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

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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)

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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)
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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.

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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.

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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.

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1.2. Fundamental Concepts

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

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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

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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.

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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

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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

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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
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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

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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)

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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

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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

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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

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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
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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

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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

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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

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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

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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.

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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

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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.

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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

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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.

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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

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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.

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1.2.8. Vapour Pressure