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)
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