Formula Sheets: Fluid Dynamics | Fluid Mechanics for Civil Engineering - Civil Engineering (CE) PDF Download

 Page 1


Fluid Dynamics Formula Sheet for GATE Civil
Engineering
1. Fluid Properties
• Density (? ): Mass per unit volume,? =
m
V
.
• Specific Weight (? ): Weight per unit volume,? =?g .
• Dynamic Viscosity (µ ): Measure of fluid’s resistance to shear, units: Pa ·s.
• Kinematic Viscosity (? ): ? =
µ ? , units: m
2
/s.
• Surface Tension (s ): Force per unit length, units: N/m.
2. Reynolds Number
• Reynolds Number (Re): Ratio of inertial to viscous forces.
Re =
?VD
µ =
VD
? where V = velocity, D = characteristic length (e.g., pipe diameter), µ = dy-
namic viscosity,? = kinematic viscosity.
3. Continuity Equation
• Continuity for Incompressible Flow:
A
1
V
1
=A
2
V
2
or Q =AV
whereA = cross-sectional area,V = velocity,Q = flow rate.
• General Form (Mass Conservation):
?? ?t
+?· (? V) = 0
4. Bernoulli’s Equation
• For steady, incompressible, inviscid flow:
P
? +
V
2
2
+gz = constant
where P = pressure, V = velocity, z = elevation, g = gravitational accelera-
tion.
• Head Form:
P
? +
V
2
2g
+z = constant
1
Page 2


Fluid Dynamics Formula Sheet for GATE Civil
Engineering
1. Fluid Properties
• Density (? ): Mass per unit volume,? =
m
V
.
• Specific Weight (? ): Weight per unit volume,? =?g .
• Dynamic Viscosity (µ ): Measure of fluid’s resistance to shear, units: Pa ·s.
• Kinematic Viscosity (? ): ? =
µ ? , units: m
2
/s.
• Surface Tension (s ): Force per unit length, units: N/m.
2. Reynolds Number
• Reynolds Number (Re): Ratio of inertial to viscous forces.
Re =
?VD
µ =
VD
? where V = velocity, D = characteristic length (e.g., pipe diameter), µ = dy-
namic viscosity,? = kinematic viscosity.
3. Continuity Equation
• Continuity for Incompressible Flow:
A
1
V
1
=A
2
V
2
or Q =AV
whereA = cross-sectional area,V = velocity,Q = flow rate.
• General Form (Mass Conservation):
?? ?t
+?· (? V) = 0
4. Bernoulli’s Equation
• For steady, incompressible, inviscid flow:
P
? +
V
2
2
+gz = constant
where P = pressure, V = velocity, z = elevation, g = gravitational accelera-
tion.
• Head Form:
P
? +
V
2
2g
+z = constant
1
5. Momentum Equation
• Linear Momentum (Newton’s Second Law):
X
F =?Q (V
2
- V
1
)
whereF = external forces,Q = flow rate, V
1
,V
2
= inlet and outlet velocities.
6. Navier-Stokes Equations
• For incompressible, Newtonian fluid (in x-direction):
? 
?u
?t
+u
?u
?x
+v
?u
?y
+w
?u
?z

=- ?P
?x
+µ 
?
2
u
?x
2
+
?
2
u
?y
2
+
?
2
u
?z
2

+?g
x
where u,v,w = velocity components, g
x
= gravitational acceleration in x-
direction.
7. Energy Equation
• Extended Bernoulli’s Equation (with losses):
P
1
? +
V
2
1
2g
+z
1
+h
p
=
P
2
? +
V
2
2
2g
+z
2
+h
L
whereh
p
= pump head,h
L
= head loss.
8. Dimensional Analysis
• Buckingham Pi Theorem: Number of dimensionless groups =n- k, where
n = number of variables,k = number of fundamental dimensions.
• Common dimensionless numbers:
– Reynolds Number: Re =
?VD
µ – Froude Number: Fr =
V
v
gL
– Weber Number: We =
?V
2
L
s – Mach Number: Ma =
V
c
, wherec = speed of sound.
9. Pipe Flow
• Darcy-Weisbach Equation:
h
f
=f
L
D
V
2
2g
whereh
f
= head loss,f = friction factor,L = pipe length,D = diameter.
2
Page 3


Fluid Dynamics Formula Sheet for GATE Civil
Engineering
1. Fluid Properties
• Density (? ): Mass per unit volume,? =
m
V
.
• Specific Weight (? ): Weight per unit volume,? =?g .
• Dynamic Viscosity (µ ): Measure of fluid’s resistance to shear, units: Pa ·s.
• Kinematic Viscosity (? ): ? =
µ ? , units: m
2
/s.
• Surface Tension (s ): Force per unit length, units: N/m.
2. Reynolds Number
• Reynolds Number (Re): Ratio of inertial to viscous forces.
Re =
?VD
µ =
VD
? where V = velocity, D = characteristic length (e.g., pipe diameter), µ = dy-
namic viscosity,? = kinematic viscosity.
3. Continuity Equation
• Continuity for Incompressible Flow:
A
1
V
1
=A
2
V
2
or Q =AV
whereA = cross-sectional area,V = velocity,Q = flow rate.
• General Form (Mass Conservation):
?? ?t
+?· (? V) = 0
4. Bernoulli’s Equation
• For steady, incompressible, inviscid flow:
P
? +
V
2
2
+gz = constant
where P = pressure, V = velocity, z = elevation, g = gravitational accelera-
tion.
• Head Form:
P
? +
V
2
2g
+z = constant
1
5. Momentum Equation
• Linear Momentum (Newton’s Second Law):
X
F =?Q (V
2
- V
1
)
whereF = external forces,Q = flow rate, V
1
,V
2
= inlet and outlet velocities.
6. Navier-Stokes Equations
• For incompressible, Newtonian fluid (in x-direction):
? 
?u
?t
+u
?u
?x
+v
?u
?y
+w
?u
?z

=- ?P
?x
+µ 
?
2
u
?x
2
+
?
2
u
?y
2
+
?
2
u
?z
2

+?g
x
where u,v,w = velocity components, g
x
= gravitational acceleration in x-
direction.
7. Energy Equation
• Extended Bernoulli’s Equation (with losses):
P
1
? +
V
2
1
2g
+z
1
+h
p
=
P
2
? +
V
2
2
2g
+z
2
+h
L
whereh
p
= pump head,h
L
= head loss.
8. Dimensional Analysis
• Buckingham Pi Theorem: Number of dimensionless groups =n- k, where
n = number of variables,k = number of fundamental dimensions.
• Common dimensionless numbers:
– Reynolds Number: Re =
?VD
µ – Froude Number: Fr =
V
v
gL
– Weber Number: We =
?V
2
L
s – Mach Number: Ma =
V
c
, wherec = speed of sound.
9. Pipe Flow
• Darcy-Weisbach Equation:
h
f
=f
L
D
V
2
2g
whereh
f
= head loss,f = friction factor,L = pipe length,D = diameter.
2
• Hazen-Williams Equation (for water):
V = 0.849CR
0.63
h
S
0.54
where C = Hazen-Williams coefficient, R
h
= hydraulic radius, S = energy
slope.
10. Hydraulic Diameter
• For non-circular ducts:
D
h
=
4A
P
whereA = cross-sectional area,P = wetted perimeter.
3