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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

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 structure’s 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 (Newton’s 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 << 1

• Kn < 0.001 -Non-slip fluid flow

- B.C.s: no velocity slip

- No temp. jump

- Classical fluid mechanics

• 0.001< Kn < 0.1 -Slip fluid flow

- Continuum with slip B.C.s

• 0.1< Kn< 10 -Transition flow

- No continuum, kinetic gas

• 10<Kn-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: M0, U«a

- Compressible flow:

- Gas dynamics

- M<1 : subsonic

- M~1 : transonic

- M>1 : supersonic (1<M<5)

- M»1 : hypersonic (5<M<40)

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 V0
- P = lim(m/V) as V V*
- V*~=R.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 @ 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 (Newton’s 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 (Newton’s 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 (Newton’s Law)

- Newton’s 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)

= µu/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, m=1
- Non-Newtonian fluid: m1
- Bingham plastic fluid:

= 0 +µu/y

1.2.5. Viscosity

- No-slip condition
- From observation of realfluid, 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 + (2R) = 0

- ∆p is the inside pressure in the droplet above that of the atmosphere

∆p=pi- 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.
- (2R) cos - g(R2)h = 0

→ h=2 cos/gR

- The pressure jump across an interface in general is

p = (1/R1 + 1/R2)

- For a free surface described by z=x3=η(x1,x2),

1/Ri= (2η/ xi2)/[1+( η/ xi)2]3/2

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.

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