Formula Sheet: Viscous Flow of Incompressible Fluids | Fluid Mechanics for Mechanical Engineering PDF Download

 Page 1


Viscous Flow of Incompressible Fluids
1. Newton’s Law of Viscosity
t =µ du
dy
Where:
• t : Shear stress
• µ : Dynamic viscosity
•
du
dy
: Velocity gradient
Kinematic viscosity:
? =
µ ? (units: m
2
/s)
2. Classification of Fluids
• Newtonian fluids: Obey Newton’s law of viscosity.
• Non-Newtonian fluids: Do not obey Newton’s law.
• Ideal fluid: No viscosity, incompressible, no heat conduction.
3. LaminarFlowBetweenParallelPlates(CouetteFlow)
Without pressure gradient:
u(y)=
U
h
y
With pressure gradient:
u(y)=
1
2µ 
- dp
dx


hy- y
2


1
Page 2


Viscous Flow of Incompressible Fluids
1. Newton’s Law of Viscosity
t =µ du
dy
Where:
• t : Shear stress
• µ : Dynamic viscosity
•
du
dy
: Velocity gradient
Kinematic viscosity:
? =
µ ? (units: m
2
/s)
2. Classification of Fluids
• Newtonian fluids: Obey Newton’s law of viscosity.
• Non-Newtonian fluids: Do not obey Newton’s law.
• Ideal fluid: No viscosity, incompressible, no heat conduction.
3. LaminarFlowBetweenParallelPlates(CouetteFlow)
Without pressure gradient:
u(y)=
U
h
y
With pressure gradient:
u(y)=
1
2µ 
- dp
dx


hy- y
2


1
4. Hagen–Poiseuille Flow (Laminar Flow in Circular
Pipe)
Velocity distribution:
u(r)=
? P
4µL
(R
2
- r
2
)
Maximum velocity:
u
max
=
? PR
2
4µL
Average velocity:
u
avg
=
1
2
u
max
Volumetric flow rate:
Q=
pR
4
8µ · ? P
L
5. Shear Stress Distribution in Pipe Flow
t (r)=
r
2
· 
? P
L

(linear variation from center to wall)
6. Reynolds Number
Re=
?uD
µ =
uD
? • Laminar: Re ¡ 2000
• Transitional: 2000 ¡ Re ¡ 4000
• Turbulent: Re ¿ 4000
7. Head Loss due to Viscosity
Darcy–Weisbach equation:
h
f
=f · L
D
· u
2
2g
For laminar flow in pipe:
f =
64
Re
8. Power Required to Overcome Viscous Resistance
P =t · A· u=µA
du
dy
u
2
Page 3


Viscous Flow of Incompressible Fluids
1. Newton’s Law of Viscosity
t =µ du
dy
Where:
• t : Shear stress
• µ : Dynamic viscosity
•
du
dy
: Velocity gradient
Kinematic viscosity:
? =
µ ? (units: m
2
/s)
2. Classification of Fluids
• Newtonian fluids: Obey Newton’s law of viscosity.
• Non-Newtonian fluids: Do not obey Newton’s law.
• Ideal fluid: No viscosity, incompressible, no heat conduction.
3. LaminarFlowBetweenParallelPlates(CouetteFlow)
Without pressure gradient:
u(y)=
U
h
y
With pressure gradient:
u(y)=
1
2µ 
- dp
dx


hy- y
2


1
4. Hagen–Poiseuille Flow (Laminar Flow in Circular
Pipe)
Velocity distribution:
u(r)=
? P
4µL
(R
2
- r
2
)
Maximum velocity:
u
max
=
? PR
2
4µL
Average velocity:
u
avg
=
1
2
u
max
Volumetric flow rate:
Q=
pR
4
8µ · ? P
L
5. Shear Stress Distribution in Pipe Flow
t (r)=
r
2
· 
? P
L

(linear variation from center to wall)
6. Reynolds Number
Re=
?uD
µ =
uD
? • Laminar: Re ¡ 2000
• Transitional: 2000 ¡ Re ¡ 4000
• Turbulent: Re ¿ 4000
7. Head Loss due to Viscosity
Darcy–Weisbach equation:
h
f
=f · L
D
· u
2
2g
For laminar flow in pipe:
f =
64
Re
8. Power Required to Overcome Viscous Resistance
P =t · A· u=µA
du
dy
u
2
9. Entrance Length in Laminar Pipe Flow
L
e
=0.05· Re· D
3