Introduction to Fluid Mechanics

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Oct 24, 2013 (4 years and 17 days ago)

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Introduction to Fluid Mechanics
Tien-Tsan Shieh
April 16,2009
What is a Fluid?
The key distinction between a uid and a solid lies in the mode of
resistance to change of shape.The uid,unlike the solid,cannot
sustain a nite deformation under the action of a shear force.
I
Hooke's law  = E holds for solids up to the proportion limit
of strain.:strain and :normal stress.
I
For most uids,Newton noted that /_
where  is a shear force and _ is the time rate of change of a
uid element's deformation.
Consider a three-dimensional element.
I
Hooke's law of shear:p
yx
= G
yx
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Newton's law of viscosity:p
yx
= 2_ = 
du
dy
Classication of Fluid Flows
I
Gases versus Liquid
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Continuum versus Discrete Fluid
I
Perfect versus Real Fluids
I
Newtonian and Non-Newtonian Fluids
I
Compressible and Incompressible Fluids
I
Steady and Unsteady Fluid Flows
I
One,Two,Three-Dimensional Flows
I
Rotational versus Irrotational Flow
Properties of Fluids
I
Mass:M
I
Density:
I
Specic Weight: weight per unit volume
I
Specic Gravity:S = g the ratio of density to the density of
pure water
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Pressure:p the normal stress
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Bulk Modulus of Elasticity:K = 

Vdp
dV

T
a measure of the
compressibility of liquids
I
Absolute or Dynamic Viscosity:
The viscosity of a gas increases with an increase of
temperature.
The viscosity of a liquid decreases with an increase of
temperature.
I
Kinematic Viscosity: =


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Surface Tension:
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Capillary Rise or Depression:h
Aerohydrostatics
For static state,the sum of all external forces acting on the uid
control colume is zero,so is the sum of all momnets of these forces.
Consider a static uid on earth.we will have
1

rp = g.
Examples:
I
Hydrostatics is the science of the static equilibrium of
incompressible iuds.
p
2
p
1
= p = h
I
Aerostatics diers from hydrostatics in the specic wight
and/or density is no longer considered constant.
I
Hally's law:p = p
0
exp

gz
RT
0

by assuming the eq of state of air  =
p
RT
(the perfect gas law)
and isothermal T = T
0
.
I
Logrithmic Law:p = p
0

1 
z
T
0

g
R
by assuming the eq of state and T = T
0
z.
(The typical value of  is 6.5 C/km)
Lagrange Description
The Lagrangian decription describes the history of the particle
exaclty.But it is rarely used in uid mechanics because of its
signicant mathematical complexities and experimental limitations.
The Lagrangian description is often used to describe the dynamic
behaviour of solids.
x = x(x
0
;y
0
;z
0
;t)
y = y(x
0
;y
0
;z
0
;t)
z = z(x
0
;y
0
;z
0
;t)
u =
@x
@t
v =
@y
@t
w =
@z
@t
a
x
=
@u
@t
=
@
2
x
@t
2
a
y
=
@v
@t
=
@
2
y
@t
2
a
w
=
@w
@t
=
@
2
z
@t
2
Euler Description
The Eulerian description is used to describe what is happening at a
given spatial location P(x;y;z) in the ow eld at a given instant
of time.
u = f
1
(x;y;z;t)
v = f
2
(x;y;z;t)
w = f
3
(x;y;z;t)
Substantive Derivative
D
Dt
:the Stoke deriv.
D
Dt


@
@t
+u
@
@x
+v
@
@y
+w
@
@z

=

@
@t
+V r

The acceration in the Eulerian Description:
a
x
=
Du
Dt
=
@u
@t
+u
@u
@x
+v
@u
@y
+w
@u
@z
a
y
=
Dv
Dt
=
@v
@t
+u
@v
@x
+v
@v
@y
+w
@v
@z
a
z
=
Dw
Dt
=
@w
@t
+u
@w
@x
+v
@w
@y
+w
@w
@z
Dierential equations of uid behaviour
6 unkown variable:three scalar velocity components,the
temperature,the pressure and the density of the uid.
Here,we use Eulerian description.
I
The equation of state (1)
I
The equation of continuity (1)
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The equation of conservation of uid momnetum (3)
I
The equation of conservation of uid energy (1)
The conservation of mass
I
The general property of balance:if  is an intensive
continuum qunatitiy of the uid.
D
Dt
Z


dx =
@
@t
Z


dx +
Z
@

V dA
D
@t
=
@
@t
+r (V)
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The equation of the conservation of mass
@
@t
+r (V) = 0
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If the uid is incompressible ( =constant),the continuity
equation is expressed as
r V = 0
Decomposition of the motion of particles(I)
Express the velocity u;v;w of a particle at Q(x;y;z) near
P(x
0
;y
0
;z
0
) in Taylor's series form:
2
4
u
v
w
3
5
=
2
4
u
0
v
0
w
0
3
5
+
2
6
4
@u
@x
@u
@y
@u
@z
@v
@x
@v
@y
@v
@z
@w
@x
@w
@y
@w
@z
3
7
5
0
2
4
x x
0
y y
0
z z
0
3
5
+O(high order)
V = V
0
+A(r r
0
) +B(r r
0
) +O(high order)
the anti-symetric part A =
1
2

DV(DV)
T

the symetric part B =
1
2

DV+(DV)
T

Decomposition of the motion of particles(II)
The velocity can be expressed as
V = V
0
(r r
0
) !+(r r
0
) 
_
S
where the angular rotation is
!=
1
2
rV
the strain rate dyadic is
_
S =
2
6
6
6
4
@u
@x
1
2

@u
@y
+
@v
@x

1
2

@u
@z
+
@w
@x

1
2

@u
@y
+
@v
@x

@v
@y
1
2

@v
@z
+
@w
@y

1
2

@u
@z
+
@w
@x

1
2

@v
@z
+
@w
@y

@w
@z
3
7
7
7
5
0
The strain rate dyadic
_
S
_
S =
@
@t

ij
=
2
4
_
xx
_
xy
_
xz
_
yx
_
yy
_
yz
_
zx
_
zy
_
zz
3
5
=
2
6
6
6
4
@u
@x
1
2

@u
@y
+
@v
@x

1
2

@u
@z
+
@w
@x

1
2

@u
@y
+
@v
@x

@v
@y
1
2

@v
@z
+
@w
@y

1
2

@u
@z
+
@w
@x

1
2

@v
@z
+
@w
@y

@w
@z
3
7
7
7
5
The strain rate dyadic
_
S involves the dilatation and shearing strain
of th uid particle at P.
The dilatation D is dened as
D = _
xx
+ _
yy
+ _
zz
= r V
The stress dyadic P
Consider the most general form of a linear relation between a
stress and a rate of strain.
P = a
_
S +bI
where tensor a contains 36 constants"a\and tensor b contains 3
constant"b".
It is called the constitutive equation of uid dynamics.
The stress dyadic P
Consider an isotropic uid (no preferred direction),
I
Incompressible uid:
P = 2
_
S pI
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Compressible uid:
P = 2
_
S (p +
2
3
r V)I
where the pressure is p = 
1
3
(p
xx
+p
yy
+p
zz
).
Newton's viscosity potulates
Consider the isotropic incompressible uid.Express the stress
tensor p
ij
as
p
ij
=
(


@u
i
@x
j
+
@u
j
@x
i

;j 6= i
p +2
@u
i
@x
i
;j = i
Comparing the stresses with the strain rate tensor S,we see
p
xy
= 2_
xy
p
xz
= 2_
xz
p
yz
= 2_
yz
These relations are called Newton's viscosity potulates.
Surface forces F
s
and Vorticity 
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Surface forces:
Normal part:
F

=
Z
A
P dA;i = j
Tangential part:
F

=
Z
A
P dA;i 6= j
I
Vorticity is dened by
 = rV
Note that  = 2!.
Cauchy's equation of motion
Applying Newton's second law:
P
F
s
+
P
F
b
= Ma
where F
s
surface forces and F
b
body forces
If there is only gravitational force acted on the body,we have
Z


adx =
Z


gdx +
Z
@

P dA
a = g +
1

r P
This is called the Cauchy's equation of motion.
@u
@t
+u
@u
@x
+v
@u
@y
+w
@u
@z
= g
x
+
1


@p
xx
@x
+
@p
yx
@y
+
@p
zx
@z

@v
@t
+u
@v
@x
+v
@v
@y
+w
@v
@z
= g
y
+
1


@p
xy
@x
+
@p
yy
@y
+
@p
zy
@z

@w
@t
+u
@w
@x
+v
@w
@y
+w
@w
@z
= g
z
+
1


@p
xz
@x
+
@p
yz
@y
+
@p
zz
@z

The Navier-Stokes Equations
Consider a compressible uid with the consitutive equation
P = 2
_
S (p +
2
3
D)I
Pluging into Cauchy's eq of motion,we obtain
@V
@t
+(V r)V = g 
1

rp +r
2
V+rD
Navier-Stokes eq for incompressible ows:
@V
@t
+(V r)V = g 
1

rp +r
2
V
Euler's equation and Stokes ow
Navier-Stoke equation for inviscid uid ow ( = 0):
@V
@t
+(V r)V = g 
1

rp
It is usually called Euler's equation.
For the case of very slow uid motion,the Navier-Sokes equation
becomes
rp = r
2
V
It is popularly called Stokes ow.
The Gromeka-Lamb form of the Navier-Stokes eq
@V
@t
+ V = g r

p

+
V
2
2

(r)
where  = rV is a vorticity vector.
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For a steady and irrotational ow,g = r

p

+
V
2
2

.
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For inviscid uid ow,
@V
@t
+ V = r

p

+
V
2
2
+


.
where g = r
.
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For a steady,inviscid and incompressible ow,
V = r

p

+
V
2
2
+


Crocco's or lamb's eq,this gives
Bernoulli's equation:
p

+
V
2
2
+
= const.
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For an irrotational ow (rV = 0),we can set V = .
Bernoullli's eq:
p

+
V
2
2
+
+
@
@t
= c(t)
Conservation of energy(I)
Specic energy:
e =
i +
V
2
2
+gz +
e
nuclear
+
e
elect
+
e
magn
+other
In the present discussion,we shall neglect all energies except
internal,kinetic and potential.
D(
e)
Dt
=
@(
e)
@t
+r (
eV)
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The rst law of thermodynamics:
dq
dt
+
dw
dt
=
D(
e)
Dt
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Conservation of heat:
dq
dt
= r q
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Fourier's Law:q = krT
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The net power:
dw
dt
=

dw
dt

mech
+

dw
dt

v
+

dw
dt

p
I
Loss of power due to viscous stress:

dw
dt

v
= 2r (V
_
S) +2(
_
S  r)  V
I
The power due to the normal stresses:

dw
dt

p
= r (pV)
Conservation of energy(II)

dw
dt

mech
=
@(
e)
@t
+r

e +
p

VkrT +2V
_
S

2(
_
Sr)V
This equation applies to any Newtonian uid in a eld where the
only transfer of heat is by conduction.
Examples
I
Steady ow:
@(
e)
@t
= 0,No heat transfer:rT = 0,Inviscid
ow: = 0,we obtain

dw
dt

mech
= r

h +
V
2
2
+gz

V
where specic enthalpy
h =
i +
p

The solution is w
mech
= 

h +
V
2
2
+gz

.
I
For a uid at rest or moving with negligible velocity and
having no mechanical energy transfer:
@(
i )
@t
= r krT
I
In paritcular,if the uid is a perfect gas,then
C
v
@
@t
(T) = kr
2
T
Dimensional Analysis
I
The Buckingham  theorem is a key theorem in dimensional
analysis.The theorem loosely states that if we have a
physically meaningful equation involving a certain number,n,
of physical variables,and these variables are expressible in
terms of k independent fundamental physical quantities,then
the original expression is equivalent to an equation involving a
set of p = n k dimensionless variables constructed from the
original variables:it is a scheme for nondimensionalization.
I
The Rayleigh Method
Dimensionless parameters
Dimensionless Navier-Stokes equation:
@V

@
+(V

 r

)V

= r

p


k
F
2
r
+
1
R
L
r

2
V

L:a constant characteristic length
U:a constant characteristic velocity
I
Reynolds number R
L
=
UL

I
Froude number F
r
=
U
p
gL
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Mach number M =
U
c
and Cauch number C =
K
U
2
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Weber number W =
U
2
L

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Euler number E =

1
U
2
,and the Pressure coecient
C
p
=
p
1=2U
2
Reynolds number R
L
R
L
=
UL

Examples where R
L
is very large or innite:
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Turbulent ows
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Inviscid ows
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Potential ows
I
Flows far removed from boundary
Examples where R
L
is very small:
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Creeping ows
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Laminar ows
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Stokes ows and lubrication theory
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Bubble ows
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Flows very close to a boundary
SOme other parameters
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Reynoolds number can be dened the ratio of the momentum
ux to the shearing stress.
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F
r
> 1:tranquil ow or rapid ow
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For large Mach number M  0:3,the eect of compressibility
must be considered.
0:3 < M < 1:subsonic ow,M > 1:supersonic ow
Mach number can be viewed as the ratio of teh intertial force
to the compressibility.
I
Cauchy number is the ratio of the compressibility force to the
intertial force.M =
1
p
C
.
I
Large Weber number W indicates surface tension is relatively
unimportant,compared to the inertial force.
Types of ows
I
Viscous Fluid Flows
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Laminar Pipe Flow
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Turbulent Pipe Flow
I
Potential Flow
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Open-Channel Flow
I
Boundary Layer Flows
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One-dimensional Compressible Flows
References
I
Robert A.Granger,Fluid Mechanics,Dover,1985.