vehicle has a soft, ragtop roof. When the car is at rest the roof is flat. When the car is cruising at highway speeds with its windows rolled up, does the roof a. bow upward b. remain flatc. bow downward?
Fluid Flow Questions1. MC - 4,14,21,42,47 2. Homework: Watch Bernoulli Video3. MOST IMPORTANTLY: Paper Airplane Competition next classGo to: http://www.funpaperairplanes.com/index.html
a. Pick a plane and build it for the start of classb. Make TWO of the same designc. Planes will be thrown in players halld. Winner will be determined by displacement from initial throw
Sports Science• Record Paper Airplane
Conservation of Energy of Fluids within a Pipe
Bernoulli's PrinciplePRESSURE plus ENERGY is CONSTANT!1. P + E = P + E2. P + U + K = P + U + K3. P + ρgh + ½ρν² = P + ρgh + ½ρν²This hold at ANY point!
P1 + ρgh1 + ½ρν1² = P2 + ρgh2 + ½ρν2²
Bernuolli Effect1. High Velocity: _____ Pressure2. Low Velocity: _____ Pressure
Bernuolli Effect1. High Velocity: LOW Pressure2. Low Velocity: HIGH Pressure
Special Case #1 – Horizontal Pipe2 2
1 1 1 2 2 2½ ½P gh v P gh v
Horizontal Pipe (h1 = h2)
2 22 1½ ½P gh v v Horizontal Pipe
QuestionSuppose the pressure in the fire hose is 350 kPa. What is the pressure in the nozzle? ν1 = 1.3 m/s
ν2 = 19.17 m/s
Special Case #2 – Constant Velocity2 2
1 1 1 2 2 2½ ½P gh v P gh v
Constant velocity (ν1 = ν2)
Notice how a difficult problem becomes easier when we remove constants!
QuestionWater flows with constant speed through a garden hose that goes up a step 20.0 cm high. If the water pressure is 143 kPa at the bottom of the step, what is its pressure at the top of the step?
ν1 = ν2
Special Case #3 – Fluids at Rest2 2
1 1 1 2 2 2½ ½P gh v P gh v
P1 - P2 = rgh2 - rgh1 DP = rg(h2 - h1)
We have already seen this!
Special Case #4 – No Change in PressureKnow as Torricelli’s Theorem
2 21 1 1 2 2 2½ ½P gh v P gh v
2v gh
h1
h2h
Torricelli’s theorem:
2v gh
v2 0
Question:A dam springs a leak at a point 20.0 m below the surface. What is the emergent velocity?
2v ghh
v = 19.8 m/s2
Summary of Hydrodynamics
1 1 2 2R v A v A Streamline Fluid Flow in Pipe:
PA - PB = rghHorizontal Pipe (h1 = h2)
2 21 2 2 1½ ½P P v v
Fluid at Rest:
Bernoulli’s Theorem:2
1 1 1½P gh v Constant
Torricelli’s theorem:
2v gh
Bernoulli’s Principal1. MC: 5,13,22,25,27,28,33,36,37,44 2. Homework: Review Free Response Questions Posted
on Website3. Next Class: Hydrodynamics Quiz
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