Chapter 2
Fluid Properties: Density, specific volume, specific weight,
specific gravity, compressibility, viscosity, measurement of
viscosity, Newton's equation of viscosity, Surface tension,
capillarity and pressure
Dr. Muhammad Ashraf Javid
Assistant Professor
Department of Civil and Environmental Engineering
1
Fluid Engineering Mechanics
Physical Properties of Fluids
2
Density
Specific Volume
Specific Weight
Specific Gravity
Compressibility
Viscosity
Surface Tension
Pressure
Buoyancy
Density
3
It is also termed as specific mass or mass density.
It is the mass of substance per unit volume .i.e., mass of fluid per
unit volume.
It is designated with symbol of ρ (rho)
ρ =mass/volume
=M/L3
Fundamental Units=kg/m3, slug/m3, g/cm3
4
2
3
2
L
FT
L
M
L
FT
a
FMMaF
Note: Density of water at 4oc=1000kg/m3, 1.938slug/ft3, 1g/cm3
Specific Volume
4
It is defined as volume of substance per unit mass.
It is designated with υ.
MLmassvolume // 3
MaF
2
4
FT
L
Fundamental Units=m3/kg, m3/slug, cm3/g
Relationship Between Density and Specific Volume
5
3
3
//
//
LMvolumemass
MLmassvolume
/1
/1
Specific Weight
6
It is the weight of substance per unit volume or say it is
the weight of fluid per unit volume.
It is designated by γ (gamma).
3L
W
volume
weight
3L
Mg MgW
2223 TL
M
TL
ML 2T
Lg
Note: Specific weight of water at 4oc=9810N/m3, 62.4lb/ft3, 981dyne/cm3
Relation Between and
7
volume
weight
volume
Mass &
g
MgW
Effect of Temperature and Pressure on
Specific Weight
8
As the equation of state for a
perfect gas is given by
Where
P =absolute pressure
υ =specific volume
T =absolute temperature
R =gas constant
For perfect gases
mR=8312 N-m/(kg-k)
Where
m=molecular weight of gas
RTP
1RT
P
RT
g
P
/ g
RT
gP
T
Pconstant)(
R
gconstant,
TP
1&
RTP
Effect of Temperature and Pressure on
Specific Weight
9
Since
Assuming constant
pressure
Assuming constant
temperature
V
W
volume
weight
nT
1
nP
1,0n
1,0n
Specific Gravity (Relative Density)
10
It is the ratio of density of a substance and density of water at 4oC.
It is the ratio of specific weight of substance and specific weight of
water at 4oC.
It is the ratio of weight of substance and weight of an equal volume
of water at 4oC.
water
fluid
water
fluid
water
fluid
W
WS
Remember: T
P1
&
CTatW
CTatW
CTat
CTat
CTat
CTatS
o
water
o
fluid
o
water
o
fluid
o
water
o
fluid
Note: Specific gravity of liquid is measured w.r.t. water while for
cases of gases it is measured w.r.t. standard gas (i.e. air)
Class Problems
11
12
Compressibility
Compressible fluids
Incompressible fluids
In fluid mechanics we deal with both compressible and
incompressible fluids of either variable or constant density.
Although there is no such thing in reality as incompressible fluid, we
use this terms where the change in density with pressure is so small
as to be negligible. This is usually the case with liquids.
Ordinarily, we consider the liquids as incompressible.
We may consider the gases to be incompressible when the pressure
variation is small compared with absolute pressure.
Compressibility
14
Compressible fluids
Fluids which can be compressed.
Fluid in which there is a change in volume with change in pressure
P1
P2
v1
v2
12 PP
12 vv
As a result of change in volume, density and specific weight of fluid also changes. Hence, for compressible fluids,
21
21
21
vv
Compressibility
15
Incompressible fluids
Fluids which can not be compressed.
Fluid in which there is no change in volume with change in pressure
P1 P2
v1
v2
12 PP
12 vv
As a result, no-change in volume, density and specific weight of fluid. Hence, for incompressible fluids,
21
21
21
vv
v2
Compressibility (Volumetric Strain)
16
Volumetric Strain is the ratio of change in volume and original volume.
P1
P2
v1
v2
12 PP
12 vv
11
21
1
21
21
1
21
/
//
v
dv
v
vv
Mv
MvMv
Volumetric strain=change in specific volume/original specific volume.
1
21
1
21
d
Compressibility
17
Bulk Modulus or Volume Modulus of Elasticity (Ev):
It is defined as ratio of volumetric stress to volumetric strain
Ev= volumetric stress/volumetric strain
Ev=change in pressure/compressibility
1v
dv
dpEv
1
d
dpEv
Viscosity
18
The viscosity of a fluid is a measure of its resistance to shear or angular
deformation.
It is the property of a fluid by mixture of which it offers resistance
to deformation under the influence of shear forces. It depends
upon the cohesion and molecular momentum exchange between
fluid layers.
It can also be defined as internal resistance offered by fluid to flow.
It is denoted by μ.
It is also termed as coefficient of viscosity or absolute viscosity or
dynamic viscosity or molecular viscosity.
Factor affecting viscosity
19
1. Cohesion
2. Molecular momentum
1. Cohesion: It is the attraction between molecules of fluid. More
the molecular attraction (cohesion) more is the viscosity (resistance
to flow) of fluid.
It is dominant in liquids.
2. Molecular momentum: Molecules in any fluid change their
position with time and is known as molecular activity. More the
molecular activity more will be viscosity of the fluid.
It is dominant in gases
A B
Effect of temperature on viscosity
20
For Liquids:
In case of liquids, cohesion (molecular
attraction is dominant). Therefore, if the
temperature of liquid is increased, its
cohesion and hence viscosity will
decrease.
For Gases:
In gases momentum exchange is
dominant. Therefore, if the temperature
of gases is increases, its momentum
exchange will increase and hence
viscosity will increase.
T
1
T
Kinematic Viscosity
21
It is ratio of absolute viscosity and density of fluid.
It is denoted by (nu)
Newton’s Equation of Viscosity
22
Consider two parallel plates, in which lower plate is fixed and upper is
moving with uniform velocity ‘U’ under the influence of force ‘F’. Space
between the plates is filled with a fluid having viscosity, μ.
F= Applied force (shearing force)
A= Contact area of plate(resisting area)
Y=gap/space between plates
U= Velocity of plate
As the upper plate moves, fluid also moves in the direction of applied force
due to adhesion.
Y
u
dy
du
U Moving plate
Fixed plate
Force, F
Newton’s Equation of Viscosity
23
Factors affecting Force, F
Hence,
Where, μ is coefficient of viscosity
Assuming linear velocity profile (as shown in figure)
YFiiiUFiiAFi
1)(;)(;)(
Y
AUF
Y
AUF
dy
du
Y
U
A
F
Newton’s Equation of Viscosity
24
At boundaries the particles of fluid adhere to wall and so their velocities are zero relative to wall. This so called non-slip condition occurs in viscous fluids
Newton’s Equation of Viscosity
25
The above equation is called as Newton’s equation of viscosity.
The equation shows that the shearing stress is directly proportional
to the velocity gradient.
In the above equation
du/dy= velocity gradient or rate of change of deformation or shear
rate
μ = absolute viscosity
τ=shear stress
dy
du
Dimensional Analysis of Viscosity
26
Viscosity
This expression is used to write
fundamental unit of viscosity
Kinematic Viscosity
2
2 )/(
L
FT
TLL
FL
U
Y
A
F
T
L
L
MLT
M2
3
LT
M
MLTFL
TMLT
2
2
2
Unit of Viscosity
27
Viscosity
Widely used unit is Poise =0.1N.s/m2
Kinematic Viscosity
Widely used unit is Stoke=10-4m2/s
LTM /
TL /2
SI BG CGS
N-s/m2 Lb-s/ft2 Dyne-s/cm2
(Poise, P)
Kg/(m-s) Slug/(ft-s) g/(cm-s)
SI BG CGS
m2/s ft2/s cm2/s
(stoke)
Problem
28
smxx
/1088.5850
105 263
Problem
29
A flat plate 200mm x 750mm slide on oil (μ =0.85N.s/m2) over a
large surface as shown in fig. What force, F, is required to drag the
plate at a velocity u of 1.2m/s if the thickness of the separating oil
film is 0.6mm?
NF
AY
UF
AY
UA
dy
duAF
dy
du
Y
U
A
F
255
7.02.01000/6.0
2.185.0
Here, t = Y
Problem
30
A space 16mm wide between two large plane surfaces is filled with
SAE 30 Western lubricating oil at 35oC(Fig). What force is required
to drag a very thin plate of 0.4m2 area between the surfaces at a
speed u=0.25m/s (a) if this plate is equally spaced between the two
surfaces? (b) if t=5mm?
Solution:
Y=16mm A=0.4m2 u=0.25m/s
T=35oC μ =0.18N.s/m2 (from figure A.1)
F=? If Y=8mm
(a)
8mm dy
du
Y
U
A
F
2121 AAFFF
Solution to Problem
31
8mm
N
Ay
uA
y
uF
AAFFF
5.4
4.01000/8
25.018.04.0
1000/8
25.018.0
21
21
2121
Solution to problem
32
(b): F=? If t =5mm
y1=11, y2=5mm
N
Ay
uA
y
uF
AAFFF
24.54.01000/11
25.018.04.0
1000/5
25.018.0
21
21
2121
5mm
33
Shear Stress ~ Velocity gradient curve
34
Ideal fluid
Newtownian Fluid
Non-Newtownian fluid
Ideal plastic
Real solid
Ideal solid/elastic solid
Real solid
Shear Stress ~ Velocity gradient curve
35
Ideal Fluid: The fluid which does not offer resistance to flow
Newtownian Fluid: Fluid which obey Newtown’s law of viscosity
slope of curve ( )is constant
Non-Newtonian fluid: Fluid which does not obey Newtown’s Law
of viscosity
slope of curve ( )changing continuously
dy
du
00
dy
du
dydu /~
dydu /~
Shear Stress ~ Velocity gradient curve
36
Ideal Solid: solid which can never be deformed under the action of
force
Real solid: solid which can be deformed under action of forces
Ideal Plastic: These are substances which offer resistance to shear
forces without deformation upon a certain extent but if the load is
further increased then they deform
Real Plastic: These are substances in which there is deformation
with the application of force and it increases with increase in applied
load.
0dy
du
Problem
37
38
Exercise Problems
39
Measurement of Viscosity
40
The following devices are used for the measurement of viscosity
1. Tube type viscometer
2. Rotational type viscometer
3. Falling sphere type viscometer
Falling Sphere Viscometer
41
It consists of a tall transparent
tube or cylinder and a sphere of
known diameter.
The sphere is dropped inside the
tube containing liquid and time of
fall of sphere between two points
(say A and B) is recorded to
estimate the fall velocity (s/t)of
sphere inside liquid.
Where ‘s’= distance between
point A and B and ‘t’ is the time
of travel.
From this velocity of fall, viscosity
is estimated from the expression
of fall sphere type viscometer.
s
A
B
Fig. Falling Tube type viscometer
W
FB
FD
Dt
D
Falling Sphere Viscometer
42
D=Ds=Diameter of sphere
Dt=Diameter of tube or cylinder
Vt=velocity of sphere in tube (s/t)
s=Distance between points A and B
t=time taken by sphere to cover
distance (s)
W=weight of sphere=γ*(Vol)
= γs(πD3/6)
FB=Force of Buoyancy
= γL(πD3/6)
FD=Drag force
= (3πμVD)
Note: V is not equal to Vt
s
A
B
Fig. Falling Tube type viscometer
W
FB
FD
Dt
D
Stoke’s Law
Falling Sphere Viscometer
43
Buoyancy: It is the resultant upward thrust exerted by the fluid on a sphere. It is the tendency of fluid to lift the body and it is equal to weight of volume of fluid displaced by the body (Archimedes Principal).
Drag Force: It is a resisting force generated by the liquid on the moving object which is acting in the opposite direction of movement .
Vt=velocity of sphere in tube with wall effect
V=velocity of sphere in tube without wall effect
V>Vt
...4
9
4
91
2
ttt D
D
D
D
V
V
3
1
tD
Dif
63
6
;0;0
33 DVD
D
WFFF
SL
DBy
Falling Sphere Viscometer
44
The above equation is governing equation for falling sphere type
viscometer.
For a particular temperature, D, γs and γL are constant. So we can write ;
Thus, velocity of fall is inversely proportional to viscosity and is indicative of viscosity in falling sphere type viscometer.
Note: This method can only be used for transparent liquids
LS
LSLS
V
D
DDV
DDVD
18
663
663
2
2233
V
1
Problem: 11.1.10
45
Problem: 11.1.10
46
Tube Type Viscometer
47
In tube type viscometer, liquid is
placed in a container to a certain
level. Valve in the bottom is opened
to fill the flask of known volume.
Time taken to fill the flask is
recorded which gives measure of
viscosity of liquid
H’=Average imposed head causing
flow=H+L-h/2
VL=volume of flask
D=Diameter of tube
L=length of tube
h=fall of liquid level in container to
fill liquid in flask
H’
h H
V
Tube Type Viscometer
48
Lets consider two points 1 and 2
and apply energy equation.
Where, are potential,
pressure, and velocity head and
HL=head loss
The head loss in tube type
viscometer is due to friction loss in
tube and is represented by
H’
h H
V
1
2 Datum
LHg
VPz
g
VPz
22
2
222
2
111
LHg
VH
20000'
2
2
g
VPz
2&,
2
VD
LHH FL 2
32
(ii)
(i)
Tube Type Viscometer
49
Eq (ii) is a Hagen Poiseulli law for laminar flows in tube.
Moreover, for laminar flow in tube viscometer V<< 1 therefore V2 can
be neglected. Hence Eq (i) becomes as
VD
LH
VD
LH
2
2
32'
3200000'
VD
LHH FL 2
32
L
HDQ
A
Q
D
LH
128
'
32'
4
2
2
4DA
AVQ
QL
HD
128
'4
Equation for tube
type viscometer
A=Cross-sectional area
of tube and V is average
velocity
Tube Type Viscometer
50
Where
kt = constant of tube type
viscometer
QL
HD
128
'4
t
V
time
VolumeQ L
tLV
HgD
L
128
'4
hdVL
2
4
t
tKt
Note: Equation of tube type viscometer is applicable for laminar
flows. For flows in pipes, flow will be laminar if Re≤2000 and flow
will be turbulent if Re>4000
2000
VDReno Reynolds
Re=2000-4000 Transition flow
Problem; 11.1.7
51
Solution:
D=0.0420in= .0420/12 ft
L=3.1in=3.1/12ft
VL=60mL=0.00212 ft3
H’=(10+9.5)/2=9.75 in
t=128.7sec
tLV
HgD
L
128
'4
sft /0000227.0/ 2
=H’
Problem
52
Rotational Type Viscometer
53
It consists of two concentric
cylinders. Small cylinder is
placed inside the bigger cylinder.
The gap (space) between
cylinders is filled with liquid up
to a height, h,.
Then either inner or outer
cylinder is fixed and other is
rotated by applying a constant
torque.
Revolution per minute (RPM) is
measured which is indicative of
viscosity.
h
r1
r2
Δr
Rotational Type Viscometer
54
r1=outer radius of inner cylinder
r2=inner radius of outer cylinder
r=mean radius=(r1+r2)/2
Δr=gap (space) between cylinders
h=height of liquid
F=shearing force
T=Applied torque (mean torque)=F x r
h
r1
r2
Δr
ArT
AF
Ardy
duT
dy
du
r
uAr
r
uT
AF
(1)
Rotational Type Viscometer
h
r1
r2
Δr
Where
N=RPM(Revolution per minute)
ω=angular frequency
hrA 2area Resisting
60
2
VelocityAngular
Nru
ru
radian/min 60/2/minrevolutionN
radian/s 2/srevolutionN
radian/s 2/srevolution 1
radian 2revolution 1
N
N
Rotational Type Viscometer
56
Now substituting the values of
A and u in Eq. (1)
Where
Kr=Rotational viscometer
constant
NKT
Nr
hrT
rrhr
rNT
r
15
260
2
32
r
hrKr
15
32
By re-arranging the formula, the
absolute viscosity using
rotational type viscometer can
be obtained as
For a particular viscometer,
both T and Kr are constant and
therefore
Nk
T
r
1
N
1
Problem: 11.1.6
57
Solution:
Rotation type viscometer
h=300mm=0.3m
OD of inner cylinder=100mm=0.1m
ID of outer cylinder=102mm=0.102m
Δr=(102-100)/2=1mm=0.001m
Torque=T= 8 N-m
N=1/4 rev/s=60/4 RPM
Neglect mechanical friction
h=300mm
r1=50mm
r2=51mm Δr
Problem: 11.1.6
58
r
hrK
NK
TNKT
r
r
r
15
32
2/4.20 mNs
Surface Tension
59
The tension force created at the imaginary
thin surface due to unbalanced-molecular
attraction is termed as surface tension.
v2 A
B
Molecule A in figure above is situated at a certain depth below the
surface. It is acted upon by equal force from all sides whereas
molecule B (situated at the surface) is acted upon by unbalanced
forces from below.
Thus a tight skin/film/surface is formed at the surface due to inward
molecular pull.
Types of molecular attraction
60
Cohesion: It is the attraction force between the molecules of same
material
Adhesion: It is the attraction force between the molecules of
different materials
Surface tension depends upon the relative magnitude of cohesion
and adhesion but primarily it depend upon the cohesion.
With the increase in temperature cohesion reduces and hence
surface tension also reduces.
Concept of surface tension is used in capillarity action
Capillarity
61
It is the rise or fall of a liquid in a small diameter (< 0.5”) tube due
to surface tension and adhesion between liquid and solid.
For capillary action diameter of tube is less than 0.5inch while for
large diameter tubes this phenomenon become negligible.
The curved surface that develops in tube is called meniscus
v2
h
σ θ
D
v2
h
σ
θ
D
Water Mercury θ<90 θ>90
Capillarity
62
D= diameter of tube
γ=specific weight of liquid
h=capillary rise/fall
θ=angle of contact or contact angle
σ=force of surface tension per unit length v2
h
σ θ
D
Derivation of expression for capillary rise/fall
Let’s take
Weight of column of liquid acting downward=
Vertical component of force of surface tension=
0Fy
hDvol 2
4
cosD
Capillarity
63
Equating both equations
The above equation is used to compute capillary rise/fall.
Note: Fall has –ve sign
hDD 2
4cos
Dhor
hD
hDD
cos4
cos4
4cos 2
Problem. 2.29
64
Solution:
At 50oF
With θ=0o
True static height=6.78-1.174in=5.61in
inft
Dh
174.10979.0
12/04.041.62
00509.04cos4
ftlbftlb /00509.0,/41.62 3
Surface Tension of water
65
Temperature
- t -
(oF)
Surface Tension of Water in
contact with Air
- σ -
(10-3 lb/ft)
32 5.18
40 5.13
50 5.09
60 5.03
70 4.97
80 4.91
90 4.86
100 4.79
120 4.67
140 4.53
160 4.40
180 4.26
200 4.12
212 4.04
Vapor Pressure of liquids
66
Vapor Pressure: It is the pressure at which liquid transforms into
vapors or it is the pressure exerted by vapors of liquid.
All the liquids have tendency to
release their molecules in the space
above their surface.
If the liquid in container have limited
space above it, then the surface is filled
with the vapors.
These vapors when released from liquid exert pressure known as
vapor pressure.
It is the function of temperature. More the temperature more will
be vapor pressure.
Vapor Pressure of liquids
67
Saturated vapor pressure: It is the vapor pressure that
corresponds to the dynamic equilibrium conditions (saturation) i.e.,
when rate of evaporation becomes equal to rate of condensation.
Boiling vapor pressure: The pressure at which vapor pressure
becomes equal to atmospheric pressure.
Effect of Temperature and Pressure on Ev
of Water
68
Pressure, dp Temp, T
Ev Ev
50oC
Ev(max)
Temperature is constant Pressure is constant
Relation Between Ev and Compressibility
69
As
1v
dv
dpEv
1
1
v
dvEv
ilitycompressibEv
1
Problem
70
1
12
1
dp
d
dpEv
/1
/1
g
Solution
71
Sample MCQs
72
1.Specific gravity of a liquid is equal to
(a). Ratio of mass density of water to mass density of liquid
(b). inverse of mass density
(c). Ratio of specific weight of liquid to specific weight of water
(d). None of all
2. What happens to the viscosity of a liquid when its temperature is raised?
(a). The viscosity of the liquid increases
(b). The viscosity of the liquid stays the same
(c). The viscosity of the liquid decreases
(d).The temperature of a liquid does not rise
3. What happens to the specific weight of a liquid when its temperature is raised?
(a). It increases
(b). It stays the same
(c). It decreases
(d). The temperature of a liquid does not rise
Sample MCQs
73
4. In a falling sphere viscometer, according to force balance we have
(a). Weight=Drag+Buoyancy
(b). Drag=Weight - Buoyancy
(c). Buoyancy=Weight +Drag
(d). None of all
5. The kinematic viscosity is
(a). Multiplication of dynamic viscosity and density
(b). division of dynamic viscosity by density
(c). Multiplication of dynamic viscosity and pressure
(d). None of the above
6. As a result of capillary action (a) liquid rise in capillary tube
(b) liquid falls down in capillary tube
© Both (a) and (b)
(d) None of all
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