Fluid Mechanics-Shear stress ,Shear stress distribution,Velocity profile,Flow Of Viscous Fluid Through The circular pipe ,Velocity profile for turbulent flow Boundary layer buildup in pipe,Velocity Distributions

1. Dynamics of Fluid Flow
Presented By…..
Abhishek Anand Thakur
Manash Pratim

2. *
u
u+ du
y
dy
𝝉 = 𝝁
𝒅𝒖
𝒅𝒚
𝜏 ∝
𝑑𝑢
𝑑𝑦
Stress is proportional to the rate of
change of velocity w.r.t y
μ is the constant of proportionality &
known as coefficient of viscosity
u
Velocity variation boundary
near a solid boundary
When two layer of fluid a distance “dy” apart.
Move one over the other at different velocity u
& u+du
Viscosity together with relative velocity cause a
shear stress acting b/w the fluid layer

3. Direction
of Flow
a
τ ∗ 2𝜋𝑟∆𝑥
A
B C
D
𝑝 +
𝜕𝑝
𝜕𝑥
𝑑𝑥 𝜋𝑟2p𝜋𝑟2
∆x
r
R
Neglecting the acceleration of fluid element ,net force =o
p 𝜋𝑟2
− 𝑝 +
𝜕𝑝
𝜕𝑥
𝑑𝑥 𝜋𝑟2 − τ ∗ 2𝜋𝑟∆𝑥 = 0

4. From it we can find out shear stress
distribution
Shear stress is maximum at the
boundary of pipe and minimum
at the center of pipe
𝝉 ∝ 𝒓
r=0
r=R
r=R
𝝉 𝒎𝒂𝒙
𝝉 𝒎𝒂𝒙
𝝉 𝒎𝒊𝒏
Shear stress increase linearly with distance
from the center

5. After integration and using boundary condition at r=R, u=0
R
r
y
a
b
c
d
y=R−𝐫 , dy =−𝐝𝐫
𝜏 = −𝜇
𝑑𝑢
𝑑𝑟
……. 2
By eq 1 and 2
𝒅𝒖 =
𝒓
𝟐𝒖
𝝏𝒑
𝝏𝒙
. 𝒅𝒓
U=−
𝟏
𝟒𝒖
.
𝝏𝒑
𝝏𝒙
𝑹 𝟐
− 𝒓 𝟐
… … … . 𝟑
Here it is clear when r=0 𝑎𝑡 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑒 𝑜𝑓 𝑝𝑖𝑝𝑒 then velocity is
maximum and minimum at boundary where 𝑟 = 𝑅

6. * At the pipe wall Velocity of the fluid will be Zero. The velocity will increase As we
move towards the Centre of the pipe. The change in velocity across the direction
of flow is known as Velocity profile.
U∝ 𝒓 𝟐 Velocity distribution across the section
of pipe is parabolic
U=−
𝟏
𝟒𝒖
.
𝝏𝒑
𝝏𝒙
𝑹 𝟐 − 𝒓 𝟐
𝜇,
𝜕𝑝
𝜕𝑥
, 𝑎𝑛𝑑 R is constant
`
𝒗 𝒎𝒂𝒙
𝒗 𝒎𝒊𝒏
𝒗 𝒎𝒊𝒏

7. *If a fluid is along way from the boundary and
all the particle moving with the same velocity
then the velocity profile look some thing like
this
• In this flow there are 4 layer
• Viscous sub layer-viscous effect are
dominant so velocity profile in this layer is
linear & flow is stream line
• Buffer layer –turbulent effect are significant
but flow is still dominated by viscous effect
• Overlap layer- turbulent effect are more
significant but still not dominated
• Turbulent layer – turbulent effect are
dominated on viscous effect
V
Velocity profile for turbulent flow

8. Pipe
Entrance
v vv
 Because of the shear force near the pipe wall, a boundary
layer forms on the inside surface and occupies a large
portion of the flow area as the distance downstream from
the pipe entrance increase. At some value of this distance
the boundary layer fills the flow area. The velocity profile
becomes independent of the axis in the direction of flow,
and the flow is said to be fully developed.
*

9. *
*Turbulence causes transfer of momentum from center of
pipe to fluid closer to the pipe wall.
*Mixing of fluid (transfer of momentum) causes the central
region of the pipe to have relatively uniform velocity
(compared to laminar flow)
*Close to the pipe wall, eddies are smaller (size proportional
to distance to the boundary)