Part II Fundamentals of Fluid Mechanics By Munson, Young, and Okiishi

WHAT we will learn

I. Characterization of Fluids

- What is the fluid? (Physical properties of Fluid)

II. Behavior of fluids - Fluid Statics: Properties of a fluid at rest (Physics of the pressure in fluids) - Fluid Dynamics: Behavior of a moving fluid

Fluid kinetics and kinematics (Bernoulli Equation & Control volume analysis)

Basic things of Fluids (Properties of Fluids) 1. How is a fluid different from a solid?

Molecular spacing: Solid < Liquid < Gases

Cohesive forces between molecules: Solid (Not easily deformed) > Liquid (Easily deformed, but not easily compressed)

> Gases (Easily deformed and compressed)

Fluid = Liquid + Gases ≡ A substance that deforms continuously when acted on by a shearing stress* of any magnitude

* Shearing stress: Tangential force per unit area acting on the surface

2. Heaviness of a fluid

Density of a fluid, ρ : Mass per unit volume

Vm

=ρ (kg/m3): Depending on pressure and temperature*

* Ideal gas law: RTp ρ= where p (T): Absolute pressure (Temp.) R : Gas constant, 287.0 m2/s2 K

Specific Weight, γ : Weight (force) per unit volume

gργ = (N/m3)

Specific Gravity, SG: Ratio of ρ of a fluid to ρ of water at 4 oC

COH o

SG4@2

ρρ

= (Unitless)

3. Compressibility of a fluid • Bulk Modulus (Compressibility of fluid, when the pressure changes)

Defined as VdV

dpEv /−= =

ρρ /ddp [lb/in2 or N/m2]

− : because p↑(dp > 0), V↓ (dV <0) - Large Ev → Hard to compress

Usually Ev of liquid: Very large, (incompressible) w.r.t. gases

4. Fluidity of a fluid [Viscosity, μ i.e. flowing feature of a fluid] Consider a situation shown

AF : Shearing stress ( AFT τ= ) (A: Area of upper plate)

aδ : Displacement of top plate δβ : Rotation angle of line AB

u(y): Fluid velocity at height y Step 1. Application of force FT (or Shearing stressτ)

- Upper plate: Moving due to a shearing stress τ [Velocity = U] = Fluid velocity in contact with upper plate = u(b)

- Bottom plate: no movement [Velocity = 0] = Fluid velocity in contact with bottom plate = u(0)

Step 2. Deformation of Fluid

If fluid velocity between two plates → Vary linearly

i.e. u = u(y) = ybU

bU

dydu

=

For a short time period tδ , line AB rotates by an small angle δβ

btU

ba δδδβδβ ==≈tan

or, γδδβ

δ&==

→ bU

tt 0lim : Shearing strain, (Function of FT )

Then, dydu

bU

AFT ==∝= γτ &)( or

dyduμτ =

b y

A

B B’

U

u

δa

δβ

FT

Special case!!

• Viscosity μ : Absolute (or dynamic) viscosity [lb⋅s/ft2 or N⋅s/m2]

- How easily (or fast) a fluid flows (deforms) due toτ - Large μ → Difficult to flow

- Depends on the temperature and type of a fluid*

* Type of a fluid

1. Newtonian Fluid: Linear relation between τ and dydu

2. Non-Newtonian: Non-linear relation

i) Shearing thinning (τ ↑, apparentμ ↓) e.g. Latex paint, suspension

ii) Shearing thickening (τ ↑, appμ ↑) e.g. water-corn starch,

iii) Bingham plastic: e.g. Mayonnaise

c.f. Kinematic viscosity, ρμν = [ft2/s or m2/s]

Newtonian Fluid

5. Speed of Sound in a fluid

Propagation of Sound Wave → Propagation of Disturbances (Oscillations) of fluid molecules → Changes of p and ρ of the fluid due to acoustic vibration

Speed of sound or Acoustic velocity, c

ρd

dpc = = ρ

vE (since ρρ /d

dpEv = )

6. Vapor pressure

Evaporation: Escape of molecules from liquids to the atmosphere

Equilibrium state of Evaporation in the closed container : Number of molecules leaving the liquid surface = No. of molecules entering the liquid surface

Vapor pressure: Pressure on the liquid surface exerted by the vapors - Property of a fluid (V. P. of gasoline > V.P. of water) - Function of Temperature (T ↑, Vapor Pressure ↑)

- High vapor pressure → Easy to be vaporized (Volatility) • Boiling (Formation of vapor bubble within a fluid) condition - When environmental (container) pressure = Vapor pressure e.g. Vapor pressure of water at 100 oC = 14.7 Psi (Standard atmospheric pressure)

7. Cohesivity of a fluid (Surface Tension, σ )

•

) ( , surfaceofboudarythealongLengthattractionularIntermolecforceCohesivetentionSurface =σ

- Property of a fluid (Especially at the boundary)

- Molecules inside a fluid: No net attraction (Balanced cohesive force by surrounding molecules)

- Molecules at the surface: Nonzero attraction toward the interior

(Unbalanced force due to lack of outside molecules = Source of Tention)

How can this unbalanced force be compensated?

∴ Tensile force along the surface ∝ Number of molecular attraction per unit length (Intensity) : Surface tension, σ )()( ForceLength =×σ → [ ] mN /=σ

Surface

Ex. 1 Spherical droplet cut in half Question: What is the inside pressure of a fluid drop? Let’s cut the drop in half, then, Force due to σ [(σ )×(Length) = σπR2 ] = Force due to the pressure difference

[ )( pΔ ×(Area)= 2RpπΔ ]

i.e. 22 RpR πσπ Δ= or

∴ ei ppp −=Δ = Rσ2 > 0

Ex. 2 Capillary action of liquid Q: Why do a liquid rise in a capillary tube?

Strong (or Weak) molecular attraction between the wall and liquid → Rise (Fall) of a liquid At the equilibrium, Vertical force due to surface tension ( θσπ cos2 R )

= Weight of a liquid column ( hRVgmg 2γπρ == )

∴ R

hγ

θσ cos2= (Radius of tube R ↓, then, h ↑)

Δpπ R2

σ R

Pi

Ex. 3 (Viscosity) The velocity distribution for the flow of a Newtonian fluid between to wide, parallel plates shown is give by the equation,

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−=

21

23

hyVu

where V is the mean velocity. The fluid has a viscosity μ of 0.04 lb⋅s/ft2. When V = 2 ft/s and h = 0.2 in, determine (a) the shearing stress )(τ acting on the bottom wall, and (b) the shearing stress acting on a plane parallel to the walls and passing through the centerline (midplane).

Sol) Shearing stress: dyduμτ = where μ = 0.04 lb⋅s/ft2

From the given equation, 2

2 312

3hVy

hyV

dyd

dydu

−=⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−=

∴ The shearing stress as a function of height, yh

V2

3μτ −=

(a) Along the bottom wall (y = - h)

)/083.0()(2.0)/2)(/04.0(33 2

inftinsftftslb

hV

×⋅

==μτ =14.4 lb/ft2

(b) Along the mid-plane (y = 0) 0=τ lb/ft2