MINISTRY OF EDUCATION AND TRAINING

NONG LAM UNIVERSITY

FACULTY OF FOOD SCIENCE AND TECHNOLOGY

Course: Physics 2

Module 2:

Fluid Mechanics

Instructor: Dr. Nguyen Thanh Son

Academic year: 2009-2010

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Contents

Module 2: Mechanics of fluid

2.1. Motion of ideal fluid

2.1.1 The concept of fluid

2.1.2 Density and pressure

2.1.3 Fluids in motion

2.2. Bernoulli’s equation and applications

2.2.1 Equation of continuity of fluid flow

2.2.2 Bernoulli’s equation

2.3. Newton’s law of viscosity – Viscosity of fluid

2.3.1 Newton's law of viscosity

2.3.2 Viscosity of fluid

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2.1. Motion of ideal fluid

2.1.1 The concept of fluid

•••• There are three “states of matter”. The simplest way to describe the differences among them is to say:

♣ A solid has a definite volume and shape.

♣ A liquid has a definite volume but no definite shape.

♣ A gas has neither a definite volume nor a definite shape.

• A fluid is a substance in which the constituent molecules are free to move relative to each other. The

fluids sub-divide further into liquids and gases.

• Fluid can be either liquid or gas and is a substance that can flow, cannot withstand shearing stress, and

conforms to container. The molecules of a fluid are not arranged in particular manner but are free to

move.

• Fluids are actually either liquids or gases. Fluids in fact are materials that flow and have no definite

shape of their own.

Liquid: A state of matter in which the molecules are relatively free to change their positions

with respect to each other but restricted by cohesive forces so as to maintain a relatively fixed

volume.

Gas: A state of matter in which the molecules are practically unrestricted by cohesive forces. A

gas has neither definite shape nor volume.

• We describe a fluid by using quantities, such as density, pressure, temperature and fluid velocity, that

can depend on location and time.

• Field of Fluid Mechanics can be divided into 3 branches:

♣ Fluid Statics: mechanics of fluids at rest.

♣ Kinematics: deals with velocities and streamlines without considering forces or energy.

♣ Fluid Dynamics: deals with the relations between velocities and accelerations and forces

exerted by or upon fluids in motion.

• Mechanics of fluids is extremely important in many areas of engineering and science. Examples are:

♣ Biomechanics

Blood flow through arteries

Flow of cerebral fluid

♣ Meteorology and Ocean Engineering

Movements of air currents and water currents

♣ Chemical Engineering

Design of chemical processing equipment

2.1.2 Density and Pressure

• We can characterize a fluid at rest by specifying its density ρ and describe the forces on the fluid in

terms of the scalar pressure P (force per unit area; the SI unit is the pascal, with 1 Pa = 1 N/m2).

a) Density

• The density ρ of a substance of uniform composition is its mass per unit volume:

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ρ = m/V (69)

• ρ is constant for an incompressible fluid.

• In the SI system, density is measured in units of kilograms per cubic meter (kg/m3).

• Since the volume of a fluid expands and contracts, the density of fluids vary with temperature. The

most common fluid, water, has maximum density of 1000 kg/m3 at 4ºC. Air, a mixture composed

principally of the gases, nitrogen (78%) and oxygen (21%), has a density of 1.29 kg/m3 at 0ºC and 1.20

kg/m3 at 20ºC.

• Density of fluid is often stated relative to water (at 4ºC), and when so stated it is called specific

gravity. For instance, the specific gravity of mercury is 13.6. This means that the density of mercury is

13.6 times that of water (at 4ºC), or 13.6 g/cm3 or 13600 kg/m

3. If a liquid has a specific gravity of 0.9,

then its density is 0.9 times that of water, or 0.9 x 1000 = 900 kg/m3.

b) Pressure

• Pressure P is defined as the (perpendicular) force per unit area

P = F/A (70)

where the force is perpendicular to the area of interest, as shown in the figure

on the left.

• Pressure is a scalar quantity measured in pascals, Pa, where 1 Pa = 1 N/m2.

• Note that 1 atmosphere (atm) = 1.01 x 105 Pa = 760 torr = 14.7 lb/in

2. Also

note that 1 atm is the value of the pressure that is equal to a 0.76 m column of mercury at T = 0°C and g

= 9.80665 m s-2.

• The force exerted by a fluid at rest acting on any rigid surface is always perpendicular to the surface.

For if there were a non-perpendicular component, then the fluid would no longer be at rest, but would

respond to the reaction force supplied by the surface and motion would result.

• For an incompressible fluid at rest in a uniform gravitational field, the pressures within the fluid can

be analyzed by applying the basic force-pressure-area relation,

F = PA (71)

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Variation of pressure with depth

In equilibrium, all points at the same depth

must be at the same pressure. Otherwise a

net force would be applied and the fluid

would accelerate (see the figure on the

right).

Consider a volume of fluid at a distance h

below the surface (see the figure on the

left):

(72)

where w = Mg is the gravitational force. Po = 1.013x105 Pa at sea level.

• (72) is the equation for pressure of fluid as a function of depth.

• The pressure at a given depth does not depend upon the shape of the vessel containing the liquid or

the amount of liquid in the vessel.

• As a result, in each of the three containers shown below, the total pressure at depth h would be the

same - it is independent of the shape of the container, volume of water above the surface, or the

exposed surface area, as shown the below figure.

• From equation (72) we see that comparing with the atmospheric pressure, P increases with depth by

an amount ρgh.

• Note that in equation (72) P is absolute (true) pressure inside the container, and P-P0 is the gauge

pressure, the pressure that is added to the atmospheric pressure to equal P. The gauge pressure is due to

the liquid alone and at a given depth depends only upon the density of the liquid ρ and the distance

below the surface of the liquid h.

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Example: Calculate the pressure at 1000 m of ocean depth. Normal atmospheric pressure is

Po = 1.013 x 105 Pa = 101,300 N/m

2. Given g = 9.8 m/s

2.

Solution: P = 1.013 x 105 Pa + (1.0 x 10

3 kg/m

3)(9.80 m/s

2) (1000 m) = 9.9 x 10

6 Pa.

Pascal's principle (Pascal's law)

• When a confined fluid is completed enclosed, a change in pressure in one location is transmitted

through the fluid. Consider a water balloon with negligible air. Squeezing one side of the balloon

transmits the pressure to all other regions usually resulting in the opposite of the balloon being pushed

outward, stretching the balloon.

•••• Pascal's principle may be stated that a change in the pressure at any point in an enclosed fluid

that is at rest is transmitted undiminished to all points in the fluid and in all directions.

•••• Pascal's principle is utilized in hydraulic systems. In Figure 13, a push on a cylindrical piston at point

a lifts an object at point b.

• Let the subscripts a and b denote the quantities at each piston. According to Pascal's principle,

the changes in pressure at a and b are equal, or ∆Pa = ∆Pb. Substitute the expression for pressure in

terms of force and area (equation 71) to obtain Fa /Aa = (Fb /Ab). Substitute πr2 for the area of a

circle, simplify, and solve for Fb: Fb = (Fa)(rb2/ra

2). Because the force exerted at point a is

multiplied by the square of the ratio of the radii and rb > ra , a modest force on the small piston a

can lift a relatively larger weight on piston b.

Figure 13: Pascal's principle is used to lift a car easily.

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2.1.3 Fluids in motion

• Fluids considered in this course move under the action of a shear stress, no matter how small that

shear stress may be (unlike solids).

• It is convenient to assume that fluids are continuously distributed throughout the region of interest.

That is, the fluid is treated as a continuum. This continuum model allows us to not have to deal with

molecular interactions directly. We will account for such interactions indirectly via viscosity (see

section 2.3.2).

Ideal fluid flow

• Motion of real fluids can be very complex. It is necessary for us to be quite restrictive and look only

at ideal fluids in motion.

• By ideal we mean that the fluid flow is (1) non-viscous, (2) steady or laminar, (3) incompressible, and

(4) irrotational.

• In other words, there are four simplifying assumptions made to the complex flow of fluids to make the

analysis easier:

(1) The fluid is non-viscous – internal friction is neglected. All real fluids (with exception of

superfluids at low temperatures) have viscosity. Viscosity is a fluid’s internal friction or resistance to

flow, e.g. compare tar, olive oil and water. We assume ideal fluids with zero viscosity.

(2) The flow is steady (or laminar): the fluid’s velocity, density and pressure do not change with

time. At any point in the moving fluid we can define a velocity of flow. Steady or laminar flow

maintains when the velocity at any point in the flow remains constant. For example, observe water

flowing from a tap. When tap is just open we see a smooth, steady flow of water – this is laminar flow.

If tap is opened wide, water gushes from the tap, the flow is irregular and the pattern of flow changes

with time – this is turbulent or non-laminar flow.

(3) The fluid is incompressible: the density has a constant value.

(4) The flow is irrotational – the fluid has no

angular momentum about any point or the objects of

interest do not spin.

• Such a fluid moves without turbulence, and no eddy

currents are present.

Streamlines and streamtubes

• A streamline is a line that is tangential to the

instantaneous velocity direction, as shown in Figure 14

(velocity is a vector that has a direction and a

magnitude).

• Each streamline traces out the trajectory of some

selected small portion or element of the fluid. To

investigate flow we might put markers in the fluid, e.g.

a dye in a liquid or smoke particles in a gas. These

Figure 14: Illustration of streamlines of

fluid flow.

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markers will follow the pattern of streamlines.

• A streamline is actually the path which the particle takes in steady flow. The velocity of the particle is

tangent to the streamline, as shown in Figure 14.

• We see that streamlines help us to visualise flow of

fluid in the same ways of electric or magnetic field

lines help us to visualize these fields.

• Flow streamlines of fluid have the following

properties:

• a tangent to a streamline at a point of interest is

in the direction of the fluid velocity at that

point;

• the density of streamlines in the vicinity of a

point is proportional to the magnitude of the

velocity at that point;

• the streamlines cannot intersect except at a point

of zero velocity, otherwise the velocity would

not be uniquely determined at that point.

• A set of streamlines is called a tube of flow or a streamtube. A streamtube is actually a tubular region

of fluid surrounded by streamlines, as shown in Figure 15. Since streamlines do not intersect, the same

streamlines pass through a streamtube at all points along its length. Let us take two cross-sections of a

streamtube, with cross-sectional areas A1 and A2 (see Fig. 15). The number of streamlines passing

through A1 is equal to that passing through A2.

• If these areas are made small enough then the fluid velocities across the cross-sections will be

constant.

2.2. Bernoulli’s equation and applications

2.2.1 Equation of continuity of fluid flow

• For an ideal fluid and in the absence of sources or

sinks, in a definite time the mass flowing into a

region must be equal to the mass flowing out. A

good analogy to such fluid flow is the flow of the

traffic of automobiles driven by perfectly behaved

drivers.

• Consider a fluid moving through a pipe of

nonuniform size (diameter), as shown in Figure 16.

The particles move along streamlines in steady

flow and assume that there are no ‘sources’ or

‘sinks’.

• Because mass is conserved and the flow is

steady, the mass that crosses A1 (lower portion of

Fig. 16) in some time interval ∆t is equal to the

Figure 15: A streamtube in a flowing fluid.

Figure 16: The mass that crosses A1 in some

time interval ∆t is equal to the mass that

crosses A2 in that time interval.

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mass that crosses A2 (upper portion of Fig. 16) in that time interval.

∆m1 = ∆m2 or ρ1A1∆x1 = ρ2A2∆x2

or ρ1A1v1∆t = ρ2A2v2∆t

or ρ1A1v1 = ρ2A2v2 (73)

or

ρAv = constant (74)

where v denotes fluid’s velocity.

• The quantity ρAv is often called the mass flow rate (mass per unit time). Equation (73) ensures that

the mass flow rate into a region equals that out of the region.

• Since the fluid of interest is incompressible, ρ is a constant; thus ρ1 = ρ2 = ρ. As a result,

A1v1 = A2v2 (75)

or Av = constant (76)

• (76 ) is called the equation of continuity for fluids which states that the cross-sectional area of the

pipe and the velocity of the fluid are inversely proportional - that is, fluids flow faster through

narrower pipes and vice versa. We can see this by the fact

that the streamlines are forced closed together whenever the

pipe narrows, as illustrated by Figure 17.

• From equation (76) we see that the product of the cross-

sectional area and the fluid speed at all points along a pipe

is constant for an incompressible fluid. The product R = Av

is called the volume flow rate or the volume flux and has

dimensions of volume per unit time (Remember that flux is

flow per unit area.).

The dimension of R is the volume per unit of time, L3/M.

The SI unit of R is m3/s.

• The condition R = Av = a constant is equivalent to the fact that the amount of fluid (the fluid

volume) enters one end of the tube in a given time interval equals the amount of fluid (the

volume) leaving the tube in the same time interval, assuming that the fluid is incompressible

and that there are no sources or sinks.

• From (76) we see that the fluid speed is high where the tube is constricted (small A) and the speed is

low where the tube is wide (large A). As the stream of fluid flows continuously, if the width of the

Figure 17: The streamlines are

forced closed together whenever the

pipe narrows.

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stream narrows the fluid speeds up and vice versa. This equation shows that where a pipe narrows the

velocity increases, e.g. jet formed when you squeeze the end of a garden hose; rivers flow faster when

narrower near their source and slower as they broaden out on the plain.

• The equation of continuity expresses the conservation of matter. It is one of the equations used for

analyzing fluid motions.

2.2.2 Bernoulli’s equation

• As a fluid moves through a region where its

speed and/or elevation above the Earth’s

surface change, the pressure in the fluid varies

with these changes.

• The relationship between fluid speed,

pressure and elevation was first derived by

Daniel Bernoulli. Consider the two shaded

segments of a flowing fluid, as shown in

Figure 18. The volumes of both segments are

equal, meaning that V1 = V2 = V.

• We can see from Figure 18 that the pipe

carrying fluid ‘up hill’ and the cross-section of

the pipe changes from A1 (at the lower end) to

A2 (at the upper portion).

• The work done on the lower end of the fluid by the fluid bedind it is

W1 = F1∆x1 = P1A1∆x1 = P1V1 = P1V,

and similarly the work done on the fluid on the upper portion is W2 = -P2V2 = -P1V.

W2 is negative because the force on the fluid on the upper portion is opposite its displacement.

• The net work done by these forces is the equal to the net work done by the fluid on the fluid segment

of interest

Wfluid = W1 + W2 = (P1 – P2)V (77)

• Part of this work goes into changing the kinetic energy and some to changing the gravitational

potential energy of the earth-fluid system.

• The change in kinetic energy is

where m = m1 = m2 due to V1 = V2 = V and ρ is constant for an ideal fluid. In other words, the

masses are the same since the volumes are the same and the densities are also the same.

• The change in gravitational potential energy:

2 22 1

1 1K mv mv (78)2 2

∆ = −

Figure 18: A fluid moving with streamline flow

through a pipe of varying cross-sectional area.

The volume of fluid flowing through A1 in a

time interval ∆t must be equal to that flowing

through A2 in the same time interval.

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∆U = mgy2 – mgy1 (79)

where y1 and y2 are the heights (elevations) at point 1 and point 2, respectively.

• According to the law of energy conservation, the work done on the fluid segment of interest equals

the change in its mechnical energy or

Wfluid = ∆K + ∆U (80)

• Combining equations (77), (78), (79), and (80) leads to

(P1 – P2)V = 2 22 1

1 1mv mv2 2

− + (mgy2 – mgy1)

• By rearranging and expressing in terms of density, we have

P1 +21

1 v2

ρ + mgy1 = P2 +22

1 v2

ρ + mgy2 (81)

• This is Bernoulli’s equation and is often expressed as

P + 21 v2

ρ + ρgy = constant (82)

• Bernoulli’s equation states that the sum of the total pressure (P), the kinetic energy per unit

volume (21 v

2ρ ), and the kinetic energy per unit volume (ρgy) has the same value at all points

along a fluid streamline.

• We can see that the Bernoulli theorem comes essentially from the conservation of energy.

• Note that each term in equation (82) is actually an expression of energy/volume and has SI units of

J/m3.

P has units of N/m2 which can be also expressed as work per unit volume, Nm/m

3 = J/m

3.

ρgy is just potential energy per unit volume, mgy/V, which is measured in J/m3.

21 v2

ρ is just kinetic energy per unit volume, 12

mv2/V, which is also measured in J/m

3.

Fluid pressure and speed

• If we examine Bernoulli’s eqation for the case y = constant, i.e. no change in height, we have from

equation (82)

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P + 21 v2

ρ = constant (83)

This equation shows that for horizontal motion of a fluid element, where the speed of the fluid

increases the pressure will decrease, and vice versa.

• When the fluid is at rest (v1 = v2 = 0), equation (81) becomes P1 – P2 = ρg(y2 – y1) = ρgh, where

h = y2 - y1 is the difference in height or P1 = P2 + ρgh which is consistent with the pressure variation

with depth we found earlier for a fluid at rest (see equation 72).

Example: From Beiser, Arthur. (1992) Physics 5th ed. Addison-Wesley Publishing Company.

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Applications

• Venturi Tube

Venturi tube is a device that consists of a gradually decreasing nozzle through which the fluid in

a pipe is accelerated, followed by a gradually increasing diffuser section that allows the fluid to nearly

regain its original pressure head, as shown in the below figure. This device can be used to measure fluid

flow rate (a venturi meter), or to draw fuel into the main flow stream, as in a carburetor.

Proportions of a Herschel-type venturi tube for standard fluid-flow measurement

As shown the above figure, a constriction (a short straight pipe section or throat between two

tapered sections) that is placed in a pipe and causes a drop in pressure as fluid flows through it.

The increase in velocity of the fluid is

accompanied by a drop in its pressure.

An understanding of the Venturi tube requires both the continuity equation and Bernoulli's

equation. The velocity of the air flowing through the tube depends on the cross - sectional area. For a

smaller area (A2 < A1), the fluid (air) velocity is greater, and from the continuity equation, we have

v2 = v11

2

AA

Applying this to Bernoulli's equation for constant y, we have

2 2 211 1 1 2

2

A1 1v P v ( ) P2 2 A

+ = +

or 2 212 1 1

2

A1P P v [1 ( ) ]2 A

= + −

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•••• Other applications

Atomizer:

A stream of air passing over a tube dipped

in a liquid causes the liquid to rise in the

tube. This effect can be found in perfume

atomizer bottles and paint sprayers.

Vascular Flutter:

The constriction in the blood vessel speeds

up going through the constriction. The

lower pressure causes the vessel to close,

stopping the flow. Without flow, there is no

Bernoulli effect, and blood pressure causes

it to re-open. The process repeats.

2.3. Newton’s law of viscosity – Viscosity of fluid

2.3.1 Newton's law of viscosity

• Consider a fluid (either a gas or a liquid) contained between two large parallel plates of area A, which

are everywhere separated by a very small distance y (the top panel of Figure 19).

The upper plate remains motionless, and a shear (parallel) force is applied to the bottom plane to

maintain it at a velocity V (the first middle panel of Figure 19).

Because of the friction in the fluid and the friction between the fluid and the plate, the fluid begins

to move with the bottom plate. The upper plate remains stationary.

• This figure also shows the coordinate systems that is usually used for momentum transfer. x

coordinate: the direction of the velocity vx and y coordinate: the direction of change of vx and the

direction for momentum transfer.

• We see that

At t < 0, the motion has not started yet. So the whole fluid is at rest (the top panel of Figure 19).

At t = 0, the part of fluid right close to the lower plate starts to move (the first middle panel of

Figure 19). But still no enough time allowed to produce the velocity profile.

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At small t values, some of the fluid is pulled along with the plate (the second middle panel of

Figure 19).

At long time (t = ∞), a linear velocity gradient develops, and flow profile reaches a steady state

(the bottom panel of Figure 19).

•••• The magnitude of this gradient (how fast the fluid speed changes with distance) is characteristic of the

fluid.

• Because of viscosity, at boundaries (walls) particles of fluid adhere to the walls, and so the fluid velocity is zero relative to the walls.

•••• Newton's law of viscosity is an empirical law that describes the behaviour of some fluids under a

limited range of conditions.

•••• It states that when a shearing stress (ττττ = F/A) acts within a fluid moving in a streamline motion, it

sets up in the liquid a velocity gradient which is proportional to the stress.

velocity gradient = (constant) x (stress)

or (stress) = (constant) x (velocity gradient)

•••• Mathematically,

Figure 19: Using a shear stress to create a steady laminar velocity

profile for a fluid contained between two plates.

F

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• The proportional constant µ is known as the coefficient of viscosity, that is characteristic of the fluid

and τ = F/A is the shear stress (parallel force per unit area) acting within the fluid. The SI unit of

viscosity is kg/m/sec.

• Application: When a fluid (e.g. air) flows past a stationary wall (e.g. table top), the fluid right close

to the wall does not move. However, away from the wall the flow speed is not zero. So a velocity

gradient exists, as shown in Figure 20.

2.3.2 Viscosity of fluid

• Viscosity can be thought as the internal stickiness of a fluid. It is a representative of internal friction

in fluids.

• Internal friction forces in flowing fluids result from cohesion and momentum interchange (transfer)

between molecules.

• Viscosity of a fluid depends on temperature:

In liquids, viscosity decreases with increasing temperature (i.e. cohesion decreases with

increasing temperature).

In gases, viscosity increases with increasing temperature (i.e. molecular interchange between

layers increases with temperature setting up strong internal shear).

• Viscosity is important, for example,

♣ in determining amount of fluids that can be transported in a pipeline during a specific

period of time.

♣ determining energy losses associated with transport of fluids in ducts, channels and pipes.

Figure 20: Velocity gradient in a stream of fluid moving past a stationary wall.

F dV (84)A dy

τ = = µ

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References

1) Halliday, David; Resnick, Robert; Walker, Jearl. (1999) Fundamentals of Physics 7th ed. John Wiley

& Sons, Inc.

2) Feynman, Richard; Leighton, Robert; Sands, Matthew (1989) Feynman Lectures on Physics.

Addison-Wesley.

3) Serway, Raymond; Faughn, Jerry. (2003) College Physics 7th ed. Thompson, Brooks/Cole.

4) Sears, Francis; Zemansky Mark; Young, Hugh. (1991) College Physics 7th ed. Addison-Wesley.

5) Beiser, Arthur. (1992) Physics 5th ed. Addison-Wesley Publishing Company.

6) Jones, Edwin; Childers, Richard. (1992) Contemporary College Physics 7th ed. Addison-Wesley.

7) Alonso, Marcelo; Finn, Edward. (1972) Physics 7th ed. Addison-Wesley Publishing Company.

8) Michels, Walter; Correll, Malcom; Patterson, A. L. (1968) Foundations of Physics 7th ed. Addison-

Wesley Publishing Company.

9) Hecht, Eugene. (1987) Optics 2th ed. Addison-Wesley Publishing Company.

10) Eisberg, R. M. (1961) Modern Physics, John Wiley & Sons, Inc.

11) Reitz, John; Milford, Frederick; Christy Robert. (1993) Foundations of Electromagnetic Theory, 4th

ed. Addison-Wesley Publishing Company.

12) Websites:

http://www.umich.edu/~amophys/125/ttwo/ttwo.html

http://www.physics.uc.edu/~sitko/CollegePhysicsIII/9-Solids&Fluids/Solids&Fluids.htm

http://galileo.phys.virginia.edu/classes/311/notes/fluids1/node1.html

http://www.cliffsnotes.com/WileyCDA/CliffsReviewTopic/Fluids.topicArticleId-10453,articleId-

10421.html

http://dev.physicslab.org/Document.aspx?doctype=3&filename=Fluids_Dynamics.xml

http://dev.physicslab.org/TOC.aspx

http://dev.physicslab.org/Chapter.aspx?cid=24

http://sfhs.sbmc.org/~thiggins/APPhysicsB/Chapter%209/notes_chapter_9.htm

http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/Pressure/BernoulliEquation.html

home.anadolu.edu.tr/~bbozan/Ch_1_visvosity.doc

http://www.answers.com/topic/venturi-tube-2#