1st Lecture

Fluid Mechanics

Dr. Mohammed Zakria Salih Xoshnaw

57:020 Fluid Mechanics 2

History

Faces of Fluid Mechanics

Archimedes(C. 287-212 BC)

Newton(1642-1727)

Leibniz(1646-1716)

Euler(1707-1783)

Navier(1785-1836)

Stokes(1819-1903)

Reynolds(1842-1912)

Prandtl(1875-1953)

Bernoulli(1667-1748)

Taylor(1886-1975)

http://www-gap.dcs.st-and.ac.uk/~history/PictDisplay/Taylor_Geoffrey.html

http://www-gap.dcs.st-and.ac.uk/~history/PictDisplay/Taylor_Geoffrey.html


Weather & Climate

Tornadoes

HurricanesGlobal Climate

Thunderstorm


Vehicles

Aircraft

SubmarinesHigh-speed rail

Surface ships


Environment

Air pollution River hydraulics


Physiology and Medicine

Blood pump Ventricular assist device


Sports & Recreation

Water sports

Auto racing

Offshore racingCycling

Surfing

Background and introduction

• Physical Characteristics of Fluids

• Distinction between Solids, Liquids, and gases

• Flow Classification

• Significance of Fluid Mechanics

• Trends in Fluid mechanics

Physical Characteristics of Fluids

Statics Dynamics

Rigid Bodies

(Things that do not change shape)

Deformable Bodies

(Things that do change shape)

Incompressible Compressible

Fluids

Mechanics

Branch of Mechanics

Physical Characteristics of Fluids

• Fluid mechanics is the science that deals with the action of forces on fluids.

. Fluid is a substance

• The particles of which easily move and change position

• That will continuously deform

• A fluid can be either gas or liquid.

• Solid molecules are arranged in a specific lattice formation and their movement is restricted.

• Liquid molecules can move with respect to each other when a shearing force is applied.

• The spacing of the molecules of gases is much wider than that of either solids or liquids and it is also variable.

Distinction Between Solids, Liquids & Gases

Flow ClassificationThe subject of Fluid Mechanics

• Hydrodynamics deal with the flow of fluid with no density change, hydraulics, the study of fluid force on bodies immersed in flowing liquids or in low speed gas flows.

• Gas Dynamics deals with fluids that undergo significant density change

• Turning on our kitchen faucets

• Flicking on a light switch

• Driving cars

• The flow of bloods through our veins

• Coastal cities discharge their waste

• Air pollution

• And so on so forth …

Significance of Fluid Mechanics

Trends in Fluid Mechanics

• The science of fluid mechanics is developing at a rapid rate.

Fluid Mechanics

FLUID PROPERTIES

Today’s subject:

Objectives of this section• Work with two types of units.

• Define the nature of a fluid.

• Show where fluid mechanics concepts are common with those of solid mechanics and indicate some fundamental areas of difference.

• Introduce viscosity and show what are Newtonian and non-Newtonian fluids

• Define the appropriate physical properties and show how these allow differentiation between solids and fluids as well as between liquids and gases.

UNIT SYSTEMS

• SI UNITS

In the SI system, the unit of force, the Newton,

is derived unit. The meter, second and

kilogram are base units.

• U.S. CUSTOMORY

In the US Customary system, the unit of mass,

the slug, is a derived unit. The foot, second

and pound are base unit.

• We will work with two unit systems in FLUID MECHANICS:

• International System (SI)

• U.S. Customary (USCS)

Basic Unit System & Units

Derived Units

There are many derived units all obtained from combination of the above

primary units. Those most used are shown in the table below:

The SI system consists of six primary units, from which all

quantities may be described but in fluid mechanics we are generally

only interested in the top four units from this table.

Derived Units

Table summarizes these unit systems.

SI System of Units• The corresponding unit of force derived from Newton’s

second law:

“ the force required to accelerate a kilogram at one meter per second per second is defined as the Newton (N)”

The acceleration due to gravity at the earth’s surface: 9.81 m/s2.

Thus, the weight of one kilogram at the earth’s surface:

W = m g

= (1) (9.81) kg m / s2

= 9.81 N

Traditional Units• The system of units that preceded SI units in several countries is the so-called English system.

Length = foot (ft) = 30.48 cm

Mass = slug = 14.59 kg

The force required to accelerate a mass of one slug at one foot per second per second is one pound force (lbf).

The mass unit in the traditional system is the pound mass (lbm).

FLUID PROPERTIES

Specific Weight

Mass Density

Viscosity

Vapour Pressure

Surface tension

Capillarity

Bulk Modules of Elasticity

Isothermal Conditions

Adiabatic or Isentropic

Conditions

Pressure Disturbances

Every fluid has certain characteristics by which its physical conditions may be

described.

We call such characteristics as the fluid properties.

Properties involving the Mass or Weight of the Fluid

Specific Weight, g

The gravitational force per unit volume of fluid, or simply “weight per unit volume”.

- Water at 20 oC has a specific weight of 9.79 kN/m3.

Mass Density, ρ

The “mass per unit volume” is mass density. Hence it has units of kilograms per cubic meter.

- The mass density of water at 4 oC is 1000 kg/m3

while it is 1.20 kg/m3 for air at 20 oC at standard pressure.

• The ratio of specific weight of a given liquid to the specific weight of water at a standard reference temperature (4 oC)is defined as specific gravity, S.

• The specific weight of water at atmospheric pressure is 9810 N/m3.

• The specific gravity of mercury at 20 oC is

Specific Gravity, S

6.133kN/m 9.81

3kN/m 133S Hg

Ideal Gas Law

• p = absolute pressure [N/m2], 14.7 psi or 101 kpa

• V = volume [m3]

• n = number of moles

• Ru = universal gas constant

• [8.314 kJ/kmol-K; 0.287 kPa·m3/kg ·K]

• T = absolute temperature [K]

• MWgas = molecular weight of gas

A form of the general equation of state, relating pressure, specific volume, and temperature

British Gravitational (BG) System. In the BG system the unit of length is the foot (ft), the time unit is the second (s), the force unit is the pound (lb), and the temperature unit is the degree Fahrenheit (°F) or the absolute temperature unit is the degree Rankine(°R) °R= °F+ 459.67where The mass unit, called the slug, is defined from Newton’s second law (Force x Acceleration ) as1 Ib = (1 Slug). (1 ft/s2)This relationship indicates that a 1-lbforce acting on a mass of 1 slug will give the mass an acceleration of 1 ft/s2

The weight, (which is the force due to gravity, g) of a mass, m, is given by the equation.W= mg and in BG unitsw(lb) = m(slugs) g (ft/s2)g = 32.2 ft/s2it follows that a mass of 1 slug weighs 32.2 lb under standard gravity

VISCOSITY

• What is the definition of “strain”?

“Deformation of a physical body under the action of applied forces”

• Solid:

– shear stress applied is proportional to shear strain

(proportionality factor: shear modulus)

– Solid material ceases to deform when equilibrium is reached

• Liquid:

– Shear stress applied is proportional to the time rate of strain

(proportionality factor: dynamic (absolute) viscosity)

– Liquid continues to deform as long as stress is applied

Example of the effect of viscosity

• Think: resistance to flow.

• V : fluid velocity

• y : distance from solid surface

• Rate of strain, dV/dy

• μ : dynamic viscosity [N.s/m2]

t: shear stress

Shear stress: An applied force per unit area needed to produce deformation in a fluid

t = μ dV/dy

Velocity distribution next to boundary

VISCOSITY µ

• Would it be easier to walk through a 1-m pool of water or oil?

– Water

Why?

– Less friction in the water

• Rate of deformation

– Water moves out of your way at a quick rate when you apply a shear stress (i.e., walk through it)

– Oil moves out of your way more slowly when you apply the same shear stress

t = μ dV/dy

Viscosity is:

• slope of the line shown above

• the ratio between shear stress

applied and rate of deformation

Kinematic Viscosity• Many fluid mechanics equations contain the variables of

- Viscosity, m

- Density, r

So, to simplify these equations sometimes use kinematic viscosity (n)

Terminology

Viscosity, m

Absolute viscosity, m

Dynamic viscosity, m

Kinematic Viscosity, n

smmkg

msN/

/

/. 2

3

2

Other viscosity highlights

• Viscous resistance is independent of the pressure in the fluid.

• Viscosity is a result of molecular forces within a fluid.

• For liquid, cohesive forces decrease with increasing temperature → decreasing μ

• For gas, increasing temperature → increased

molecular activity & shear stress: increasing μ

Kinematic viscosity for air & crude oilIncreasing temp → increasing

viscosity

Increasing temp → decreasing

viscosity

Newtonian vs. Non-Newtonian Fluids

• Newtonian fluid: shear stress is proportional to shear strain

– Slope of line is dynamic viscosity

• Shear thinning: ratio of shear stress to shear strain decreases as shear strain increases (toothpaste, catsup, paint, etc.)

• Shear thickening: viscosity increases with shear rate (glass particles in water, gypsum-water mixtures).

Surface tension

• What’s happening here?

– Bug is walking on water

• Why is this possible?

– It doesn’t weigh much

– It’s spreading its weight out

– The downward forces are less than the effects of surface tension

Surface Tension

• A molecules in the interior of a liquid is under attractive force in all direction.

• However, a molecule at the surface of a liquid is acted on by a net inward cohesive force that is perpendicular to the surface.

• Hence it requires work to move molecules to the surface against this opposing force and surface molecules have more energy than interior ones

• Higher forces of attraction at surface

• Creates a “stretched membrane effect”

Surface Tension• Surface tension, σs: the force resulting from

molecular attraction at liquid surface [N/m]

• surface tension varies with temperature

Fs= σs L

Fs= surface tension force [N]

σs = surface tension [N/m]

L = length over which the surface tension acts [m]

CapillarityRise and fall of liquid in a capillary tube is caused by surface tension.

Capillarity depends on the relative magnitudes of the cohesion of the liquid

to walls of the containing vessel.

When the adhesive forces between liquid and solid are larger than the

liquid's cohesive forces, the meniscus in a small diameter tube will tend to

be concave

If adhesive forces are smaller than cohesive forces the meniscus will tend

to be convex, for example mercury in glass.

water mercury

concaveconvex

Differences between adhesive & Cohesive

A distinction is usually made between an adhesive force,

which acts to hold two separate bodies together (or to stick

one body to another)

and

a cohesive force, which acts to hold together the like or unlike

atoms, ions, or molecules of a single body.

h=height of capillary rise (or depression)

s=surface tension

q=wetting angle

G=specific weight

R=radius of tube

If the tube is clean, qo is 0 for water

Capillary EffectFor a glass tube in a liquid…

0, WF z

hRCosR 22

r

Cosh

2

Vapor PressureVapor pressure: the pressure at which

a liquid will boil.

Vapor pressure ↑ when

temperature increases

At atmospheric pressure,

water at 100 °C will boil

Water can boil at lower

temperatures if the

pressure is lower

When vapor pressure > the

liquid’s actual pressure

Fundamentals of Fluid Mechanics 41

Coefficient of Compressibility

• How does fluid volume change with P and T?

• Fluids expand as T ↑ or P ↓

• Fluids contract as T ↓ or P ↑



• Need fluid properties that relate volume changes to changes in P and T.

– Coefficient of compressibility

– k must have the dimension of pressure (Pa or psi).

– What is the coefficient of compressibility of a truly incompressible substance ?(v=constant).

T T

P Pv

v

09:10

(or bulk modulus of compressibility

or bulk modulus of elasticity)

is infinity



• A large implies incompressible.

• This is typical for liquids considered to be incompressible.

• For example, the pressure of water at normal atmospheric conditions must be raised to 210 atmto compress it 1 percent, corresponding to a coefficient of compressibility value of = 21,000 atm.

09:10



• Small density changes in liquids can still cause interesting phenomena in piping systems such as the water hammer—characterized by a sound that resembles the sound produced when a pipe is “hammered.” This occurs when a liquid in a piping network encounters an abrupt flow restriction (such as a closing valve) and is locally compressed. The acoustic waves produced strike the pipe surfaces, bends, and valves as they propagate and reflect along the pipe, causing the pipe to vibrate and produce the familiar sound.

09:10



• Differentiating = 1/v gives d = - dv/v2; therefore, d/ = -dv/v

• For an ideal gas, P = RT and (∂P/∂)T = RT = P/, and thus

ideal gas = P (Pa)

• The inverse of the coefficient of compressibility is called the isothermal compressibility a and is expressed as

09:10


09:10

Coefficient of Volume Expansion

The density of a fluid depends

more strongly on temperature

than it does on pressure.

To represent the variation of

the density of a fluid with

temperature at constant

pressure. The Coefficient of

volume expansion (or volume

expansivity) is defined as

1 1

P P

v

v T T

(1/K)


Coefficient of Volume Expansion

• For an ideal gas, ideal gas = 1/T (1/K)

• In the study of natural convection currents, the condition of the main fluid body that surrounds the finite hot or cold regions is indicated by the subscript “infinity” to serve as a reminder that this is the value at a distance where the presence of the hot or cold region is not felt. In such cases, the volume expansion coefficient can be expressed approximately as

• where is the density and T is the temperature of the quiescent fluid away from the confined hot or cold fluid pocket.

09:10

Fundamentals of Fluid Mechanics 4809:10


The combined effects of pressure and temperature

changes on the volume change of a fluid can be

determined by taking the specific volume to be a

function of T and P. Differentiating v = v(T, P) and

using the definitions of the compression and expansion

coefficients a and give

P T

v vdv dT dP

T P

= (dT - adP)v

• What is the weight of a pound mass on the earth’s surface, where the acceleration due to gravity is 32.2 ft/s2, and on the moon’s surface, where the acceleration is 5.31 ft/s2.

• Solution by Newton’s second law

W=Mg (lbf = slug*ft/s2)

Example 2.1:

2.32

1

/2.32

11 slugs

sluglbm

lbm

g

lbmM

c

Example 2.1: Cont…….

Therefore, the weight on the earth’s surface is

And on the moon’s surface is

lbf1s

ft32.2x

32.2

1slugsW

2

lbf165.0s

ft5.31x

32.2

1slugsW

2

Example 2.2: Capillary Rise Problem

• How high will water rise in a glass tube if the inside diameter is 1.6 mm and the water temperature is 20°C?

Answer: 18.6 mm

• Hint: for water against glass is so small it can be assumed to be 0.N/m 073.0

Example 1• A) calculate the density , specific weight and specific volume of Oxygen at 100 °F and 15 Psi.

• B) what would be the temperature and or pressure of this gas if it were compressed isentropically to 40 percent of its origin volume.

• C) if the process described in (b) had been isothermal , what would the temperature and pressure have been.

Solution

a) ρ= P/RT = 15*144/(1552)*(100+460) = 0.00248 slug/ft3

γ= ρ g= 0.00248(32.2)= 0.0799 Ib/ ft3

Vs= 1/ ρ = 1/0.00248 = 403 ft3/slug

b) P1(Vs1)K = P2(Vs2)

k =P2= 54.1 Psi

P2 = ρ2 RT2 (54.1)*144= (0.00218/0.4)*1552*(T2+460)

T2 = 350 °F

c) If its isothermal , T2=T1= 100 °F

15*144*403= P2(144*0.4*403) =

• P2= 37.5 Psi

Example 2• What is specificweight of air at 70 psi and 70°F , R= 53.3 ft/°R

• γ = 70(144) /(53.3)*(70+460)= 0.357 Ib/ft3

Example 3• A cylinder contains 12.5 ft3 of air at 120 °F and 40 Pisa, The air compressed to 2.5 ft3

A) assuming isothermal condition what a pressure at the new volume and bulk modules of elasticity

B) assuming adiabatic conditions, what are the final pressure and temperature and the bulk modules of elasticity for isothermal condition

Solution

A) P1V1= p2V2 for isothermal P2 = 200 Pisa

K = ( ∆P/∆V/V)= ( 40- 200)/(12.5-2.5)/12.5= 200 psi

b) P1 (V1)k = P2 (V2)

k, k= 1.4 ,

P2 381 Pisa

T2/T1= (P2/P1) k-1/k,

T2 = 1104 °R or 644 °F ,

K = bulk modules= k* P2 = 583 Ps

Example 4

Example 5

Example 6

Example 7

Example 8

Documents

1st Lecture