1-Considerações Básicas

Sandro R. Lautenschlager

Mecânica dos Fluidos

Aula 1

Mecânica dos Fluidos

• Definição– Estudo dos líquidos e gases no qual não há

movimento (estáticos) e naqueles no qual há movimentos (dinâmica)

HistoryFaces of Fluid Mechanics

Archimedes(C. 287-212

BC)

Newton(1642-1727)

Leibniz(1646-1716)

Euler(1707-1783)

Navier(1785-1836)

Stokes(1819-1903)

Reynolds(1842-1912)

Prandtl(1875-1953)

Bernoulli(1667-1748)

Taylor(1886-1975)

Weather & ClimateTornadoes

HurricanesGlobal Climate

Thunderstorm

VehiclesAircraft

Submarines

High-speed rail

Surface ships

Environment

Air pollution River hydraulics

Physiology and Medicine

Blood pump Ventricular assist device

Sports & Recreation

Water sports

Auto racing

Offshore racingCycling

Surfing

Aplicação em engenharia

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Aplicação em engenharia

Aplicação em engenharia

Aplicação em engenharia

Propriedades dos Fluidos

• Peso específico

g Onde:

g-gravidade local (9,8 m/s2);

-massa especifica (kg/m3);

-peso específico (kg/m2s2) ou N/m3

Propriedades dos Fluidos

• Densidade

águaágua

S

Temperatura de referência 40C

Variação de com a temperatura

e

180

41000

2

2

TOH

18

49800

2

2

TOH

Massa Específica

992.0993.0994.0995.0996.0997.0998.0999.0

1000.01001.0

-15 -10 -5 0 5 10 15 20 25 30 35 40 45

Temperatura (0C)

Mas

sa e

spec

ifica

(kg/

m3) Massa Especifica (H2O)

180

41000

2

2

TOH

)4( )0( 02

02 COHCOH

9720.0

9730.0

9740.0

9750.0

9760.0

9770.0

9780.0

9790.0

9800.0

9810.0

-15 -10 -5 0 5 10 15 20 25 30 35 40 45

Temperatura (0C)

Mas

sa e

spec

ific

a (k

g/m

3 )

Peso Especifico (H2O)

Peso Específico

18

49800

2

2

TOH

Massa Especifica

(kg/m3)

Peso Especifico

(N/m3)

Densidade

Ar 1,23 12,1 0,00123

Água 1000 9810 1

Nas condições Normais

Some Simple Flows• Flow between a fixed and a moving

plateFluid in contact with the plate has the same velocity as the plate

u = x-direction component of velocity

u=VMoving plate

Fixed plate

y

x

V

u=0

B yB

Vyu )( Fluid

Some Simple Flows

• Flow through a long, straight pipeFluid in contact with the pipe wall has the same velocity as the wall

u = x-direction component of velocity

rx

R

21)(

R

rVru

VFluid

Fluid Deformation• Flow between a fixed and a moving

plate• Force causes plate to move with

velocity V and the fluid deforms continuously.

u=VMoving plate

Fixed plate

y

xu=0

Fluid

t0 t1 t2

u=V+VMoving plate

Fixed plate

y

x

u=V

Fluid

Fluid Deformation

t t+t

x

y

L

t Shear stress on the plate is proportional

to deformation rate of the fluid

V

Lt

y

L

y

V

t

y

V

Shear in Different Fluids

• Shear-stress relations for different typesof fluids

• Newtonian fluids: linear relationship

• Slope of line (coefficient of proportionality) is “viscosity”

dy

dV

dy

dV

Viscosity• Newton’s Law of Viscosity

• Viscosity

• Units

• Water (@ 20oC)– = 1x10-3 N-s/m2

• Air (@ 20oC)– = 1.8x10-5 N-s/m2

• Kinematic viscosity

dydV /

2

2

//

/

m

sN

msm

mN

V

V+dv

dy

dV

Flow between 2 plates

u=VMoving plate

Fixed plate

y

x

V

u=0

B yB

Vyu )( Fluid Force acting

ON the plate

21

21

222111

AA

FAAF

221

1 dy

du

dy

du

Thus, slope of velocity profile is constant and velocity profile is a st. line

Force is same on topand bottom

Flow between 2 plates

u=VMoving plate

Fixed plate

y

x

V

u=0

B yB

Vyu )(

B

V

dy

du Shear stress anywherebetween plates

Shearon fluid

mB

smV

CSAEmsN o

02.0

/3

)38@30(/1.0 2

2

2

/15

)02.0

/3)(/1.0(

mN

m

smmsN

Flow between 2 plates

• 2 different coordinate systems

rx

B

21)(

B

rVru

Vy

x yByCyu )(

Example: Journal Bearing

• Given– Rotation rate, = 1500 rpm– d = 6 cm– l = 40 cm– D = 6.02 cm

– SGoil = 0.88

– oil = 0.003 m2/s

• Find: Torque and Power required to turn the bearing at the indicated speed.

Example: cont.

• Assume: Linear velocity profile in oil film

2/1242/)0002.0(

)2/06.0(1500*602

)003.0*1000*88.0(

2/)(

)2/( StressShear

mkN

dD

d

dy

dV

mN

dl

dM

2812

06.0)4.0*

2

06.0*000,124*2(

2)

22(Torque

kWsmNMP 1.44/100,441.157*281Power

Compressibilidade

TT

TTV

p

V

pV

p

VV

pB

/

lim/

lim00

Módulo de elasticidade volumétrica

Módulo de elasticidade para água

• B=2100MPa ou 21000x atm

Mudança 1% OH2

21Mpa (210atm)

Líquidos podem ser considerados incompressíveis

Cálculo velocidade do som

Bp

cT

Exemplo para água c = 1450m/s (condições normais )

Data da Provas

• P1 17/08/11 Cap. 1 e 2;

• P2 05/10/11 Cap. 3 e 4;

• P3 21/11/11 Cap 4 e 6.

• Exame 5/12/11

final avaliação realizar Deverá 0,6Média

direta Aprovação 0,6Média3

NPNPNPMédia 321

Avaliação final Cap. 1, 2, 3, 4 e 6