The document summarizes flow of viscous fluid through a circular pipe. It describes that flow can be laminar, transitional, or turbulent depending on the Reynolds number. It presents the Hagen-Poiseuille law which describes laminar flow in a circular pipe. The law states that velocity distribution varies with the square of the radial distance from the center, and that maximum velocity is twice the average velocity. It also provides the equation for pressure drop along the length of the pipe based on flow properties and pipe dimensions.

1. Flow of Viscous Fluid
Through Circular Pipe
FLUID MECHANICS

2. • SUBJECT :- FLUID MECHANICS
• CLASS :- 2nd year , 4rd semester B.E Mechanical
• TOPIC :- Flow of Viscous Fluid Through Circular Pipe
• NAME :- Jadav Parth S. 170093119007
Joshi Utsav P. 170093119009
Mehta Nishit H 170093119015
Punjabi Rahul V. 170093119026
Tailor Vaibhav K. 170093119029
• GUIDED BY :- PROF.

3. Types of fluid flow
• The flow in pipe is differ from this occur in open channel where the
flow in pipe is at a pressure & it does not have a free surface.
• The flow in pipe can be demonstrated such as
• Laminar flow
• Transitional flow
• Turbulent flow
• Reynolds number (Re = ρVD/μ) can be used to classify if a flow
is laminar or turbulent. For pipe flow, the flow is laminar when Re <
2000, and it is turbulent when Re > 4000. The flow is referred to as
transition flow when Reynolds number is in between 2000 and 4000.

4. VISCOUS FLOW IN CIRCULAR PIPE OR
HAGEN POISEILLE LAW.
• The most simple and practical case of laminar flow is the steady
laminar flow through out along straight pipe of a circular cross
section.
• Consider a pipe of radius R and of straight length L throughout which
a liquid of viscosity µ is flowing at a steady volumetric rate of Q.
• Let us consider the equilibrium of forces on a small concentric
cylinderical fluid element of radius r and length dX

5. Viscous Flow Through a Circular Pipe
Where ,p=pressure intensity
𝜏=shear stress
𝑝 +
𝜕𝑝
𝜕𝑥
𝑑𝑥=pressure intensity at right face

6. The shear stress(𝜏)on the Periphery of the cylindrical element will be
acting in the direction opposite to that of the flow of fluid.
For equilibrium
Shear Stress distribution:
at the centre of pipe, r=0
At the wall r=R
)1...(..........
2
0)2(
0)2()()(
2
22
r
x
p
dxrrdx
x
p
dxrrdx
x
p
prp




















2
max
R
x
p




7. Velocity Distribution
To obtain velocity distribution across the prime using Newton's
law of viscosity for laminar flow
(-ve sign indicates direction)
Substituting this value in eq.1
dr
du
dy
du
 
x
pr
dr
du
r
x
p
dx
dr
du








2
2

8. Integrating with respect to r,
Value of c can be obtained by the boundary conditions:
When r=R, u=0,
Substituting value of c in eq 2
Here velocity u varies with the square of r, distribution is shown above.
)2.....(..........
4
1 2
cr
x
p
u 




22
4
1
4
1
0 R
x
p
ccR
x
p







  )3......(..........
4
1
4
1
4
1
22
22
rR
x
p
u
R
x
p
r
x
p
u


















9. Ratio of maximum velocity to average velocity:
velocity is maximum where r=0,(at the centre)
Here
considering small element
Integrating it we get,
)3.1&3)(2.3.........(..........1
)1.3......(..........
4
1
2
max
2
max
from
R
r
Vu
R
x
p
V


























 22
2
4
1
)2(
)2(
discharge
velocityAverage
rR
x
p
rdrdQ
urdrdQ
R
Q
Area
















10. Pressure drop
)5.........(..........2
2
8
1
V
velocityaverage
)4......(..........
8
max
max
2
2avg
4
avg
avg
VV
V
V
R
x
p
R
Q
R
x
p
Q
























2
2
avg
8
8
1
V
R
V
x
p
R
x
p
avg















11. Integrating with respect to x,
Equation 8 is called hagen - poiseille formula
)7....(..........
128
)p-(p
32
)p-(p
)6..(..........
V32
)p-(p
421
2
4
221
2
avg
21
D
QL
L
D
Q
D
D
L








w
p21
f
p
)head(hPressureofLoss


)8(..........
32
p
2
21
gD
LV
g
p
avg





