15. Fluid Motion 流體運動

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15. Fluid Motion 流體運動. Density &amp; Pressure 密度 &amp; 壓力 Hydrostatic Equilibrium 流體靜力平衡狀態 Archimedes’ Principle &amp; Buoyancy 阿基米德原理 &amp; 浮力 Fluid Dynamics 流體力學 Applications of Fluid Dynamics 流體力學的應用 Viscosity &amp; Turbulence 粘性 &amp; 亂流.

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### 15. Fluid Motion 流體運動

Density & Pressure 密度 & 壓力

Hydrostatic Equilibrium 流體靜力平衡狀態

Archimedes’ Principle & Buoyancy 阿基米德原理 & 浮力

Fluid Dynamics 流體力學

Applications of Fluid Dynamics 流體力學的應用

Viscosity & Turbulence 粘性 & 亂流

Why is only the “tip of the iceberg ” above water?

iceis only slightly less than water.

ice 只比 water 小一點點。

Fluid = matter that flows under external forces

= liquid or gas.

= 液體 或 氣體

• Examples of fluid motion: 流體運動的例子
• Airflight: plane supported by pressure on wings. 飛行: 飛機受翼上壓力支撐
• Gas from giant star being sucked into a black hole. 巨星的氣體被黑洞吸收
• Brake fluid in a car’s braking system. 汽車剎車系统裏的剎車液
• Breathing: air into lung & blood stream. 呼吸: 空氣進入肺和血液
15.1. Density & Pressure 密度&壓力

Avogadro’s number 阿伏伽德羅常數NA = 6.022  1023 / mol .

1 mole = amount of substance containing NA basic elements.

( with NA = number of atoms in 12 g of 12C ).

1 摩爾 = 內含NA個基本單元的物質 ( NA =12 g 碳 -12內的原子個數)。

thousands of molecules

Fluid: average position of molecules not fixed.

Macroscopic viewpoint: deformable continuum.

dV

fluid point

dV  0

Density  = mass / vol, [  ] = kg / m3 .

Incompressible = density unchanged under pressure 不可壓縮的 = 密度在壓力下不變

Liquid is nearly incompressible (molecules in contact). 液體近乎不可壓縮 (分子已踫到)

Gas is compressible. 氣體可壓縮

Pressure 壓力

Pressure = normal force per unit area

A n

F

Fluid point

Pressure is a scalar. The pressure at a point in a fluid is the magnitude of the radial force per unit area acting on a fluid point at that position.

15.2. Hydrostatic Equilibrium 流體靜力平衡狀態

Hydrostatic equilibrium 流體靜力平衡狀態:

Fnet = 0 everywhere in fluid 流體內任何位置

Fluid is at rest. 流體處於靜止狀態

Fnet 0 gives rise to pressure differences. 產生壓力差

x

Force experienced by the fluid element:

Let f be the force density within the fluid :

(f is the force per unit volume experienced by a small fluid element due to pressure differences )

( f 是一個小流體元素中，每單位體積因壓力差而受到的力 )

Hydrostatic Equilibrium with Gravity重力下的流體靜力平衡狀態

Fluid element : area A, thickness dh, mass dm.

Net pressure force on fluid element:

Gravitational force on fluid element:

Hydrostatic Equilibrium :

Liquid (~incompressible):

Example 15.1. Ocean Depths 海洋深度
• At what water depth is the pressure twice atmospheric pressure?
• 水中甚麼深度的壓力是大氣壓的兩倍？
• What’s the pressure at the bottom of the 11-km-deep Marianas Trench, the deepest point in the ocean?
• 在11公里深的馬里安納海溝底(海洋最深處)的壓力是多少?
• Take 1 atm = 100 kPa & water = 1000 kg/m3 .

(a)

• Pressure increases by 1 atm per 10 m depth.
• 水壓每 10m 深就增大 1 atm.

Measuring Pressure 量壓力

Barometer 氣壓計

= device for measuring atmospheric pressure

vacuum inside tube:

For p = 1 atm = 101.3 kPa :

Cf. h = 10 m for a water barometer 水式氣壓計

Manometer 壓力計

Manometer = U-shaped tube filled with liquid to measure pressure differences.

equal p

Gauge pressure

= excess pressure above atmospheric.

Used in tires, sport equipments, etc.

E.g., tire gauge pressure 胎壓 = 30 psi

 absolute tire pressure = 44.7 psi

Pascal’s law帕斯卡定律 :

An external pressure applied to a fluid in a closed vessel is uniformly transmitted throughout the fluid.

Example 15.2. Hydraulic Lift 液壓升降機

In a hydraulic lift, a large piston supports a car.

The total mass of car & piston is 3200 kg.

What force must be applied to the smaller piston to support the car?

Pascal’s law

fluid element in equilibrium

Buoyancy force 浮力:

Upward force felt by an object in a fluid

Archimedes’ Principle 阿基米德原理 :

The buoyancy force on an object is equal to the weight of the fluid it displaces.

Neutral buoyancy 中性浮力:

average density of object is the same as that of fluid.

Fb unchanged after replacement

Example 15.3. Working under Water 水中作業

To set up a raft, you need to move a 60-kg block of concrete on the lake bottom.

What’s the apparent weight of the block as you lift it underwater?

The density of concrete is 水泥的密度是 2200 kg / m3 .

~ ½ weight on land

Example 15.4. Tip of the Iceberg 冰山一角

Average density of a typical iceberg is 0.86 that of seawater.

What fraction of an iceberg’s volume is submerged?

Arctic sea ice is melting as a result of global warming.

Does this contribute to rising sea levels?

No.不

Volume of the melted ice (which becomes water) is the same as that displaced by the floating ice.

Making the Connection 連起來

The land based Greenland ice cap occupies some 3 million km3 ,

while some 15,000 km3 of ice are afloat in the Arctic Ocean.

Compare the approximate rise in the world’s oceans that would result from complete melting of these two ice volumes.

• Melting sea ice doesn’t change sea level. 海冰融化不會改變海平面。
• Melting land-based ice adds water of volume ~ 86% that of ice.
• 陸上的冰塊融化後，海洋約增加冰體積 86% 的水。
• Melting Greenland ice cap adds 格陵蘭的冰帽融化後，海洋約增水
• 0.86  3  106km3 = 2.58  106km3
• of water to ocean.
• Ocean covers 71% of earth 海洋約涵蓋地球 71%
• Ocean surface海洋面積 = 0.71  4  (6.37  103 km)2 = 3.62  108 km2 .
• Sea level increased by 海平面升高 2.58  106km3 / 3.62  108 km2 = 7.1 m
Center of Buoyancy 浮心

Buoyancy force acts at the center of buoyancy (CB),

which coincides with the CM of the displaced water.

CM must be lower than CB to be stable.

15.4. Fluid Dynamics

Moving fluid is described by its flow velocity v( r, t ). 流動的流體以其流速 v( r, t ) 描述。

Streamlines = Lines with tangents everywhere parallel to v( r, t ).

Spacing of streamlines is inversely proportional to the flow speed.

fast 快

slow 慢

Small particles (e.g., dyes) in fluid move along streamlines.

e.g., calm river. 例: 平靜的河

Example of unsteady flow: blood in arteries ( pumped by heart ).

Fluid dynamics 流體力學:

Newton’s law + diffusing viscosity  Navier-Stokes equations

GOT IT? 15.1.
• Photo shows smoke particles tracing streamlines in a test of a car’s aerodynamic properties.
• 圖片顯示在測試汽車的空氣動力性能時，煙粒子描出的流線。
• Is the flow speed greater 流速較大的區域是在
• over the top, or 車頂，還是在
• at the back? 車後?

Flow tube : small region with sides tangent, & end faces perpendicular, to streamlines.

• flow tubes do not cross streamlines.
• 流管不會與跨越流線

Mass entering tube:

Mass leaving tube:

Conservation of mass:

Equation of continuityfor steady flow 穩流的連續性方程 :

Mass flow rate =

[  v A ] = kg / s

Liquid 液體:

Volume flow rate =

[ v A ] = m3 / s

Liquid : flows faster in constricted area. 液體: 狹窄區域流速較大.

Gas with v < vs ound: flows faster in constricted area. 氣體 v < vs ound : 狹窄區域流速較大.

Gas with v > vsound : flows slower in constricted area. 氣體 v > vs ound : 狹窄區域流速較小.

Example 15.5. Ausable Chasm

The Ausable river in NY is about 40 m wide. 紐約卅的鄂西埗河大概有 40 m 寬。

In summer, it’s usually 2.2 m deep & flows at 4.5 m/s. 夏天時它通常有 2.2 m 深 & 流速 4.5 m/s.

Just before it reaches Lake Champlain, it enters Ausable Chasm, a deep gorge only 3.7 m wide.

If the flow rate in the gorge is 6.0 m/s, how deep is the river at this point?

Assume a rectangular cross section with uniform flow speed.

Same fluid element enters & leaves tube:

Work done by pressure upon its entering tube:

Work done by pressure upon its leaving tube:

Work done by gravity during the trip:

W-E theorem:

Incompressible fluid:

Bernoulli’s Eq.

Viscosity & other works neglected

15.5. Applications of Fluid Dynamics流體力學的應用
• Strategy 策略
• Identify a flow tube.
• 認定一個流管。
• Draw a sketch of the situation, showing the flow tube.
• 把現場畫成一個簡圖，其中要包括上述流管。
• Determine two points on your sketch.
• 在簡圖中選兩點。
• Apply the continuity equation and Bernoulli’s equation.
• 使用連續性方程和伯努利方程。
Example 15.6. Draining a Tank 水塔放水

A large open tank is filled to height h with liquid of density .

Find the speed of liquid emerging from a small hole at the base of the tank.

At top surface 在水面:

At hole 在小洞:

Venturi Flows 文丘里(文氏)流

Venturi : a tube with constricted central region C.

Eq. of continuity 連續性方程  v larger in C.

Bernoulli eq. 伯努利方程 p lower in C.

• Venturi flow meter measures flow speed by measuring pressure drop in C.
• 文丘里(文氏)流量計以量 C 的壓降來量流速。
Example 15.7. Venturi Flowmeter 文丘里流量計

Find the flow speed in the unconstricted pipe of a Venturi flowmeter.

Bernoulli’s eq. 伯努利方程:

Continuity eq. 連續性方程

Bernoulli Effect 伯努利効應

Bernoulli Effect: p   v 

• Example: Prairie dog’s hole 草原犬鼠的洞
• Dirt mound forces wind to accelerate over hole
• 土堆使風在流經洞口時加速
• low pressure above hole 洞上方氣壓變低
• natural ventilation 天然通風糸統

A ping-pong ball supported by downward-flowing air. 乒乓球由向下的氣流撑住。

High-velocity flow is inside the narrow part of the funnel. 高速氣流是在漏斗較窄處。

GOT IT? 15.2.

A large tank is filled with liquid to level h1 . 一個大水槽裝了 h1 高的液體。

It drains through a small pipe whose diameter varies. 它的泄口是一條口徑有變化的細管。

Emerging from each section of pipe are vertical tubes open to the atmosphere.

Although the picture shows the same liquid level in each pipe,

they really won’t be the same. 其實它們都不一樣。

Rank order the levels h1 through h4. 把 h1 到 h4 從大到小排序。

ordered inversely by flow speed

Flight & Lift 飛行升力

Air pushes up (3rd law)

Faster flow, lower P : uplift.

Aerodynamic lift

Blade pushes down on air旋槳把空氣往下推

Top view on a straight ball : no spin

Top view on a curved ball : spin

Application: Wind Energy 風能

A chunk of air, of speed v & density ,

passing thru a turbine of area A in time t, has kinetic energy

• available power per unit area =
• 可用的單位面積功率

Better analysis 

For 設

Present tech gives 80% of this.

15.6. Viscosity & Turbulence 粘性 & 亂流

flow with no viscosity

Viscosity: friction due to momentum transfer between adjacent fluid layers or between fluid & wall.

• B.C.: v = 0 at wall 邊際條件
• drag on moving object. 拖住移動物件
• provide 3rd law force on propellers.
• 為螺旋槳提供第三定律力
• stabilize flow. 穩定流量

flow with viscosity

Smooth flow becomes turbulent.