me305 chapter 1

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ME 305 Fluid Mechanics I

Chapter 1

Introduction

These presentations are prepared by

Dr. Cneyt Sert

Mechanical Engineering Department

Middle East Technical University

Ankara, Turkey

[email protected]

They can not be used without the permission of the author.

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Classification of Matter

One classification of matter depends on the magnitude of intermolecular attraction forces.

IntermolecularAttraction Forces

MoleculesVolume and

Space

Solid StrongRelative positionsare rather fixed

Definite volume

Definite shape

Liquid MediumFree to change their

relative positions

Definite volume

Indefinite shape

Gas WeakPractically

unrestricted

Indefinite volume

Indefinite shape

There are two other states of matter, plasma and Bose-Einstein condensate.

www.edinformatics.com/math_science/states_of_matter.htm
http://www.edinformatics.com/math_science/states_of_matter.htmhttp://www.edinformatics.com/math_science/states_of_matter.htm

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At the microscopic scale, fluids are composed of molecules which are in constant motion andcollision.

For an exact analysis, the action of each molecule (or group of molecules) should be studied.

This is done in the kinetic theory of gasses for example, but it is not practical for most engineering

problems.

We study most engineering problems at the macroscopic scale, where we are concerned with thebehaviour of matter in the large.

That is we treat fluids as continuum and do not concern with the behavior of individual molecules.

The Concept of Continuum

Microscopic level: Each molecule istravelling at a different speed.

Macroscopic level: The speed at pointA is 10 m/s. This is an average value.

A

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Continuum assumes that the fluid characteristics, such as density, pressure, temperature, varycontinuously throughout the fluid.

In continuum, the smallest element of a fluid is called a fluid particle, which contains enoughnumber of molecules to make statistical averages.

In the previous example, the speed at point A, measured as 10 m/s, is actually the averagespeed of molecules in the small volume surrounding point A. We can say that the fluid particlelocated at point A is moving with a speed of 10 m/s.

Question: Is continuum a reasonable assumption?

Practical Answer: Yes, in many engineering problems

The Concept of Continuum (contd)

Air at atmospheric conditions

Number of molecules in 1 cm3 ~ 3 x 10 19

Mean free path (m) ~ 8 x 10 8

Distance between molecules (m) ~ 3 x 10 9

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fieldflowtheofdimensionsticCharacteri

)pathfree(meancollisionsbetweenmoleculesbytraveleddistanceAverageKn =

More realistic answer: Depends on the Knudsen number.

Continuum is said to be valid for Kn < 0.01.

In this course we will always treat fluids as continuum.

Kn is typically large (continuum approach should be questioned) for Flows at high attitudes and low pressures (numerator is large).

Flows at macro or nano geometries (denominator is small).

Although not mentioned in this detail, continuum assumption is also used in solid mechanics.

Actually continuum mechanics, a branch of physics, is the study of matter as a continuum. Itdoes not differentiate between solids and fluids.

The Concept of Continuum (contd)

www.lab-on-a-chip.com

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Two Different Ways to Describe Continuum

As a fluid particle moves in a flow field, its properties change from point to point in space andfrom time to time.

p = p (x, y, z, t ) , = (x, y, z, t) , etc.

These properties can be described in two different ways.

Material (Lagrangian) Description: identified fluid particles are followed in the course of time asthey move in the flow field.

Spatial (Eulerian) Description: attention is focused at fixed points in the flow field and thevariation of properties at these points is determined as fluid particles pass through these points.

Lagrangian description is suitable for solid mechanics,where relatively small and simple motion of individualparticles can be followed.

Fluid motion however is much more complex andkeeping track of large number of fluid particles is adifficult/impossible task; therefore the Euleriandescription is more suitable for fluid mechanics.

Karman vortex street behind a cylinderhttp://www2.icfd.co.jp/examples/karman/kr2.htm

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Properties of the Continuum

Mass Density: () [ kg/m3 ]

Mass per unit volume of a fluid. Consider the point P inside a continuous fluid of volume . According to the continuum assumption, the density at point P can be defined as the average

density within the small volume of surrounding point P.

P

=m

Question: How small can the differential volume be ?Answer: Large enough such that the continuum assumption is satisfied. That is, it can not be

too small such that there are only a few molecules inside it. Then the molecules passing in andout of will not give a reliable averaging.

Domain ofmolecular affects

m Domain of

contnuum

=

m

lim '

Therefore a better definition for density is

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Properties of the Continuum (contd)

Specific gravity (relative density): (s) [ unitless ] Ratio of density of a substance to the density of water. water

s =

Specific weight: () [ N/m3 ] Weight per unit volume of a substance. g

=

Fluid velocity: ( ) [ m/s ]

For fluids considered as continuum, similar to density, velocity at a point P of a flow field isactually the average velocity of the molecules within a small volume surrounding point P.

Vr

Compressibility:

A fluid can be considered as incompressible when its density is constant over the range of workingconditions.

Water is nearly incompressible, but it does compress a little.

At 25 oC, 1 atm water = 999.84 kg/m3

It takes 400 kPa (4 atm) pressure to get water = 1000 kg/m3

Air is much more compressible (think about the ideal gas law to understand -p relationship of air)

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Forces acting on a body of fluid: ( ) [ N ]

Surface forces act on the boundaries of a medium through direct contact.

Body forces are distributed over the volume of a fluid, and defined per unit mass.

A surface force can be decomposed into a normal force acting perpendicular to the surface andtangential (shear) force acting parallel to the surface.

Fr

External Forces

=

=5

1kk 0FmequilibriuAt

rr

x

y

z

F5

F4

F3

F2F1

Internal Forces

0FFFmequilibriuAt i43rrrr

=++

tFnFF itinir

r

r

+=

x

y

z

F4

F3

Fi

Fin

Fit

nrt

r

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Pressure: ( p ) [ Pa = N/m2 ]

is the normal component of the force acting on an area divided by that area.

Pressure always acts perpendicular and towards the surface.

For a fluid at rest, pressure at a point is independent of the direction (see section 1.5.7).

Note that it is also possible to think pressure as a thermodynamic property.

Atmospheric pressure: 1 atm = 760 mmHg = 101.3 kPa = 14.7 psi

Crashing can experiment:

Take a soda can.

Put very little water into it, just enough to cover the bottom.

Heat the can for about 1 minute so that the water evaporates.

Water vapor will push the air inside the can out. The can will be somewhat vacuumed.

Quickly invert the can and insert it into water.

It will be crushed immediately.

For a movie of this experiment and similar ones

http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/patm.html#atm
http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/patm.html#atmhttp://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/patm.html#atm

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Viscosity: () [N.s/m2]

If a similar experiment is performed using fluid between the plates, a vertical fluid element AB

will deform continuously as long as the shear force is applied.

Experiment: Consider a solid block firmly attached to two parallel plates.

A

B

Fixed plateA

B

A

BF

F

A

B

to t1

Afluid can also be defined as a substancethat deforms continuously under the

application of a shear (tangential) force, nomatter how small the force is.

The block deforms slightly if a force F is appiled to the upper plate.

B

t2

B

Measure of a fluids resistance to shear and angular deformation. It is sometimes defined as the fluidity of a fluid.

Movie 1.1: Viscous fluids
http://movies/v1_1%20viscous%20fluids.pdf

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Now consider an infinitesimal (differential) fluid element of dimensions dxand dy.

The constant force F will cause the upper plate to move with a constant velocity of Uo. Note that the lower plate is not moving.

Uo

dy

dxxy

Viscosity (contd):

Due to the no-slip condition (fluid particles stick to the solid walls and move with them), speedof the fluid particles adjacent to the lower and upper plates will be zero and Uo, respectively.

We will observe a linear velocity profile between the two plates (why linear ?).

Uou=Uo

Linear

velocity profile

u=0

P i f h C i ( d)

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The deformation of this fluid element will be studied in detail during the class.

Uo

dy

dxx

y

x

y

A

B

A

B C

D

C

D

Viscosity (contd):

P ti f th C ti ( td)

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Sign convention for shear stress:

On a surface whose normal is acting in the positive direction of the coordinate axis,

shear stress is positive if it is acting in the positive direction of the coordinate axis.

it is negative if it is acting in the negative direction of the coordinate axis.

On a surface whose normal is acting in the negative direction of the coordinate axis,

shear stress is positive if it is acting in the negative direction of the coordinate axis.

it is negative if it is acting in the positive direction of the coordinate axis.

Newtons Law Of Viscosity:

dt

d=

y

x coefficient of viscosity

absolute viscosity

dynamic viscosity

viscosity

Relates shear stress () to shear strain rate (d/dt ). For Newtonian fluids, the shear stress on a surface tangent to the flow direction is proportionalto the rate of shear strain or to the velocity gradient on the surface (change of velocity in adirection normal to the surface).

dy

du =

P ti f th C ti ( td)

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Sign convention for shear stress (contd):

Surface normal direction Shear stress direction Shear stress is

+ + +

+ - -

- + -- - +

y

x

Sign convention example:

1

Surface 1: snd: + ssd: - ss: -

2

Surface 2: snd: - ssd: + ss: -

4

Surface 4: snd: - ssd: - ss: +

3

Surface 3: snd: + ssd: + ss: +

Therefore shear stress is positive on the lower

fluid and it is negative on the upper fluid.

Example

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Example

Determine the

(a) velocity profile in the oil.

(b) shear stress distribution.

(c) force required to pull the plate.

(d) power required to pull the plate.

Uo = 0.3 m/s (constant)

oil

= 3 Pa.syx

h = 2 mm

Area = 0.3 m2

Pressure is constant everywhere.

Oil is a Newtonian fluid.

(solution will be given during the lecture)

Classification of Fluids According to the Behavior of Their Viscosity

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Classification of Fluids According to the Behavior of Their Viscosity

Fluids

Inviscid(ideal)

Viscous

Newtonian Non-Newtonian

Time independent Time dependent

Pseudoplatic(shear thinning)

Binghamplastic

Dialatant(shear thickening)

Thixotropic Rheopetic

dy

du

Pseudoplastic

Dialatant

Bingham plastic

Inviscid (ideal)

Elasticsolid

Newtonian

1

Movie 1.2: Non-Newtonian fluids

Classification of Fluids According to the Behavior of Their Viscosity (contd)
http://movies/v1_4%20nonnewtonian%20behavior.pdf

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Newtonian behaviour simple (it is linear), but not all fluids are Newtonian.

Fortunately the most common ones, water and air behave as Newtonian.

Classification of Fluids According to the Behavior of Their Viscosity (cont d)

Inviscid (ideal) fluids do not exist in real world. They have = 0. This might be a useful

simplification for some analytical analysis.

Viscous fluids are the real fluids. They have 0. Dialatant fluids become thicker and thicker under increased shear stress. (printing ink).

Bingham plastics do not flow below a certain amount of shear stress. (toothpaste).

Pseudoplastics become thinner and thinner under increased shear stress. (wall paint, blood).

For thixotropic fluids viscosity decreases with time (lipstick).

For rheopetic fluids viscosity increases with time (betonite solution).

www reference: http://www.answers.com/topic/non-newtonian-fluid

Movie 1.3: Capillary tubeviscometer

Variation of Viscosity with Temperature
http://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://movies/v1_3%20capillary%20tube%20viscometer.pdfhttp://www.answers.com/topic/non-newtonian-fluid

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Temperature dependence of viscosity of gases and liquids are affected by different mechanisms.

Liquid molecules have limited mobility compared to gases. As temperature increases average distance between the molecules increase,

and intermolecular attraction forces decrease.

Therefore fluidity increases, meaning that viscosity decreases.

Gas molecules have greater mobility, but weak intermolecular attraction forces.

Viscosity is due to the collision of molecules.

As temperature increases mobility of the molecules increase, resulting in more collisions.

This decreases fluidity, meaning that viscosity increases.

Variation of Viscosity with Temperature

A, B, C, D are fluiddependent constants

Variation of Viscosity with Pressure

Affect of pressure on viscosity is usually small.

BTeA= General formula for liquids: Andrades equation

DTTC 2

3

+= General formula for gases: Sutherlands equation


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Properties of the Continuum (cont d)

Kinematic viscosity: () [m2/s]

=

Surface tension: () [N/m]

Surface tension is due to the

asymmetric cohesive forces acting onthe molecules of a free surface(interface between a liquid and a gas).

This asymmetry will result in ahypothetical skin (membrane) all

around the surface

Surface tension exists whenever there is a density discontinuity

between a liquid and another liquid or a gas or a solid.

More about surface tensionhttp://www.funsci.com/fun3_en/exper2/exper2.htm

Movie 1.4: Magic sand

http://www.funsci.com/fun3_en/exper2/exper2.htmhttp://www.funsci.com/fun3_en/exper2/exper2.htm

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A more general analysis for a double curvature surface (not the surface of a sphere) is given in

the textbook.

Gaspo

Liquidpi

pi po

Surface tension (contd)

Surface tension creates a pressure difference across a curved interface of two fluids (one ofthem is always a liquid).

Consider a spherical liquid droplet in a gas.

Force due to surface tension: 2r

r

2p-p

r2rprp

oi

2o

2i

=

+=

Force balance:


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Liquid c

Fluid a

Liquid b


Surface Tension (contd):

If ab cos() + bc cos() = ac

then liquid b does not spread over liquid c

Example: a: air, b: oil, c: water

If ac > ab + bc

then there is no equilibrium. Liquid b will spread over liquid c.

Example: a: air, b: gasoline, c: water

Liquid + Liquid + Fluid meet at a point:

ac

bc

ab


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Liquid b

Fluid a

Solid c


Surface Tension (contd):

(a) Liquid wetting a surface

: wetting angle (angle of contact). It is alwaysmeasured from solid/liquid interface to liquid/fluid

interface.

< /2

For equilibrium: bc + ab cos() = ac

Example: a: air, b: water, c: glass

bc

(b) Liquid not wetting a surface

> /2

Example: a: air, b: mercury, c: glass

ac

Liquid b

Fluid a

Solid c

ac

ab

bc

ab

Liquid + Fluid + Solid meet at a point:


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Vapor pressure:

At this equilibrium, the pressure exerted on the free surface is called the vapor pressure of theliquid.

Vapor pressure increases with temperature (high temperature increased molecular activity more molecules escaping from the free surface increased vapor pressure).

Volatile liquids have high vapor pressure (easy to evaporate).

Nonvolatile liquids have low vapor pressure (hard to evaporate).

Visit http://en.wikipedia.org/wiki/Vapor_pressure for a discussion about vapor pressure of solids.

Consider a liquid in a confied space kept at constant temperature.

Even at temperatures below the boiling point, some of the liquid moleculeswill have enough kinetic energy to escape from the free surface to the spaceabove.

After some time, there will be enough molecules in the vapor above the liquid

and they will start exerting pressure so that some of the liquid molecules willrejoin the free surface.

At some point, there will be a balance of molecules escaping from the freesurface and molecules rejoining the free surface.

Vapor

Liquid

http://en.wikipedia.org/wiki/Vapor_pressurehttp://en.wikipedia.org/wiki/Vapor_pressure

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Boiling and vapor pressure:

When the pressure above the free surface of a liquid is brought slightly below the vapor pressure,it starts boiling.

Boiling can be achieved by

raising the temperature of the liquid, therefore its vapor pressure rises.

lowering the pressure of the space above the free surface below the vapor pressure of theliquid.

How does a pressure cooker work?http://missvickie.com/workshop/howdoesit.html
http://missvickie.com/workshop/howdoesit.htmlhttp://missvickie.com/workshop/howdoesit.html

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