Southern Methodist UniversityBobby B. Lyle School of EngineeringCEE 2342/ME 2342 Fluid Mechanics

Roger O. Dickey, Ph.D., P.E.

I. INTRODUCTIONB. Physical Characteristics of FluidsC. The Science of Fluid MechanicsD. Units and DimensionsE. Fluid Properties

Reading Assignment:

Chapter 1, following along with daily lecture material.

B. Physical Characteristics of Fluids

In engineering applications, matter exists in one of three physical states:

• Solid

• Liquid

• Gas

The term fluid encompasses both liquids and gases.

Fluids

Qualitative characteristics of a solid:

• Essentially has constant volume

• Maintains a fixed shape

• Molecules are held in a more less “rigid”

structure

Qualitative characteristics of a liquid:

• Molecules are relatively free to change

their positions with respect to each other

but they are restricted by cohesive forces

to a practically constant volume

• Assumes the shape of a container

Qualitative characteristics of a gas:

• Molecules are unrestricted by cohesive

forces

• Has no definite volume

• Expands to fill a container

Definitions of shear force/shear stress and

normal force/normal stress (pressure):

(i) A shear force on a surface is the

component of the total applied force that is

tangent to the surface. Shear stress is the

shear force per unit area of the surface.

(ii) A normal force on a surface is the

component of the total applied force that is

perpendicular to the surface. Normal stress

or pressure is the normal force per unit

area of the surface.

Consider a solid subjected to shear forces,

Initial solid shape in static equilibrium

Apply shear force

Deformation of the solid takes place until internal forces (spring-like forces due to molecular cohesion) build up to resist further deformation resulting in a deformed solid shape in static equilibrium

A fluid can be defined as follows:

Fluid a substance that continuously

deforms (i.e., flows) under the action of a

shear stress.

Consider a fluid element subjected to shear forces,

Initial fluid element shape in static equilibrium

Apply shear force

Continuous deformation occurs until the shear stress is removed

Fluid as a Continuum -

It is virtually impossible to evaluate the forces

acting on, and the behavior of individual

molecules when studying the behavior of fluids

because the number of molecules in a tiny

volume is astronomical (approximately 1021

molecules/mm3 for liquids and 1018

molecules/mm3 for gases).

In the fluid sciences, behavior of fluids is

characterized by considering the average, or

macroscopic value of physical quantities of

interest where the average is taken for an

extremely large number of molecules. These

macroscopic fluid properties can be perceived

by human senses, and measured by instruments.

In essence, the derivation and application of the

basic principles of fluid mechanics requires that

the actual molecular structure of fluids be

replaced by a hypothetical continuous medium

called the continuum.

C. The Science of Fluid Mechanics

The sub-disciplines of the science of fluid

mechanics are summarized on the following

flow chart:

Fluid Mechanics

Fluid Dynamics(Flow of Fluids)

Fluid Statics(Fluids at Rest)

Hydrodynamics(Incompressible Flow - liquids and low speed flow of gases)

Gas Dynamics(Compressible Flow - gases)

Aerodynamics

Hydraulics(Flow of Liquids in Pipes and Channels)

Examples of the practical importance of fluid

mechanics:

(1) Hydrology

• Flow of groundwater

• Flow of water in streams and lakes

• Ocean currents and tides

• Weather (flow of air in the atmosphere)

(2) Closed Conduit Flow

• Water supply transmission pipelines

• Water distribution networks

• Cold and hot water plumbing for buildings

• Oil and gas pipelines

• Air and oxygen distribution piping (e.g., in waste treatment systems, hospitals, etc.)

• Blood flow in human circulatory system

(3) Open Channel Flow

• Storm sewer networks and channels

• Sanitary sewer networks

• Wastewater plumbing in buildings (e.g.,

industrial wastewater and sewage

collection networks)

• Irrigation networks

(4) Turbomachinery

• Turbines (power generation)

• Pumps

• Compressors and blowers

(5) Aerodynamics

• Airframe design (lift and drag)

• Propulsion systems (propellers, jet

engines, rocket engines)

(6) Naval Architecture – design of boats and

ships

Consider the example of an automobile:

• Pneumatic tires

• Gasoline and engine coolant pumped through closed conduits

• Hydraulic shock absorbers

• Hydraulic brakes

• Aerodynamic body shapes increase fuel efficiency by reducing drag

Interest in fluid behavior dates back to ancient

civilizations based on the necessity for

development of water supply and irrigation

systems, design of boats and ships, and

propelling projectiles through the air (e.g., the

addition of feathers at the tail of an arrow,

creating drag to true the flight of the arrow).

The beginnings of the formalization of the

science of fluid mechanics began with the need

to control water in large irrigation systems in

ancient Egypt, Mesopotamia (Iraq), and India as

early as 3000 BC.

However, the first quantitative scientific laws in

the field of fluid mechanics were not developed

until 250 BC when the Greek scientist,

mathematician, and inventor Archimedes

discovered and recorded the principles of

hydrostatics and flotation.

The fundamental principles of hydrodynamics

were not set forth until the 17th and 18th centuries

with the greatest contributions made by Isaac

Newton, Daniel Bernoulli, and Leonhard Euler.

Refer to Table 1.9, pp. 28-29, in the textbook

for a chronological listing of some major

contributors to the science of fluid mechanics, an

excerpt is shown on the following two slides:

Table 1.9,pp. 28-29

Table 1.9(Continued)

D. Units and DimensionsBoth SI and British Gravitational (BG), also called U.S. Customary (USC), units are used:

Physical UnitsQuantity Dimension SI USCForce F N lbMass M kg slugLength L m ftTime T sec sec

A consistent set of physical units involves

either: (1) defining the units of F-L-T as the

basic units, then deriving the unit of M from

Newton’s Second Law of Motion (F = ma), or

(2) defining the units of M-L-T, and deriving

the unit of F from Newton’s Second Law.

Using SI units, M-L-T units are defined, and the

unit of F is derived:

2

2

secmkg 1 N 1

)m/sec kg)(1 (1 N 1

ma F

1 N is derived as the force required to accelerate a1 kg mass at 1 m/sec2

SI Force Dimensions:

22 or -MLT

TLM

Using USC units, F-L-T units are usually

defined, and the unit of M is derived:

ftseclb 1 slug 1

ft/sec 1lb 1 slug 1

aF m

2

2

1 slug is derived as the mass accelerated at 1 ft/sec2 by a force of 1 lb

USC Mass Dimensions:

122

or -LFTLTF

Reconsider the derived unit of force in the SI

system of units,

Multiply both sides of this equation by 1 sec2,

2secmkg 1 N 1

mkg 1 secN 1 2

Divide both sides of the equation by 1 m and

rearrange,

msecN 1 kg 1

2

These are the same physical dimensions derived for slugs, the unit of mass in the USC system of units:

122

or -LFTLTF

A comprehensive listing of dimensions for

common physical quantities encountered in

science and engineering is shown in Table 1.1

Dimensions Associated with Common

Physical Quantities, p. 5 in the textbook:

Table 1.1 (Continued)

Table 1.1 (Continued)

Homework No. 1 Dimensional consistency of

mathematical expressions describing fluid

phenomena.

E. Fluid Properties

Properties vary from fluid-to-fluid. Properties for

a given fluid frequently vary with temperature.

Refer to Tables 1.5 and 1.6, inside the front

cover of the textbook, as shown on the

following two slides:

Measures of Mass and Weight

(1) Specific Weight, ,

weight per unit volume

SI Units - N/m3

USC Units - lb/ft3

3LF

(2) Density, ,

mass per unit volume

SI Units - kg/m3

USC Units - slug/ft3

3LM

(3) Relationship Between and ,

Weight and mass are related by Newton’s Second Law,

F = ma

Weight, w, is the gravitational force exerted on a body of mass, m, by the earth and thus,

w = mg

where, g = gravitational acceleration

Divide both sides of the previous equation by

the volume, , of the body of weight w and

mass m yields,

By definition,

gVm

Vw

Vmρ and

Vwγ

V

Substituting,

Standard gravitational acceleration at mean sea

level (MSL) is,

g = 9.81 m/sec2

g = 32.2 ft/sec2

ρ gγ [Equation (1.6), p. 12]

(4) Specific Gravity, S,

S ratio of the density of a fluid to

the density of water at 4° C

S is dimensionless (one of many dimensionless

ratios encountered in fluid mechanics) and,

thus, the numerical value of S does not depend

on the system of units.

Viscosity

Contrary to the case of solid bodies, specific

weight and density are insufficient to uniquely

characterize the dynamic behavior of fluids

when acted upon by external forces. For

example, two fluids with similar density, such

as certain oils and water, can behave quite

differently when flowing.

There is need for an additional physical

property to describe the “fluidity” of gases and

liquids. This property is called viscosity.

Viscosity is the fluid property that offers

resistance to shear stresses.

Film Clip Textbook film, Segment V1.3: Viscous Fluids.

Isaac Newton made pioneering experimental

observations of fluids upon which shear forces

were applied. He discovered that the resulting

velocity gradient created within the fluid was

directly proportional to the applied shear stress.

His discovery is called Newton’s Law of

Viscosity, and for one-dimensional flow in the

x-direction it is written as,

where,

dydu

[Equation (1.9), p. 15]

= shear stress applied to the fluid [F/L2]

u = velocity in the x-direction [L/T]

y = distance along the y-axis [L]

(i.e., distance above the x-axis)

= velocity gradient perpendicular to the

direction of flow

= viscosity

dydu

LTL

22 or

FTL

LTF

is given the name viscosity but is also called

the absolute viscosity or dynamic viscosity.

Typical units are,

SI Units — (no special name)

USC Units — (no special name)

c-g-s — (named Poise, P)

2msecN

2ftseclb

2cmsecdyne

Consider an experiment where a fluid film of

thickness “b” is located between a moving top

plate having velocity U in the +x-direction, and

a parallel but fixed bottom plate as shown in

Figure 1.5, p. 15 in the textbook, slightly

modified on the following slide:

Figure 1.5, p. 15 –Modified

bU

dydu

x

1

The constant velocity of the top plate, U, is

induced by applying a constant horizontal force,

P, to the plate. For a plate having planar area A,

the shear stress (i.e., force per unit area)

applied to the fluid by the top plate is simply:

2

LF

AP

The fluid between the plates deforms continuously

under the action of the shear stress, . The fluid

motion may be conceptualized as many thin

horizontal layers sliding one over another at

differing rates yielding a velocity gradient ,

which is also the rate of angular deformation of

the fluid as illustrated in Figure 1.5. Hence, is

sometimes called the rate of shearing strain.

dydu

dydu

Experimental observations reveal that fluids

“stick” to solid boundaries due to molecular

adhesion forces. Thus, fluid particles in direct

contact with moving solid boundaries move with

the same velocity as the solid surface, and fluid

particles contacting stationary solid boundaries

have zero velocity. This is called the no-slip

condition for flowing fluids.

Film Clip Textbook film, Segment V1.4: No-Slip Condition.

Experimental observations for the geometry of the current experiment reveal that the fluid velocity u increases linearly when proceeding upward in the +y-direction, hence is constant, i.e.,

The no-slip condition establishes the boundary conditions: (i) u = 0, y = 0 at the bottom plate, and (ii) u = U, y = b at the top plate. Thus,

bU

dydu

bU

dydu

00

dydu

yu

dydu

Viscosity is also a constant, i.e., a characteristic

physical property of the specific fluid used in the

experiment. Therefore, according to Newton’s Law

of Viscosity, , the shear stress must

remain constant throughout the fluid, ,

being transmitted from one sliding fluid layer to

the next.

bU

dydu

Most fluids adhere to Newton’s Law of Viscosity

—shear stress varying as a linear function of

velocity gradient and having viscosity, μ, as the

constant of proportionality. These fluids are

termed Newtonian fluids. Fluids that do not

adhere to the law are classed as non-Newtonian

fluids. Figure 1.7, p. 16 in the textbook

graphically compares Newtonian and several

common classes of non-Newtonian fluids:

Figure 1.7, p. 16 –Modified

μ1

Film Clip Textbook film, Segment V1.6: Non-Newtonian Behavior – Shear-Thickening

It is sometimes convenient in fluid mechanics to

use the kinematic viscosity, , which is defined

as the viscosity divided by the density of the

fluid:

M

TLFLMLTF3

2

From Newton’s Second Law of Motion, force has

equivalent dimensions of which yields the

common dimensions for kinematic viscosity, ,

that is:

TL2

2TLM

TL

MTLTLM

MTLF 22

Common units for kinematic viscosity are,

SI Units - m2/sec (no special name)

USC Units - ft2/sec (no special name)

c-g-s - cm2/sec (named Stoke, St)

Viscosity does not vary significantly with pressure and, as such, is assumed solely a function of temperature. Figure 1.8, p. 17 in the textbook is reproduced on the following slide, showing that:

(i) Liquids – viscosity decreases dramatically with increasing temperature

(ii) Gases – viscosity increases slightly with increasing temperature

(iii) Viscosities for common liquids are several orders of magnitude higher than those of gases.

Figure 1.8,p. 17

Dyn

amic

Vis

cosi

ty, μ

(N·s

/m2 )

Temperature, C

Homework No. 2 Newton’s Law of Viscosity.

Vapor Pressure

Vapor pressure, pv , is defined as,

pv the absolute pressure at which a

liquid will boil at a given temperature

Vapor pressure increases as the temperature of the liquid increases, refer to Handout — I.E. Properties of Common Fluids. A liquid can be made to boil at low temperature by reducing the pressure.

In many common liquid flow situations (e.g., in

some hydraulic machinery such as pumps and

turbines, inside valves, and along the face of

dam spillways) pressures at or below the vapor

pressure of the liquid can occur. Under these

conditions, the liquid flashes to vapor forming

tiny bubbles or gas pockets.

As the bubbles are carried along by the flow,

they can enter zones of higher pressure where

the vapor condenses and the bubbles collapse.

As the liquid rushes into the vacuum left by the

collapsed bubble, high transient forces are

generated (like tiny hammer blows).

Where the collapsing bubbles are in contact

with solid boundaries these “hammer blows”

can cause pitting and erosion of the solid

surface. The phenomena of vapor pocket

formation and collapse is called cavitation.

Cavitation

Worker inspecting cavitation damage to spillway

Cavitation – Glen Canyon Dam Spillway near Page, AZ

Workers repairing cavitation damage to dam spillway

Cavitation – Dworshak Dam Spillway near Ashaka, Idaho

Bulk Modulus of Elasticity

Bulk Modulus of Elasticity, Ev , often simply

called the Bulk Modulus, is a measure of the

elasticity of a liquid.

Let,

= volume of liquid [L3]

p = pressure [F/L2]

V

Then,

vEV

dpVd

Rate of change ofvolume with respect to pressure

The minus sign indicates that volume decreases with increasing pressure

Rearranging yields,

SI Units - N/m2 (called the Pascal)

USC Units - lb/ft2 which is frequently

converted to lb/in2 or psi

2

LF

VVddpEv

PressureDimensions

[Equation (1.12), p. 20]

Since a decrease in volume (i.e., ) for a

given mass, , causes an increase in

density (i.e., d > 0), the previous equation may

also be written:

2

LF

ddpEv

PressureDimensions

[Equation (1.13), p. 20]

0Vd

Vm

Values of Ev tend to be very large for common

liquids, e.g., approximately 300,000 psi for

water. For other examples refer to Handout —

I.E. Properties of Common Fluids,

emphasizing that liquids can be assumed

incompressible for the vast majority of applied

fluid mechanics problems.

However, there are a few engineering

applications involving extremely large pressure

changes where it becomes necessary to consider

the compressibility of liquids:

• Water hammer – rapid valve closure can

cause large pressure transients that

generate loud “bangs”, shaking, and even

rupture of piping.

• Pumping of groundwater from deep

confined aquifers.

Valve Closure Water Hammer

Water Hammer Damage

Water Hammer Shock Alleviation

Wells in Confined Aquifers

Surface Tension

At interfaces between liquids and gases,

between two immiscible liquids, and between

liquids and solids unbalanced cohesive forces

develop that cause the liquid surface to appear

and behave as if it were a “skin” or “membrane”

stretched over the fluid mass.

Consider a liquid-gas interface, e.g., the

interface between water in a drinking glass and

the surrounding air. Molecules within the

interior of the liquid mass experience balanced

cohesive forces because they are surrounded by

like molecules that are attracted to each other

equally.

Conversely, liquid molecules residing at the liquid-

gas interface are subjected to a net force toward the

interior of the liquid mass because the cohesive

forces between liquid molecules are much stronger

than the adhesive forces between the liquid

molecules and the overlying gas. The unbalanced

cohesive forces along the interface create the

appearance of a taut skin over the liquid surface.

A tensile force acts in the plane of the surface

“skin,” along any line in the surface, much like

the tensile force in a drum head. The intensity of

the tensile force per unit length is called surface

tension, σ [F/L].

Surface tension is a characteristic property of a given liquid, as illustrated in Handout — I.E. Properties of Common Fluids for various liquids in contact with air (e.g., 0.466 N/m for mercury compared to 0.022 N/m for gasoline). However, surface tension also depends on the other substance—liquid, gas, or solid—forming the interface with the liquid. Furthermore, surface tension varies with temperature, tending to decrease with increasing temperature.

Surface tension forces are usually negligible compared to inertial and viscous shear forces in flowing fluids. However, there are a number of interesting and/or important phenomena involving surface tension:

• Steel needles, razor blades, and water walking insects can float on water under the right conditions, because the tensile force in the taut skin balances their weight.

• Liquids can form tight, compact droplets when placed on smooth solid surfaces.

Film Clip Textbook film, Segment V1.9:

Floating Razor Blade.

Water Strider Insect Mercury Droplets

• Formation of liquid droplets and gas bubbles, e.g., break up of liquid jets into discrete droplets through fuel injectors and atomizing spray nozzles, and diffusing compressed air into aeration tanks.

• Formation of a meniscus and the associated capillary rise (wetting liquids) or fall (non-wetting liquids) in small diameter tubes; involves liquid-gas-solid interfaces

Atomizing Spray Nozzles Fuel Injectors

Ceramic Disk, Fine BubbleAir Diffuser

Figure 1.10, p. 25Effect of capillary action in small tubes. (a) Rise of column for a liquid that wets the tube, e.g., water in a glass tube. (b) Free-body diagram for calculating column height. (c) Depression of column for a non-wetting liquid, e.g., mercury in a glass tube.

• Formation of a capillary fringe above ground

water aquifers, where the tiny interstitial

spaces between soil grains (soil porosity)

create natural “capillary tubes.”

Refer to Handout I.E. Surface Tension –

Examples Problems.

Homework No. 3 Surface tension.