Fluid Mechanics (Part III)
Objectives:-
1. To know about the continuity equation
2. To study the Bernoulli’s equation
3. To get an idea about the Venturimeter
4. To have an idea about laminar and turbulent flows
Module 1
The Continuity Equation:-
THE CONTINUITY EQUATION
The continuity equation is the mathematical expression of the law of
conservation of mass in fluid mechanics. Let us consider an arbitrary volume
element. Now, the mass of the fluid flowing outward per second through the
surface enclosing the volume element is the scalar product .v ds reveals
automatically that inward flow is negative. The quantity flowing out must be
equal to decrease per second, in
the amount of fluid within the volume element, provided there are no sources or
sinks present within the volume. This means we have,
.sv ds =
vdV
t
(1)
Using Gauss’s divergence theorem, equation (1) takes the form,
.( )v
v V =vdV
t
(2)
Since equation (2) must hold well for every element of the volume, one finds
.( )v = t
(3)
This equation is called the hydronomic equation of continuity. One can express
this equation in the component form as
( )( ) ( )yx zvv v
x y z t
(4)
The continuity for an incompressible fluid 0pt
is .( ) 0v
or( )( ) ( ) 0yx zvv v
x y z
(5)
With a steady motion, the flow liquid through a cross section of a stream filament
does not depend on the location of the cross section. For two arbitrary cross
section 1ds and 2ds of an elemental filament the following conditions hold:
1 1 1v ds = 2 2 2v ds (6)
Further, if the liquid is incompressible, the density is
same everywhere and equation (6) reduces to
1 1 2 2v ds v ds (7)
or
vds a constant (8)
It predicts that in the steady compressible flow the speed of flow varies inversely
with the cross sectional area, being larger in the narrower parts of the tube.
Module 2
THE BERNOULLI’S EQUATION:STEADY FLOW OF FLUIDS
When dealing with the motion of liquids we can often consider that the
displacement of some portions of a liquid relative to others is not associated with
the appearance of forces of friction. A liquid in which internal friction (viscosity)
is completely absent is called ideal (or non-viscous)
Consider now a region in a fluid where there is a stationary or steady flow as
shown in figure1. During a short time interval t , the fluid that was initially
passing through surface 1A , has advanced to the surface 11A a distance
1 1( )x v t , while the fluid at 2A has advanced a distance 2x to 12A . Since the
reminder of the volume between surfaces 1A and 2A , remains unchanged. One
can focus attention on two (equal) volumes that are shown in the shaded area in
figure no: 1. these two volumes are equal because we assume the fluid is
incompressible and equation of continuity holds. Let 1F and 2F are the forces
exerted in the surface 1A and 2A as a result of the pressure within the fluid.
Because of these forces, work is done on or by the fluid in moving the two
volumes: at 1A the surface is pushed by the fluid, and the work done by the fluid
is 1 1F x ; while at 2A the fluid is pushing the surface, and the work done by the
fluid is 2 2F x . Obviously, the net work done on the volume of the fluid between
1A and 2A is
Figure No: 1
1 1 2 2extW F x F x 1 1 1 2 2 2extW PA x P A x (9)
Let 1P and 2P be the pressure in the fluid at 1A and 2A respectively. We have
F pA . Since 1 1 2 2A x A x according to the equation of the continuity and the
assumption of incompressibility, one finally gets
1 2 1 1( )extW P P A x (10)
The external work on a system changes the proper energy of the system, extW v
. For the fluid volume shown in figure no: 1 the proper energy of the shaded
volume is comprised of kinetic and gravitational potential energy. The fluids
between 1A and 2A gains the energy in the volume 2 2A x and loses the energy in
1 1A x .
Let the two volumes have masses M , which are equal, again due to the equation
of continuity. Thus, the next gain energy is
2 1U U U
Since we have assumed an in compressible fluid, the density of fluid p is the same
everywhere and M may be replaced by 1pA 1x for both ends. Thus
1
2 12 2 2 1 1
1 12 2
U V gZ V gZ A x (11)
Combining equation (10) and (11), one finds
Or
(12)
Since equation (12) refers to quantities at two arbitrary points along a stream line,
one may generalize to
212V gZ p
a constant (c) (13)
This result is known as Bernoulli’s equation for steady, non-viscous,
incompressible flow. This expresses conservation of energy in a fluid. The first
term in equation (13) is the kinetic energy per unit volume while the second term
is potential energy per unit volume. Thus one may consider pressure also in an
energy per unit volume.
If the flow is in a horizontal plane 1 2Z Z only, the gravitational potential
energy remains constant and equation (13) reduces to
212V p a
constant (14)
This shows that in a horizontal pipe, the greater the velocity of flow, the lower the
pressure and conversely. In other words, where the velocity of flow is less, the
pressure is larger and vice versa. This effect is used to produce the lift of an
airplane (figure no: 2). The profile of the wing is so designed that the air has a
greater velocity above the wing surface than below it. This greater velocity then
produces a lower pressure above the wing than below it, and this different result is
a net upward force.
If the wing area is A, the upward force is
2 22 1 1 2
12
F A p p A V V where the subscripts 1 and 2 refer to the
condition above and below to the wings respectively. Since
2 21 2 1 2 1 2
1 12 2V V V V V V , and one may approximate 1 2
12V V
equal to the plane’s air speed V the resultant upward force of the lift may be
expressed as 1 2F A V V V
Figure 2
A second example of Bernoulli’s theorem is a fluid at rest or moving with a
constant velocity in a pipe. In such circumstances the kinetic energy of the fluid is
a constant and may be dropped from equation (13), which then reduces to
gZ p a constant. Writing the constant by 0p , one then has the pressure in
an incompressible fluid in equilibrium given by
0p p pZ (15)
For example, the pressure at the surface of a lake ( 0Z ) is that caused by the
atmosphere above it (i.e., atmospheric pressure), and the pressure then increases
linearly as one goes below the surface (because z is negative) as long as the lake
water is considered incompressible. In the same way, as we increase our altitude
above sea level. The atmospheric pressure decreases linearly as long as one may
ignore variations in air density with altitude.
One can also extend discussion to cases in which the fluid is compressible or the
forces are not conservative. (This latter situations arises, for example, when a
fluid does shaft work in driving mechanism like a turbine in a hydroelectric
installation, or when is exchanged with the surroundings, as in an industrial
chemical plant).
The constant c appearing in equation (13) may in general be different for
different stream lines.
However, if the flow of fluid is irrotational, V vanishes throughout the
entire fluid
When the fluid is placed in the gravitational field, we have f g
or u V
or U V
where V is gravitational potential.
Equation (12) now assumes the form
2 21 1 2 2
1 22 2V p V pV V
a constant
or 2 2
1 1 2 21 22 2
V p V pgZ gZ
a constant
or2
2V p Zg g
= a constant (16)
This is same as equation (13). Every term equation (16) has the dimensions of
length, and is called the head; 2
2Vg
is the velocity of the head,pg is the pressure
head and Z is the elevation head. The condition (16) is observed in a steadily
flowing ideal liquid along any stream line. Equation (12) or Equation (13)
equivalent to it is called Bernoulli’s equation. Although we obtained this equation
for an ideal liquid, it is obeyed sufficiently well for real liquids in which the
internal friction is not very great. One can also derive Bernoulli’s equation
directly from energy conservation principle
Module 3
VENTURIMETER
This is an instrument, based upon Bernoulli’s principle and used for
determining the velocity of a fluid in a pipe. It consists of two horizontal
truncated pipes a and b (figure 3) connected together by a short length of
cylindrical tube c. Two pressure gauges 1G and 2G measures the presence in the
pipe and at a contraction inserted in it.
Let 1A be the cross sectional area of the main pipe (a and b) and 2A that
of the restricted throat c. Let 1p and 2p be the pressures and 1V and 2V be the
velocities of water in a and b respectively. If there is no friction between the pipe
and the flowing water, then the pressure and velocity
in b will be same as in a. The equation of continuity gives
1 1 2 2AV AV
or 12 1
2
AV VA
(17)
Also if pipe is horizontal Bernoulli’s theorem gives us
2 21 1 2 2
1 12 2V p V p (18)
Substituting the value of 2V from equation (17) in equation (18) solving for 1V ,
one finally obtains
1 21 1 22
1
2
2
1
p pV K p p
AA
(19)
Where K is a constant depending on the pipe and on the density of the fluid. The
amount of fluid passing through the any section of the pipe per unit time is
1 1 1 1 2V AV KA p p (20)
Figure 3- Venturimeter
The actual discharge is, however, slightly less than that obtained from equation
(20) due to the friction between the pipe and the water, the viscous forces and
eddy motions. Relation (20) specifying the dependence of the flow rate on the
pressure difference serve as the theoretical foundation for the construction of a
device known as water flow meter which makes it possible to determine (from the
pressure difference) the amount of water passing through the cross section of the
pipe in unit time.
Module 4
LAMINAR AND TURBULENT FLOW
An ideal fluid is a fluid which is non viscous and incompressible.
However, such an ideal fluid does not exist around us. According to Neumann
such an ideal fluid is white water. In the preceding analysis we have not
considered an essential property of real fluids, the internal friction or viscosity.
This is a property inherent to some extent or other in all real fluids (liquids and
gases). Viscosity manifests itself in that motion set up in a fluid gradually stops
after the action of the reasons causing the motion is discontinued. If the internal
friction or viscous force per unit volume in a fluid in motion is visf , the equation
of motion becomes visdVf f pdt
and equation
read as
Although, the approximation of considering the fluid as incompressible is often
quite good, but internal friction or viscosity is significant.
Figure 4- Laminar flow of a fluid (viscous)
Two kinds of flow of a fluid (liquids or gas) is observed. In some cases,
the liquids separate, as it were, into layers that slide relative to one another
without mixing such flow is called laminar (from the Latin word “lamina”
meaning
plate or strip) as shown in figure (4). The parallel layers in the fluid flow part one
another with different velocities. As a result of viscosity, the more rapidly moving
layer tends to drag the adjacent layer along with it, and thus accelerate it.
Reciprocally, we can say that the slower tends to retard the faster one. If we
introduce a colored stream into a laminar flow, its retained without being washed
out over the entire length of the flow because the liquid particle in a laminar flow
do not pass over from one layer to another. A laminar flow is steady.
Turbulent Flow:-
With an increase in the velocity or cross sectional dimension of a flow, its
nature changes quite appreciably vigorous stirring of the liquid appears such a
flow is called turbulent. In a turbulent flow, the velocity of particles at each given
place constantly changes chaotically- flow is not steady. If we introduce a colored
stream into a turbulent flow, already at a small distance from the place of its
introduction, the colored stream into a turbulent flow, already at a small distance
from the place of its introduction, the colored liquid will be uniformly distributed
over the entire section
Conclusion:-
The continuity equation is the mathematical expression of the law of
conservation of mass in fluid mechanics. The Bernoulli’s equation for
steady, non viscous, incompressible flow is given by
212 V gZ p a constant c.
Bernoulli’s equation is valid for an ideal barotropic fluid in a conservative
force field. Venturimeter is an instrument, based upon Bernoulli’s
principle and used for determining the velocity of a fluid in a pipe. Two
kinds of flow of a fluid are laminar and turbulent flows. In laminar flow
the liquid separates, as it were, into layers that slides relative to one
another without mixing. In turbulent flow the velocity of the particle at
each given place constantly changes chaotically – flow is not steady.
Assignments:-
1. What is the basic problem of fluid dynamics?
2. State and prove the principle of continuity in the flow of liquids.
3. State and prove Bernoulli’s theorem and deduce Bernoulli’s equation.
4. Explain the principle and working of a venturimeter to determine the flow of a liquid.
5. Give some simple illustrations of Bernoulli’s theorem.
Reference:-
1. A Text Book of Fluid Mechanics by R.K Rajput.
2 Fluid Mechanics A Course Reader by Jermy M.
3. An Introduction to Fluid Dynamics by G.K. Batchelor
4. Elementary Fluid Dynamics by D.J Acheson.
FAQs:-
1. What is the physical significance of the continuity equation?
This is the mathematical expression for the law of conservation of mass
in fluid mechanics.
2. What does the Bernoulli’s theorem express?
This expresses conservation of energy in a fluid. However, the
Bernoulli’s equation is valid for an ideal barotropic fluid in a conservative force
field.
3. What is venturimeter? What does it consist of?
Venturimeter is an instrument, based on Bernoulli’s principle and need
for determining the velocity of a fluid in a pipe. It consist of two horizontal pipes
a and b connected together by a short length of cylindrical tube c. two pressure
gauges 1G and 2G measures the pressure in the pipe and at a contraction inserted
in it.
4. Distinguish laminar a turbulent flow of fluids?
In laminar flow, the liquid separates, as it were, into layers that slide
relative to one another without mixing. The laminar flow is steady. In turbulent
flow, the velocity of particle at each given place constantly changes chaotically.
The turbulent flow is not steady.
Quiz:-
1. Bernoulli’s equation can be designed directly from ----------------- conservation principle
a. energy b. momentum c. angular momentum
2. A liquid in which viscosity is completely absent is called -----------------------
a. viscous b. ideal c. compressible
3. Venturimeter is an instrument based upon ---------------------- equation
a. Torricelli’s b. Bernoulli’s c. continuity
4. -------------------- flow is steady
a. Laminar b. Turbulent c. none of these
Quiz answers:-
1. a 2. b 3. b 4. a
Glossary:
Buoyant force:
The upward force exerted by the fluid when an object is placed in the fluid.
Nozzle:
A nozzle is a device dessigned to controll the direction or charecteristics of a fluid flow
as it exits or enter an enclosed chamber or pipe via an on fice.
Pneumatic tyre: A tyre made of reinforced rubber and filled with compressed air, used on motor
vehicles and bicycles etc.