FLUID DYNAMICS
LEARNING OBJECTIVES
At the end of this chapter the students will be able to:
1. Understand that viscous forces in a fluid cause a retarding force on an object moving through it.
2. Use Stokes’ law to derive an expression for terminal velocity of a spherical body falling through
a viscous fluid under laminar conditions.
3. Understand the terms steady (laminar, streamline) flow, incompressible flow, non-viscous flow
as applied to the motion of an ideal fluid.
4. Appreciate that at a sufficiently high velocity, the flow of viscous fluid undergoes a transition
from laminar to turbulence conditions.
5. Appreciate the equation of continuity Av = Constant for the flow of an ideal and incompressible
fluid.
6. Appreciate that the equation of continuity is a form of the principle of conservation of mass.
7. Understand that the pressure difference can arise from different rates of flow of a fluid
(Bernoulli effect).
8. Derive Bernoulli’s equation in form P + ½ρv2 + ρgh = constant.
9. Explain how Bernoulli effect is applied in the filter pump, atomizers, in the flow of air over an
aerofoil, Venturimeter and in blood physics.
10. Give qualitative explanations for the swing of a spinning ball.
INTRODUCTION:
The study of fluids in motion is relatively complicated, but analysis can be simplified by making
a few assumptions. The analysis is further simplified by the use of two important conservation
principles; the conservation of mass and the conservation of energy. The law of conservation of mass
gives us the equation of continuity while the law of conservation of energy is the basis of Bernoulli’s
equation. The equation of continuity and the Bernoulli’s equation along with their applications in
aeroplane and blood circulation are discussed in this chapter.
FLUID:
“Any thing that can flow is called fluid”.
Example:
Water and air are the examples of fluid. S.I unit of η is Poiseuille or Pl.
FLUID DYNAMICS:
“The branch of physics that deals with fluid in motion and their effects is called fluid
dynamics”.
243
,244 Physics Intermediate Part-I
Conservation Principle of Fluid Dynamics:
Following are the two principles that deal with motion of fluid.
Law of Conservation of Mass:
Law of conservation of mass of fluid is given by the equation of continuity.
Law of Conservation of Energy:
Law of conservation of energy of fluid is given by Bernoulli’s equation.
VISCOUS DRAG AND STOKE’S LAW:
Viscosity and Viscous:
“The frictional force existing between the different layers of fluid that opposes its motion is
called viscosity”.
The substance possessing this property is said to be viscous.
Co-efficient of Viscosity (η
η):
“The tangential force that produces a velocity difference of 1 ms−1 between the adjacent layers
that are one metre apart is known as co-efficient of viscosity”. It is represented by η.
Substances having Large Co-efficient of Viscosity:
Substances that cannot flow easily are said to have large co- Viscosities of Liquids and
efficient of viscosity. Gases at 30°C
Viscosity
Examples: Material
10− 3 (Nsm− 2)
Honey and thick tar are the substances having large co-efficient Air 0.019
of viscosity. Acetone 0.295
Methanol 0.510
Substances having Small Co-efficient of Viscosity: Benzene 0.564
Water 0.801
Substances that flow easily are said to have small co-efficient of Ethanol 1.000
viscosity. Plasma 1.6
Glycerin 6.29
Examples:
Air, water, milk are the examples of substances having small co-efficient of viscosity.
Drag Force (Fluid Friction):
“When a substance moves through a fluid it experiences a retarding force that opposes its
motion. This retarding force is called drag force”.
Effect of Speed:
The drag force increases as the speed of an object increases through a fluid but at very high
speed drag force may no longer be proportional to the speed.
Examples:
(i) The force offered by the air to the motion of cyclist.
(ii) The retarding force offered by air to the car moving at high speed.
,[Chapter-6] Fluid Dynamics 245
Stoke’s Law:
The law that expresses the drag force in mathematical form, according to which drag force ‘Fd’
on a sphere of radius ‘r’ moving with velocity ‘v’ through a fluid of viscosity ‘η’ is given by
Fd = 6πηrv
Limitation:
The expression of drag force is valid for slow moving objects. At high speed drag force is
proportional to squared value of speed and is given by
1
Fd = ρC Av2
2 D
Where, ρ = Density
CD = Drag co-efficient
A = Frontal area
TERMINAL VELOCITY:
“The maximum constant velocity of an object falling vertically downward through a fluid
under the action of gravity is called terminal velocity”.
Symbol:
It is represented by ‘vt’.
Mathematical Form:
Mathematically terminal velocity is given by
2ρgr2
vt =
9η
Derivation:
Suppose a fog droplet of radius ‘r’ is falling down due to its
weight ‘W’, through a medium of co-efficient of viscosity ‘η’ and it
experiences a drag force ‘Fd’ in opposite direction to that of weight. A table tennis ball can be made
The net force acting on the droplet is suspended in the stream of air
coming from the nozzle of a hair
F = ω – Fd …… (1) dryer.
Now F = ma , ω = mg and Fd = 6 πηrv
So eq. (1) becomes Drag force
ma = mg − 6πηrv …… (2)
As the speed of droplet continues to increase, the drag force ‘Fd’ Fog droplet
also increases till it attains a magnitude which will be equal to its
weight. In this case the velocity of fog droplet will be maximum and Weight
constant so acceleration ‘a’ will be zero. So eq. (2) becomes
m(0) = mg – 6 πηrvt
0 = mg – 6 πηrvt
6 πηrvt = mg
mg
vt = …… (3)
6πηr
, 246 Physics Intermediate Part-I
As we know that:
m = ρV …… (a) Viscosity of gases rises with
temperature.
Where m = Mass of droplet
ρ = Density of droplet
4 3
Volume of sphere (droplet) = πr
3
So eq. (a) becomes
4 3
m = ρ πr
3
Put value of ‘m’ in eq. (3)
4 3
ρ πr g
3
vt =
6πηr
2
vt = (ρgr2)
9η
2ρgr2
vt =
9η
vt ∝ r2
Result:
The above expression shows that terminal velocity is directly proportional to square of radius of
sphere.
FLUID FLOW:
The fluid passing through the cross-sectional area of pipe per unit time is called fluid flow.
Types of Fluid Flow:
Following are the two types of fluid flow:
Laminar Flow Turbulent Flow
The regular and steady flow of fluid is called The irregular and un-steady flow of fluid is called
laminar or streamline flow. turbulent flow.
OR OR
The flow in which every particle follows the same The flow in which each particle does not
path that has already been followed by other necessarily follow the same path that has already
particles is called laminar flow. Velocity of been followed by other particles. Velocity of fluid
streamlines remains same. changes abruptly.
LEARNING OBJECTIVES
At the end of this chapter the students will be able to:
1. Understand that viscous forces in a fluid cause a retarding force on an object moving through it.
2. Use Stokes’ law to derive an expression for terminal velocity of a spherical body falling through
a viscous fluid under laminar conditions.
3. Understand the terms steady (laminar, streamline) flow, incompressible flow, non-viscous flow
as applied to the motion of an ideal fluid.
4. Appreciate that at a sufficiently high velocity, the flow of viscous fluid undergoes a transition
from laminar to turbulence conditions.
5. Appreciate the equation of continuity Av = Constant for the flow of an ideal and incompressible
fluid.
6. Appreciate that the equation of continuity is a form of the principle of conservation of mass.
7. Understand that the pressure difference can arise from different rates of flow of a fluid
(Bernoulli effect).
8. Derive Bernoulli’s equation in form P + ½ρv2 + ρgh = constant.
9. Explain how Bernoulli effect is applied in the filter pump, atomizers, in the flow of air over an
aerofoil, Venturimeter and in blood physics.
10. Give qualitative explanations for the swing of a spinning ball.
INTRODUCTION:
The study of fluids in motion is relatively complicated, but analysis can be simplified by making
a few assumptions. The analysis is further simplified by the use of two important conservation
principles; the conservation of mass and the conservation of energy. The law of conservation of mass
gives us the equation of continuity while the law of conservation of energy is the basis of Bernoulli’s
equation. The equation of continuity and the Bernoulli’s equation along with their applications in
aeroplane and blood circulation are discussed in this chapter.
FLUID:
“Any thing that can flow is called fluid”.
Example:
Water and air are the examples of fluid. S.I unit of η is Poiseuille or Pl.
FLUID DYNAMICS:
“The branch of physics that deals with fluid in motion and their effects is called fluid
dynamics”.
243
,244 Physics Intermediate Part-I
Conservation Principle of Fluid Dynamics:
Following are the two principles that deal with motion of fluid.
Law of Conservation of Mass:
Law of conservation of mass of fluid is given by the equation of continuity.
Law of Conservation of Energy:
Law of conservation of energy of fluid is given by Bernoulli’s equation.
VISCOUS DRAG AND STOKE’S LAW:
Viscosity and Viscous:
“The frictional force existing between the different layers of fluid that opposes its motion is
called viscosity”.
The substance possessing this property is said to be viscous.
Co-efficient of Viscosity (η
η):
“The tangential force that produces a velocity difference of 1 ms−1 between the adjacent layers
that are one metre apart is known as co-efficient of viscosity”. It is represented by η.
Substances having Large Co-efficient of Viscosity:
Substances that cannot flow easily are said to have large co- Viscosities of Liquids and
efficient of viscosity. Gases at 30°C
Viscosity
Examples: Material
10− 3 (Nsm− 2)
Honey and thick tar are the substances having large co-efficient Air 0.019
of viscosity. Acetone 0.295
Methanol 0.510
Substances having Small Co-efficient of Viscosity: Benzene 0.564
Water 0.801
Substances that flow easily are said to have small co-efficient of Ethanol 1.000
viscosity. Plasma 1.6
Glycerin 6.29
Examples:
Air, water, milk are the examples of substances having small co-efficient of viscosity.
Drag Force (Fluid Friction):
“When a substance moves through a fluid it experiences a retarding force that opposes its
motion. This retarding force is called drag force”.
Effect of Speed:
The drag force increases as the speed of an object increases through a fluid but at very high
speed drag force may no longer be proportional to the speed.
Examples:
(i) The force offered by the air to the motion of cyclist.
(ii) The retarding force offered by air to the car moving at high speed.
,[Chapter-6] Fluid Dynamics 245
Stoke’s Law:
The law that expresses the drag force in mathematical form, according to which drag force ‘Fd’
on a sphere of radius ‘r’ moving with velocity ‘v’ through a fluid of viscosity ‘η’ is given by
Fd = 6πηrv
Limitation:
The expression of drag force is valid for slow moving objects. At high speed drag force is
proportional to squared value of speed and is given by
1
Fd = ρC Av2
2 D
Where, ρ = Density
CD = Drag co-efficient
A = Frontal area
TERMINAL VELOCITY:
“The maximum constant velocity of an object falling vertically downward through a fluid
under the action of gravity is called terminal velocity”.
Symbol:
It is represented by ‘vt’.
Mathematical Form:
Mathematically terminal velocity is given by
2ρgr2
vt =
9η
Derivation:
Suppose a fog droplet of radius ‘r’ is falling down due to its
weight ‘W’, through a medium of co-efficient of viscosity ‘η’ and it
experiences a drag force ‘Fd’ in opposite direction to that of weight. A table tennis ball can be made
The net force acting on the droplet is suspended in the stream of air
coming from the nozzle of a hair
F = ω – Fd …… (1) dryer.
Now F = ma , ω = mg and Fd = 6 πηrv
So eq. (1) becomes Drag force
ma = mg − 6πηrv …… (2)
As the speed of droplet continues to increase, the drag force ‘Fd’ Fog droplet
also increases till it attains a magnitude which will be equal to its
weight. In this case the velocity of fog droplet will be maximum and Weight
constant so acceleration ‘a’ will be zero. So eq. (2) becomes
m(0) = mg – 6 πηrvt
0 = mg – 6 πηrvt
6 πηrvt = mg
mg
vt = …… (3)
6πηr
, 246 Physics Intermediate Part-I
As we know that:
m = ρV …… (a) Viscosity of gases rises with
temperature.
Where m = Mass of droplet
ρ = Density of droplet
4 3
Volume of sphere (droplet) = πr
3
So eq. (a) becomes
4 3
m = ρ πr
3
Put value of ‘m’ in eq. (3)
4 3
ρ πr g
3
vt =
6πηr
2
vt = (ρgr2)
9η
2ρgr2
vt =
9η
vt ∝ r2
Result:
The above expression shows that terminal velocity is directly proportional to square of radius of
sphere.
FLUID FLOW:
The fluid passing through the cross-sectional area of pipe per unit time is called fluid flow.
Types of Fluid Flow:
Following are the two types of fluid flow:
Laminar Flow Turbulent Flow
The regular and steady flow of fluid is called The irregular and un-steady flow of fluid is called
laminar or streamline flow. turbulent flow.
OR OR
The flow in which every particle follows the same The flow in which each particle does not
path that has already been followed by other necessarily follow the same path that has already
particles is called laminar flow. Velocity of been followed by other particles. Velocity of fluid
streamlines remains same. changes abruptly.
