Elementary Mechanics of Fluids

CE 319 F

Daene McKinney

Control Volumes

Approaches to Solving Fluids Problems

• Experimental Analysis

• Differential Analysis

• Control Volume Analysis– Single most valuable tool available (White, Ch. 3)

Systems

• Laws of Mechanics– Written for systems

– System = arbitrary quantity of mass of fixed identity

– Fixed quantity of mass, m

0dt

dm

dt

md )( VF

dt

dW

dt

dQ

dt

dE

• Conservation of Mass– Mass is conserved and

does not change

• Momentum– If surroundings

exert force on system, mass will accelerate

• Energy– If heat is added to

system or work is done by system, energy will change

Control Volumes

• Solid Mechanics– Follow the system, determine

what happens to it• Fluid Mechanics

– Consider the behavior in a specific region or Control Volume

• Convert System approach to CV approach– Look at specific regions, rather

than specific masses• Reynolds Transport Theorem

– Relates time derivative of system properties to rate of change of property in CV

)(extensiveenergymomentum,mass,

CVCV

dbbdmB

)(intensivemassunitperofamount Bdm

dBb

CV Inflow & Outflow

Area vector always points outward from CV

AV

Q

CS

inout AVAVQQ

AV

AVAV

1122

1122

CV Inflow & Outflow

CSCS

inoutinoutnet mbbmbmbBBB

AV

Bmb

inoutttCVttsys

inoutttCVttsys

BBBB

MMMM

,,

,,

Reynolds Transport Theorem

CSCV

sys

netCV

inout

t

tCVttCV

t

tCVinoutttCV

t

tCVtt

t

sys

bdbdt

d

dt

dB

Bdt

dBt

BB

t

BBt

BBBBt

BB

dt

dB

AV

0

,,

0

,,

0

,

0

limlim

lim

lim

Steady vs. Unsteady CV

CSCV

sysbdb

dt

d

dt

dBAV

CS

sysb

dt

dBAV

Continuity Equation

• Reynolds Transport Theorem

)(extensivesysMB

)(intensive1dm

dM

dm

dBb

sys

CSCV

sysbdb

dt

d

dt

dBAV

CSCV

ddt

dAV

0 CS

AV

0

Unsteady Case Steady Case

Example (4.57)

• Continuity equation

smV

gVx

AVAVdt

dhA

AVAVhAdt

d

ddt

d

in

in

outoutinintank

outoutinintank

CSCV

/47.4

)0025.0(1*2)0025.0(101.0*1.0

)(

0

2

AV

Example (4.61)• Select a CV that moves up

and down with the water surface

• Continuity Equation

CS

VA=2VB VB

risingissurface0

)(20

;6;3

20

0

21

41

412

42

4

BB

BBBB

BABA

BBABCV

CSCV

AVdt

dhA

AVAVdt

dhA

AAAA

AVAVddt

d

ddt

d

AV

h

HW (4.80)

HW (4.81)

HW (4.82)