Properties of Matter - U3 - Viscosity-PPT 1

Unit III - Viscosity
Ms Dhivya R
Assistant Professor
Department of Physics
Sri Ramakrishna College of Arts and Science
Coimbatore - 641 006
Tamil Nadu, India
1

Coefficient of Viscosity
Viscosity:
Viscosity is the resistance of a fluid (liquid or gas) to
a change in shape or movement of neighboring
portions relative to one another. Viscosity
denotes opposition to flow.
2

3

The viscosity is calculated in terms of the coefficient
of viscosity.
It is constant for a liquid and depends on its liquid’s
nature.
The Poiseuille’s method is formally used to estimate
the coefficient of viscosity, in which the liquid
flows through a tube at the different level of
pressures.
4

5

Streamline Flow
Streamline flow in fluids is defined as the flow in
which the fluids flow in parallel layers such that
there is no disruption or intermixing of the layers
and at a given point, the velocity of each fluid
particle passing by remains constant with time.
(Laminar Flow)
In steady (Streamline/Laminar) flow, the density of
the fluid remains constant at each point.
In unsteady (Turbulent) flow, the velocity of the fluid
varies between any given two points.
6

Turbulent Flow
Turbulent flow is a type of fluid (gas or liquid) flow
in which the fluid undergoes irregular
fluctuations, or mixing, in contrast to laminar
flow, in which the fluid moves in smooth paths or
layers.
In turbulent flow the speed of the fluid at a point is
continuously undergoing changes in both
magnitude and direction.
7

Streamline & Turbulent Flow
8

Critical Velocity & Reynolds
number
9
Critical velocity is the speed and direction at which
the flow of a liquid through a tube changes from
smooth to turbulent.
Determining the critical velocity depends on
multiple variables, but it is the Reynolds number
that characterizes the flow of the liquid through a
tube as either turbulent or laminar.
The Reynolds number is a dimensionless variable,
which means that it has no units attached to it.

Critical Velocity & Reynolds
number
10
Critical Velocity:
𝑣𝑐 =
𝑘 ⋅ 𝜂
𝜌𝑟
Reynolds number
𝑘 =
𝑣𝑐𝜌𝑟
⋅ 𝜂

Poiseuille’s formula for the flow
of liquid through a capillary tube
11

Now the velocity of the liquid at a distance 𝑟 from the
axis is 𝑣 and at a distance 𝑟 + 𝑑𝑟 is 𝑣 − 𝑑𝑣.
So, the velocity gradient=− 𝑑𝑣/𝑑𝑟
Surface area of the cylinder, A=2𝜋𝑟𝑙
According to Newton’s law viscosity, the viscous Force
between two layer is given by
𝐹1 = −𝜂𝐴𝑑𝑣/𝑑𝑟 = −𝜂 × 2𝜋𝑟𝑙 ×( 𝑑𝑣/𝑑𝑟) ----------(1)
Where 𝜂= coefficient of viscosity,
Now, the forward push due to the difference of
pressure P on two sides of the tube of radius r is
𝐹2 = 𝑃 × 𝐴𝑟𝑒𝑎 = 𝑃 × 𝜋𝑟2 ---------- (2)
12

For steady flow, F1= F2
𝑃 × 𝜋𝑟2 = 𝜂 × 2𝜋𝑟𝑙 ×( 𝑑𝑣/𝑑𝑟)
Or
ⅆ𝒗 =
−𝒑
𝟐𝜼𝒍
𝒓𝒅𝒓------------(3)
Integrating
ⅆ𝒗 =
𝟎
𝒓
−𝒑
𝟐𝜼𝒍
𝒓𝒅𝒓
13

ⅆ𝒗 =
𝟎
𝒓
−𝒑
𝟐𝜼𝒍
𝒓𝒅𝒓
𝒗 =
−𝒑𝒓𝟐
𝟒𝜼𝒍
+ 𝑪 ----- (4)
Where r=a & v=0
𝟎 =
−𝒑𝒂𝟐
𝟒𝜼𝒍
+ 𝑪
𝐂 =
𝒑𝒂𝟐
𝟒𝜼𝒍
𝐂 =
𝒑𝒂𝟐
𝟒𝜼𝒍
----- (5)
∴ 𝐯 =
𝒑
𝟒𝜼𝒍
(𝒂𝟐 − 𝒓𝟐) – (6)
14

Volume of the liquid flowing per second
𝑑𝑉 =
𝐴𝑟𝑒𝑎 𝑜𝑓 𝑐𝑟𝑜𝑠𝑠 𝑆𝑒𝑐𝑡𝑖𝑜𝑛
𝑜𝑓 𝑡ℎ𝑒 𝑠ℎ𝑒𝑙𝑙 𝑜𝑓 𝑟𝑎𝑑𝑖𝑢𝑠 𝑟
𝑎𝑛𝑑 𝑡ℎ𝑖𝑐𝑘𝑒𝑛𝑒𝑠𝑠 𝑑𝑟
× 𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑙𝑜𝑤
𝑑𝑉 = 2𝜋𝑟𝑑𝑟
𝑝
4𝜂𝑙
(𝑎2 − 𝑟2) =
𝜋𝑝
2𝜂𝑙
(𝑎2𝑟 − 𝑟3) 𝑑𝑟
𝑉 =
0
𝑎
𝜋𝑝
2𝜂𝑙
(𝑎2𝑟 − 𝑟3) 𝑑𝑟 =
𝜋𝑝
2𝜂𝑙
𝑎2
𝑟2
2
−
𝑟4
4 0
𝑎
15

𝑉 =
𝜋𝑝
2𝜂𝑙
𝑎2
𝑟2
2
−
𝑟4
4 0
𝑎
=
𝜋𝑝
2𝜂𝑙
𝑎4
4
𝑉 =
𝜋𝑝𝑎4
8𝜂𝑙
16

Corrections to Poiseuille’s
Formula
Correction for pressure head:
Less Effective pressure due to KE acquired
𝑝1 = 𝑝 −
𝑉2
𝜌
𝜋2𝑎4
The KE 𝐸′
= 0
𝑎 1
2
2𝜋𝑑𝑟𝑣𝜌 𝑣2
= 𝜋𝜌 0
𝑎
𝑟𝑣3
𝑑𝑟
But 𝑣 =
𝒑
𝟒𝜼𝒍
(𝒂𝟐 − 𝒓𝟐)
∴ 𝐸′= 𝜋𝜌
0
𝑎
𝑟
𝒑
𝟒𝜼𝒍
3
(𝒂𝟐 − 𝒓𝟐)3𝑑𝑟
17

Formula
𝐸′
= 𝜋𝜌
𝒑
𝟒𝜼𝒍
3
𝑎8
8
=
𝜋𝑝𝑎4
8𝜂𝑙
𝜌
𝜋2𝑎4
𝐸′
=
𝑉3
𝜌
𝜋2𝑎4
Correction for pressure head:
𝐩𝐕 = 𝐩𝟏𝐕 +
𝑉3𝜌
𝜋2𝑎4
Less Effective pressure due to KE acquired
𝒑𝟏 = 𝒑 −
𝑽𝟐
𝝆
𝝅𝟐𝒂𝟒
= 𝒈𝝆 𝒉 −
𝑽𝟐
𝝅𝟐𝒂𝟒𝒈
18

Formula
Correction for length of the tube:
Not streamlined flow for some distance
Effective length is increased from 𝑙 𝑡𝑜 𝑙 + 1.64𝑎.
𝜂 =
𝜋𝑝𝑎4
8𝑉𝑙
𝜼 =
𝝅𝒂𝟒
𝟖𝑽(𝒍 + 𝟏. 𝟔𝟒𝒂)
𝒉 −
𝑽𝟐
𝝅𝟐𝒂𝟒𝒈
𝒈𝝆
19

Terminal Velocity
The uniform velocity acquired by a body while moving
through a highly viscous liquid is called terminal
velocity.
F = k 𝑣𝑎𝑟𝑏𝜂𝑐r
F = MLT-2;
𝑣=LT-1;
𝑟=L;
𝜂=ML-1T-1;
20

Stoke’s Formula
MLT-2=(LT-1)a (L)b (ML-1T-1)c
MLT-2=Mc La+b-c T-a-c
C = 1 ; a+b+c = 1; -a-c=-2
F=k𝑣𝑟𝜂; 𝑘 = 6𝜋
F= 6𝜋𝑣r𝜂;
21

Expression for Terminal Velocity
Weight of the ball =
4
3
𝜋𝑟3𝜌𝑔
Weight of the displaced liquid =
4
3
𝜋𝑟3𝜌′𝑔
Apparent weight of the ball =
4
3
𝜋𝑟3𝜌𝑔-
4
3
𝜋𝑟3𝜌′𝑔
=
4
3
𝜋𝑟3(𝜌 − 𝜌′)𝑔
When the ball attains its terminal velocity,
6𝜋𝑣r𝜂=
4
3
𝜋𝑟3(𝜌 − 𝜌′)𝑔
𝑣 =
2
9
𝑟2
𝜂
(𝜌 − 𝜌′)𝑔
22

Assumptions made by Stoke
 The medium through which the body falls
is infinite in extent
 The moving body is perfectly rigid and
smooth
 There is no slip between the moving body
and the medium
 There are no eddy current or waves set up.
The object moves very slowly in the
medium
23

Stoke’s Method
Terminal velocity 𝑣 =
𝑥
𝑡
𝜂 =
2
9
𝑟2
𝑣
(𝜌 − 𝜌′)𝑔
24

Searle’s Viscometer: Rotating
Cylinder Method
Angular velocity = ω
Linear velocity = v = r ω
Velocity Gradient =
𝑑𝑣
𝑑𝑟
=
𝑑
𝑑𝑟
𝑟𝜔 = 𝜔
𝑑𝑟
𝑑𝑟
+ 𝑟
𝑑𝜔
𝑑𝑟
=𝜔 + 𝑟
𝑑𝜔
𝑑𝑟
25

Cylinder Method
Since 𝜔 is necessary for preventing slip of a layer, this is
not involved in viscous drag. Hence the effective
Velocity Gradient =𝑟
𝑑𝜔
𝑑𝑟
According to newtons formula,
F = η𝐴
𝑑𝑣
𝑑𝑟
F = η2𝜋𝑟𝑙𝑟
𝑑𝜔
𝑑𝑟
Moment of this force = C = η2𝜋𝑟𝑙𝑟
𝑑𝜔
𝑑𝑟
× 𝑟
26

Cylinder Method
Moment of this force = C = 2𝜋𝑙η𝑟3 𝑑𝜔
𝑑𝑟
𝑜𝑟 C
𝑑𝑟
𝑟3
= 2𝜋𝑙η𝑑𝜔
C
𝑏
𝑎
𝑑𝑟
𝑟3
= 2𝜋𝑙η
0
𝜔1
𝑑𝜔
C −
1
2𝑟2
𝑏
𝑎
= 2𝜋𝑙η𝜔1
C
2
1
𝑏2
−
1
𝑎2
𝑏
𝑎
= 2𝜋𝑙η𝜔1
𝐶 =
4𝜋𝜂𝜔1𝑎2𝑏2
𝑎2 − 𝑏2
𝑙
27

Cylinder Method
28

Properties of Matter - U3 - Viscosity-PPT 1

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