Advanced_Fluid_Mechanics__Course_Notes_.pdf

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Advanced Fluid Mechanics
Chapter 1 Introduction
1.1 Classification of a Fluid (A fluid can only substain tangential force when it moves)
1.) By viscous effect: inviscid & Viscous Fluid.
2.) By compressible: incompressible & Compressible Fluid.
3.) By Mack No: Subsonic, transonic, Supersonic, and hypersonic flow.
4.) By eddy effect: Laminar, Transition and Turbulent Flow.
The objective of this course is to examine the effect of tangential (shearing) stresses
on a fluid.
Remark:
For a ideal (or inviscid) flow, there is only normal force but tangential force between
two contacting layers.
1.2 Simple Notation of Viscosity
(tangential force required to move upper
plate at velocity of U )
U
F
y
h
Fluid (e.g. water)
x
u(y) = y/h U
From observation, the tangential force per unit area required is proportional to U/h, or
du/dy. Therefore
τ ≡ shear stress = tangential force per unit area (F/A)
U
∝
or
τ =
µ =
µ 〝 Newton’s Law of function 〞 (1.1)
∂
u
h
∂
y
µ: Constant of proportionality
The first coefficient of viscosity
Remark :
E.g. (1.1) provides the definition of the viscosity and is a method for measuring the
viscosity of the fluid.
Chapter1- 1
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Advanced Fluid Mechanics
In generally, if
ε represent the strain rate, then
XY
τ ε
= (1.2)
()
xy
plastic
yielding fluids
Dilatent fluid
Pesudoplastic fluid
τ
Non- Newtonian
fluid
Newtonian fluid
Yield stress
ε
Newtonian fluid: linear relation between τ and ε
Pesudoplastic fluid: the slope of the curve decrease as ε increase (shear-thinning) of
the shear-thinning effect is very strong. The fluid is called plastic
fluid .
Dilatent fluid: the slope of the curve increases as ε increases (shear-thicking).
Yielding fluid: A material, part solid and part fluid can substain certain stresses before
it starts to deform.
Note
1 Pa (Pascal) ≡
Newton (Pascal, a French philosopher and Mathematist)
m
2
(a unit of pressure )
kg
⋅
m
s
2
kg
⋅
g
[µ] = [pa · sec] (=
⋅
s
=
= 10
)
m
2
m
s
cm
⋅
s
The metric unit of viscosity is called the poise (p) in honor of J.L.M. Poiseuille (1840),
who conducted pioneering experiment on viscous flow in tubes.
1 P ≡ (()
1
= 0.1
pa
sec
⋅
cm
s
Chapter1- 2
f
xy
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Advanced Fluid Mechanics
The unit of viscosity:
τ
F
F
2
[ µ =
L
2
=
=
T
← (Old -English Unit: F-L-T)
α
L
L
∂ y
T
L
or
[ µ =
ML
T
2
M
⋅ T
=
← (international system SI unit: M-L-T)
L
2
LT
Denote :
M
N ≡ Pa, then
2
µ
=
1
01
×
10
3
Pa
⋅
sec
water
,
20
°
c
µ
=
283
Pa
⋅
sec
(liquid): T → µ
water
,
100
°
c
µ
air
,
20
°
c
=
17
.
9
Pa
⋅
sec
µ
=
22
.
9
Pa
⋅
sec
(gas): T → µ
air
100
°
c
For dilute gas:
µ
T
n
≈
(Power- law)
µ
T
°
µ
T
3
2
T
+
S
≈
°
(Sutherland’s law)
µ
T
T
+
S
°
°
Where µ, T and S depends on the nature of the gases.
Kinematics Viscosity
µ
υ≡
ρ
Chapter1- 3
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Advanced Fluid Mechanics
Exp: (Effect of Viscosity on fluid)
Flow past a cylinder
Foe a ideal flow:
U
∞
u
θ
R
w
( )
R
2
u
r
,
θ
=
U
cos
θ
−
1
∞
r
2
R
2
1
( )
v
r
,
θ
=
U
sin
θ
+
1
∞
r
2
At r = R, u=0,
v
= U
2
∞
sin
θ
-3
The Bernoulli e.g. along the surface is:
1
ρ
U
2
+
P
=
1
ρ
v
2
+
p
(Incompressible flow)
2
∞
∞
2
P
−
P
v
2
1
C p
=
∞
=
1
−
=
1
−
sin
2
θ
1
U
2
4
2
ρ
U
∞
2
∞
D’Albert paradox: No Drag.
For a real flow: (viscous effect in)
Re
=
ρ VD
µ
(前後幾乎對稱)
Re=0.16 (fig 6)
(
前後不對稱
) Re=1.54 (fig 24)
Chapter1- 4
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Advanced Fluid Mechanics
Separation occur
(pair of recirculating eddies)
Re=9.6 (fig 40) (6 < Re <40)
d
d ∼
Re
R=26 (fig. 26)
L ∼
Re
L
supercritical
θ>°
90
Subcritical
θ < °
sep
90
Chapter1- 5
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