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PPT – MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 1: INTRODUCTION PowerPoint presentation | free to view - id: 70f934-YmY1N

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

INTRODUCTION

- Instructor Professor C. T. HSU

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)

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)

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.

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.

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.

1.2. Fundamental Concepts

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

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

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.

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

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

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

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

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)

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

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

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

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

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

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

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

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

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.

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

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.

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

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.

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

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.

1.2.8. Vapour Pressure