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Electronics - Devices

Oct 10, 2013 (4 years and 1 month ago)

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Chapter 1

Theorem 3.
1
(
Convection Theorem
)

If
,

(3.5)

Corollary 3

Let
,

(3.6)

Corollary 4.1

If

then

(4.4)

Theorem 6.1

Irrotational motion of incompressible fluid on an open, bounded region
Ω

with
regular boundary

is unique if

1.

is prescribed for all

2.

circulations of all irreducible, basic generating circuits are prescribed.

Theorem 6.2

Irrotational moti
on of incompressible fluid on an open, bounded region
Ω

with
regular boundary

is unique if

1.

is prescribed for all

2.

is prescribed at 1 point

of each connected component of
.

Theorem 6.3

(
Mean Value Theorem of Potential Theory
)

Let

be a sphere with
.

If

1.

and single
-
valued.

2.

on
Ω
,

then

Corollary 6.1

(
Maximum Principle of Potential Theory
)

A maximum or minimum of

cannot occur in
Ω
.

Corollary 6.2

In nonuniform, irrotational motion of incompressible fluid, a maximum of

cannot occur in the interior of
Ω

if
.

Corollary 6.3

In steady, irrotational motion of ideal fluid, a minimum of the hydrodynamic
pressure cannot occur in the interior of
Ω

if
.

Stokes

Theorem

Let
S

be a regular surface with a single circuit boundary
.

If
, the Stokes

theorem says

For
, we have

Vortex flux through
S

Circulation around
.

Corollary

7.1

Since

if
S

is a closed regular surface, we have

for
S

closed.

Let the interior of the closed surface
S

be
R
, we have

ie

Corollary

7.2

Vortex flux

has the same value for all regular
S

bounded by the same
single circuit.

Corollary

7.3

Stokes

theorem applies to the case where

is a union of a finite number of
circuits that are consistently oriented.

Corollary

7.4

Any homologous circuits that are subsets of a vortex tube have the same circulation.

Corollary

7.5

If a circuit in a vortex tube has nonz
ero circulation, it is irreducible in the tube.

Chapter 2

Lemma 10.1

in a fluid in uniform motion.

Lemma 10.2

is continuous across a regular surface
S

with unit normal
n

if its normal velocity

is continuous on
S
.

Corollary 10.2

is continuous across a free vortex sheet with unit normal
n
.

Lemma 12.1

In steady incompressible flow of inviscid fluid,

is constant
along each streamline.

If the flow is
also
irrotational
,

on any connected fluid domain.

Theorem 13.1
(
d

Alembert

)

Consider a finite impermeable body immersed in a steady stream of ideal fluid
flowing through a lo
ng straight pipe. Let the flow be uniform far away from the body.
The fluid then exerts no force on the body in the axial direction.

Lemma 13.1

In a steady flow of ideal fluid with
, if there are bounded, open
subsets

&

of

Ω

such that

1.

flow on

is arbitrarily close to uniform but not stagnant,

2.

every streamline intersecting

also intersects
,

th
en the flow on

is irrotational.

Theorem 14.1
(
Kelvin

s Theorem
)

In ideal fluid under a potential body force field, any circuit moving with the fluid
conserves its circulation.

Corollary 14.1

If motion of ideal fluid starts f
rom a uniform one at time
, then every
circuit

has zero circulation for all
t
.

Corollary 14.2

If motion of ideal fluid starts from an irrotational one at time
, then
it
rem
ains irrotational

for all
t
.

Corollary 14.3

Vortex lines in ideal fluid are convected with the motion of the fluid.

Theorem 15.1

Postulate 8 is equivalent to the condition that the stress tensor is symmetric.

Theorem 15.2

In the absence of body forces or boundary conditions dependent on
the direction
of velocity, ideal fluid motion is reversible.

Chapter 3

Theorem 17.1

If

is

1.

symmetric

2.

linear in

3.

independent of
v
,
,
, and higher derivatives of
v

4.

relation between

and

is isotropic (invariant under rotation)

then

where

Chapter 5

Bjerknes

Theorem

Taylor
-
Proudman Theorem

Chapter 6

Theorem 33.1

In continuous motion of inviscid gas, entropy is convected with the fluid in its
motion.

Corollary 33.1

If a body of inviscid gas in c
ontinuous motion has uniform entropy at
, its
entropy will remain uniform at all times.

This kind of motion is called homentropic.

Lemma 33.1

For steady homentropic flow of perfect gas in ducts:

If
,

then

for all
x
.

Here

(
a

is the speed of sound)

A

= area of cross section.