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.

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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 three-dimensional 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 Non-Newtonian Fluids

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Compressible and Incompressible Fluids

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Steady and Unsteady Fluid Flows

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One,Two,Three-Dimensional 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 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

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 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 = 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 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

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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.

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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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Open-Channel Flow

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Boundary Layer Flows

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One-dimensional Compressible Flows

References

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Robert A.Granger,Fluid Mechanics,Dover,1985.

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