part ii fundamentals of fluid mechanics by munson,...
TRANSCRIPT
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