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