Transcript Fluid Mechanics

Lecture no 1 to 10
Basic Fluid Mechanics
Summary of introductory concepts
By Engr Sarfaraz Khan Turk
Introduction


What is Fluid Mechanics?
Fluid mechanics deals with the study of all fluids
under static and dynamic situations. Fluid
mechanics is a branch of continuous mechanics
which deals with a relationship between forces,
motions, and statically conditions in a continuous
material. This study area deals with many and
diversified problems such as surface tension, fluid
statics, flow in enclose bodies, or flow round bodies
(solid or otherwise), flow stability, etc.
History
Faces of Fluid Mechanics
Archimedes
(C. 287-212 BC)
Navier
(1785-1836)
Newton
(1642-1727)
Stokes
(1819-1903)
Leibniz
(1646-1716)
Reynolds
(1842-1912)
Bernoulli
Euler
(1667-1748)
(1707-1783)
Prandtl
Taylor
(1875-1953)
(1886-1975)
Introduction contd

In fact, almost any action a person is doing
involves some kind of a fluid mechanics problem.
Furthermore, the boundary between the solid
mechanics and fluid mechanics is some kind of
gray shed and not a sharp distinction for the
complex relationships between the different
branches which only part of it should be drawn in
the same time.). The fluid mechanics study involve
many fields that have no clear boundaries between
them.
Introduction contd

Fluids omnipresent
 Weather
& climate
 Vehicles: automobiles, trains, ships, and
planes, etc.
 Environment
 Physiology and medicine
 Sports & recreation
 Many other examples!
Introduction
Field of Fluid Mechanics can be divided into 3
branches:
 Fluid Statics: mechanics of fluids at rest
 Kinematics: deals with velocities and
streamlines w/o considering forces or energy
 Fluid Dynamics: deals with the relations
between velocities and accelerations and forces
exerted by or upon fluids in motion
Streamlines
A streamline is a line that is tangential to the
instantaneous velocity direction (velocity is a
vector that has a direction and a magnitude)
Instantaneous streamlines in flow around a cylinder
Intro…con’t
Mechanics of fluids is extremely important in many
areas of engineering and science. Examples are:

Biomechanics & Bio-fluid mechanics.
 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
Intro…con’t

Mechanical Engineering
 Design
of pumps, turbines, air-conditioning
equipment, pollution-control equipment, etc.

Civil Engineering
 Transport
of river sediments
 Pollution of air and water
 Design of piping systems
 Flood control systems
Intro…con’t

Fluids essential to life
 Human
body 65% water
 Earth’s surface is 2/3 water
 Atmosphere extends 17km above the earth’s surface

History shaped by fluid mechanics
 Geomorphology
 Human
migration and civilization
 Modern scientific and mathematical theories and
methods
 Warfare

Affects every part of our lives
Dimensions and Units
Before going into details of fluid
mechanics, we stress importance of units
 In U.S, two primary sets of units are used:

 1.
SI (Systeme International) units
 2. English units
Unit Table
Quantity
SI Unit
English Unit
Length (L)
Meter (m)
Foot (ft)
Mass (m)
Kilogram (kg)
Time (T)
Second (s)
Slug (slug) =
lb*sec2/ft
Second (sec)
Temperature (  ) Celcius (oC)
Farenheit (oF)
Force
Pound (lb)
Newton
(N)=kg*m/s2
Dimensions and Units con’t
1 Newton – Force required to accelerate a
1 kg of mass to 1 m/s2
 1 slug – is the mass that accelerates at 1
ft/s2 when acted upon by a force of 1 lb
 To remember units of a Newton use F=ma
(Newton’s 2nd Law)

 [F]
= [m][a]= kg*m/s2 = N
More on Dimensions
To remember units of a slug also use
F=ma => m = F / a
 [m] = [F] / [a] = lb / (ft / sec2) = lb*sec2 / ft


1 lb is the force of gravity acting on (or
weight of ) a platinum standard whose
mass is 0.45359243 kg
Weight and Newton’s Law of Gravitation

Weight
 Gravitational

attraction force between two bodies
Newton’s Law of Gravitation
F = G m1m2/ r2
G
- universal constant of gravitation
 m1, m2 - mass of body 1 and body 2, respectively
 r - distance between centers of the two masses
 F - force of attraction
Weight
m2 - mass of an object on earth’s surface
 m1 - mass of earth
 r - distance between center of two masses
 r1 - radius of earth
 r2 - radius of mass on earth’s surface
 r2 << r1, therefore r = r1+r2 ~ r1
 Thus, F = m2 * (G * m1 / r2)

Weight

Weight (W) of object (with mass m2) on surface of earth
(with mass m1) is defined as
W = m2g ; g =(Gm1/r2) gravitational acceleration
g = 9.31 m/s2 in SI units
g = 32.2 ft/sec2 in English units

See back of front cover of textbook for conversion tables
between SI and English units
Properties of Fluids - Preliminaries


Consider a force, F, acting on a 2D region of
area A sitting on x-y plane

F
z
y
A
x
Cartesian components:

F  Fx (  i)  Fy (  j )  Fz (  k)
Cartesian components
 i - Unit vector in  x-direction
 j
- Unit vector in  y-direction
 k - Unit vector in  z-direction

Fx - Magnitude of F in  x-direction (tangent to surface)

Fy - Magnitude of F in  y-direction (tangent to surface)

Fz - Magnitude of F in  z-direction (normal to surface)
- For simplicity, let
Fy  0
• Shear stress and pressure
Fx

A
Fz
p
A
( shear stress)
(normal stress ( pressure))
• Shear stress and pressure at a point
 Fx 
 
 A  lim A 0
 Fz 
p  
 A  lim A 0
• Units of stress (shear stress and pressure)
[F] N
 2  Pa ( Pascal ) in SI units
[ A] m
[ F ] lb
 2  psi ( pounds per square inch) in English units
[ A] in
[ F ] lb
 2  pounds per square foot ( English units)
[ A] ft
Properties of Fluids Con’t



Fluids are either liquids or gases
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.
More on properties of fluids

Fluids considered in this course move
under the action of a shear stress, no
matter how small that shear stress may be
(unlike solids)
Continuum view of Fluids




Convenient to assume 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
A good way to determine if the continuum model is
acceptable is to compare a characteristic length ( L) of the
flow region with the mean free path of molecules, 
If L   , continuum model is valid

Mean free path (  ) – Average distance a
molecule travels before it collides with
another molecule.
Density and specific weight
Density (mass per unit volume):
Units of density:
m

V
[m] kg
[ ] 
 3
[V ] m
Specific weight (weight per unit volume):
(in SI units)
  g
Units of specific weight:
kg m
N
[ ]  [ ][ g ]  3 2  3
m s
m
(in SI units)
Viscosity (  )




Viscosity can be thought as the internal stickiness of a fluid
Representative of internal friction in fluids
Internal friction forces in flowing fluids result from cohesion
and momentum interchange 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)
More on Viscosity

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
No slip condition
Because of viscosity, at boundaries (walls)
particles of fluid adhere to the walls, and
so the fluid velocity is zero relative to the
wall
 Viscosity and associated shear stress may
be explained via the following: flow
between no-slip parallel plates.

Flow between no-slip parallel plates
-each plate has area A
Moving plate
 
F, U
y
Y
x
Fixed plate
z

F  Fi
Force

F

U  Ui
induces velocity
At bottom plate velocity is
0

U
on top plate. At top plate flow velocity is

U
The velocity induced by moving top plate can be sketched as follows:
y
u( y  0)  0
U
u( y  Y )  U
Y
u( y)
The velocity induced by top plate is expressed as follows:
U
u( y )    y
 Y
For a large class of fluids, empirically,
More specifically,
AU
F 
;
Y
Shear stress induced by
F
is
From previous slide, note that
Thus, shear stress is
AU
F
Y
 is coefficient of vis cos ity
F
U
 
A
Y
du U

dy Y
du

dy
In general we may use previous expression to find shear stress at a point du
inside a moving fluid. Note that if fluid is at rest this stress is zero because
0
dy
Newton’s equation of viscosity
du
Shear stress due to viscosity at a point:   
dy

- viscosity (coeff. of viscosity)
 - kinematic

 viscosity
fluid surface
y
e.g.: wind-driven flow in ocean
u( y) (velocity profile)
Fixed no-slip plate
As engineers, Newton’s Law of Viscosity is very useful to us as we can use it to
evaluate the shear stress (and ultimately the shear force) exerted by a moving
fluid onto the fluid’s boundaries.
 du 
 at boundary    
 dy  at boundary
Note y is direction normal to the boundary
Viscometer
Coefficient of viscosity

can be measured empirically using a viscometer
Example: Flow between two concentric cylinders (viscometer) of length
r
r
h
R
L
- radial coordinate
y
Moving fluid
O
Fixed outer
cylinder
Rotating inner
cylinder

, T
x
z

Inner cylinder is acted upon by a torque, T  T k , causing it to
rotate about point O at a constant angular velocity  and
causing fluid to flow. Find an expression for T

T  T k

Because
is constant,
is balanced by a resistive torque
exerted by the moving fluid onto inner cylinder
 res
T  T res (  k)
T  T res

res
The resistive torque comes from the resistive stress
exerted by the
moving fluid onto the inner cylinder.
 res This stress on the inner cylinder leads
to an overall resistive force F , which induces the resistive torque about
point
 res
res

y
z

T
x

R
F

T


T
O
 res
T
Compressibility
• All fluids compress if pressure increases resulting in an
increase in density
• Compressibility is the change in volume due to a
change in pressure where V is volume and p is pressure.
• A good measure of compressibility is the bulk modulus
(It is inversely proportional to compressibility often
denoted K sometimes B).
dp
E   
d

1

( specific volume)
p is pressure
Vertical, drained compressibility's
β (m²/N or Pa−1)
Material
Plastic clay
2×10–6 – 2.6×10–7
Stiff clay
2.6×10–7 – 1.3×10–7
Medium-hard clay
1.3×10–7 – 6.9×10–8
Loose sand
1×10–7 – 5.2×10–8
Dense sand
2×10–8 – 1.3×10–8
Dense, sandy gravel
1×10–8 – 5.2×10–9
Rock, fissured
6.9×10–10 – 3.3×10–10
Rock, sound
<3.3×10–10
Water at 25 °C (undrained)
4.6×10–10
Compressibility
• From previous expression we may write
( final  initial )
initial

( p final  pinitial )
E
• For water at 15 psia and 68 degrees Farenheit, E  320,000 psi

• From above expression, increasing pressure by 1000 psi will compress
the water by only 1/320 (0.3%) of its original volume
• Thus, water may be treated as incompressible (density
( )
is constant)
• In reality, no fluid is incompressible, but this is a good approximation for
certain fluids The degree of compressibility of a fluid has strong
implications for its dynamics.
Vapor pressure of liquids
• All liquids tend to evaporate when placed in a closed container
• Vaporization will terminate when equilibrium is reached between
the liquid and gaseous states of the substance in the container
i.e. # of molecules escaping liquid surface = # of incoming molecules
• Under this equilibrium we call the call vapor pressure the saturation
pressure
• At any given temperature, if pressure on liquid surface falls below the
the saturation pressure, rapid evaporation occurs (i.e. boiling)
• For a given temperature, the saturation pressure is the boiling pressure
Properties of Liquids: Surface
Tension
http://www.visionlearning.com/library/module_viewer.php?mid=57
Water Strider Video
Surface Tension
Surface Tension-a force that tends to pull adjacent parts
of a liquid’s surface together, thereby decreasing surface
area to the smallest possible size.
~The higher the attraction forces (intermolecular forces),
the higher the surface tension. Surface tension causes
liquid droplets to take a spherical shape.
 The surface of any liquid behaves as if it was a stretched
membrane. This phenomenon is known as surface
tension
 Surface tension is caused by intermolecular forces at the
liquid’s interface with a gas or a solid.
more bugs that think they’re all that and a
bag of chips: the Water Strider
Surface Tension



Surface tension depends on the nature of the
liquid, the surrounding media and temperature.
Liquids that have strong intermolecular forces
will have higher values of surface tension than
liquids that have weak intermolecular forces.
A. Beading of rain water on a waxy surface,
such as a leaf. Water adheres weakly to wax
and strongly to itself, so water clusters into
drops. Surface tension gives them their nearspherical shape, because a sphere has the
smallest possible surface area to volume ratio.

B. Formation of drops occurs when a mass of liquid is
stretched. The animation shows water adhering to the
faucet gaining mass until it is stretched to a point where the
surface tension can no longer bind it to the faucet. It then
separates and surface tension forms the drop into a sphere.
If a stream of water was running from the faucet, the stream
would break up into drops during its fall. Gravity stretches
the stream, then surface tension pinches it into spheres.

C. Flotation of objects denser than water occurs when the
object is nonwettable and its weight is small enough to be
borne by the forces arising from surface tension. For
example, water striders use surface tension to walk on the
surface of a pond. The surface of the water behaves like an
elastic film: the insect's feet cause indentations in the
water's surface, increasing its surface area.


D. Separation of oil and water (in this case, water and
liquid wax) is caused by a tension in the surface between
dissimilar liquids. This type of surface tension is called
"interface tension", but its chemistry is the same.
E. Tears of wine is the formation of drops and rivulets on
the side of a glass containing an alcoholic beverage. Its
cause is a complex interaction between the differing
surface tensions of water and ethanol; it is induced by a
combination of surface tension modification of water
by ethanol together with ethanol evaporating faster than
water.
Even a piece of steel can do this trick
if it is small (steel  ~ 8x water)
4 H2 O
molecules
separated
in space
from each
other
 have partial
+ and –
charges
 what
would they
do???

but what’s surface tension, really?
4 H2 O
molecules



they clump
together
+ and –
charges
snuggle up
close
potential
energy of
system has
dropped




Surface Tension
water in bulk has
many binding
partners
water at surface
has less, has
exposed charges
left over
potential energy
of water at
surface is higher
deforming
droplet to
increase surface
area takes work
Contact Angles

here’s a droplet on a surface -
Contact
Angle



here’s a slice of
it –
tangent to
droplet edge is
“contact angle”
why is theta
theta?
Contact
Angle




balance of
forces
surface tension
pulls up
gravity &
adhesion pulls
down
what are the
other two?
Remember this?

water at surface
has less binding
partners

energy at surface
is higher
What if 
what if the circles
are aluminum
atoms in a solid?

what if the space
above it is liquid
ethanol?
Contact
Angle

F = dE/dX

surface/air &
surface/water
interfaces also have
“surface tension”, in
ergs/cm2

moving water edge
back and forth incurs
energy costs/profits

but units of F are
energy/distance, not
area?! what’s the
deal?
Obtuse contact
Angles



hydrophobic
surface
“gravity &
adhesion” is
now “gravity &
repulsion”
if no gravity,
drop leaves
http://citt.ufl.edu/Marcela/Sepulveda/html/en_tension.htm