. Fluid Statics 1. 流體靜力學 (Fluid Statics): 係探討流體處於靜止狀態或流體內彼此無相對運動情況下之流體受力狀況 因無相對速度, 故無 du 速度梯度, 亦即無剪力, 流體受力主要為 dy 壓力與重力. 流體靜壓力之等向性 (Isotropic) 取一個很小的 element d dy d 因靜止 F = 0 : ( 力平衡 ) ( ) F = 因 ds sinθ = d = 0 : ( ds d y) 3. 流體靜力學之基本方程式 : 1 ds d y sinθ = d y d 1 cosθ = d d y 1 3 因 ds cosθ = d = 1 3 1 = = 3 故流體靜壓力具有等向性, 亦即在同一點任何方向之壓力均相等 1
因靜止 F = 同理 F y = F = 0: 1 d dyd d dyd + 1 = 0 = ddyd 0 = 0 f ( ) 0 : y = 0 f ( y) 0: 1 d ddy d ddy + 1 dw = 0 ddyd= dw= ρ g ddyd = ρ g 只隨 而變 結論 :(1) 由 與 ressure is constant in a horiontal plane in a static fluid. 1 1 h () 由 : dp= ρ gd p p1 (3) p p = γ ( ) dp= ρg d 1 ( p p ) = ρ g( ) 1 1 p= p1 p = ρ gh ( 壓力差僅與高度差有關 ) 1 1 Hydraulic Head= p 1 p + 1 = + = const. γ γ 故靜止之流體, 其 Hydraulic head 為一常數 ( 第一章已提過 )
4. Absolute ressure ( 絕對壓力 ) Gage ressure ( 計壓 )(=Relative ressure, 相對壓力 ) Gage ressure: gage Absolute ressure: abs atm gage = abs abs, atm atm abs = 絕對大氣壓力, page 一般取為 0 一般提到 pressure, 指的是 gage pressure (relative pressure) Liquid ressure Gage Manometer ( 測壓計 ) E1: p γ l l 1 γ 1 h p y p 4 p 5 γ 3 p = γ l + p = p = p + γ l + γ h 4 1 1 5 y 3 p p = γ l + γ h γ l y 3 11 E:If pressure is very small use "inclined gage" to "enlarge" the reading. p γ 1 l 1 h p 1 p γ p = p + γ l = p = γ h 1 1 1 p γ h γ l = 1 1 3
5. ressure Force on lane or Curved Surface 合力 : F = df = p da 合力作用點 <i> p = pda/ F <ii> yp = y pda/ F 6. Fluid Mass Subjected to Acceleration ρ( ) p d p + a y d p p p d p W dy a d p p d p d + p d p d F = ma = dd yd a = p dyd p dyd + ρ a = p p = ρ a p d p d F = ma = p ddy p ddy W + = p ρ( ddyda ) ( d d yd) ρ g( d d yd) p = ρ( g+ ) p d a p d a d [ g a ] d dp= + = ( ρ ) + ρ ( + ) along a constant p (e.g. free surface) dp= 0 = ρa d+ ρ g+ a d d d [ ] ( ) ( ) a = g + a 4
7. Buoyancy ( 浮力 ) Law of Buoyancy ( 浮力定律 ) p A h V b W F ' F 1 ' W 1 For a submerged body Upper ortion: pa 1 F = F W p A = 0 Lower ortion: F = F 1 + W1 p A= 0 1 Eqn -: F F F = p p A W + W B = ( ) ( ) 1 1 1 ( ) = γ ha W + W 1 ( ) = W W + W Total 1 = γ V b 同理可證 :for a floating body F Stability ( 穩定性 ) B = γ V ' b V b = volume in the liquid dependent on the "relative location" of Buoyancy and Weight For the submerged bodies: C.B. above C.G. Stable For floating bodies: C.G. above C.B. Stable C. B.= Center of Buoyancy ( 浮心 ) C. G.= Center of Gravity ( 重心 ) 5
1. 流體靜力學 (Fluid Statics):. Fluid Statics 係探討流體處於靜止狀態或流體內彼此無相對運動情況下之 流體受力狀況 因流體間無相對速度, 故無速度梯度, 亦即剪應力為零, 流體之受力主要為重力與壓力 1 ressure ( 壓應力 ) and ressure Force ( 壓力 ): -A normal force eerted by a fluid per unit area is normal stress, 法向應力 ) -This normal force is pressure force, and the normal stress is pressure. 1N/m = 1a(pascal) 10 3 a = 1 ka F F N F S A FN / A σ n τ FS / A
Absolute ressure ( 絕對壓力 ), Relative ressure ( 相對壓力 ) Absolute ressure ( 絕對壓力 ), abs Gage ressure ( 計壓 ) (=Relative ressure, 相對壓力 ), gage, ( 絕對大氣壓力 ) gage abs atm atm (+) (-) 一般提到 pressure, 通常指的是 gage pressure (relative pressure) 3. 流體靜壓之等向性 (Isotropic property) 靜止流體中取一點 dd yd 因靜止 0 :( 力平衡 ) F d s d y d y d 1 sin 因 ds sin d 1 0 : F d s d y d d y 3 1 cos 因 ds cos d 1 3 1 3 故流體靜壓具有等向性, 亦即在靜止流體中同一點任何方向壓力均相等 (pressure 為純量 ) F p A 4
3. 流體靜力學之基本方程式 : 5 1 1 因靜止 F 0: d d yd d d yd 0 d d yd 0 0 f () 同理 F y 0 : 0 y f ( y) 1 1 F 0: d d d y d d d y dw 0 d d yd dw g d d yd g 只隨 而變 6
結論 :(1) 由 與 ressure is constant in a horiontal plane in a static fluid. () 由 : d p gd d p g p p1 p p g 1 1 p p1 p gh ( 壓力差僅與高度差有關 ) 1 d 7 5. Total ressure Force F on lane ( 平板 )or Curved Surface ( 曲面 ) 靜水壓力合力 : F df da 合力作用點 : < i > < ii > A F df da A F y ydf yda y A A A da F y da F
Curved Surface : Use free-body diagram ( 自由體受力圖 ) 將水體切出, 分析其受力 9 6. Fluid Mass Subjected to D Accelerations (in - and -directions) 10
p d p d p p a a F ma d d yd a p d yd p d yd p d p d F ma p d d y p d d y W p p d d yd a d d yd g d d yd a g a p p d p d d a d g a along a constant p (e.g. free surface) d p 0 d d a d g a a g a d d 11 7. Buoyancy ( 浮力 ) 1
Law of Buoyancy ( 阿基米德浮力定律 ) p A For a submerged body Upper ortion: F W p F A 0 h V b W F ' F 1 ' Lower ortion: F 1 1 p A 1 F W p 0 W 1 p 1 A Eqn -: F F p p A W F B 1 1 1 W ha W Total W 1 W W 1 W Vb 同理可證 :for a floating body FB V b V b volume in the liquid 13