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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 Statics2. Fluid Statics
1. 1. 流體靜力學流體靜力學 (Fluid Statics)(Fluid Statics):
係探討流體處於係探討流體處於靜止狀態靜止狀態或或流體內流體內彼此無相對運動彼此無相對運動情況下之情況下之
流體受力狀況。流體受力狀況。流體受力狀況流體受力狀況
因無相對速度,故無因無相對速度,故無速度梯度,,亦即剪應力為零,流體之受力
主要為壓力與重力。
1
Pressure (壓力):A normal force exerted by a fluid per unit area ( = normal stress,法向應力)
1 N/m2 = 1 Pa (pascal)
103 Pa = 1 kPa
F FN
F
2
FS A
AFP N / σn
AFS /τ
2
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)
(+)
(-)
3◎一般提到 pressure,通常指的是 gage pressure (relative pressure)。
靜止流體中取一點 zdydxd
因靜止 : (力平衡) 0xF zdydPydsdP 21 sin
i PPdd 因
2. 2. 流體靜壓之等向性流體靜壓之等向性 (Isotropic(Isotropic property) property)
21sin PPzdsd 因
: 0zF ydxdPydsdP 31 cos
31cos PPxdsd 因
4
321 PPP
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 (壓力pressure為純量)。
3
3. 3. 流體靜力學之基本方程式:流體靜力學之基本方程式:
5
因靜止 02
1
2
1 0
zdydxd
x
PPzdydxd
x
PPFx
:
0 zdydxdx
P
P0
x
P
)(xfP
同理 0 : 0 y
PFy
)( yfP
02
1
2
1 : 0
Wdydxdzd
z
PPydxdzd
z
PPFz
6
22 zz
zdydxdgWdzdydxdz
P
gz
P
∴ P 只隨 z 而變
4
◎結論:(1)由與 Pressure is constant in a horizontal plane in a static fluid.
(2)由: zdgpd
2
1
2
1
p
p
z
zzdgpd
1212 zzgpp 1212 zzgpp
hgppp 21 (壓力差僅與高度差有關)
7
Liquid Pressure Gage
Manometer (測壓計)
Ex1:
?x yp p
8
hlppplp yx 3225114
11322 lhlpp yx
5
Ex2:If pressure is very small
use the ‘inclined gage’ to ‘enlarge’ the pressure reading.
?xp
9
hplpp x 22111
112 lhpx
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 /p
FdApyy p /< ii >
10
6
Curved Surface : Use freeCurved Surface : Use free--body diagram body diagram
11
6. Fluid Mass Subjected to 2D Accelerations (6. Fluid Mass Subjected to 2D Accelerations (in in xx-- and and zz--directionsdirections))
12
7
zdydxd
x
ppzdyd
xd
x
ppazdydxdmaF xxx
22
x
pa x
xa
x
p
dd Wydxd
zd
z
ppydxd
zd
z
ppmaF zz
22
zdydxdgzdydxdz
pazdydxd z
zag
z
p
zdagxdazdp
xdp
pd zx
13
zgzzx
p zx
along a constant p (e.g. free surface)
zdagxdapd zx 0
z
x
ag
a
xd
zd
7. Buoyancy (7. Buoyancy (浮力浮力) )
14