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.