Formula Sheet: Fluid Dynamics | Fluid Mechanics for Mechanical Engineering PDF Download

 Page 1


F ormula Sheet: Fluid Dynamics
Introduction to Fluid Dynamics
• Definition : Fluid dynamics studies the behavior of fluids in motion, con-
sidering forc es, energy , and momentum.
• Assumptions : Fluid is a continuum, Newtonian (unless specified), and flow
can be laminar or turbulent.
• K ey Equations : Continuity , Bernoulli’ s, and Navier-Stok es equations.
Continuity Equation
• Gener al F orm :
??
?t
+?·(?
?
V) = 0
where? = fluid density ,
?
V = velocity vector .
• F or Incom pressible Flow (? = constant) :
?·
?
V =
?u
?x
+
?v
?y
+
?w
?z
= 0
• F or Stea dy Flow in Pipes :
A
1
V
1
=A
2
V
2
=Q
where A = cross-sectional area, V = aver age velocity , Q = volumetric flow
r ate.
Bernoulli’ s Equation
• F or Incom pressible, Steady , Inviscid Flow :
p
?
+
V
2
2
+gz = constant
wherep = pressure,V = velocity ,z = elevation,g = gr avitational acceler ation
(9.81 m s
-2
).
• Energy F orm (per unit mass) :
p
?
+
V
2
2
+gz = constant
• Head F orm :
p
?g
+
V
2
2g
+z =H
whereH = total head.
1
Page 2


F ormula Sheet: Fluid Dynamics
Introduction to Fluid Dynamics
• Definition : Fluid dynamics studies the behavior of fluids in motion, con-
sidering forc es, energy , and momentum.
• Assumptions : Fluid is a continuum, Newtonian (unless specified), and flow
can be laminar or turbulent.
• K ey Equations : Continuity , Bernoulli’ s, and Navier-Stok es equations.
Continuity Equation
• Gener al F orm :
??
?t
+?·(?
?
V) = 0
where? = fluid density ,
?
V = velocity vector .
• F or Incom pressible Flow (? = constant) :
?·
?
V =
?u
?x
+
?v
?y
+
?w
?z
= 0
• F or Stea dy Flow in Pipes :
A
1
V
1
=A
2
V
2
=Q
where A = cross-sectional area, V = aver age velocity , Q = volumetric flow
r ate.
Bernoulli’ s Equation
• F or Incom pressible, Steady , Inviscid Flow :
p
?
+
V
2
2
+gz = constant
wherep = pressure,V = velocity ,z = elevation,g = gr avitational acceler ation
(9.81 m s
-2
).
• Energy F orm (per unit mass) :
p
?
+
V
2
2
+gz = constant
• Head F orm :
p
?g
+
V
2
2g
+z =H
whereH = total head.
1
Navier-Stok es Equations
• Gener al F orm (Incompressible, Newtonian Fluid) :
?
 
?
?
V
?t
+(
?
V ·?)
?
V
!
=-?p+µ?
2
?
V +?? g
whereµ = dynamic viscosity ,? g = gr avitational bo dy force.
• F or Stea dy , 2D Flow (x-direction) :
?

u
?u
?x
+v
?u
?y

=-
?p
?x
+µ

?
2
u
?x
2
+
?
2
u
?y
2

+?g
x
Pipe Flow and Losses
• Reynolds Nu mber :
Re =
?VD
µ
=
VD
?
whereD = pipe diameter ,? =
µ
?
= kinematic viscosity .
– Laminar: Re< 2300
– Tr ansitional: 2300=Re= 4000
– Turbulent: Re> 4000
• Friction F actor (Darcy-W eisbach) :
h
f
=f
L
D
V
2
2g
whereh
f
= head loss due to friction ,f = friction factor ,L = pipe length.
• Laminar Flo w (Hagen-Poiseuille) :
f =
64
Re
h
f
=
32µLV
?gD
2
• Turbulent Flo w : Use Moody chart or Colebrook equation:
1
v
f
=-2 log
10

?/D
3.7
+
2.51
Re
v
f

where? = pipe roughness.
• Minor Loss es :
h
m
=K
V
2
2g
whereK = loss coefficient (e.g., for bends, valves, fittings).
2
Page 3


F ormula Sheet: Fluid Dynamics
Introduction to Fluid Dynamics
• Definition : Fluid dynamics studies the behavior of fluids in motion, con-
sidering forc es, energy , and momentum.
• Assumptions : Fluid is a continuum, Newtonian (unless specified), and flow
can be laminar or turbulent.
• K ey Equations : Continuity , Bernoulli’ s, and Navier-Stok es equations.
Continuity Equation
• Gener al F orm :
??
?t
+?·(?
?
V) = 0
where? = fluid density ,
?
V = velocity vector .
• F or Incom pressible Flow (? = constant) :
?·
?
V =
?u
?x
+
?v
?y
+
?w
?z
= 0
• F or Stea dy Flow in Pipes :
A
1
V
1
=A
2
V
2
=Q
where A = cross-sectional area, V = aver age velocity , Q = volumetric flow
r ate.
Bernoulli’ s Equation
• F or Incom pressible, Steady , Inviscid Flow :
p
?
+
V
2
2
+gz = constant
wherep = pressure,V = velocity ,z = elevation,g = gr avitational acceler ation
(9.81 m s
-2
).
• Energy F orm (per unit mass) :
p
?
+
V
2
2
+gz = constant
• Head F orm :
p
?g
+
V
2
2g
+z =H
whereH = total head.
1
Navier-Stok es Equations
• Gener al F orm (Incompressible, Newtonian Fluid) :
?
 
?
?
V
?t
+(
?
V ·?)
?
V
!
=-?p+µ?
2
?
V +?? g
whereµ = dynamic viscosity ,? g = gr avitational bo dy force.
• F or Stea dy , 2D Flow (x-direction) :
?

u
?u
?x
+v
?u
?y

=-
?p
?x
+µ

?
2
u
?x
2
+
?
2
u
?y
2

+?g
x
Pipe Flow and Losses
• Reynolds Nu mber :
Re =
?VD
µ
=
VD
?
whereD = pipe diameter ,? =
µ
?
= kinematic viscosity .
– Laminar: Re< 2300
– Tr ansitional: 2300=Re= 4000
– Turbulent: Re> 4000
• Friction F actor (Darcy-W eisbach) :
h
f
=f
L
D
V
2
2g
whereh
f
= head loss due to friction ,f = friction factor ,L = pipe length.
• Laminar Flo w (Hagen-Poiseuille) :
f =
64
Re
h
f
=
32µLV
?gD
2
• Turbulent Flo w : Use Moody chart or Colebrook equation:
1
v
f
=-2 log
10

?/D
3.7
+
2.51
Re
v
f

where? = pipe roughness.
• Minor Loss es :
h
m
=K
V
2
2g
whereK = loss coefficient (e.g., for bends, valves, fittings).
2
Dr ag and Lift F orces
• Dr ag F orce :
F
D
=
1
2
?V
2
C
D
A
whereC
D
= dr ag coefficient, A = projected area,V = free stream velocity .
• Lift F orce :
F
L
=
1
2
?V
2
C
L
A
whereC
L
= lift coefficient.
• Coefficients : C
D
,C
L
depend on shape, Re , and flow conditions (obtained
from empirical data or charts).
Boundary La yer
• Boundary L a yer Thickness (d ) :
d˜
5x
v
Re
x
(laminar flow over flat plate)
wherex = distance from leading edge,Re
x
=
?Vx
µ
.
• Shear Str ess at Surface :
t
w
=µ
?u
?y




y=0
• Displacement Thickness (d
*
) :
d
*
=
Z
8
0

1-
u
U

dy
whereu = local velocity ,U = free stream velocity .
Applications
• Used in designing pipelines, pumps, turbines, and aerodynamic structures.
• Essential for GA TE problems on pipe flow losses, Bernoulli’ s applications,
and dr ag/lift c alculations.
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