15. Fluid Motion 流體運動 Density & Pressure 密度 & 壓力

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1 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

2 Why is only the “tip of the iceberg ” above water?
為甚麼只看到〝冰山一角〞? ice is only slightly less than water. ice 只比 water 小一點點。

3 流體 = 在外力下會流動的物體 = 液體 或 氣體 solid 固體 liquid 液體 gas 氣體 inter-mol forces
Fluid = matter that flows under external forces = liquid or gas. 流體 = 在外力下會流動的物體 = 液體 或 氣體 solid 固體 liquid 液體 gas 氣體 inter-mol forces 分子間的作用力 strongest 最強 medium 中等 weakest 最弱 volume 體積 fixed 固定 variable 可變 shape 形狀 variable 可變 Examples of fluid motion: 流體運動的例子 Tornadoes 龍捲風 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

4 15.1. Density & Pressure 密度&壓力
Avogadro’s number 阿伏伽德羅常數 NA =  / 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 幾千個分子 Fluid: average position of molecules not fixed. 流體：分子平均位置不是固定的。 Macroscopic viewpoint: deformable continuum. 巨觀: 可變形的連續體 dV fluid point 流體點 dV  0 Density  = mass / vol, [  ] = kg / m3 . 密度  = 質量 / 體積 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

5 Pressure 壓力 A n F Pressure = normal force per unit area 壓力 = 單位面積的正交力
壓力 = 單位面積的正交力 A n F Fluid point 流體點 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. 壓力是一個純量。流體內一點的壓力，等於在該處一個流體點，每單位面積所受的徑向力的大小。

6 15.2. 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 定壓 壓力增加

7 x Force experienced by the fluid element: 流體元素所受力： Let f be the force density within the fluid : 使 f 為流體內的力密度： 壓力增加 ( 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 )

8 Hydrostatic Equilibrium with Gravity 重力下的流體靜力平衡狀態
Fluid element : area A, thickness dh, mass dm. 流體單元: 面積 A, 厚度 dh, 質量 dm. Net pressure force on fluid element: 流體單元所受淨壓力: Gravitational force on fluid element: 流體單元所受重力: Hydrostatic Equilibrium : 流體靜力平衡:  Liquid (~incompressible): 液體 (近乎不可壓縮)

9 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. 

10 Measuring Pressure 量壓力
真空 Barometer 氣壓計 = device for measuring atmospheric pressure 量大氣壓的儀器 vacuum inside tube: 管內真空: 水銀 For p = 1 atm = kPa : Cf. h = 10 m for a water barometer 水式氣壓計

11 Manometer 壓力計 Manometer = U-shaped tube filled with liquid to measure pressure differences. 壓力計 = 內填液體的U 形管，用來量壓力差。 equal p Gauge pressure = excess pressure above atmospheric. 規壓 = 超過大氣壓的壓力 Used in tires, sport equipments, etc. 用於車胎，體育器材等。 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. 對一封閉容器內的流體所施的壓力，必均勻的傳遞到流體中的任一部份

12 Example 15.2. Hydraulic Lift 液壓升降機
In a hydraulic lift, a large piston supports a car. 一個液壓升降機的大活塞上放了一部汽車 The total mass of car & piston is 3200 kg. 汽車&活塞的總質量為3200 kg. What force must be applied to the smaller piston to support the car? 小活塞上要加多少力才能撑得住? Pascal’s law 巴斯葛定律 

13 15.3. Archimedes’ Principle & Buoyancy 阿基米德原理 & 浮力
fluid element in equilibrium 流體單元在平衡狀態 Buoyancy force 浮力: Upward force felt by an object in a fluid 物體在流體中受到的提升力 Archimedes’ Principle 阿基米德原理 : The buoyancy force on an object is equal to the weight of the fluid it displaces. 一個物體在流體中受到的浮力， 等於它所排開的流體的重量。 Neutral buoyancy 中性浮力: average density of object is the same as that of fluid. 物體的平均密度和流體的一樣。 Fb unchanged after replacement 在置換後不變

14 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. 在建一個浮台時，你必需在湖底搬動一塊６０公斤的水泥。 What’s the apparent weight of the block as you lift it underwater? 你在水中舉起水泥塊時，它感覺上的重量是多少? The density of concrete is 水泥的密度是 kg / m3 . 自由體圖 ~ ½ weight on land 陸上重量

15 Example 15.4. Tip of the Iceberg 冰山一角
Average density of a typical iceberg is 0.86 that of seawater. 一個典型冰山的平均密度是海水的 0.86 倍。 What fraction of an iceberg’s volume is submerged? 冰山有幾部份沈在水下? 冰山簡圖 自由體圖

16 Conceptual Example 15.1. Shrinking Arctic 縮小中的北極
Arctic sea ice is melting as a result of global warming. 北極的海冰因地球暖化而融化。 Does this contribute to rising sea levels? 這會造成海面上升嗎 ? No.不 Volume of the melted ice (which becomes water) is the same as that displaced by the floating ice. 海冰融化(變成水)後，體積和被冰排開的水一樣。

17 Making the Connection 連起來
The land based Greenland ice cap occupies some 3 million km3 , 格陵蘭陸上的冰帽約占三百萬 km3， while some 15,000 km3 of ice are afloat in the Arctic Ocean. 北極海的浮冰則約占 15,000 km3。 Compare the approximate rise in the world’s oceans that would result from complete melting of these two ice volumes. 粗估一下這兩組冰塊全部融化時，海平面會上升多高。 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  106 km3 = 2.58  106 km3 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  106 km3 / 3.62  108 km2 = 7.1 m

18 Center of Buoyancy 浮心 Buoyancy force acts at the center of buoyancy (CB), which coincides with the CM of the displaced water. 浮力的作用點在浮心， 亦即被排的水的重心。 重心要低於浮心才會穩定。 CM must be lower than CB to be stable.

19 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 ). 流線 = 每點的切線都跟 v( r, t ) 平行的曲線。 Spacing of streamlines is inversely proportional to the flow speed. 流線的間隔與流速成反比。 fast 快 slow 慢 Small particles (e.g., dyes) in fluid move along streamlines. 流體中的粒子(如染料) 都隨流線走 Steady flow 穩流 : e.g., calm river. 例: 平靜的河 Example of unsteady flow: blood in arteries ( pumped by heart ). 不穩流的例子: 動脈裹的血液 (由心臟抽吸) Fluid dynamics 流體力學: Newton’s law + diffusing viscosity  Navier-Stokes equations 牛頓定律 + 擴散性粘度

20 GOT IT? 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?

21 Conservation of Mass: The Continuity Equation 質量守恆: 連續性方程
Flow tube : small region with sides tangent, & end faces perpendicular, to streamlines. 流管: 管狀小區域，其側邊與流線相切，前後兩端截面則與流線垂直。 flow tubes do not cross streamlines. 流管不會與跨越流線

22 Equation of continuity for steady flow 穩流的連續性方程 :
Mass entering tube: 質量進入管中 Mass leaving tube: 質量離開流管 Conservation of mass: 質量守恆 Equation of continuity for steady flow 穩流的連續性方程 : Mass flow rate = 質量流動速率 [  v A ] = kg / s Liquid 液體: Volume flow rate = 體積流動速率 [ 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.

23 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. 剛在流入尚普蘭湖之前，它通過一個衹有 3.7 m 寬，名叫鄂西埗深坑的峽谷。 If the flow rate in the gorge is 6.0 m/s, how deep is the river at this point? 如果峽谷中的流速為 6.0 m/s，河水在此處有多深? Assume a rectangular cross section with uniform flow speed. 假定截面為長方形，且截面上各點流速都一樣。 

24 Conservation of Energy: Bernoulli’s Equation 能量守恆: 柏努利方程式
Same fluid element enters & leaves tube: 進入和離開流管的是同一個流體單元: Work done by pressure upon its entering tube: 進入流管時壓力對它做的功: Work done by pressure upon its leaving tube: 離開流管時壓力對它做的功: Work done by gravity during the trip: 走一遭重力對它做的功: W-E theorem: 功-能定理:  Incompressible fluid: 不可壓縮流體: Bernoulli’s Eq. 柏努利方程式  Viscosity & other works neglected 忽略粘性和其它的功

25 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。

26 Example 15.6. Draining a Tank 水塔放水
A large open tank is filled to height h with liquid of density . 一個上空的大水塔存了高度為h ，密度為  的液體。 Find the speed of liquid emerging from a small hole at the base of the tank. 求從塔底小洞中噴出的液體的速率。 At top surface 在水面: At hole 在小洞: 

27 Venturi Flows 文丘里(文氏)流
Venturi : a tube with constricted central region C. 一條中間部份 C 較窄的管子。 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。

28 Example 15.7. Venturi Flowmeter 文丘里流量計
Find the flow speed in the unconstricted pipe of a Venturi flowmeter. 找出文丘里流量計中管子較寬處的流速。 Bernoulli’s eq. 伯努利方程: 量p的錶 Continuity eq. 連續性方程  

29 Bernoulli Effect 伯努利効應
Bernoulli Effect: p   v  伯努利効應 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。

30 GOT IT? A large tank is filled with liquid to level h 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. 細管的每一段都有一條外通大氣的直立的管。 Although the picture shows the same liquid level in each pipe, 雖然圖中這些管的液面都一樣高， they really won’t be the same 其實它們都不一樣。 Rank order the levels h1 through h 把 h1 到 h4 從大到小排序。 ordered inversely by flow speed 依流速反排序

31 Flight & Lift 飛行升力 Aerodynamic lift 空氣動態升力
Air pushes up (3rd law) 空氣往上推 (第三定律) Faster flow, lower P : uplift. 流速較快，P 較低：升力 Aerodynamic lift 空氣動態升力 氣流 Blade pushes down on air旋槳把空氣往下推 Top view on a straight ball : no spin 從上往下看一個直球: 無自旋 Top view on a curved ball : spin 從上往下看一個彎球: 有自旋

32 Application: Wind Energy 風能
A chunk of air, of speed v & density , 一團空氣，其速度為 v ，密度為 , passing thru a turbine of area A in time t, has kinetic energy 在 t 時間內通過一部面積為 A 的渦輪機。其動能為 available power per unit area = 可用的單位面積功率 Better analysis  好一點的分析 For 設 Present tech gives 80% of this. 現在的科技可達此值的80% 。

33 15.6. Viscosity & Turbulence 粘性 & 亂流
flow with no viscosity 無粘性流 Viscosity: friction due to momentum transfer between adjacent fluid layers or between fluid & wall. 粘性: 因相隣流層或流體與容璧之間的動量交換而產生的阻力。 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 粘性流 Smooth flow becomes turbulent. 平順的氣流變成亂流