Short Version : 15. Fluid Motion 短版: 15. 流體運動

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

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 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

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。

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

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 )

Hydrostatic Equilibrium with Gravity 重力下的流體靜力平衡狀態 流體靜力平衡: ( Down is positive) 朝下為正 f < 0 : upward 朝上  p increases with depth. p 隨深度而增。 Liquid (~incompressible): 液體 (近乎不可壓縮)

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 = 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. 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，必均勻地傳遞到流體中的任一部份

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 

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

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

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.

15.4. Fluid Dynamics 流體動力學 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 流體動力學: Newton’s law + diffusing viscosity  Navier-Stokes equations 牛頓定律 + 擴散性粘度  纳维－斯托克斯方程

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. 流管不會跨越流線

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

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: 走一趟重力對它做的功: 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

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: 

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  

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。

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

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% 。

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 Laminar flow 層流 Reynolds number : 雷諾數 : Re < 2000 laminar 2000 < Re < 4000 transition Re > 4000 turbulence