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## MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 1: INTRODUCTION

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### MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 1: INTRODUCTION Instructor: Professor C. T. HSU – PowerPoint PPT presentation

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Title: MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 1: INTRODUCTION

1
MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 1
INTRODUCTION
• Instructor Professor C. T. HSU

2
1.1. Historic Background
• Until the turn of the century, there were two
main disciplines studying fluids
• Hydraulics - engineers utilizing empirical
formulas from experiments for
practical applications.
• Mathematics - Scientists utilizing analytical
methods to solve simple
problems.
(Aero/Hydrodynamics)

3
1.1. Historic Background
Prandtl (1875-1953)
• Fluid Mechanics is the modern science developed
mainly by Prandtl and von Karman to study fluid
motion by matching experimental data with
theoretical models. Thus, combining
Aero/Hydrodynamics with Hydraulics.
• Indeed, modern research facilities employ
mathematicians, physicists, engineers and
technicians, who working in teams to bring
together both view points experiment and theory.

Von Karman (1881-1963)
4
1.1. Historic Background
• Some examples of fluid flow phenomena-
• Aerodynamics design the engagement of a wing
from static state using a suitable angle of
attack will produce a start vortex. The strength
of it is very important for the airplane to
obtain high upwards lift force, especially in
aircraft takeoff on carrier. This photo shows a
model wing suddenly starts its motion in a wind
tunnel.
• Waves motion Much of the propulsive force of a
ship is wasted on the wave action around it. The
distinctive wave patterns around a ships is the
source of this wave drag. The study of these
waves, therefore, is of practical importance for
the efficient design of ship.

5
1.1. Historic Background
• Hydraulic Jump
• A circular hydraulic jump in the kitchen sink.
Hydraulic jump is a fluid phenomenon important to
fluid engineers. This is one type of
supercritical flow, which is a rapid change of
flow depth due to the difference in strength of
inertial and gravitational forces
• Structure-Fluid interaction
• Vortices generated due to motion in fluid is of
great important in structural design. The
relation of a structures natural frequency with
the shedding spectrum affect many fields of
engineering, e.g. building of bridges and piers.
Photo shows the vortex resembling the wake after
a teaspoon handle when stirring a cup of tea.

6
1.1. Historic Background
• Tidal Bore
• Tidal bore is a kind of hydraulic jump, and can
be regarded as a kind of shockwave in fluid. The
knowledge of its propagation is critical in some
river engineering projects and ship scheduling.
The photo shows the famous tidal bore in Qiantang
River, China.
• Droplets dynamics
• Fluid dynamics sometimes is useful in
microelectronic applications. Droplets dynamics
is crucial to the bubblejet printing and active
cooling technology. Photo shows a drop of water
just hitting a rigid surface, recorded by high
speed photography.

7
1.2. Fundamental Concepts
• The Continuum Assumption
• Thermodynamical Properties
• Physical Properties
• Force Acceleration (Newtons Law)
• Viscosity
• Equation of State
• Surface Tension
• Vapour Pressure

8
1.2.1. The Continuum Assumption
• Fluids are composed of many finite-size molecules
with finite distance between them. These
molecules are in constant random motion and
collisions
• This motion is described by statistical mechanics
(Kinetic Theory)
• This approach is acceptable, for the time being,
in almost all practical flows

9
1.2.1. The Continuum Assumption
• Within the continuum assumption there are no
molecules. The fluid is continuous.
• Fluid properties as density, velocity etc. are
continuous and differentiable in space time.
• A fluid particle is a volume large enough to
contain a sufficient number of molecules of the
fluid to give an average value for any property
that is continuous in space, independent of the
number of molecules.

10
1.2.1. The Continuum Assumption
• Characteristic scales for standard atmosphere
• - atomic diameter 10-10 m
• - distance between molecules 10-8 m
• - mean free path, ? (sea level) 10-7 m
• ??? ? const. 100,000m ? .000006 m
• 250,000m ? 0.0012 m
• Knudsen number Kn ?/ L
• ? - mean free path
• L - characteristic length

11
1.2.1. The Continuum Assumption
• For continuum assumption Kn ltlt 1
• Kn lt 0.001 - Non-slip fluid flow
• - B.C.s no velocity slip
• - No temp. jump
• - Classical
fluid mechanics
• 0.001lt Kn lt 0.1 - Slip fluid flow
• -
Continuum with slip B.C.s
• 0.1lt Knlt 10 - Transition flow
• - No
continuum, kinetic gas
• 10ltKn - Free molecular flow
- Molecular dynamics

12
1.2.2. Thermodynamical Properties
• Thermodynamics - static situation of equilibrium
• ?n - mean free time
• a speed of molecular motion ( speed of sound
c)
• ?n ?/a microscopic time scale to equilibrium

Liquid
Gas
13
1.2.2. Thermodynamical Properties
• Convection time scale ?s L / U
• - L characteristic length
• - U fluid velocity (macroscopic scale)
• Local thermodynamic equilibrium
• assumption ?n?s
• - ?/a L/U ? (?/L).(U/a) 1 ? Kn.M 1

14
1.2.2. Thermodynamical Properties
• Mach number M U / a
• - Incompressible flow M?0, Ua
• - Compressible flow
• - Gas dynamics
• - Mlt1 subsonic
• - M1 transonic
• - Mgt1 supersonic (1ltMlt5)
• - M1 hypersonic (5ltMlt40)

15
1.2.3. Physical Properties
• Example density ? at point P
• ? density, mass/volume kg/m3
• ? specific weight N/m3
• ? g
• average density in a small volume ?V
• ?m / ?V

16
1.2.3. Physical Properties
• ?P ? lim(?m/?V) as ?V ?0
• ?P lim(?m/?V) as ?V ? ?V
• ?VR.E.V. (representative elementary volume)
• Fluid particle with volume ?V(1 ?m)3 109
particles
• Specific gravity, S.G.
• the ratio of a liquid's density to that of
pure water at 4oC (39.2oF)
• H2O _at_ 4oC
• ? ? 1000 kg/m3
• 1 g/cm3

17
1.2.3. Physical Properties
• Similarly, other macroscopic physical properties
or physical quantities can be defined from this
microscopic viewpoint
• Momentum M,
• Velocity u
• Acceleration a
• Temperature T
• Pressure, viscosity, etc

18

19

20
1.2.4. Force Acceleration (Newtons Law)
• The force on a body is proportional to the
resulting acceleration
• ? F ma unit 1N 1kg . 1m/s2
• The force of attraction between two bodies is
• proportional to the masses of the bodies
• ?

r Distance
G Gravitational Constant
21
1.2.4. Force Acceleration (Newtons Law)
• Various kinds of forces
• Static pressure
• Dynamic pressure
• Shear force
• Body force (weight)
• Surface tension
• Coriolis force
• Lorentz force, etc

22
1.2.4. Force Acceleration (Newtons Law)
• Newtons law is a conservation law. It describes
the conservation of linear momentum in a system.
• Different kinds of conservation Laws, e.g.
• Conservation of mass
• Conservation of linear momentum
• Conservation of energy, etc
• Continuity equation
• Navier-Stokes equations
• Energy equation, etc

23
1.2.5. Viscosity
• The shear stress on an interface tangent to the
direction of flow is proportional to the strain
rate (velocity gradient normal to the interface)
• ? µ?u/?y
• µ is the (dynamic) viscosity kg/(m.s)
• Kinematic viscosity ? µ/? m2/s

24
1.2.5. Viscosity
• Power law
• ? k (? u/? y)m
• Newtonian fluid k µ, m1
• Non-Newtonian fluid m?1
• Bingham plastic fluid
• ? ?0 µ?u/?y

25
1.2.5. Viscosity
• No-slip condition
• From observation of real fluid, it is found that
it always stick to the solid boundaries
containing them, i.e. the fluid there will not
slip pass the solid surface.
• This effect is the result of fluid viscosity in
real fluid, however small its viscosity may be.
• A useful boundary condition for fluid problem.

26
1.2.6. Equation of State (Perfect Gas)
• Equation of state is a constitutive equation
describing the state of matter
• Ideal gas the molecules of the fluid
have perfectly elastic
collisions
• Ideal gas law p ? R T
• R is universal gas constant
• Speed of sound c(dp/d?)1/2

27
1.2.7. Surface Tension
• At the interface of a liquid and a gas the
molecular attraction between like molecules
(cohesion) exceed the molecular attraction
between unlike molecules (adhesion). This results
in a tensile force distributed along the surface,
which is the surface tension.

28
1.2.7. Surface Tension
• For a liquid droplet in gas in equilibrium
• -(?p)?R2 ? (2?R) 0
• ?p is the inside pressure in the droplet above
that of the atmosphere
• ?ppi- pe 2? / R

29
1.2.7. Surface Tension
• For liquids in contact with gas and solid, if the
adhesion of the liquid to the solid exceeds the
cohesion in the liquids, then the liquid will
rise curving upward toward the solid. If the
adhesion to the solid is less than the cohesion
in the liquid, then the liquid will be depressed
curving downward. These effects are called
capillary effects.

30
1.2.7. Surface Tension
• The capillary distance, h, depends for a given
liquid and solid on the curvature measured by the
contact angle ?, which in turn depends on the
internal diameter.
• ? (2?R) cos? - ?g(?R2)h 0
• ? h2? cos?/?gR
• The pressure jump across an interface in general
is
• ?p ? (1/R1 1/R2)
• For a free surface described by zx3?(x1,x2),
• 1/Ri (? 2?/? xi2)/1(? ?/? xi)23/2

31

32
1.2.8. Vapour Pressure
• When the pressure of a liquid falls below the
vapor pressure it evaporates, i.e., changes to a
gas. If the pressure drop is due to temperature
effects alone, the process is called boiling. If
the pressure drop is due to fluid velocity, the
process is called cavitation. Cavitation is
common in regions of high velocity, i.e., low p
such as on turbine blades and marine propellers.

33
1.2.8. Vapour Pressure