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1
2. 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
速度梯度d ud y
,亦即無剪力,流體受力主要為
壓力與重力。
2.流體靜壓力之等向性(Isotropic)
取一個很小的element d x d y d z⋅ ⋅
因靜止 Fx =∑ 0 : (力平衡) ( )P d s d y P d y d z1 2⋅ ⋅ = ⋅ ⋅sinθ
因 d s d z⋅ =sinθ P P1 2=
Fz =∑ 0 : ( )P d s d y P d x d y1 3⋅ ⋅ = ⋅ ⋅cosθ
因d s d x⋅ =cosθ P P1 3=
∴ = =P P P1 2 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。
3.流體靜力學之基本方程式:
2
因靜止 Fx =∑ 0: P Pxd x d yd z P P
xd x d yd z−
− +
=
12
12
0∂∂
∂∂
− =∂∂Pxd xd yd z 0
∂∂Px= 0 ∴ ≠P f x( )
同理 Fy =∑ 0: ∂∂Py= 0 ∴ ≠P f y( )
Fz =∑ 0: P Pzd z d xd y P P
zd z d xd y dW−
− +
− =
12
12
0∂∂
∂∂
− = = ⋅∂∂
ρPzd xd y d z dW g d xd y d z
∂∂
ρPz
g= − ∴ P只隨 z而變
◎結論:(1)由 與 Pressure is constant in a horizontal plane in a static fluid.
∇
z2
P2
hz1 P1
(2)由 : d p gd z= −ρ
d p g d zz
z
p
p= − ∫∫ ρ
1
2
1
2
( ) ( )p p g z z2 1 2 1− = − −ρ
∆p p p gh= − =1 2 ρ (壓力差僅與高度差有關)
(3) ( )p p z z2 1 2 1− = − −γ
Hydraulic Head= p z p z const11
22γ γ
+ = + = .
故靜止之流體,其Hydraulic head 為一常數。(第一章已提過)
3
4. Absolute Pressure (絕對壓力)
Gage Pressure (計壓)(=Relative Pressure,相對壓力)
Gage Pressure: Pgage
Absolute Pressure: Pabs
P P Pgage abs absatm= − , Pabs
atm =絕對大氣壓力, Ppageatm 一般取為0。
◎一般提到pressure,指的是gage pressure (relative pressure)。
Liquid Pressure Gage
Manometer (測壓計)
Ex1:
pypx
γ 2 l 2
l 1 γ 1 h
p4 p5
γ 3
p l p p p l hx y4 1 1 5 2 2 3= + = = + +γ γ γ
∴ p p l h lx y− = + −γ γ γ2 2 3 1 1
Ex2:If pressure is very small use "inclined gage" to "enlarge" the reading.
px
γ 1 l 1
h
p1 p2γ 2
p p l p hx1 1 1 2 2= + = =γ γ
∴ p h lx = −γ γ2 1 1
4
5. Pressure Force on Plane or Curved Surface
合力:F dF p dA= = ⋅∫ ∫
合力作用點 <i> x x pdA Fp = ⋅∫ /
<ii> y y pdA Fp = ⋅∫ /
6. Fluid Mass Subjected to Acceleration
ppzdz+ ∂
∂ 2 az
z y
x dz
pp p
xdx+ ∂
∂ 2
p pxdx− ∂
∂ 2 W dy ax
dx
p pzdz− ∂
∂ 2
( )F ma d x d y d z a p pxd x d y d z p p
xd x d y d zx x x= = ⋅ = −
− +
∑ ρ ∂
∂∂∂2 2
ρ ∂∂
a pxx = −
∂∂
ρpx
ax= −
F ma p pzd z d x d y p p
zd z d xd y Wz z= = −
− +
−∑ ∂
∂∂∂2 2
( ) ( ) ( )ρ ∂∂
ρd xd yd z a pzd xd yd z g d xd yd zz = − −
( )∂∂
ρpz
g az= − +
( ) ( )[ ]d p pxd x p
zd z a d x g a d zx z= + = − + − +
∂∂
∂∂
ρ ρ
along a constant p (e.g. free surface)
( ) ( )[ ]d p a d x g a d zx z= = − + − +0 ρ ρ
∴ d zd x
ag a
x
z
= −+
5
7. Buoyancy (浮力) Law of Buoyancy (浮力定律)
p A2
W2 F2
' h Vb
F1'
W1
p A1
For a submerged body
Upper Portion:
F F W p Az =′− − =∑ 2 2 2 0
Lower Portion:
F F W p Az =′ + − =∑ 1 1 1
0
Eqn - :
( ) ( )F F F p p A W WB =′ − ′ = − − +1 2 1 2 1 2
( )= − +γ h A W W1 2
( )= − +W W WTotal 1 2
= ⋅γ Vb
同理可證:for a floating body
F VB b= ⋅γ ' Vb′ = volume in the liquid
Stability (穩定性)
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)
1
2. Fluid 2. Fluid StaticsStatics
1. 1. 流體靜力學流體靜力學(Fluid (Fluid StaticsStatics)):
,ydud
係探討流體處於係探討流體處於靜止狀態靜止狀態或流體內彼此或流體內彼此無相對運動無相對運動情況下之流體受情況下之流體受
力狀況。力狀況。
因無相對速度,故無因無相對速度,故無速度梯度 亦即無剪力,流體受力主要為
壓力與重力。
2
Pressure:a normal force exerted by a fluid per unit area ( = normal stress)
1N/m2 = 1Pa (pascal)
103Pa = 1kPa
3
Absolute Pressure (絕對壓力),
Gage Pressure (計壓)(=Relative Pressure,相對壓力), gageP
absP
=−= atmatmabsgage P,PPP ( 絕對大氣壓力)
◎ 一般提到pressure,指的是gage pressure (relative pressure)。
4
取一個很小的element zdydxd ⋅⋅
因靜止 : (力平衡) ∑ = 0xF ( ) zdydPydsdP ⋅⋅=⋅⋅ 21 sinθ
21sin PPzdsd =→=⋅ θ因
:∑ = 0zF ( ) ydxdPydsdP ⋅⋅=⋅⋅ 31 cosθ
31cos PPxdsd =→=⋅ θ因
321 PPP ==∴
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。
2. 2. 流體靜壓力之等向性流體靜壓力之等向性(Isotropic) (Isotropic)
5
3. 3. 流體靜力學之基本方程式:流體靜力學之基本方程式:
6
因靜止 021
21 0 =
+−
−=∑ zdydxd
xPPzdydxd
xPPFx ∂
∂∂∂ :
0 =−→ zdydxdxP
∂∂
0 =→xP
∂∂ )(xfP ≠∴
同理 0 : 0 ==∑ yPFy ∂
∂ )( yfP ≠∴
021
21 : 0 =−
+−
−=∑ Wdydxdzd
zPPydxdzd
zPPFz ∂
∂∂∂
zdydxdgWdzdydxdzP
⋅==− ρ∂∂
gzP ρ
∂∂
−=→ ∴ P 只隨 z 而變
7
◎ 結論:(1)由 與 Pressure is constant in a horizontal plane in a static fluid.
(2)由 : zdgpd ρ−=
∫ ∫−=2
1
2
1
p
p
z
zzdgpd ρ
( ) ( )1212 zzgpp −−=− ρ
hgppp ρ=−=∆⇒ 21 (壓力差僅與高度差有關)
8
Liquid Pressure Gage
Manometer (測壓計)
Ex1:
hlppplp yx 3225114 γγγ ++==+=
11322 lhlpp yx γγγ −+=−∴
9
Ex2:If pressure is very small
use "inclined gage" to "enlarge" the reading.
hplpp x 22111 γγ ==+=
112 lhpx γγ −=∴
10
5. Pressure Force on Plane or Curved Surface 5. Pressure Force on Plane or Curved Surface
合力: ∫∫ ⋅== dApdFF
合力作用點 < i > ∫ ⋅= FdApxx p /
∫ ⋅= FdApyy p /< ii >
11
12
6. Fluid Mass Subjected to Acceleration 6. Fluid Mass Subjected to Acceleration
13
( ) zdydxdxppzdydxd
xppazdydxdmaF xxx
+−
−=⋅==∑ 22 ∂
∂∂∂
ρ
xpa x ∂
∂ρ −= xa
xp ρ
∂∂
−=→
Wydxdzdzppydxdzd
zppmaF zz −
+−
−==∑ 22 ∂
∂∂∂
( ) ( ) ( )zdydxdgzdydxdzpazdydxd z ρ
∂∂
ρ −−= ( )zagzp
+−=→ ρ∂∂
( ) ( )[ ] zdagxdazdzpxd
xppd zx +−+−=+= ρρ
∂∂
∂∂
along a constant p (e.g. free surface)
( ) ( )[ ] zdagxdapd zx +−+−== ρρ0
z
x
aga
xdzd
+−=∴
14
7. Buoyancy (7. Buoyancy (浮力浮力) )
15
Eqn - : ( ) ( )212121 WWAppFFFB +−−=′−′=
( )21 WWAh +−= γ
( )21 WWWTotal +−= bV⋅= γ
同理可證:for a floating body
bB VF ⋅= γ liquid the in volume=′bV
Law of Buoyancy (浮力定律)
For a submerged body
Upper Portion:
0222 =−−′
=∑ ApWFFz
Lower Portion:
∑ =−+′= 0111 ApWFFz
p A2
W2 F2
' h Vb
F1'
W1
Ap1
…
…
16
Stability (穩定性)
dependent on the "relative location" of C.B. and C.G.
C.B. = Center of Buoyancy (浮心)
C.G. = Center of Gravity (重心)
17
For the submerged bodies: C.B. above C.G. Stable
18
For floating bodies: GM > 0 Stable