COURSE NUMBER: ME 321

Fluid Mechanics I

Fluid: Concept and Properties

Course teacher

Dr. M. Mahbubur Razzaque

Professor

Department of Mechanical Engineering

BUET

1

What is Fluid Mechanics?

Fluid mechanics is a branch of mechanics. It is the study of

fluids either in motion (fluid dynamics) or at rest (fluid

statics) and the subsequent effects of the fluid upon the

boundaries, which may be either solid surfaces or interfaces

with other fluids.

Both gases and liquids are classified as fluids, and the number

of fluids engineering applications is enormous: breathing,

blood flow, swimming, pumps, fans, turbines, airplanes, ships,

rivers, windmills, pipes, missiles, icebergs, engines, filters,

jets, and sprinklers, etc.

Everything on this planet either is a fluid or moves within or

near a fluid.2

What is Fluid?

From the point of view of fluid mechanics, all matter consists

of only two states, fluid and solid.

The technical distinction lies with the reaction of the two to an

applied shear or tangential stress. A solid can resist a shear

stress by a static deformation; a fluid cannot.

Any shear stress applied to a fluid, no matter how small, will

result in motion of that fluid. The fluid moves and deforms

continuously as long as the shear stress is applied.

As a corollary, we can say that a fluid at rest must be in a state

of zero shear stress, a state often called the hydrostatic stress

condition in analysis.3

What is Fluid? …

There are two classes of fluids: liquids and gases.

The distinction technically lies in the effect of cohesive forces.

A liquid, being composed of relatively close-packed molecules

with strong cohesive forces, tends to retain its volume and will

form a free surface in a gravitational field if unconfined from

above.

Gas molecules are widely spaced with negligible cohesive

force, a gas is free to expand until it encounters confining

walls. A gas has no definite volume and when left to itself

without confinement, a gas forms an atmosphere which is

essentially hydrostatic. Gases cannot form a free surface.

4

What is Fluid? … …

5

Model of Fluids

Fluids are aggregations of molecules, widely spaced for a gas, closely

spaced for a liquid. The distance between molecules is very large compared

with the molecular diameter.

The molecules are not fixed in a lattice but move about freely relative to

each other. Thus fluid density or mass per unit volume, has no precise

meaning because the number of molecules occupying a given volume

continually changes.

This effect becomes unimportant if the unit volume is large compared with

the molecular spacing, when the number of molecules within the volume

will remain nearly constant in spite of the enormous interchange of particles

across the boundaries.

If, however, the chosen unit volume is too large, there could be a noticeable

variation in the bulk aggregation of the particles.

6

The limiting volume δυ* is about 10-9 mm3 for all liquids and

for gases at atmospheric pressure.

10-9 mm3 of air at standard conditions contains approximately

3x107 molecules, which is sufficient to define a nearly constant

density.7

Fig. The limit definition of fluid density: (a) an elemental volume in a fluid

region of variable density; (b) calculated density versus size of the

elemental volume.

Fluid as a Continuum

Most engineering problems are concerned with physical dimensions

much larger than this limiting volume, so that fluid density is

essentially a point function and fluid properties can be thought of as

varying continually in space. Such a substance is called a

continuum, which simply is a mathematical idealization of fluids.

Although any matter is composed of several molecules, the concept

of continuum assumes a continuous distribution of mass within the

matter or system with no empty space and the properties of the matter

are continuous functions of space variables. It means that the

variation in properties is so smooth that the differential calculus

can be used to analyze the substance.

This approximation is invalid for low pressure gases where the

molecular spacing and mean free path are comparable to, or larger

than, the physical size of the system. Classical fluid mechanics is not

applicable in such cases.8

Fluid Properties

Density and Specific Weight

Fluid density is defined as mass per unit volume. The units

of density are Kg/m3 or slug/ft3.

r = m/V

A fluid property directly related to density is the specific

weight. Specific weight is defined as the weight per unit

volume.

g = W/V = mg/V = (m/V)g = rg

Where g is the local gravitational acceleration. The units of

specific weight are N/m3 or lb/ft3.

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Fluid Properties…

Specific gravity

The specific gravity is used to determine the specific weight

or density of a fluid (usually a liquid). It is defined as the ratio

of the density of a substance to that of water at a reference

temperature of 4oC.

For example, the specific gravity of mercury is 13.6, a

dimensionless number; means the mass of mercury is 13.6

times that of water for the same volume.

10

Fluid Properties…

The density, specific weight and specific gravity of air and

water at standard conditions are given in Table 1.4.

11

12

Fluid Properties…

Viscosity

Viscosity is the most important fluid property in the study of

fluid flows.

-It can be thought of as the internal stickiness of a fluid.

-It is one of the properties that controls the fluid flow rate in a

pipeline.

-It accounts for the energy losses associated with the transport

of fluids in ducts, channels, and pipes.

-It plays a primary role in the generation of turbulence.

-The rate of deformation of a fluid is directly linked to the

viscosity of the fluid. For a given stress, a highly viscous

fluid deforms at a slower rate rhan d fluid with a low

viscosity.13

Viscosity

Consider a flow in which the fluid particles move in the x-direction

at different speeds, so that particle velocities u vary with the y-

coordinate. The Figure shows two particle positions at different

times. For such a simple flow field, in which u = u(y), we can

define the viscosity m of the fluid by the relationship

where t is the shear stress and u is the velocity in the x-direction.

The units of are N/m2 or Pa, and of m are N.s/m2. The quantity

du/dy is a velocity gradient and can be interpreted as a strain rate.

This equation is known as the Newton’s Law of Viscosity.14

Example

Consider a fluid within the small gap between two concentric

cylinders. A torque is necessary to rotate the inner cylinder at

constant speed while the outer cylinder remains stationary.

This resistance to the rotation of the cylinder is due to

viscosity.

The shear tress that resists the applied torque for this simple

flow depends directly on the velocity gradient in the fluid

film in the gap between the cylinders, i.e.

15

Example

For a small gap h<<R, this gradient can be approximated by

assuming a linear velocity distribution in the gap.

Thus using the Newton’s Law of viscosity, the shear stress on the

surface of the inner cylinder may be written as

16

Example

We can then relate the applied torque T to the viscosity and other

parameters by the equation

Here the shearing stress acting on the ends of the cylinder is

neglected; L represents the length of the rotating cylinder. Note that

the torque depends directly on the viscosity, thus the cylinders

could be used as a viscometer, a device that measures the viscosity

of a fluid. 17

18

Fluids which follow the

linear pattern of the

Newton’s law of

viscosity are called

Newtonian fluids.

There are many non-

Newtonian fluids and

they are treated in

rheology.

The figure compares

four examples of non-

Newtonian fluids with

the behaviour of

Newtonian fluids.

Figure: Rheological behavior of various materials

Stress vs. Strain

19

A dilatant (or shear-thickening)

fluid increases resistance with

increasing applied stress.

A pseudoplastic (or shear-

thinning) fluid decreases resistance

with increasing stress.

If the thinning effect is very strong

(the dashed curve) the fluid is

termed plastic.

The limiting case of a plastic substance is one which requires a finite

yield stress before it begins to flow. The linear-flow Bingham plastic

idealization is shown. The flow behavior after yield may also be

nonlinear.

An ex ample of a yielding fluid is toothpaste, which will not flow out

of the tube until a finite stress is applied by squeezing.

A further complication of non-newtonian behavior is the transient

effect shown in the following figure.

Some fluids require a gradually increasing shear stress to maintain

a constant strain rate and are called rheopectic.

The opposite case of a fluid which thins out with time and requires

decreasing stress is termed thixotropic.20

Figure: Rheological behavior of

various materials

Effect of time on applied stress

Example:

A 60-cm-wide belt moves as shown. Calculate the horsepower

requirement assuming a linear velocity profile in the 10oC water.

Sol:

du/dy = 10*1000/2 = 5000 s-1

m for 10oC water = 1.308 x 10-3 N.s/m2

t = m. du/dy = 5000*1.308 x 10-3 N/m2

F = t.A = 5000*1.308 x 10-3 *4*0.6 N

Power = F.U = 5000*1.308 x 10-3 *4*0.6*10 Nm/s

= 0.21 Hp

21

Example:

A 1.2 m long, 2 cm diameter shaft rotates inside an equally long

cylinder that is 2.06 cm in diameter. Calculate the torque required

to rotate the inner shaft at 2000 rpm if SAE-30 oil at 20oC fills the

gap. Also, calculate the horsepower required. Assume symmetric

motion.

Sol:

N = 2000 rpm

w = 2*3.142*N/60 = 209.5 rad/s

du = wr - 0 = 209.5 * 0.01 = 2.095 m/s

dr = 0.06/2 = 0.03 cm = 0.03 x 10-2 m

du/dr = 6982.22 s-1

m for SAE-30 oil at 20oC = 0.4 N.s/m2

t = m. du/dr = 0.4* 6982.22 = 2792.89 N/m2

F = t.A = 2792.89 *3.142*2 x 10-2 *1.2 = 210.61 N

T = r . F = 0.01* 210.61 = 2.1061 Nm

Power = T. w = 2.1061 * 209.5 Nm/s = 441.15 watt

= 0.6 Hp22

Example:

A 25-cm-diameter horizontal disk rotates a distance of 2 mm above

a solid surface. Water at 10oC fills the gap. Estimate the torque

required to rotate the disk at 400 rpm.

Sol:

N = 400 rpm, h = 0.002 m

w = 2*3.142*N/60 = 41.9 rad/s

m for 10oC water = 1.308 x 10-3 N.s/m2

du = wr - 0 = wr; du/dy = wr /h

t = m. du/dy = mwr /h

dF = t.dA = (mwr /h)*2prdr = 2pmwr2dr/h

dT = r*dF = 2pmwr3dr/h

= 3.142*1.308 x 10-3 *41.9/(2*2 x 10-3)* 0.1254 = 0.0105 Nm

Power = T. w = 0.44 watt23

dA

w

r

4

R

0

4 R

0

3 Rh24

r

h

2drr

h

2T

pm

pm

pm

24

Viscosity

Examples of Newtonian fluids: air, water, and oil, etc. Examples

of non-Newtonian fluids: liquid plastics, blood, slurries, paints,

and toothpaste.

An important effect of viscosity is to cause the fluid to adhere to

the surface; this is known as the no-slip condition. This was

assumed in the previous examples.

The viscosity is very dependent on temperature in liquids.

Viscosity of liquids decreases with increased temperature. For a

gas, the viscosity increases as the temperature increases.

The CGS physical unit for viscosity or dynamic viscosity is the

poise (P), named after Jean Leonard Marie Poiseuille. It is more

commonly expressed, as centipoise (cP). Water at 20 °C has a

viscosity of 1.0020 cP.

1 P = 0.1 Pa·s, 1 cP = 1 mPa·s = 0.001 Pa·s = 0.001 N·s/m2.

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Kinematic Viscosity

Viscosity is often divided by the density in the derivation of

equations, it has become useful and customary to define

kinematic viscosity to be

n = m/r

Where the units of n are m2/s (ft2/sec). Note that for a gas, the

kinematic viscosity will also depend on the pressure since the

density is pressure sensitive.

The SI unit of kinematic viscosity is m2/s. The CGS physical unit

for kinematic viscosity is the stokes (St), named after George

Gabriel Stokes. It is sometimes expressed in terms of centistokes

(cSt).

1 St = 1 cm2·s−1 = 10−4 m2·s−1. 1 cSt = 1 mm2·s−1 = 10−6 m2·s−1.

Water at 20 °C has a kinematic viscosity of about 1 cSt.

26

EXAMPLE:

A viscometer is constructed with two 30-cm-long concentric

cylinders, one 20.0 cm in diameter and the other 20.2 cm in

diameter. A torque of 0.13 N-m is required to rotate the inner

cylinder at 400 rpm (revolutions per minute). Calculate the

viscosity.

Ans: 0.00165 Ns/m2

EXAMPLE:

Express the above result in cP and cSt.

Ans: 1 cP = 1 mPa·s = 1cSt

0.00165 Ns/m2 = 1.65 mPa·s = 1.65 cP = 1.65 cSt

27

Compressibility

In the preceding section we discussed the deformation of fluids that

results from shear stresses. In this section, we discuss the

deformation that results from pressure changes.

All fluids compress if the pressure increases, resulting in an

increase in density. A common way to describe the compressibility

of a fluid is by the following definition of the bulk modulus of

elasticity B:

In words, the bulk modulus is defined as the ratio of the change in

pressure (Dp) to relative change in density (Dr/r) while the

temperature remains constant. The bulk modulus obviously has the

same units as pressure.

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Compressibility

The bulk modulus for water at standard conditions is approximately

2100 MPa (310,000 psi), or 21 000 times the atmospheric pressure.

For air at standard conditions, B is equal to 1 atm. In general, B for

a gas is equal to the pressure of the gas.

To cause a 1% change in the density of water a pressure of 21 MPa

(210 atm) is required. This is an extremely large pressure needed to

cause such a small change; thus liquids are often assumed to be

incompressible.

For gases, if significant changes in density occur, say 4%, they

should be considered as compressible; for small density changes

they may also be treated as incompressible.

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Compressibility

Small density changes in liquids can be very significant when large

pressure changes are present.

For example, they account for "water hammer," which can be heard

shortly after the sudden closing of a valve in a pipeline.

When the valve is closed an internal pressure wave propagates

down the pipe, producing a hammering sound due to pipe motion

when the wave reflects from the closed valve.

The bulk modulus can also be used to calculate the speed of sound

in a liquid; it is given by

This yields approximately 1450 m/s (4800 ft/sec) for the speed of

sound in water at standard conditions.

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Vapor Pressure

When a small quantity of liquid is placed in a closed

container, a certain fraction of the liquid will vaporize.

Vaporization will terminate when equilibrium is reached

between the liquid and gaseous states of the substance in the

container - in other words, when the number of molecules

escaping from the water surface is equal to the number of

incoming molecules. The pressure resulting from molecules in

the gaseous state is the vapor pressure.

The vapor pressure is different from one liquid to another. For

example, the vapor pressure of water at standard conditions

(15oC, 101.3 kPa) is 1.70 kPa absolute and for ammonia it is

33.8 kPa absolute.

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Vapor Pressure

The vapor pressure is highly dependent on pressure and

temperature; it increases significantly when the temperature

increases. For example, the vapor pressure of water increases to

101.3 kPa (14.7 psi) if the temperature reaches 100oC.

In general, a transition from the liquid state to the gaseous state

occurs if the local absolute pressure is less than the vapor pressure

of the liquid.

In liquid flows, conditions can be created that lead to a pressure

below the vapor pressure of the liquid. When this happens, bubbles

are formed locally. This phenomenon is called cavitation.

Cavitation in a flow can be very damaging when bubbles are

transported by the flow to high pressure regions and collapse. It has

the potential of damaging a pipe wall or a ship‘s propeller.

Surface Tension

Suface tension is a property that results from the attractive forces

between molecules. As such, it manifests itself only in liquids. The

forces between molecules in the bulk of a liquid are equal in all

directions, and as a result, no net force is exerted on the molecules.

However, at the surface, the molecules exert a force that has a

resultant in the surface layer. This force holds a drop of water

suspended on a rod and limits the size of the drop that may be held.

It also causes the small drops from a sprayer or atomizer to assume

spherical shapes.

Surface tension has units of force per unit length, N/m (lb/ft). The

force due to surface tension results from a length multiplied by the

surface tension; the length to use is the length of fluid in contact

with a solid, or the circumference in the case of a bubble.

32

Surface Tension

A surface tension effect can be illustrated by considering the free

body diagrams of half a droplet and half a bubble as shown in Fig.

1.11. The droplet has one surface and the bubble is composed of a

thin film of liquid with an inside surface and an outside surface. The

pressure inside the droplet and bubble can now be calculated.

The pressure force ppR2 in the droplet balances the surface tension

force around the circumference. Hence

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Surface Tension

Similarly, the pressure force in the bubble is balanced by the surface

tension forces on the two circumferences. Therefore,

34

So, we can conclude that the

internal pressure in a bubble is

twice as large as that in a droplet

of the same size.

Figure 1.12 shows the rise of a liquid

in a clean glass capillary tube due to

surface tension. The liquid makes a

contact angle b with the glass tube.

Surface Tension

Experiments have shown that

this angle for water and most

liquids is zero.

There are also cases for which

this angle is greater than 90o (e.g.

mercury); such liquids have a

capillary drop.

If h is the capillary rise, D the

diameter, and r the density, s

can be determined from equating

the surface tension force to the

weight of the liquid column.

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EXAMPLE

A 2-mm-diameter clean glass tube is inserted, as shown, in water at

l5oC. Determine the height that the water will climb up the tube. The

water makes a contact angle of 0o with the clean glass.

36

Solution

A free-body diagram of

the water shows that

the upward surface-

tension force is equal

and opposite to the

weight. Writing the

surface-tension force

as surface tension

times distance, we

have

=>

EXAMPLE

Solving for h, we get,

The numerical values for s and r were obtained from Table of water

properties. Note that the nominal value used for the density of water

is 1000 kg/m3.

37

Contact Angle

Another important surface effect is the contact angle b which

appears when a liquid interface intersects with a solid surface, as in

the above Fig. The force balance would then involve both s and b. If

the contact angle is less than 90o, the liquid is said to wet the solid;

if b > 90o, the liquid is termed nonwetting.

For example, water wets soap but does not wet wax. Water is

extremely wetting to a clean glass surface, with b = 0o. Like s, the

contact angle b is sensitive to the actual physico-chemical conditions

of the solid-liquid interface. For a clean mercury-air-glass interface,

b = 130o.

38

bb