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Fluid MechanicsFluid Mechanics

FluidFluid

Liquid GasGas

DensityDensitym massVm

=densitymass

volumedensity SI kg/m3

or

m = Vm = V

density of water

w = 1.00 x 103 kg/m3 = 1.00 g/cm3 = 1.00 g/c.c. = 1.00 kg/Ldensity

1 c.c. = 1 mL = 1 cm31 L = 103 cm31 m3 = 103 L = 106 cm3

Volume Volume 1 cm3 (c.c. or mL) water = 1 g water

1 L water = 1 kg water

( ) g

1000

1 L = 103 cm3 (c.c. or mL)

1 cm3 (c.c. or mL) = 10-3 L

m = 10-3

x 103 =

1 L water = 1 kg water

1 m

1 m3 = 103 L1 Lg

1 m3 water = 103 kg water

1000

1 m 1 m1 m3 = 103 L

1 kg water

103 kg water103 kg water

1 m

1 m1 m

Specific GravitySpecific Gravity

w

grsp

=..

specific gravity

= (w )(sp. gr.)

sp. gr.

density density density kg/m3 sp. gr.

water 1.00 x 103 1.00sea water 1.03 x 103 1.03ice 0.917 x 103 0.917mercury 13.6 x 103 13.6gold 19 3 x 103 19 3gold 19.3 x 103 19.3copper 8.92 x 103 8.92iron 7.86 x 103 7.86 air 1.29 1.29 x 10-3

pressure buoyant force buoyant force fluid dynamics

continuityflow rate

Bernoullis Equation pressure

pressurepressure

AFP =pressure

force

pressure pressure SI N/m2Pa

F = PA

or P

FF = PA F

AF = PA

A

P0 pressure P0 pressure

h

P pressuredepth

P0A

P0A + W = PA

m = V V = Ah P0A + Ahg = PAA

h

W = mg = Vg = Ahg

P = P0 + gh

PA

A B pressure

Atmospheric PressureAtmospheric Pressure

Patm = 1.013 x 105 Pa (N/m2)

pressure= pressure

pressure= pressurep

1 29 k / 3 a = 1.29 kg/m3 1 atm = 1.013 x 105 Pa P0 = 0

P = agh => h = P/(ag) = 1.013 x 105/1.29/9.8 =

8000 m

Pressure Measurement Pressure Measurement P0 = 0

barometerP

Patm + gh

manometer

P

pressure

pressure

P = Patm + gh

0 + gh = Patm density

P - Patm = gh

gauge pressure

gauge pressure

density

Barometer Barometer

1 atm = 760 mmHgBarometer 760 mm 0.76 m Hg g

Patm = 1 atm = (13.6)(103)(9.8)(0.76) = 1.013 x 105 N/m2(Pa) g105 N/m2(Pa)

Hg sp. gr. w

g

Hg

barometer barometer 21 atm = 1.013 x 105 N/m2 = wghw

1.013 x 105 N/m2 = (103)(9.8)hw( )( ) w

( )( ) m31010013.1 5

=

=h ( )( ) m3.108.9103wh pressure P0 pressure P 0

13.6 0.76 x 13.6 P0 = 0 > 10.3 m

Patm = P0 + gh

10.3 m

ManometerManometer

pressure P2 = P1 + ghp 2 1 g

P2 P1

b l t absolute pressure 0 P

gauge pressureg g p = -1 atm gauge pressure :

> > 0 < < 0

density

pressure

pressure

pressure

p

P0 + oilgL

Oil 0 750

P0 + wg(L h)

Oil = 0.750 L = 5.00 cm h =

oilgL = wg(L h)h =

( )( ) Oil

( )( ) cm25.1cm00.575.011 oiloil ==

== LLh

ww

w

Pascals LawPascal s Law

P1 = P2

2

2

1

1

AF

AF

=2

1

2

1

AA

FF

=pressure

pressure

P2 = P1 + gh ghAF

AF +=

1

1

2

2

p

AA 1221

96 96

U A = 0.01 m2

il = 500 kg/m3 t = 1000 kg/m3oil 500 kg/m water 1000 kg/m F =

WWWWF airwaterairoil +=++

AA

wateroil WWF =+ h

AhgAhgF woil =+

( ) ( )( )( )( ) N9489100105001000 === AhgF ( ) ( )( )( )( ) N9.48.91.001.05001000 === AhgF oilw

Buoyant Force Buoyant Force P1 pressure P1 pressure

h

P2 pressuredepth

P1A

P A P A

Ah buoyant force B =

dispalcement volume

W

P2A P1A = (P1 + gh)A P1A= ghA V = Ah

d sp ce e vo u e

g= Vg= mg

m = V

P2A

P2 = P1 + gh Archemedes principlebuoyant force =

= B = V g

densitydisplacement volume = object volume

B Vog W = oVog

density

F= B W = ( o)Vog > > F >0

> o > F >0

< o < F

= < B = V'g displacement volume = volume d i B V g W = oVog

volume

density

densityVo

F = B W = 0

density

V'

B = W => V' = oVo

1

= buoyant forcebuoyant force

buoyant force

Fluid DynamicsFluid Dynamics incomprissible

compressible compressible nonviscous

viscous steady

unsteady laminar

turbulent flowirrotational irrotational

rotational

streamlinestreamline

t = 0

t = t

AAAv = const

A2x2 = A1x1A2v2t = A1v1t

2211 vAvA =

or

21

AA

vv

=12 Av

x2 = v2tx1 = v1t

A2 > A1v2 < v1 2 1 2 1 A2 < A1v2 > v1

or flow rate

or flow rate t = 0 t V

V xAxA 2211

v1 v2

t

VQ

= txA

txAQ

=

= 2211

m3/s

V = Qt 2211 vAvAQ ==

V = vAt

A

v

vt

Q = V/t = Av

v

Q = Av A = Q/vQ = Av

A Qv = Q/A

Q1

Q = Q + QQ3 Q1 + Q2

QQ3 = Q1 + Q2

Q

Q1

Q2Q2

pressure Bernoullis Equation

Pressure Pressure

y2 y1

y 0y 0

2

t = 0 t V V

2

Kinetic energy K2 = 22

22 2

121 Vvmv =

Potential energy Ug2 = 22 Vgymgy =

pressure work W2= VPxAPxF 222222 ==

force 1

Kinetic energy K1 =

Potential energy U = Vgymgy =

21

21 2

121 Vvmv =

Potential energy Ug1 = 11 Vgymgy =

pressure work W1= VPxAPxF 111111 ==

force

11

volume

work kinetic energy potential energy

K + Ug = W 22221

211 2

121 gyvPgyvP ++=++

pressure V or

constant1 2 =++ gyvP

pressure V

Bernoullis equation

2gy

Patm

22221

211 2

121 gyvPPgyvPP atmatm ++=++

atm

22

1 gauge pressure 2 gauge pressure

Bernoullis equation P absolute pressure gauge pressure

Bernoullis Equation Bernoulli s Equation

22221

211 2

121 gyvPgyvP ++=++ 222111 22

gygy

g

222

211 11 vPvP

222

111

21

21 y

gv

gPy

gv

gP

++=++ y

= = w

v = v = 0 v1 = v2 = 0

2211 gyPgyP +=+ ( )2112 yygPP += P = P2P = PP = P0 + gh

P0 = P1h = y1- y2

h y

22 11 PP y1 = y2 222211 2

121 vPvP +=+

pressure -> kinetic energy

kinetic energy -> pressure

Pitote tube

pressurePA = P0

P0 pressure

B = 0

Ba PvP =+2

0 21

PAP P = ghBa0 2

21 AB PP 2 AB PPv = 2

a

PB PA = gh density pressure

ABa PPv =2

21

a

AB PPv

= 22a

v

= 2

v

1 y1 = 0 m

v1

y = 0P = Patm

g = 10 0 m/s2

25.00 m

g 10.0 m/s

2

34

5A2 = 0.02 m2

A = 0 04 m23.00 m 4A3 = 0.04 m2

A4 = 0.01 m2

1.00 m

gauge pressure

y y1 = 0 m y2 = -5.00 m y3 = -8.00 my3 y4 = -9.00 m y5 = -5 00 my5 5.00 m

gauge pressure P 0 P1 = 0

P4 = 0

00 0 0

42441

211 2

121 gyvPgyvP wwww ++=++

v1

v1

( ) ( )( )( ) m/s4.1318000.91022 44 ==== ygvh = 9.00hh

Bernoullis Equation continuity

( )( ) /sm134.04.1301.0 3444 === vAQBernoullis Equation pressure

/sm134.0 3432 === QQQ

m/s71.602.0

134.0

2

22 === A

Qv m/s35.304.0134.0

3

33 === A

Qv

22 11 PP

0 0 v2 = 6.71 y2 = -5.000

22221

211 2


P 0 ga ge press re

( )( ) ( )( )( ) Pa107520051010002/71610001 422222 === gyvP

w = 1000 kg/m3 P1 = 0 gauge pressure gauge pressure

( )( ) ( )( )( ) Pa1075.200.51010002/71.610002 222

gyvP ww

P2 = wgh2 = (1000)(10)(5.00) = 5.00 x 104 Pa2 wg 2 ( )( )( )

0

22 11 PP

0 0 v3 = 3.35 y3 = -8.000

32331

211 2


P 0 ga ge press re

( )( ) ( )( )( ) Pa104470081010002/35310001 423233 === gyvP

P1 = 0 gauge pressure gauge pressurew = 1000 kg/m3

( )( ) ( )( )( ) Pa1044.700.81010002/35.310002 333

gyvP ww

P3 = wgh3 = (1000)(10)(8.00) = 8.00 x 104 Pa3 wg 3 ( )( )( )

0

22 11 PP

0 0 0 0 y5 = -5.00

52551

211 2


( )( )( ) Pa1000.500.5101000 455 === gyP w

P5 = wgh5 = (1000)(10)(5.00) = 5.00 x 104 Pa

2 pressure

0 hP 21

y2 = h

v2

0

v = 0 ghvP wwatm ++222

01 2 ++ vP

ghv 21 =v1 = (2gh)1/2

02 1

++ vP watm y1 = 0

v = 0

(2 h)1/2

v 0

v1 = (2gh)1/2

continuity

A2211 vAvA =

continuity

Bernoullis equation

21

21 vA

Av =

021

21 2

122 ++=++ vPghvP watmwwatm

Bernoulli s equation

22

22 11 Ah

( )[ ] 22212 1/2 vAAgh =2

21

222 2

121 v

AAghv

=+

( )[ ] 212g

O !

A1

continuity Bernoullis Equation A1 = A2 ti it1 2v1 = v2P1 = P2

continuityBernoullis Equation

fluid mechanicsfluid mechanics 流體機械 -...

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