Introduction to Fluid Mechanics
TienTsan 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.
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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 threedimensional element.
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Hooke's law of shear:p
yx
= G
yx
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Newton's law of viscosity:p
yx
= 2_ =
du
dy
Classication of Fluid Flows
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Gases versus Liquid
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Continuum versus Discrete Fluid
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Perfect versus Real Fluids
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Newtonian and NonNewtonian Fluids
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Compressible and Incompressible Fluids
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Steady and Unsteady Fluid Flows
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One,Two,ThreeDimensional Flows
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Rotational versus Irrotational Flow
Properties of Fluids
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Mass:M
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Density:
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Specic Weight: weight per unit volume
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Specic 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
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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.
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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:
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Hydrostatics is the science of the static equilibrium of
incompressible iuds.
p
2
p
1
= p = h
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Aerostatics diers from hydrostatics in the specic wight
and/or density is no longer considered constant.
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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
.
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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
signicant 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
Dierential 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.
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The equation of state (1)
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The equation of continuity (1)
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The equation of conservation of uid momnetum (3)
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The equation of conservation of uid energy (1)
The conservation of mass
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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 antisymetric 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
rV
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 dened 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),
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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
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Vorticity is dened by
= rV
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
adx =
Z
gdx +
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 NavierStokes 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
NavierStokes eq for incompressible ows:
@V
@t
+(V r)V = g
1
rp +r
2
V
Euler's equation and Stokes ow
NavierStoke 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 NavierSokes equation
becomes
rp = r
2
V
It is popularly called Stokes ow.
The GromekaLamb form of the NavierStokes eq
@V
@t
+ V = g r
p
+
V
2
2
(r)
where = rV 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 (rV = 0),we can set V = .
Bernoullli's eq:
p
+
V
2
2
+
+
@
@t
= c(t)
Conservation of energy(I)
Specic 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 (
eV)
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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
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Loss of power due to viscous stress:
dw
dt
v
= 2r (V
_
S) +2(
_
S r) V
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The power due to the normal stresses:
dw
dt
p
= r (pV)
Conservation of energy(II)
dw
dt
mech
=
@(
e)
@t
+r
e +
p
VkrT +2V
_
S
2(
_
Sr)V
This equation applies to any Newtonian uid in a eld where the
only transfer of heat is by conduction.
Examples
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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 specic enthalpy
h =
i +
p
The solution is w
mech
=
h +
V
2
2
+gz
.
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For a uid at rest or moving with negligible velocity and
having no mechanical energy transfer:
@(
i )
@t
= r krT
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In paritcular,if the uid is a perfect gas,then
C
v
@
@t
(T) = kr
2
T
Dimensional Analysis
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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.
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The Rayleigh Method
Dimensionless parameters
Dimensionless NavierStokes 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
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Reynolds number R
L
=
UL
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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 coecient
C
p
=
p
1=2U
2
Reynolds number R
L
R
L
=
UL
Examples where R
L
is very large or innite:
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Turbulent ows
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Inviscid ows
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Potential ows
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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 dened 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 eect 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
.
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Large Weber number W indicates surface tension is relatively
unimportant,compared to the inertial force.
Types of ows
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Viscous Fluid Flows
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Laminar Pipe Flow
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Turbulent Pipe Flow
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Potential Flow
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OpenChannel Flow
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Boundary Layer Flows
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Onedimensional Compressible Flows
References
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Robert A.Granger,Fluid Mechanics,Dover,1985.
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