Fluid Mechanics Formulas for Civil Engineering Exam | Fluid Mechanics for Civil Engineering - Civil Engineering (CE) PDF Download

 Page 1


Civil ENGINEERING – FLUID MECHANICS
Pressure (P): 
? If F be the normal force acting on a surface of area A in contact with liquid, then
pressure exerted by liquid on this surface is: A F P / ?
? Units :
2
/ m N or Pascal (S.I.) and Dyne/cm
2
 (C.G.S.)
? Dimension : ] [
] [
] [
] [
] [
] [
2 1
2
2
? ?
?
? ? ? T ML
L
MLT
A
F
P 
? Atmospheric pressure: Its value on the surface of the earth at sea level is nearly
2 5
/ 10 013 . 1 m N ? or Pascal in S.I. other practical units of pressure are atmosphere,
bar and torr (mm of Hg)
? torr 760 bar 01 . 1 10 01 . 1 1
5
? ? ? ? Pa atm
? Fluid Pressure at a Point:  
dF
dA
? ?
Density ( ? ): 
? In a fluid, at a point, density ? is defined as: 
dV
dm
V
m
V
?
?
?
?
? ? 0
lim ?
? In case of homogenous isotropic substance, it has no directional properties, so is a
scalar.
? It has dimensions ] [
3 ?
ML and S.I. unit kg/m
3
 while C.G.S. unit g/cc with
3 3
/ 10 / 1 m kg cc g ?
? Density of body = Density of substance
? Relative density or specific gravity which is defined as :
of water  Density
of body Density
? RD
? If 
1
m mass of liquid of density 
1
? and 
2
m mass of density 
2
? are mixed, then as 
2 1
m m m ? ? and ) / ( ) / (
2 2 1 1
? ? m m V ? ? [As ? / m V ? ] 
) / ( ) / ( ) / (
2 2 1 1
2 1
i i
i
p m
m
m m
m m
V
m
?
?
?
?
?
? ?
? ?
?
If 
2 1
m m ? , ?
?
?
2 1
2 1
2
? ?
? ?
? Harmonic mean 
? If
1
V volume of liquid of density 
1
? and 
2
V volume of liquid of density 
2
? are 
mixed, then as: 
2 2 1 1
V V m ? ? ? ? and 
2 1
V V V ? ? [As V m / ? ? ] 
If V V V ? ?
2 1
2 / ) (
2 1
? ? ? ? ? = Arithmetic Mean 
Page 2


Civil ENGINEERING – FLUID MECHANICS
Pressure (P): 
? If F be the normal force acting on a surface of area A in contact with liquid, then
pressure exerted by liquid on this surface is: A F P / ?
? Units :
2
/ m N or Pascal (S.I.) and Dyne/cm
2
 (C.G.S.)
? Dimension : ] [
] [
] [
] [
] [
] [
2 1
2
2
? ?
?
? ? ? T ML
L
MLT
A
F
P 
? Atmospheric pressure: Its value on the surface of the earth at sea level is nearly
2 5
/ 10 013 . 1 m N ? or Pascal in S.I. other practical units of pressure are atmosphere,
bar and torr (mm of Hg)
? torr 760 bar 01 . 1 10 01 . 1 1
5
? ? ? ? Pa atm
? Fluid Pressure at a Point:  
dF
dA
? ?
Density ( ? ): 
? In a fluid, at a point, density ? is defined as: 
dV
dm
V
m
V
?
?
?
?
? ? 0
lim ?
? In case of homogenous isotropic substance, it has no directional properties, so is a
scalar.
? It has dimensions ] [
3 ?
ML and S.I. unit kg/m
3
 while C.G.S. unit g/cc with
3 3
/ 10 / 1 m kg cc g ?
? Density of body = Density of substance
? Relative density or specific gravity which is defined as :
of water  Density
of body Density
? RD
? If 
1
m mass of liquid of density 
1
? and 
2
m mass of density 
2
? are mixed, then as 
2 1
m m m ? ? and ) / ( ) / (
2 2 1 1
? ? m m V ? ? [As ? / m V ? ] 
) / ( ) / ( ) / (
2 2 1 1
2 1
i i
i
p m
m
m m
m m
V
m
?
?
?
?
?
? ?
? ?
?
If 
2 1
m m ? , ?
?
?
2 1
2 1
2
? ?
? ?
? Harmonic mean 
? If
1
V volume of liquid of density 
1
? and 
2
V volume of liquid of density 
2
? are 
mixed, then as: 
2 2 1 1
V V m ? ? ? ? and 
2 1
V V V ? ? [As V m / ? ? ] 
If V V V ? ?
2 1
2 / ) (
2 1
? ? ? ? ? = Arithmetic Mean 
Civil ENGINEERING – FLUID MECHANICS
? With rise in temperature due to thermal expansion of a given body, volume will
increase while mass will remain unchanged, so density will decrease, i.e.,
) 1 ( ) / (
) / (
0
0 0
0 0
? ? ?
?
? ?
? ? ?
V
V
V
V
V m
V m
[As ) 1 (
0
? ? ? ? ? V V ] 
 or 
) 1 ( –
~
) 1 (
0
0
? ? ?
? ?
?
? ? ?
? ?
? 
? With increase in pressure due to decrease in volume, density will increase, i.e.,
V
V
V m
V m
0
0 0
) / (
) / (
? ?
?
?
[As
V
m
? ? ] 
? By definition of bulk-modulus:
V
p
V B
?
?
? ?
0
i.e., 
?
?
?
?
?
? ?
? ?
B
p
V V 1
0
 
?
?
?
?
?
? ?
? ? ?
?
?
?
?
? ?
? ?
?
B
p
B
p
1
~
1
0
1
0
? ? ?
Specific Weight ( w ): 
? It is defined as the weight per unit volume.
? Specific weight 
.
.
Weight m g
g
Volume Volume
? ???
Specific Gravity or Relative Density (s): 
? It is the ratio of specific weight of fluid to the specific weight of a standard fluid.
Standard fluid is water in case of liquid and H
2
 or air in case of gas.
.
.
w w w
g
s
g
? ? ?
? ? ?
? ? ?
Where, 
w
? ? Specific weight of water, and 
w
? ? Density of water specific. 
Specific Volume ( v ): 
? Specific volume of liquid is defined as volume per unit mass. It is also defined as the
reciprocal of specific density.
? Specific volume 
1 V
m ?
??
Inertial force per unit area = 
A
dt dm v
A
dt dp ) / ( /
? = 
A
Av v ?
 = ?
2
v 
Page 3


Civil ENGINEERING – FLUID MECHANICS
Pressure (P): 
? If F be the normal force acting on a surface of area A in contact with liquid, then
pressure exerted by liquid on this surface is: A F P / ?
? Units :
2
/ m N or Pascal (S.I.) and Dyne/cm
2
 (C.G.S.)
? Dimension : ] [
] [
] [
] [
] [
] [
2 1
2
2
? ?
?
? ? ? T ML
L
MLT
A
F
P 
? Atmospheric pressure: Its value on the surface of the earth at sea level is nearly
2 5
/ 10 013 . 1 m N ? or Pascal in S.I. other practical units of pressure are atmosphere,
bar and torr (mm of Hg)
? torr 760 bar 01 . 1 10 01 . 1 1
5
? ? ? ? Pa atm
? Fluid Pressure at a Point:  
dF
dA
? ?
Density ( ? ): 
? In a fluid, at a point, density ? is defined as: 
dV
dm
V
m
V
?
?
?
?
? ? 0
lim ?
? In case of homogenous isotropic substance, it has no directional properties, so is a
scalar.
? It has dimensions ] [
3 ?
ML and S.I. unit kg/m
3
 while C.G.S. unit g/cc with
3 3
/ 10 / 1 m kg cc g ?
? Density of body = Density of substance
? Relative density or specific gravity which is defined as :
of water  Density
of body Density
? RD
? If 
1
m mass of liquid of density 
1
? and 
2
m mass of density 
2
? are mixed, then as 
2 1
m m m ? ? and ) / ( ) / (
2 2 1 1
? ? m m V ? ? [As ? / m V ? ] 
) / ( ) / ( ) / (
2 2 1 1
2 1
i i
i
p m
m
m m
m m
V
m
?
?
?
?
?
? ?
? ?
?
If 
2 1
m m ? , ?
?
?
2 1
2 1
2
? ?
? ?
? Harmonic mean 
? If
1
V volume of liquid of density 
1
? and 
2
V volume of liquid of density 
2
? are 
mixed, then as: 
2 2 1 1
V V m ? ? ? ? and 
2 1
V V V ? ? [As V m / ? ? ] 
If V V V ? ?
2 1
2 / ) (
2 1
? ? ? ? ? = Arithmetic Mean 
Civil ENGINEERING – FLUID MECHANICS
? With rise in temperature due to thermal expansion of a given body, volume will
increase while mass will remain unchanged, so density will decrease, i.e.,
) 1 ( ) / (
) / (
0
0 0
0 0
? ? ?
?
? ?
? ? ?
V
V
V
V
V m
V m
[As ) 1 (
0
? ? ? ? ? V V ] 
 or 
) 1 ( –
~
) 1 (
0
0
? ? ?
? ?
?
? ? ?
? ?
? 
? With increase in pressure due to decrease in volume, density will increase, i.e.,
V
V
V m
V m
0
0 0
) / (
) / (
? ?
?
?
[As
V
m
? ? ] 
? By definition of bulk-modulus:
V
p
V B
?
?
? ?
0
i.e., 
?
?
?
?
?
? ?
? ?
B
p
V V 1
0
 
?
?
?
?
?
? ?
? ? ?
?
?
?
?
? ?
? ?
?
B
p
B
p
1
~
1
0
1
0
? ? ?
Specific Weight ( w ): 
? It is defined as the weight per unit volume.
? Specific weight 
.
.
Weight m g
g
Volume Volume
? ???
Specific Gravity or Relative Density (s): 
? It is the ratio of specific weight of fluid to the specific weight of a standard fluid.
Standard fluid is water in case of liquid and H
2
 or air in case of gas.
.
.
w w w
g
s
g
? ? ?
? ? ?
? ? ?
Where, 
w
? ? Specific weight of water, and 
w
? ? Density of water specific. 
Specific Volume ( v ): 
? Specific volume of liquid is defined as volume per unit mass. It is also defined as the
reciprocal of specific density.
? Specific volume 
1 V
m ?
??
Inertial force per unit area = 
A
dt dm v
A
dt dp ) / ( /
? = 
A
Av v ?
 = ?
2
v 
Civil ENGINEERING – FLUID MECHANICS
Viscous force per unit area: 
r
v
A F
?
? /
Reynold’s number: 
area unit per force Viscous
area unit per force Inertial 
?
R
N
?
?
?
? r v
r v
v
? ?
/
2
Pascal’s Law: 
x y z
p p p ?? ; where, 
x
p ,
y
p and 
z
p are the pressure at point x,y,z respectively. 
Hydrostatic Law: 
? 
p
pg
z
?
?
?
or dp pg ? dz
? 
ph
oo
dp pg dz ?
??
? p pgh ? and 
p
h
pg
? ; where, h is known as pressure head. 
Pressure Energy Potential energy Kinetic energy 
It is the energy possessed by a 
liquid by virtue of its pressure. It 
is the measure of work done in 
pushing the liquid against 
pressure without imparting any 
velocity to it. 
It is the energy possessed by 
liquid by virtue of its height or 
position above the surface of 
earth or any reference level 
taken as zero level.  
It is the energy possessed by a 
liquid by virtue of its motion or 
velocity. 
Pressure energy of the liquid PV Potential energy of the liquid 
mgh 
Kinetic energy of the liquid 
mv
2
/2
Pressure energy per unit mass of 
the liquid P/ ? 
Potential energy per unit mass of 
the liquid gh 
Kinetic energy per unit mass of 
the liquid v
2
/2
Pressure energy per unit volume 
of the liquid P 
Potential energy per unit volume 
of the liquid ?gh 
Kinetic energy per unit volume 
of the liquid ? v
2
/2
Quantities that Satisfy a Balance Equation 
Quantit
y 
mass x momentum y momentum z 
momentum 
Energy Species 
? ?
m mu mv mw E + mV
2
/2 m
(K)
? ?
1 u v w e + V
2
/2 W
(K)
In this table, u, v, and w are the x, y and z velocity components, E is the total 
thermodynamic internal energy, e is the thermodynamic internal energy per unit mass, 
and m
(K)
 is the mass of a chemical species, K, W
(K)
 is the mass fraction of species K.
Page 4


Civil ENGINEERING – FLUID MECHANICS
Pressure (P): 
? If F be the normal force acting on a surface of area A in contact with liquid, then
pressure exerted by liquid on this surface is: A F P / ?
? Units :
2
/ m N or Pascal (S.I.) and Dyne/cm
2
 (C.G.S.)
? Dimension : ] [
] [
] [
] [
] [
] [
2 1
2
2
? ?
?
? ? ? T ML
L
MLT
A
F
P 
? Atmospheric pressure: Its value on the surface of the earth at sea level is nearly
2 5
/ 10 013 . 1 m N ? or Pascal in S.I. other practical units of pressure are atmosphere,
bar and torr (mm of Hg)
? torr 760 bar 01 . 1 10 01 . 1 1
5
? ? ? ? Pa atm
? Fluid Pressure at a Point:  
dF
dA
? ?
Density ( ? ): 
? In a fluid, at a point, density ? is defined as: 
dV
dm
V
m
V
?
?
?
?
? ? 0
lim ?
? In case of homogenous isotropic substance, it has no directional properties, so is a
scalar.
? It has dimensions ] [
3 ?
ML and S.I. unit kg/m
3
 while C.G.S. unit g/cc with
3 3
/ 10 / 1 m kg cc g ?
? Density of body = Density of substance
? Relative density or specific gravity which is defined as :
of water  Density
of body Density
? RD
? If 
1
m mass of liquid of density 
1
? and 
2
m mass of density 
2
? are mixed, then as 
2 1
m m m ? ? and ) / ( ) / (
2 2 1 1
? ? m m V ? ? [As ? / m V ? ] 
) / ( ) / ( ) / (
2 2 1 1
2 1
i i
i
p m
m
m m
m m
V
m
?
?
?
?
?
? ?
? ?
?
If 
2 1
m m ? , ?
?
?
2 1
2 1
2
? ?
? ?
? Harmonic mean 
? If
1
V volume of liquid of density 
1
? and 
2
V volume of liquid of density 
2
? are 
mixed, then as: 
2 2 1 1
V V m ? ? ? ? and 
2 1
V V V ? ? [As V m / ? ? ] 
If V V V ? ?
2 1
2 / ) (
2 1
? ? ? ? ? = Arithmetic Mean 
Civil ENGINEERING – FLUID MECHANICS
? With rise in temperature due to thermal expansion of a given body, volume will
increase while mass will remain unchanged, so density will decrease, i.e.,
) 1 ( ) / (
) / (
0
0 0
0 0
? ? ?
?
? ?
? ? ?
V
V
V
V
V m
V m
[As ) 1 (
0
? ? ? ? ? V V ] 
 or 
) 1 ( –
~
) 1 (
0
0
? ? ?
? ?
?
? ? ?
? ?
? 
? With increase in pressure due to decrease in volume, density will increase, i.e.,
V
V
V m
V m
0
0 0
) / (
) / (
? ?
?
?
[As
V
m
? ? ] 
? By definition of bulk-modulus:
V
p
V B
?
?
? ?
0
i.e., 
?
?
?
?
?
? ?
? ?
B
p
V V 1
0
 
?
?
?
?
?
? ?
? ? ?
?
?
?
?
? ?
? ?
?
B
p
B
p
1
~
1
0
1
0
? ? ?
Specific Weight ( w ): 
? It is defined as the weight per unit volume.
? Specific weight 
.
.
Weight m g
g
Volume Volume
? ???
Specific Gravity or Relative Density (s): 
? It is the ratio of specific weight of fluid to the specific weight of a standard fluid.
Standard fluid is water in case of liquid and H
2
 or air in case of gas.
.
.
w w w
g
s
g
? ? ?
? ? ?
? ? ?
Where, 
w
? ? Specific weight of water, and 
w
? ? Density of water specific. 
Specific Volume ( v ): 
? Specific volume of liquid is defined as volume per unit mass. It is also defined as the
reciprocal of specific density.
? Specific volume 
1 V
m ?
??
Inertial force per unit area = 
A
dt dm v
A
dt dp ) / ( /
? = 
A
Av v ?
 = ?
2
v 
Civil ENGINEERING – FLUID MECHANICS
Viscous force per unit area: 
r
v
A F
?
? /
Reynold’s number: 
area unit per force Viscous
area unit per force Inertial 
?
R
N
?
?
?
? r v
r v
v
? ?
/
2
Pascal’s Law: 
x y z
p p p ?? ; where, 
x
p ,
y
p and 
z
p are the pressure at point x,y,z respectively. 
Hydrostatic Law: 
? 
p
pg
z
?
?
?
or dp pg ? dz
? 
ph
oo
dp pg dz ?
??
? p pgh ? and 
p
h
pg
? ; where, h is known as pressure head. 
Pressure Energy Potential energy Kinetic energy 
It is the energy possessed by a 
liquid by virtue of its pressure. It 
is the measure of work done in 
pushing the liquid against 
pressure without imparting any 
velocity to it. 
It is the energy possessed by 
liquid by virtue of its height or 
position above the surface of 
earth or any reference level 
taken as zero level.  
It is the energy possessed by a 
liquid by virtue of its motion or 
velocity. 
Pressure energy of the liquid PV Potential energy of the liquid 
mgh 
Kinetic energy of the liquid 
mv
2
/2
Pressure energy per unit mass of 
the liquid P/ ? 
Potential energy per unit mass of 
the liquid gh 
Kinetic energy per unit mass of 
the liquid v
2
/2
Pressure energy per unit volume 
of the liquid P 
Potential energy per unit volume 
of the liquid ?gh 
Kinetic energy per unit volume 
of the liquid ? v
2
/2
Quantities that Satisfy a Balance Equation 
Quantit
y 
mass x momentum y momentum z 
momentum 
Energy Species 
? ?
m mu mv mw E + mV
2
/2 m
(K)
? ?
1 u v w e + V
2
/2 W
(K)
In this table, u, v, and w are the x, y and z velocity components, E is the total 
thermodynamic internal energy, e is the thermodynamic internal energy per unit mass, 
and m
(K)
 is the mass of a chemical species, K, W
(K)
 is the mass fraction of species K.
Civil ENGINEERING – FLUID MECHANICS
The other energy term, mV
2
/2, is the kinetic energy.
? z y x
t t
z y x
t
m
t
Storage ? ? ?
?
?
?
?
? ? ? ?
?
?
?
?
?
? ?
?
) ( ) ( ) ( ?? ? ? ?
? x y w z x v z y u Inflow
z y x
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 
? x y w z x v z y u Outflow
z z y y x x
? ? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? z y x Source ? ? ? ?
?
S 
? 
?
? ? ? ? ? ? ? ?
? ? ? ?
??
S ?
?
?
?
?
?
?
?
?
?
?
?
? ?
? ?
? ?
z
w w
y
v v
x
u u
t
z z z
y y y
x x x
? 
*
?
? ? ? ? ? ? ??
S
z
w
y
v
x
u
t
?
?
?
?
?
?
?
?
?
?
?
?
? 
? ?
S
z y x
Lim
0
S
*
? ? ? ?
?
The Mass Balance Equations: 
? 0 ?
?
?
?
?
?
i
i
x
u
t
? ?
? 0 ?
?
?
?
?
?
?
?
?
?
?
?
z
w
y
v
x
u
t
? ? ? ?
 
? 0 ?
?
?
?
?
?
?
?
?
i
i
i
i
x
u
x
u
t
?
? ?
? 0 ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
z
w
y
v
x
u
z
w
y
v
x
u
t
? ? ?
?
?
 
? 
i
i
x
u
t Dt
D
or
z
w
y
v
x
u
t Dt
D
?
? ?
?
?
? ?
?
?
?
? ?
?
?
? ?
?
?
? ?
?
?
? ?
?
?
? 0 0 0 ? ? ? ?
?
?
? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
Dt
D
x
u
Dt
D
z
w
y
v
x
u
Dt
D
i
i
? 0 0 ?
?
?
? ? ?
?
?
?
?
?
?
?
?
? ?
i
i
x
u
or
z
w
y
v
x
u
Page 5


Civil ENGINEERING – FLUID MECHANICS
Pressure (P): 
? If F be the normal force acting on a surface of area A in contact with liquid, then
pressure exerted by liquid on this surface is: A F P / ?
? Units :
2
/ m N or Pascal (S.I.) and Dyne/cm
2
 (C.G.S.)
? Dimension : ] [
] [
] [
] [
] [
] [
2 1
2
2
? ?
?
? ? ? T ML
L
MLT
A
F
P 
? Atmospheric pressure: Its value on the surface of the earth at sea level is nearly
2 5
/ 10 013 . 1 m N ? or Pascal in S.I. other practical units of pressure are atmosphere,
bar and torr (mm of Hg)
? torr 760 bar 01 . 1 10 01 . 1 1
5
? ? ? ? Pa atm
? Fluid Pressure at a Point:  
dF
dA
? ?
Density ( ? ): 
? In a fluid, at a point, density ? is defined as: 
dV
dm
V
m
V
?
?
?
?
? ? 0
lim ?
? In case of homogenous isotropic substance, it has no directional properties, so is a
scalar.
? It has dimensions ] [
3 ?
ML and S.I. unit kg/m
3
 while C.G.S. unit g/cc with
3 3
/ 10 / 1 m kg cc g ?
? Density of body = Density of substance
? Relative density or specific gravity which is defined as :
of water  Density
of body Density
? RD
? If 
1
m mass of liquid of density 
1
? and 
2
m mass of density 
2
? are mixed, then as 
2 1
m m m ? ? and ) / ( ) / (
2 2 1 1
? ? m m V ? ? [As ? / m V ? ] 
) / ( ) / ( ) / (
2 2 1 1
2 1
i i
i
p m
m
m m
m m
V
m
?
?
?
?
?
? ?
? ?
?
If 
2 1
m m ? , ?
?
?
2 1
2 1
2
? ?
? ?
? Harmonic mean 
? If
1
V volume of liquid of density 
1
? and 
2
V volume of liquid of density 
2
? are 
mixed, then as: 
2 2 1 1
V V m ? ? ? ? and 
2 1
V V V ? ? [As V m / ? ? ] 
If V V V ? ?
2 1
2 / ) (
2 1
? ? ? ? ? = Arithmetic Mean 
Civil ENGINEERING – FLUID MECHANICS
? With rise in temperature due to thermal expansion of a given body, volume will
increase while mass will remain unchanged, so density will decrease, i.e.,
) 1 ( ) / (
) / (
0
0 0
0 0
? ? ?
?
? ?
? ? ?
V
V
V
V
V m
V m
[As ) 1 (
0
? ? ? ? ? V V ] 
 or 
) 1 ( –
~
) 1 (
0
0
? ? ?
? ?
?
? ? ?
? ?
? 
? With increase in pressure due to decrease in volume, density will increase, i.e.,
V
V
V m
V m
0
0 0
) / (
) / (
? ?
?
?
[As
V
m
? ? ] 
? By definition of bulk-modulus:
V
p
V B
?
?
? ?
0
i.e., 
?
?
?
?
?
? ?
? ?
B
p
V V 1
0
 
?
?
?
?
?
? ?
? ? ?
?
?
?
?
? ?
? ?
?
B
p
B
p
1
~
1
0
1
0
? ? ?
Specific Weight ( w ): 
? It is defined as the weight per unit volume.
? Specific weight 
.
.
Weight m g
g
Volume Volume
? ???
Specific Gravity or Relative Density (s): 
? It is the ratio of specific weight of fluid to the specific weight of a standard fluid.
Standard fluid is water in case of liquid and H
2
 or air in case of gas.
.
.
w w w
g
s
g
? ? ?
? ? ?
? ? ?
Where, 
w
? ? Specific weight of water, and 
w
? ? Density of water specific. 
Specific Volume ( v ): 
? Specific volume of liquid is defined as volume per unit mass. It is also defined as the
reciprocal of specific density.
? Specific volume 
1 V
m ?
??
Inertial force per unit area = 
A
dt dm v
A
dt dp ) / ( /
? = 
A
Av v ?
 = ?
2
v 
Civil ENGINEERING – FLUID MECHANICS
Viscous force per unit area: 
r
v
A F
?
? /
Reynold’s number: 
area unit per force Viscous
area unit per force Inertial 
?
R
N
?
?
?
? r v
r v
v
? ?
/
2
Pascal’s Law: 
x y z
p p p ?? ; where, 
x
p ,
y
p and 
z
p are the pressure at point x,y,z respectively. 
Hydrostatic Law: 
? 
p
pg
z
?
?
?
or dp pg ? dz
? 
ph
oo
dp pg dz ?
??
? p pgh ? and 
p
h
pg
? ; where, h is known as pressure head. 
Pressure Energy Potential energy Kinetic energy 
It is the energy possessed by a 
liquid by virtue of its pressure. It 
is the measure of work done in 
pushing the liquid against 
pressure without imparting any 
velocity to it. 
It is the energy possessed by 
liquid by virtue of its height or 
position above the surface of 
earth or any reference level 
taken as zero level.  
It is the energy possessed by a 
liquid by virtue of its motion or 
velocity. 
Pressure energy of the liquid PV Potential energy of the liquid 
mgh 
Kinetic energy of the liquid 
mv
2
/2
Pressure energy per unit mass of 
the liquid P/ ? 
Potential energy per unit mass of 
the liquid gh 
Kinetic energy per unit mass of 
the liquid v
2
/2
Pressure energy per unit volume 
of the liquid P 
Potential energy per unit volume 
of the liquid ?gh 
Kinetic energy per unit volume 
of the liquid ? v
2
/2
Quantities that Satisfy a Balance Equation 
Quantit
y 
mass x momentum y momentum z 
momentum 
Energy Species 
? ?
m mu mv mw E + mV
2
/2 m
(K)
? ?
1 u v w e + V
2
/2 W
(K)
In this table, u, v, and w are the x, y and z velocity components, E is the total 
thermodynamic internal energy, e is the thermodynamic internal energy per unit mass, 
and m
(K)
 is the mass of a chemical species, K, W
(K)
 is the mass fraction of species K.
Civil ENGINEERING – FLUID MECHANICS
The other energy term, mV
2
/2, is the kinetic energy.
? z y x
t t
z y x
t
m
t
Storage ? ? ?
?
?
?
?
? ? ? ?
?
?
?
?
?
? ?
?
) ( ) ( ) ( ?? ? ? ?
? x y w z x v z y u Inflow
z y x
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 
? x y w z x v z y u Outflow
z z y y x x
? ? ? ? ? ? ? ? ?
? ? ? ? ? ?
? ? ? ? ? ?
? z y x Source ? ? ? ?
?
S 
? 
?
? ? ? ? ? ? ? ?
? ? ? ?
??
S ?
?
?
?
?
?
?
?
?
?
?
?
? ?
? ?
? ?
z
w w
y
v v
x
u u
t
z z z
y y y
x x x
? 
*
?
? ? ? ? ? ? ??
S
z
w
y
v
x
u
t
?
?
?
?
?
?
?
?
?
?
?
?
? 
? ?
S
z y x
Lim
0
S
*
? ? ? ?
?
The Mass Balance Equations: 
? 0 ?
?
?
?
?
?
i
i
x
u
t
? ?
? 0 ?
?
?
?
?
?
?
?
?
?
?
?
z
w
y
v
x
u
t
? ? ? ?
 
? 0 ?
?
?
?
?
?
?
?
?
i
i
i
i
x
u
x
u
t
?
? ?
? 0 ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
z
w
y
v
x
u
z
w
y
v
x
u
t
? ? ?
?
?
 
? 
i
i
x
u
t Dt
D
or
z
w
y
v
x
u
t Dt
D
?
? ?
?
?
? ?
?
?
?
? ?
?
?
? ?
?
?
? ?
?
?
? ?
?
?
? 0 0 0 ? ? ? ?
?
?
? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
Dt
D
x
u
Dt
D
z
w
y
v
x
u
Dt
D
i
i
? 0 0 ?
?
?
? ? ?
?
?
?
?
?
?
?
?
? ?
i
i
x
u
or
z
w
y
v
x
u
Civil ENGINEERING – FLUID MECHANICS
? 
?
?
?
? ?
?
?
? S ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
i
i
i
i
x
u
x
u
t t
? 
?
?
?
?
? S ?
?
?
?
?
?
i
i
x
u
t
Momentum Balance Equation: 
? 
j
i
ij
j
j j j
B
x
B
x x x
term source direction j Net ?
?
?
? ? ?
?
?
?
? ?
?
?
?
?
?
?
?
?
? ?
3
3
2
2
1
1
? 3 , 1 ? ? ?
?
?
?
?
?
?
?
?
j B
x x
u u
t
u
j
i
ij
i
j i j
?
? ? ?
 
? For a Newtonian fluid, the stress, s
ij,
 is given by the following equation:
ij
i
j
j
i
ij ij
x
u
x
u
P ? ? ? ? ? ? ? ? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ? ? )
3
2
(
? 3 , 1 )
3
2
( ? ? ?
?
?
?
?
?
?
?
?
? ? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
j B
x
u
x
u
P
x x
u u
t
u
j ij
i
j
j
i
ij
i i
j i j
? ? ? ? ? ?
? ?
 
? 3 , 1 )
3
2
( ? ? ?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
j B
x x
u
x
u
x x
P
x
u u
t
u
j
j i
j
j
i
i j i
j i j
? ? ? ?
? ?
 
? 
x
u uu vu wu
B
t x y z
? ? ?
?
? ? ? ?
? ? ? ?
? ? ? ?
? 
2
2 ( )
3
P u v u w u
x x x y x y z x z x
? ? ? ? ?
? ?
? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
?? ? ? ? ? ? ? ??
??
? ? ? ? ? ? ? ? ? ?
? ? ? ? ? ?
?? ??
??
Energy Balance Equation: 
? This directional heat flux is given the symbol q
i
: 
i
i
x
T
k q
?
?
? ?
? 
x
q q
z y
z y x
q q
Volume Unit
heat xDirection Net
x
x
x x
x
x
x
x x
x
?
?
? ? ? ?
? ? ?
?
? ?
? ? ? ?
? 
x
q
Volume Unit
source heat xDirection Net
x
Limit
x
?
?
? ?
? ? 0