An introduction to the general equations of fluid dynamics

..

TECHNISCHEHOGESCHOO' VLIEGTUIG1l0UWK U ~'DE Lma a lstra at 10 - DELFT

"

J

.1

AN INTRODUCTION

Ta

THE

GENERAL EQUATIONS OF FLUID DYNAMICS

BY

G. N. PATTERSON

10-

10

I

SUMMARY TEC HNISCHE HOGESCHOO VLIEGTUI GBOUWKU, D KlIDaa Jltraat 10 - DftFT , IJ

0

J

~

.

1

This review has been written for the research worker in the general field of fluid dynamics. In teaching this subject. the

!

usual procedure is to increase gradually its .complexity by pestulating a number of simplifying-assumptions and then progressively removing them as more diff-icultproblems are considered. On the other hand. the research student requires the general form of the equatfons 50 that he can carefully assess the effect of neglecting varióus te rms in order to make ahe problem mathematically tractable.

The relevant portions of vector algebra and vector calculus used in developing the general equations of motion are first r eviewed, Attention is also given to generalized coordinates so that the research student can make a suitable choice of coordinates for a particular prob lern , General forms are finally derived for the equation of motfon,

CONTENTS SUMMARY

I. VECTOR ALGEBRA

1. Definitions

2. Addition of Vectors

3. Scalar Product of Two Vectors 40 Vector Product of Two Vectors 5. Triple Scalar Product

1 1 2 3 3

6. Triple Vector Product 4

7. Products Involving Four Vectors - Resolution of a Vector 5

,

8. Linear Vector Function of a Vector 6

9. Vector Equation of a Straight Line and a Plane 7 10. Vector Equation of a Central Surface of the Second Dagr-ee 8

n,

THE VECTOR OPERATOR

"V"

11. Notation

12. Curl of a Vector

13. Divergence of a Vector

14. Gradient of a Scalar Function

15. The Operator

\l

and lts Vector Properties

16. The Operation of

'1

on

V

x

a.

V.

a.

and

\7

~ 17. Some Properties of the Scalar Operator (

a

.\1)

18. Operations on a Pr-oduct by the Operator

19. Same Formulae Derived by the Application of

\l

to Products of Tw o Vectors

20. Special Formulae Used in Fluid Dynamics 21. Stokes Theorem 9 9 10 10 12 13 13 14 15 17 19

CONTENTS (Concluded)

Page

22. Gauss's Theorem

23. Green' s Theorem

nr.

ORTHOGONAL COORDINATE SYSTEMS 24. Types of Orthogonal Coordinates

25. Generalized Orthogonal Coordinates

26. Generalized Coordinate Expressions Invo lvirig

9

21 23 24 25 28 . ",

2 7. Generalized Coordinate Forms for Some Expressions 34 Used in Gas Dynamics

28. Cartesian Coordinates 36

29. Spherical Polar Coordinates 38

30. Cylindrical Co-ordinates 40

IV.: BASIC EQUATIONS OF FLUID DYNAMICS

31. Taylor's Theorem in Vector Form

32. Differentiation Following the Motion

33. The Equation of Continuity

34. Deformation of the Fluid Element

35. Viscous Forces and Fluid Pressure

41

42

45

46

51

36. Rate of Change of Linear Momentum 55

37. Equations of Motion 56

38. Application of the law of conservation of energy to the 59

fluid element

39. Other Forms for the Dissipation Function 63

40. Equations Arising from the Properties of the Fluid 64

•

41. The Basic Equations for an Ideal Gas

42. Integration of Special Forms of the Basic Equations

43. Dynamic Similarity

66

67 68

(1

I." VECTOR ALGEBRA 1. Definitions

The basic quantities required for a consideration of vector algebra are:

Scalar quantities (or simply scalars) - pure numbers and physical quantities whic h do not re q u i r e direction in space for their complete speci-fication. Examples: volume , density, mas s, energy.

Ve ctor quantity (or simply vector ) - a quantity for which both mag-nitude and direction must be specified e. g, velocity, linear momentum, force, angular velocity, angula r momentum..

Localize d vector - a ve ct o r wh i c h is considered as localized in a

line e. g. line of action of the forc e wh e n calculating a moment of the force. :Free vectors - completely specified by their magnitude and direct -ion and which may therefore be drawn in convenient positions e. g. use of a

polygon. of force s to determi ne ihe magn it ude and direction of the resultanti

r-res pe c tive of the actual pos i tion s of the line s of action of the forces in space ,

Unit vector - a vec t o r wh ose magnitude is unity (denoted by i. j.

k in the Ca r t es i an sy ste m .)

Vecto r field - a field wiith ea c h point of whichther-e is associated a magnitude and direction, e.g. field of fluid velocity .

Ze r o ve cto r - one havi ng zer o magnitude and direction (0).

Length - the ma gnitu d e of a vector is indicated by its length, 2. Addition of Vectors

Two vectors are added by the parallelogram law, Thus in figure 1.

if the vector

a

is re pre s ented in magnitude and direction by OA and

simi-la rly the vector

b

by OB. then

a

+

b is defined as the vector represented by the re s u ltant OC where OACB is the completed parallelogram.

...~ /~".

a.

r---i~----...,c

B

a.

o

r FIG URE 1. :

(2)

It is evident from..the above defintttonthat (a) the commutative law and

(b) the associative law

are valid for the addition of ve ctor-s ,

a

+b=b+ä

ä

+

(b

+ ê) = (ä +

b)

+

ë

( 1)

Special forms of addition are illustrated by figure 2.

Using the parallelogram law it can be seen that the follow-ing re lations hold:

m(nä) = n(mä) = nma (m + n)ä = mä + nä m(ä +

b)

= mä +mb FIGURE 2

d

'O

-/.)(

6

o-a

.

.

(2) . ".

It is also possible to express a vector in tèrm s of a number of components. Thus by the r-ules-of·v e ct or·-ad di tion :

-

-

_

.

..,

r=a+b+ •••• r

3. Scalar Product of Two Vectors

(3)

The scalar product of two vectors a, b having magnitudes a, b is defined as

o

.~ _M

ä.

b

= ab cos .g (4)

where g is the angle between the tW0 vector-a, The scalar product has the following characteristics, (a) the order of the two vectors is irre levant i. e •

--

_{b.a = ba cos (-g) = ab cos g '" a. b.}

..

--

₍₅₎

(b) When

a,

bare perpendicular, cos g = 0 and hence

ä.

b.

=

o.

(c) The scalar product is negative when g is an obtuse angl.e ,

(d) If

ä

is a unit vector, then

ä.

b

= b cos ~ while if

b

is a unit vector also, then

ä.

b

= cos g.. When

ä

and

b

are unit vectors at right angles,

a.

b

= 1.

(3)

(e) Sc a l a r ·product s are distributive i. e.

ä

.

(1)

+

ë)

=

ä.

ij.

+

·

ä.

ë

An example.of a scalar pr od u c t is the product

F.

v

whi c h is the r-a te at whic h.t h e .force F.is doing wo r k on a pa r ticle moving with ve lo cit y v ,

4. Vector Product of Tw o Vectors

..

.

axb

-FIG URE 4

The ve c tor produ ct of two vecto rs

ä,

b having magnit udes a, band an

angle 9 betwe en them is de fine d'a s .

ä

x

b

= a b sin 9

p

(6)

&'

wh ere p is a unit

v

ecto~perpendicular

to·both

a

and b,whose sense is such

that rotati on from

ä

to b would move a right -hand ed sc rew in the direction

of

p

o

The vecto r product has the followi ng cha r-acte r-Istt c sr

(a) It is not commutative i. e ,

b

x

a·

=

ba sin (-9 ) p

= -

a b si n 9p

=

-äxb (7)

eb) When

ä

,

b

ar e parallel. 9 = 0 or

1i

and ä .x

b

..

0 Cc) Ve ctor multiplication is dtstr-ib u ttv e i.e ,

_

_{a x (}

..

_b

₊

-

_{c) ::}

-

_{a x}

- - -

_b

₊

_{a x c (8)}

5. Tr ipl e Sc a lar Pr o duc t

Let

a,

E,

ë

be thre e ve cto rs

having magn itudes a, b, c, The n th e co m bination

ä.

(

b

x ë) is called

their triple scalar pr o d u ct. It can be repre sented by the volume of a

parallel epipe d as sh own in figure 5.

b

Thus the area of th e fac e ha ving

b

and

ë

for edges is b c sin 9. i. e,;

ij x

ë

is a vec t o r normal to this face

having a ma gnitud e bc sin 9 and a FIGURE 5

direc tion in dicate d by the ooit ve Ctor

p and

a.

b c sin

9p

=

[b c sin Q)(a co s

fJ

)

=

area of a face x perpendicular width

and

(4)

The properties of-s ca l a r a nd veetor-s products-a p p l y i. e.

ä.

(b

x ë) =

{ij

x ê),

a

a.

(b

x ë) =

-s,

(ë x

b)

(9)

.(10)

Note that the volume can be evaluated in three ways, area of base X perpendicular

height , area of side face X perpendicular width, andar-ea of end X perpendicular

le ngth i. e.

s.

(b

x ë)

= b.

(ë x ä)

=

ë.

(a

x

b)

( 11)

Thus we have the cyclic rule: the triple scalar product changes its's i gn, only

whe n an alteration in the cycli c order is made. Reversal.of the symbols X and

produces no change i.e. ,

.

'à.(b

x ë) = ë.(ä xb) ::: (a xb).ë (12) 6. Triple Vector Product

The combination

ä

x (b

x

ë) is called the triple-v e ctor- product of the ve etor-s

ä,

b,

ë.

As wr-itten, it is the vector·product of the two vectors ä.>

and ij x

ë.

It will now be shown that the triple vector product satisfies the follow ing important relation

ä

x (b x ë) =

b(ë.ä

)

-

ë(b.ä) ( 13)

The vector

b

x ë is perpendicular to the plane containing vectors b, c, But the vector

ä

x (b x ë) is perpendicular to the vector b x

c.

It must therefore

lie in the plane containingij and

c.

i.e. ä x (b x

c)

is coplanar with band

c, Thus we can resalve

ä

x (ij x c) in terms of ij and

&

and write .

ä x

(b

x ê] = r

b -

s

ë

whe re r-, s are scalars.

To evaluate r, s we form the scalar product of

ä

and

ä

x

(b

x ë)

i.e, sinc e they are per-pendicular-,

o :::r(b.ä) - s(ë.ä) (14) from whic h r

ë.a

s = =

b.ä

i.e. r = À(ë.ä.) and s =À(b.ä) ( 15)

FIGURE 6

-

_cl

(5)

-Consi de r a"vector d coplanar with

b a nd

ë

and perpendicular to

ë

(fi gur e 6);-th en ë.

d.

= 0

and from equations (14 and 15)

ä

x (b x C)

=

À (ë . ä)b - À (b.1.)ë ( 16)

,,"

Formi ng a scalar pr o d uc t of ä x

(

ij

x ë with

d

,

we ha ve

a

,[

ä "x

(

b

x

ëU

=

À.

(

ë.

ä

.

l

b .

cl

=

ä

~

[ (b x"ë) x

dJ

usi ng equa tion (il) and r-egar-di ng

b

x ë as a si ngle ve c t o r . Now

(S

x ë)x

d

is cop anar wit h

15

,

ë

and pe rpend i cu lar to

d

and therefore must be a vector

"di rected along

ë

.

Th u s, referring to Fi gur e 6,

(

b

x'Ç)"xd has the magnitude bc d si n Q

=

bd cos (0/2. -9) c and hence "

(

b

x ë)x

cr

=

I

-

cos

(

~

~

-9))"cp =

(

b

.

d)ë

whe re

p

is the unit vector di re cted along

ë

.

"

Then

i.e.

)..

(

b .

d

)(

ë.

a

)

=

"

(

b

ed

Hä

.

ë)

).. ="1 whi c h proves equa tion (13)

Fr om the proper ties of th e vector product

-

_{a x (}

- -

_{b x c) = -}

-

_{a x (}

.. -

_{c x b]}

₌

0

₍

- __

_{c x b)x a ( 17)}

we obta i n th e centric rule: the tr iple ve ct or pro du ct change s its s ign only

with a chang e of the centre ve cto r , the bra cket co ntaining th e same two vec t o r s. Note the id entity

ä

x

(

ij

x ë) +

b

x (ë x ä +

ë

x (ä x

b

)

::

0

which can be verified with the help of equation (13 ).

7. Product s Involvi ng :Four Vecto rs - Re s o l ution of a Vector

_

....

_-

.

_

...

--Let a, b, c, d be fourvectors , then (a x b). (c x d} is a triple scalar

pr-edu c t of (ä x

b

)

,

ë

and

cl

or

s,

b and

ë

x

a.

We ~ve "

(a x

b

)

.

(ë x (1) =(ë x

d

l

.

(ä x

b

)

=

b.

(ë x d)x

ä

=

b

x (ë x

d

).

ä

·

=

{(

b

.

d

)ë

~

(

b .

ë

)

d

}

.

a

[(ë

x

d

).ä]

b

+

[(a

x

b).

~

G

"

(ä

x

d).cJ b

+«ä

x

b).dJ

ë'

[ (c

x

a).

dj b

+ (

(ä x

b).

'dj

~

(6) In par-ticul.ar-, if ë =

ä,

cl

=

b

then

(ä x b)2

=

1;2 ä"2 -

(b.

13)2

i. e, a2b2 sin2 Q

~

a2b2 - a2b2 _cos2 Q

For the same vectors

ä,

b

,

s,

d

we have also

(a

x h) x (ë x d)

= [

(ä x

b) .

ëi

J

c -

[(ä

x

b) •

cJ

cl

=

(ëxd) x (bx ä)

=

[(ë

x

d)

.aJ

b -

[(ex

d).b]

ä

From this we deduce the identity

"

[

(äx

b).ë]

cl - L(ë

"Xd).'b]

ä

- (J

x

h

).:ë]

ä

+

_ [

(b

x

ë)

.r:"d]

ä

+

(19) (20) (21) (22)

This re lati on shows how a vector

cl

may be related in a linear combination wit h any thre e given non - c o p lanar veetor-s

ä.

b,

ë.

8. Linear Vector Function of a Vector

Let

ä,

b,

ë,

;1,

b1,

~1 be given vectors; then

f(r

)

=

ä

(ä.lr)

+

b(bJ;

r

)

+

ë(2r)

(23)

is called a homogeneous lin e ar function of the vector

'F,

e , g. equation (22) ab o v e , Wemay regard the symbe.l f( ) as a type of operator which converts

r

int o another vector (f(r).

"L in e a r homogeneous functions have the following :pr o pe r t i e s : (a) If

F,s

are arbitrary ve ctór-s , then

f(r

+

5)

=

f(r)

+

f(s) (24)

a relati on wh i c h can be used to determine whether a given function is

homo-gene ous. In particular, if h is an arbitrary scalar, f(hF)

=

hf(r).

(b) The soalar- products

r.

f(s) and s~ f(r) are in general

differ-ent Inva l u e but if ,"

F.f(s)

=

s.f(r)

then the line a r vector function is self-conjugate.

(25)

(7)

(c) A mèr-e general form for f(F) is not obtained by including,t e r m s such as kF,

r.

(éi x h),

F

x ë 'a s can be seen by resolution along three fixed vectors or by use of the triple vector product.

p

o

Vector Equation of a Straight Line and a Plane

bt

'

9.

.

'

FIGURE 7.

Let ä be a vector denoting the position of'a n arbitr-ary point A·re la tiv e the origin O. Also 'ij is a vector parallel to the straight 1ine AP and hence ht is the position vector for P with res pect to A where t is a scalar quantity.

Then the position vector for

.p

re la tive to 0 is given as

r = ä +

bt

(26)

which is the vector equation for the given straight Ii rie ,

FIGURE 8.

Now cönsider the case where AP lies in a plane having a normal vector n. Then since Ïi is perpendicular to F -

ä,

.

'

(27)

(8)

Iü , Vector Equation of a Central Surface of the Second Degree

EUIPSolb - A ï'/P\c:AL.

cFN11(A\.. uc!~,:lA~])~----_

.FIGURE 9.

We shall now consider a group of surfaces called centra1 surfaces of.

the seco nd degreej ofcentral quadrics. These surfaces include the spher-e,

elltps oi.d, conej and the hyperboloids of one or tW9-sheets. This type of

sur-face is characterized by the fact that every straight Iine meets it in two points

only (r-eal, coincident or imaginary).

The equation of the surface (see figure 9) is

r.

f(r)

=

c (28)

wher e f(r) is a s e lf-conjugate, homogeneous , linear vector function and c .

is a scalar constant. We wi s h to know now where the straight line given by equation (2 6 ) intersects the surface. By substitution, we get, at the point of inters e c tio n where

r

·

is common to line and surface,

(ä + bt).f(á

+

bt)

=

c Si n ce f(r) is linear and homogeneous

(ä + bt) •[ f(ä )

+

f(ht)}

=

c

or (ä

+

b

t) •tf(ä) +t

f(iJ)}

= c

M

ultiplying out

ä

.

f(ä)

+

t

ä.

f(b)

+

tb.f(ä)

+

t 2

b.

f(b)

=

c

Usin g th e self-conjugate property

ä.

f(b)

=

b.

f(ä), the above equation becomes

'

.

,

-t 2

b'

.

f(h )

+

2t

a.

f(b)

+

ä.

f(a) = c (29) ··1'. ( •

which is aquadratic equation in t which gives two values of t

correspond-In g to two intersections. Hence equatîon (28) is the expression for a

sur-face of the second degree. Als o, if ä

=

0, the values of t are equal and

opposite and every chord through 0 is bisected at 0 and hence the surface

(9)

Consider now the 'c a s e for which the straight line defined

bY

equation

(26) is tangential to the surface at the extremity of the vector

r

=

a

(figure 9)

i. e. equation (29) has equal roots and must yield t

=

0 since the straight line touches the surface at

r

=

ä. According to equation (28), ä.f(ä) = c and hence . fr-om equation (29) ti.feE)

=

0 or since the function is self- conjugate, b."f(á) = O.

Substituting

'E

=

r

t- ä from equation"(2 6 ), then

(r - ~) f(a) = 0 (30)

•

which is the equation of the tangent plane to the surface defined by equation (28)

at the extremity of the vector 'f =

a.

Comparing this with equation (27) we see that the normal position at F

=

ä is in the direction of f(á) i.e , in genez-at f(r) is in the direction of the normal at the extremity of F, the position vector for the central quadr-Ie

r.

f(F)

=

c ,

n.

THE VECTOR OPERATOR

'V"

Il. Notation

In developing the properties of the vector operator fQ (pronounced nabla) the following general notation is used,

A a point in a fluid,

S a closed surface surrounding A in the fluid, V the volume of fluid within the closed surface S,

P a point on the surface S.

&8

an element of area of'the surface S with centre at P,

9,

the fluid velocity at P,

~ a unit vector in the directicn of the ontward dr'awn normal to the surface S at Point P,

tl'

element of volume

Other symbols wiU be defined.as required.

,

12. Curl of a Vector

The surface Scan be divided into infinitesimal areas and we can form the following vector sum per unit volume

~

z

&s

-n

x'i

where ~ ')(~ Is a vector directed tangentially to the surface at P. In the limit, as

bS ...

o,

(10)

..Lf:vlxï

J S

v

Jes>

The surface S is now allowed to shrink while always enclosing the point "

A 50 that the volume V tends to zero. The curl ofthe vector

ï

at the point

A is therefore defined as the vector

( 1-)

It is"assumed on the ground of physical continuity that this limit wiU involve only continuous and differentiable functions .

13. Divergence of a Ve ctor ,

The scalar product

~.

Ci,

bS

measures the rate at which a homo-geneous, incompressible fluid flows out across the element ~S The

ra te of outflow, or flux, across S is

-1

;:;:,-gJs

Ccs)

The dave r-g e n e e of the vector

ï

at the point A is therefore defined as the scalar

--

ds.

(2) _•

It is to be noted that the definitions of curl and div. while given here for flui d ve loc ity, can be applied to any vector.

14. Gradient of a Scalar Function

If f i s a scalar function having a value at every point in space , then

we may sum the product .:;;:

f

Ss

over the surface S"and in the limit

Th e 'gr a dt e rrt of the function

<:p

is therefore defined as

(3)

•

The meaning of this definition can be made clear in"t h e foUowing

wa y . Let V be the volume of a cylinder of length ~)'\ , having end sections of infinitesimal area over which

f

is constant i. e , th~y form part of two

(11)

.

'

T

_~V\

1

... ... ~_~~f:..C~ ,yt

~

bb

FIGURE 10

surfaces over which

f

=

constant, The values of

t:f

at the ends are

'P

and

~+ ~

d"" .

The contrtbuttons tb grad

c{)

made by the circular andflat

surfaces of the cylinder are r espectfvély:

(a) Zero from the curved surface due to symmetry, (b)

pn,

(~+ ~ ~'f\

)

&SJ

bY\' S

from the top s e ction, (c)-[

Y\

efJ

&

SJ

~'(\Its

from the lower end s ection, Therefore

rOod.

~

=

fY\

iIco

(~+ ~

-

~)

&5

(4)

Thus we see that gr~z

cf

is a vector- perpendicu!:r to the surface

{J

=

constant, having a magnitude ~ taken along the normal Y\ .

FIGURE 11

Consider now two adjacent points, A on the sur:faS1

'f

=

constant and B on the surface ~

+

b

~

=

constant (figure 11). Let AB

=

J

~

.

'T h e n (since ~ is a unit vector) .

to the first order of infinitesimp.ls, I, e , we may write

(12)

(6)

15... The Operator

'V

and lts Vector Properties

The importance of the operator

'7

(called nabla) is that it effect'ively r-educes the above three definitions to a single comprehensive.one , Let F be any unspecified vector or scalar function of pos ition in space, Then we make'

the following definition

~

1

VF : "...

0

V

(~),;;.

F

JS.

(7)

wh e r e the form of operation,of

\1

or ~ with F must be specified. Now it

will be seen that the right hand side of equation (7) can be changed to the right hand side of equations (1). (2) and (3) by substituting xä, •

ä

and

ti

for the function F in turn. (Note that, for generality. ~ is replacedby

ä

here). Thus ~UJJ ~

-

7'lt

ët

..

~.;"..

I

Yi)tQ:

ds

,

c.)

,

-

v..

o

~ 0-

•

'r,a

-

-

~_V-i'O

s;

. : ; \ I

;:cls.

r

Ml

_~

~t()

--

_v~o

~

S

(,)

~~

ds.

(8) (9) (10)

all of wh i ch apply to the point A in space ,

lt will be seen from the above'defin i t i o ns that

\l

is both a distribu-tive and a commutative operator, the latter with respect to constant scalar quantiti e s i. e.

=

vF

+

Ç7q

(11)

-_

.

c

\1

F

.

(12)

Equations (8), (9) and (10) show that

V)(

Q. and

V~

are vectors

while V~

'[.

is a s calar, Inother words

V

may be said to have the proper-ties of a symbolic vector. By itself,

'V

has no absolute vector character-istics but when used in combination y\ith other symbols it is governed by the ord i nary vector laws , Thus 7'(VX~) is a scalar product of two "vector-s ",

~....

I

, ₍₁₃₎

V

and

7)(

~

,

which·are ·perpendicular·and

v'(\7)(a:)

mustthere

-fore be zero • . Note however that '7 is an operator and this must also he apparent in the combihation.Thus

(7)(

8:) •

V

has no meaning in itself. 16. The Operation of

V

on

~

.ä:

>

'V.

tr \

'\l

~

The properties of

V

as an operator may now be extended by applying it to the forms

7xa,

7'

.

a and

'Q'fJ

making fultuse of the laws of scalar and vector. multiplication. The following expr-ess ions are thus obtain-ed:

( 14) (15)

( v

.

and

9'

are paraHe 1 and

~

is

scalar) (13)

(b) curl (curl

~

)

=

\l'X.

(V~

"a:.

)

.

=

triple vector product

-

V(V'~).

-

(V-V)Ö:

·

= grad (div ä ) -

72.~

(a) curl (grad

f)

=

V'K

7{J

·

= 0

•

where "2." 1..5 called the Laplacian operator•. This mayalso be written

I /

7

·2.

(i"" ::

V

(V'.

~)

-

7)(

(V~

a)

(c) div (grad

f.

)

= " .

v~

•

v'1.f

(d) div (curlä) =

v'

(V)(it)

'= 0 .

(16)

(17)

17. Some Properties of the Scalar Operator

(ä.7

)

It is convenientat this stage to investigate the properties of an impor-tant operator which is obtained by a combination of the vector

ä

and the op-erator V . Two applications of this operator are considered here.

FIGURE 12

(a) Operation on a scalar function

t :

we have

(14)

.. I

(20) us ing equations.(4) and (10) and.figure 12. _Note that

~

denotes a different-ia ti on-of t f i n the direction of"Y\. perpendicular to the surface

tf

C.:x'i

~l}-)

=

co nst, If we take an element of the vector a such that

d

0..

c..~

S

.=

ei

"fl-c

~

tV)

4

=

0-

~

...

then

direction of ä)

= a x {rate of change of

f

in the (21)

(b) Operation on a vector quality, b: The above result may be used to

determine the significance of

(lr.

7)!

for

b

can be

res~ved

along three arbitrary non - coplanar vectors (see section 7). Equation (20) applies to each of these components and, therefore

(ä.'7

)

b

=

ä

(ra t e of change of

b

in the direction ä) (22)

It will be seen that

(

a.

V ) is a scalar operator and will the r efor-e act the same as the ordinary differential operator usually indicated by D.

Th us the application of (ä.'7 ) to scalar and vector products gives respectively.

(ä.

'7

)(

b.

ë) =

t

,

[

(

a.

V

)

ëJ

+

ë

.

[(à

.V

)

b]

(

ä.7)(b

x

ë) =

[Cä.

'7

)

b]

x

e

+

b

x(ä.

v

)

e]

(23)

(24)

18. Operations on a Product by the Operator

Let us consider twosymbols having the values F, G at the point A inside the su rface S and values F1,G1 at a point on the surface S (see section 11). We shall now make use of the following identity

F1G 1 ::(F

+

(F L F il [ G

+

(G 1-G)]

:: FG

+

F(G1-G)

+

(F 1- F)G

+

(F 1- F )( G L G ) (25)

wh e r e the order of the te rms is pres erved thr o ughout.

Mu ltiplying by the unit vecto r

V\.

and inte gratin g over the surface S then

=fY\

~c;ds+f~F(a'_q)ds + f~(F''-F)4r:JS

.

+J~

(F'-F)(Q'-q)d5

.

(26)

We shall now allo w the su rfac e S to shrink to Infirrite s irnal size.

Then FLF an d GLG beoome infinite s im ally small and the product is a

(15) 1 f· ."-

.=

0

I (27)

=-0

-.A l s o FI Gare constante in this process since they apply to the point A.

Hence

Therefore

J

-t\

F' Cl

J

d

s

e

f;

F

(C/-

c,)

d&

+

f-n-(F'-F)

Cl

dS,

(29)

or using equation.(28) and dividirig.by.V

~ .r~

F'C;

'J

s ..

V

S'~

FC'

cis

+

&-

{:Yi

F'C;

cis

(30)

In the limit as V~O, according to the definition of

'V

(see equation (7», then

'·

7 (Fq) :

f\;/

[cF)C;l

+

7[F(Ci)]

(3~

)

where the bracket ( ) im plie s that

V

does not operate.on the bracketted symbo l,

A comparison of equation (31) with the correspdnding expression far the differentiation ope.ratord,e.

D(FG}

=

F(DG) + (DF.)G ( 32)

shows that this property, cansidered in conjunction with thegradient pr-oper-ty

in section 14. gives

\1

the characte r-istics.of a generalized differential oper

'-ator-,

19" Some Formulae De r ived by the Appltcafion of-

ti'

.

to Pr-oducts of Two Vectors

(33)

(34)

.

.

The follawing formulae are derived by the application of

'V

to scalar

and vector products keeping in mind (a ) the vector prcpe

r

ty

of

V

(b) the operator proper-ty

GiJ

!

\J

(c) the arrangement must be such that "never occurs la st, Wè first summarize the formulae far s calar and vector products i. e •

p ·

(fl<;r) "'."

(~~

ï)

""-

ï'

W)<:;:.)

( 16)

Th e latte r equation mayalso be written

p(

f

» ) ...

f

x

(~x

:r)

T

(ï'~).r

(35)

using equation (33) /

Th ese relations along with characteristics (a), (b), (c) above give us the

follow-ing formulae: (a)

V·CO:)l

t)

=

ç. [

'6:)(

(bJ

1

+

9'.

[ro )(.

"b]

=

(~)

,

('Q'j.

~) -

(äJ •

C

\J

X

b)

(37) (36) (38) (from equation (31» (b)

Therefore, dropping the bracket

". (axh)

':=

1:

I

(Vy. -;.) -

Q'

(vx

b).

7

X

(~)(Ç)

:\7

x[;->t(b)]-t-

~)([(i)

)l-çJ

(see equation (31)

::[(D

I~Ö:

-

L~)[7~1-ra).~b+{ä)~.t](USing

equation (34)

Therefore, dropping the bracket ( ),

\7x(o:)ll) "

(l~, \7)~

-(0:,7)

b

-l

_(V,

~

+

~

(\7,"b)

(c)

7

(~I

b)

·

~v

KaJ

.b]

ot

v[

i.

,c\)]

or

V

~

16) :

ëi)(

(vx

t)

t(~I'V)~

+

~x~;;]4·~ti"'(frOm

equation (35»

(cl) Equations (37) and (38) may be rearranged as follows:

(i..

9Y[

~

-

71l

~

1)

+(b

IV)~

-

I

(t]-~)

+

a

Cv-

't)

(~.

v)

t

~

V

6r,1)

-(ho

v)a -

h

(7)/1) -

t

)<

('7I<;r)

(39)

(40)

(41)

or

In pa rticular, if

ä

=

b, then, by substitution in equation (39)

.1.~

I

9)

~

:.

V' (;.- •

ër) -

'<'0:

X

(\])L"i:)

(O:I'V~

:

±\7tta.

-6:)(

(7~a)

(42)

. (17)

(43)

now

-(e) We shall now consider the case of a vector-

.

1

.

which is constant with respect to the operator

.

'V

•

Equation (3~) gives

(~ \~)

t

:r

·-VX

.(~)(

b)

+(Ï

tV)~

-

t(71~)+~

(VI

r:)

..,. ï

" 0 and

cr.

v)~

" 0·by equation (22). Therefore

(~1~1 ~-7x (rx1),.~

(V."h)

(44) (45) (46) Also there fo re

Similarly, from equation (40) .

(.if

,'V)1 "

'7

(~.1)

-

'ijx

(?xt) -

r)(

(Vi(

~)

-

(b

I

~

i'

= '?'

(Yf '

t) -

1 )(

(V)i

t)

Therefore

l~

.v)

~

::"V

(11-bJ

.

-t-

.

(7x

b)

x

~

(f) ~e~t we have the operation of

'7

on the product of a vector and a scalar\~o..t'i. e.

~,~tf).

Q;(V,

7;)

+~,

'

(V'~)

\7X

(ët~)

==

'P)(

(~ ,~+

"'7

,x

(;r

~)

:::

[

\7

x

La)

J

cr

--I- [\7X

.

ëtJ

({)J

\7

X

(<3:4') ::

-

'Q:

~

\7

cf

.

-+

4'(V

X

tt)

(47) -(48) and

(g) Finally we shall consider the operation of

V

en the product of two. scalars )

(~f)

.

Then .

"l

(

toP,,)

=

cP

rrtj;

-+r

V

cf

,,"2.(tfr)

~

'V.rK

f~)

:>

Vf'

Vr

4-

(J""

'1t.f

+-

'?

if-

vtf

+

tf

V-

T7f

Hence

'l7Wf)

=

r

'y>

'cP

+

~

(r,tf)-

(il~)

.".t/

'1

1

tf

using equations (4 5) and (47)

.

'

20. Special Forrnulae Used in Fluid Dynamics

Certain special forms of the equatrons in section 19 above are used in the mathematical the ory of fluid dynamics. These are listed below. As re-gards notation le t us wr ite Ä=~'l)

g

=-V){ï

and

f

(representing any one of

hï,;-,

T ) is a scalar.q ua .Iit y ,

(19)

(g) Using first equation (4 5) and then equation (47) with

f-=-

/ f j

~'(~î)

=:

v,(gf)

·

ftv'f)

+ï

,Vtf

=-jJ4(V/f)+ï'

~va

-+..617/)

so that

v- (;.

A'i

~

=-/ '

A{'7-g

)-t/"

'i

.VA

+-

Ai

I

~

(58)

Equatio n (4 7) also gives

vç...

Al

'")<i711

+6~

where it is to be note d tha t

\16

and ~ are vectors. 21. Stokes Theorem

(59)

SI

pu.

,

(o-t.4

~C\). f\~

~

~

b~ ~

.

...-:;;=

----,-~--'. '

-:

~

bOlA"

J4.Y~

FIGURE 13

I C.lA~uQ..,

Conside r an eleme nt of area

~~

having the unit

vector~

direct-ed along the normal to elS I its di rection being related to the direction of'

circulation along C by th e right hand screw r-ule , Then Stokes theorem sta-tes that

S;n ·

('(\I.'ï)

cl

s

=

i

ï ·

s

.Is

t.s)

_

Ce)

S

--

~~...;..

...

.,..-~

-(Jsys

_.,

p

o

FIGURE 14 (60)

(-2.0)

wh e r e

5

is a unit vector drawn tangentially to· Cat P (see figure sense shown, Note that by the addition of vector-a;'appr-oximately

,i=

~ (d~)

S -

()=

-+

&r)

~O

.

14} in the

.i.~.

,

S-J s

~

.

~

=-

d:;:

.

Figure 13 shows tha t the circulation around C is equal tothe s urn of the circulations in the interstices. It is sufficient therefore to prove the the-«, or-em for a typical infinitesimal me sh formed by the rietwork, for éxa m pl e , .

a parallelogram. (The choice.of this shape does not affect the gen~~a:lity of the re s ult) . Let

AB

=

a. .A5

=

b

.

Now since

'7~1;

'

,

may be taken as .

constant over an infinitesimal mesh and

(62) then

--

(63)

P pi

~

1\'\~t'Y\1~~tW\~

l

~Q.S~.

FIGURE 15

Now take

p~::.

&:y::.

and complete the construction shown in

figufe 15. If

q

is the field (e . g, )velOCity) vector at P, then the value of this vector a't

~

I

is

f

+

ft",

'V

i

byequation (22) above , The contr-ib utiora.of

PP>

I and

~ ~t

to the circulation are ther-efor-e

approx-imat ely

f'

b;=

+[ii+(b,V)i].(-è:r):= - (b

.V>f,

b~

and henc e the total contribution from sides AB and CD is

Similarly for the sides

Be

and DA the contr-ibution.. is

-(2 1) The refo r e

=-

t·

(~

.V)~

-:

;;:(bl)

î

•

[

b·

(~

.

ti)

-

~

.

(b

~

vIi

:. (17

)(

(b

)l

Q

n'

i

:

(~xl) (~r1)

.(64)

(s ee se c tion 6) Compa r ing eq uation s (63 and (64) it will be seen that Stoke s relation (60) has been proven for the infinitesimal mesh and hence for t_. he Z

_,

whole ne tw er-k.

The following relations may be deduced from Stokes theorern,

(a) Using the pr ope rties of the triple scalar product (see section 5),

~ )C'V)I~

.=

~

I(V)l

~J

so th a t

~l'il)(

V

).

i

d

:;

"=

J./i ·

i :

S

(65 )

(b) Let us now writ e

i"':-

t~

wh e r e b is a constant vector and

f

a

scalar function. Then 4 ;

[fs

C~ )lv) 1c1s J-b

-

[JcefSJ

S] -t

and si nce b is arbitrary

(66 )

and

(c) A third va r iati on of th e form of Stokes theorem may be obt a ined by

sub stituting ii)t

b

::

ïï

wh ere ij'

=

constant vector. Then by the pr opertïes

of the tr iple.scalar pro duct )

(,:;'

X"V) .

(à)L6

)

..,

[(~~

V

)l~

J- I"

s

.

(

~)l

I

)

=-

(S

>t

4

) ·

r

-Therefore, if b is arbitrary (67) 22. Gauss 's Theor em

In th is se ction we shall use the notation given in section 11. Gauss 's th eorem states that if F is a scalar or vector function of position, then

(22)

wh ere it is .t o be noted here that

"n.

lies in the direction of the inward drawn norm al to

d..

s .

The volume

V

cao be divided up into infinitesimal elements of volume

(

\1

figure 16). For an elementary volume. equation (7) above will have the ap proxi m a t e form

CV

V

b

~

:.. -

S~"'d~

F

ds

.

FIGURE 16

wher e the integral is taken over the surface of the elementary volume. Then,

su

mmingr

ê~;)d:;

=

'€.:..

~

(v

ç:)

f,1'

=

-~S~

Vi

FJs.

V

c!'4a()

.

It is to be noted now that two neighboring elements of volume have a common boundary and the normals (fa cin g inward) have opposite s igns , Hence only the contr-ibuttons of those surfaces not forming a common boundary remain i. e.

_~

n

~F

eLs

Jr'l

whi c h proves equation (68).

--

P

Y\Fds

J

_S

Various substitutions for F will yield the following deductions from Gauss'5 theorem:

(b)

V F ::

"l

'f :

(

c) V~ =-~Vf:.

(69)

(71)

(a)

ti'

F=

_{- n -}

_-

_y-o..

Iv

é'Y •

;r-

)cI-y

== -

S

s

11 •

cr cl

s

J

_V

('1)(;:) d'r' ':

-~ ~o:

Js

(70)

1v

vcfJ,

e _

Ss:;t:fds

SI'

v..

tlcfd"r

==

~

7"1

J7' -=fv(p·p)

-I

Jt

~~ (72)

=-

fl~

.

r;7

)cfd>

=-

~ ~

d

s

(75)

(77)

(78)

. (23)

(dj

V'

~

,,(

v~~

);. -::

J:

6r

~

y)

ëi:ï/,

~~

11'

l

a-clt

~

{("fr.9

)0.

-

els

(73)

(e)

'v

I=~'Y-

(.të{J..

.À

V·

(lI"4VJr-

l/efV-

Q

'+

en

v~)

eh"

_{(74)_}

V

=

-Ie

~ (~_~)ds

.

~

-(f) Equation (74) ca n be ex tende d if we con s i de r a vector b which is

resol-ved along thr e e non- c ap lanar unit ve c t o r-a, the magnitudes of the components being

-bl_{l b 2 a}nd b₃₀ The n equation (74) applies to each component and hence for b we

have

tv

[r('17~o:-)+~ I~r

Jd-r

:-~~

t;

(~-D:)dS

23~ Gr e e n' s Theor em

(76) Since the left-hand side of this equation is not altered by intercha ngi ng ,{) and

lIJ

the n

r

T

S~Vtf~

'1

rfJ1'

:::

-1

rft7

l

_ePd"r'

_-

trt~ds

Equation s (76) and (7 7) ar e the ma t hema ttcal statement of Green's theorem. These

. xpressions hold only when bath

cl

and r a r e single-valued. If circulation exists

the abov e stateme nt of the th e orem require s modifi cation. In th e present for-m ,

equations (76)1 (77) apply to pot e ntial fields . In par-ttcular , if ~-e

r

(24)

lIIORTHOGONAL COORDINATE SYSTEMS 24: Types of Orthogonal Coordinates

The types of orthogonal coordinate systems

are:-.

~

-~

/1

~PI

FIGURE 17

(a) Cartesian coordinates - the position of the point P is determined by the intersection of three mutually perpendicular planes x . const, , y '"' CQ$t.,

z = const,

FIGURE 18

(b).Spherical coordinates - the position -of the point is defined by the

inters e ction of:

,.

a sphere a plane

(mea s u r e d from x-y plane) and a cone

r :IC const;

Cl) = const,

9 = cons

t,

measured from Ote.

(c) Cylindrical coordinates - the position of the point is fixed by the

(25)

FIGURE 19

It will be seenthat the position of the point in all cases is determined

by the inte r s e c t i on of three surfaces.

25. Ge ne ra lized Orthogonal Coordi nates

Let

oe

= con st ••

f>

= const,• }- = const, be three surfaces intersecting

ort h o gonally. The orthog onal coordina t e s in the Cartesion system are given

by

(1 )

Let us conside r two adjacent points P; Q defined by the planes

oe.

= const ••

~

=

const .•

r

=

con st , and Cl. ~tll.

=

const, • ~+ ~~

=

const•• }+

b6""

= const res pectively and let

bS

re pre sen t the distance PQ. Then in carte

-sion coordinate s

(2)

and fr-om equation (1) above

~:x..=::>~~~

:)C~

5

~ ~.x~

~à-~)

-::.

~

()/.~Oi.

_4-

i~~~

_--\c

'i~~;t

(3)

&~

;

)0l~0l.

~

)

~ ~ ~

-+"o

_t

~d"

Substitution in (2) gives

~

s,"- :

(Kd-

~~

4-

)(~ ~ ~ ~)(

6

.

~ ~

')

l.

-+

(10l

~d- ~"i ~~~

..j,

~ ~

6

Ö'

'j

1-+ ("'ot>{

~C>t ~ 'à-~

&

~

'+

'0-

<f

SJ')

L-(26)

Multiplying out. we have .

.

S ,

~

''S.'1.

...

:c;i

~O('L

-+

:>c~bf' +-x~~~'1.

+

~:)e".,)~~ol.\~ ~ 'l.'C$.X'~bO( ~~ +':l..x~

,X~

~ ~

of- r-1lI'L

~

Ol\.

+4-

~ ~ ~L

+?i;

&

\~

+)

~~ ~~olá

(!> T ::L

~ol

'1k

&ot

~'t

-+-

.21~

"1--__

~~ ~

y.

~

,.,';. bet"\.

-f. ")

~ ~~L

-t )

~ ~~

'I.. +.:L

)ot)~

boi &f-> ....

"4.)01

~'t~

ó

t

-t-:l

~Ç>)- ~Ó(J> ~~

~ (~1.~~'I.-+xt)bO('-

+

(~4- i~4 ~;-) ~L

-+

(r~ ~~~~)~)~~'l­

+:l

(~)C~

+

ibl

'3

('>

-+:

"cr

0(

~

r--)

~ &~

+:l.

(':XbL

')C'r +

'toL

i~

+)()I.

is

~

')

b.0I.

~'a

oT 2.

(0~ ~ö-

-T

'i~

'i

6' -\-

)-~ )-~) ~~lç

(4)·

We shall now show that the last three terms on the right handside of equation (4) are zero. Cons ider- a point P lo ca t e d at the intersection ofthree orthogonal surfaces for which ex..

=

cl' ~

=

c2.

Cr

=

c3. (figure 20).

We require to know the condition for the perpendicularity of the lines of inter-section of the three surfaces , Figure 21 shows two infinitesimal elements of,

.

P

FIGURE .2 1

length PR'and PQ. The application of the cosine law of trigonometry tothe elementary triangle PQR gives

(2 7)

Now we have in Cortes ion coord ina te s

~î..

=

hSI~

=- h?t','L +

b~:

+-

ç

\:-PRl.. :::

<;'s~ ~

bx;

+

~~~ +~}~

and with th e he lp of figure 22 \.. RQÎ.

-=-

(b~(

, -'-:~:>c'\..)").+(h~,-b~1.)

4-~

TI

R

,J

~

FIGURE Th e r efor e co s Q =

=

PG:l"\.. ~

p

Rl -

R

CQ'-a,

PQ. PR

'bX , tb~"l. +

è'a,

bi~ ~ ~'h S~ b&l ÇS"l. (6)

Th us the condition for pe rpendic u larity is

è:>(,b~'l.. t~OI ~'ä\...~ ~~

I

è

21 '\.

=-0

₍₇₎

.Ifthe planes passing thr ough P are ort h o gonal then the curves of

inte rsection are mut u ally perpéndicu la r at P. Consider the two lin e s

com m on to the surface

r

= cg, For the first Line, being in the planes

~ = cl .

r

=

c3.

then~:Ol~f 0 while for the second line ~~ .= O, J.(r = o,

Thu s equations (3) giv e . . '

~x,

... 'X.(?>

&~

\ è.::>t'\... "-

~ ~Cl(l.

S'il

~ ~~ b~,

~'i1. ~isCll ~C)("1.

(28)

Substitution in Equation (7) gives

(9)

Now·since

'bo<\.

I

b~,

are arbitrary, the bracket must be ze rov

Simil-arly, by considering the lines common to each of the planes ~

=

cl' ~

=

c2' it can be shown that the remaining two terms on the right hand side of equa-tion (4) are zero. Therefore equation (4) becomes

,

( 10)

where

(11)

26. Generalized Coordinate Expressions Invo lvirig

\7

Let

J. \..;;

I':;; he unit vectors in the directions of the lines of

inter-section of the planes ~

=

cl' ~

=

c2'

cr

=

c3 (figure 20). Let us now con-sider the geometrical figure formed by the intez-s ectmg lines of the surfaces corresponding to lll, ~ I ~ and ~+tlJ. I ~ + ~~I i"~ ~

r

(figure 23). To a first

order of approximation this figure may be regarded as a rectangular parallele-piped with edges

k,

bol , "-\.

~~I

k\

r

~ where hl' _{h 2, h3 are in general} func-tions of the coordinates

(2 9)

.Ac card ing..to.equatian 11(.20), .we hav e ..

)( rate of change of , ~ce

~d;

f

in th e directianof

..e.

(13)

Naw

Ix.:;;.

:.

'V\

,

.::m )(

-)j

=

-e.

J

~

x-l

-:-

~

)

Therefore equa t i o n (13 reduces ta .

Fina lly, subs t ituting fram (12)

(14)

_ J

"l

ef

=

.e

h

_,

₍₁₅₎

i.e, in te rms of gen e ralized caordina tes,

V

has the farm

I ~

+~-

_~~

-b

6-'

_(1~

It is im portant ta nate that the gene ra lized uni t ve eto r-s

1.~

)':;,..

are

funct..!?ns of th e coordinates. We require te find the ex pres sion s far ~~

X

)

""1

"

.e

etc. Applying the ope rat o r 7 to fX , then ~ ~:r~

.

Sin c e

byequatia n 11 (13),

'V

x:

Vol ::- 0 ) the n

(17)

Also

using equationII (36) .. By substitution from equation (18) above

/

-- /..., J

1.

Thus we have the three equations

( 19)

t~

(

~),l.·1

)

1;(

L,

L~)

-

-The above equations can now be used to determine

\7,

ä:"

.

and

Vx

~

·wh e r e

(20)

Then

\7. -;:

=

et,"'7-:;

+

..,i-..

V

1>...,

+

'l

~

V..

...);:j

or;:;;

f

'7Q

~

-+

q 1

"\7..

A

-4-l\' -

'7

~

a.

(31)

~

a-t

t

(~L

k\)

+

T,

~~

J..,

t)~

o~

0

+

ctt.

..L

(~

t. )

+..L

~

~,

tol

(~ d~

~

I

A..."

d

~

+

a~

'

.L

(L,

k

_{2 )}

+

t:

lG1.~

A./

t:

i

t.

~

d

d'"

)

J

r

[ q,

tJ/..,

/..~)

+

t..J?,

~

.,

;

4- ' l

à

(

l

I

J

I

Ida.\.,.

~ ~

\' , "

;-

~a,

1\.,

-rr:

+"'-~ ~

UI

L.J

+

1..

1

k

1

cl;;

J.

Ther-efore

As a particula r ca se of equation (2 1), put

G:~ V

tP

where , as befor e

tP

is a sca la r. Compar i ng equation s (15) and (20), it will be seen that

)

j

~

al.

:

-

T:

!ja.

)

(22)

'--- - - --

-_ We now obtain an ex pr-e s sion for

V

X

0: .

Using equation (20) above and equation II (4 6~.

. (32)

.

V"-f ;:

.

:

~

x

(-7

Q: , )

-r

'v

x

(~ ~"l.)

+

)?

x

(~ ~ '~J

=-

-.:i

')(\]

'~I

*

~,

Vx

~ -~

')t

~~'\..

+q\

~'>I ~

-

i1')(

~q~

+tt

~ \7~ ~

Subs~ituti

ng

frox:n equations (18) and also equatton (15) with al' a2' a3 in place

of.

f

Collecting terms

The refore

lIL')?,

[Á,-i

[~~l

L)-

tJ"

À,)1

+

~,;:;;;

f

t

(~/kl)

-

&~J'C~ l~)J

(33)

Equation (23) mayalso be written in the determinentalform

.

kiE

h,&;

k~::;

'

I

L

~

L

-

=

l,

'1.

L\

TI?>

~t

(24) V~C\.. _~Ol..

Ot,k,

~\.kl.

~~l~

.. The above results for

'1-

a

,~

-=

can be used in

eval~ti~g

the quantity "l:;71.a- If

V

replaces ~ in equation II (43) (also

b"'-

ët' ) then

(25)

8ubstitution from equations (21) and (23) will give the expression for-V 2

ä

in generalized orthogonal coordinates. It is evident that a longcumbersome expres sion will he obtained and this illu s tra t e s , as do other expressions above , the economy effected by the use of vector-s ,

. Two further forms used frequently in gas dynamics are

(~-V)~

and

(~ I~)

0:

•

Using equations (2 0) and (16) '.

l

Cï-~)~

.\1J

'=

'

[ (

q,"'C

7'

+

a,l.~ +'\1\

-.

-),(wlf/t:+

n

Q~

~,+~+,,~t\lq?

" ...

~

-~..

1\,

~~)J

Therefore

The evaluation of

(0:-'1

ft;

i.e.

can be made through the use of equation II (42)

I

::'7

(27)

(34)

27. Generalized Coordinate Forms for Some Expr-es stons Used in Gas Dyn-amics.

The following relations are given here since they are of importance in the theory-of gas dynamics. They are all special forms of the expressions deduced.in section 26 above:

(a) Dtvergence (b) Vorticity

i"

9It

t

~,,~f

l,l

f~(lrL\)-t}fv"j~)j

+k..

~

f

~

(Llk,)-

t

(4r~l)1

+i

1;

r

t.

(v~.)

-

trJt.~,)3_

(29) (c) Scalar functions,

4

-, JI

"14

=

.-'[

_I

~

(30) (d) Application of

V

to

1.:::--=

_I

~I r ' - see equation (18) above,

(31) (e) The Laplace operator applied to a s cala r and vector for

't:;/Lf

see equation (23) above , Also according to equation (25)

V1.

g "

{17-V)f

:=

~(17'1)_r?Y {~\lfJ

=-

\74

-

\?

y:

J

. \

. . (35).. . . . ." . . .

ft-~

·

..{

~

/~

~]-~J~($1q -t~

., .

{Stu1

~;

Therefore

(fr .Ap p lica tion of the operator

f·

V • Equation (26) gives directly

( _

_Z..

_{VI .,- ;:}

.l J)

"

J:i

_h,

~

jf

~

_1;

V-

~

d~ ~ ~

WO'

~

_Q

r

(33) From Equation (27)

CftV){

.

=fVi~-f)( (~~f)

_ ...l...

~-1.

- ;;

~

" - ~ Y

b b

.

S

(34) ·1 , Thus

.

(~'V)f ~

frf4f+-T..

~"\.~~J ~r(.I~H'Ir+ii~)X

. .

(-/1,

-+-~

f

'L...

~i~

Jl

I

[

'.-f.-

~

t

,;r!r

;;

à

-1.:J

=

z:

T,

J~

4-

T...

d

~

+1;

-ij -[

i'

/..<

j

e:

)Öl"

"SI

- i"V-!,

+-Tv-j,.

+.w,

~!,-J~f~J

.

..~~ :

.

::~ :; 'f;' :~.?~i~ -: -,:

(36) (35) 28. Cartesian Coordinates

T,

T

~

1 ·

t ,

1.

T:

.0 .,... - ~ ... -r ~ )(~=\, I ~)l,

=d .

In this syatem of coordinates we use the unit veetors

I,

J

I

k

p

àra-llel to the axes O~\ c~ I O~ • The unit vectors combine amongst

them-selves as follows:- ~

__

- -

-. l

-:"'"1. -:-2:

L\

.

T.

i

::

~

I

~

=-

k.

~

::.J I r

t -:..

R=-I

;

~~4.~

L>ll :-J'lCT)

T)(T

=

T>t1 :.

tllt :

0 j

Tv.1:\l)

The scalar and vector products take the forms

ä:.

b

,

=

(ï

o.)t

~

r

~

+

t

4~)

,(7

bit

+

~ 6r~

·1

b~)

~

q'W:

~

4- 0.

'

!

b1

+

a. \

b).

(36)

Also, the vector product become s

~

'IC

t

=.

(fc.~ ~1~~

-tiQà-)

><-0'

~lt

-+

~ b~~

t

t))

,.

T"(Q~~\~

"'\

~~

+1;(o.\b. -

~ ~~+~

p."

I.~ -o.~ ~~)

(37) In the following results the velocity vector

ï

_{~ ~}

Ll .j.

d

IJ""

+

I

ur

is used, but the results apply generally to any vector. Now since, for Célrrte"':·._.~­

sian coordinates

~ ~"\.

.=.

b

:Ir"\.

~ ~ ~l

4-

s

~

'l.

th e n hl

=

_{h 2}

=

h3

=

land

rX \

~l

X

~

x)

:,!-

I

~

• Then the results given in section 3 above becornè , ror Cartesian

dSordinates:-(37) 50 that (38) (39)

-"Y.',

=

'\l "

I

--

.

'V)(Î

=

\)7')(6

-:: 9·

R

<c

J

I _ "C"

17.><'

R. .::

0

₍₄₀₎

Then the operator

(ëf;v)

has the form

(0

,\7)

=

u.~ ~ \r-

'~

+

W-~

b ·

wc

u'5

~}-AlSG (41) (42) (44) (45)

(38)

50 that

(46)

(47)

29. Spherical Polar Coordinates

(49)

\

~:. -r'p~ ,.~w

i

à

=

.."..fl""",g

p~\,U)

Referring to figure 18 it will be seen that for spherical polar co-ordinates for which ~::

--r

R

=

è

\., -::.

IJ,)

.

,

) r

)

0 __ :::>-'("'""

ell")~

d.x :.

('d'~

_

~

-'"'

_"I"'p""e -

~

:::

0 Jo- è.. I u~ ,

~ ~ fl~~(r:fiJ,j ~ ~

=

.:r-~

~(.,,~

ï

~

--

--r..,~~~~

CoC

~

:

f;"'~~

lJJ

i

~

fT

;-(O?~

"';ko-Wi

~-"ÏP'w/[)(lf)W

pi, E quations (11) now gi ve ~} '..'

(39)

h~ ~

'IJ

I'L.v..'~ (/:)~

"'l

W

+

((f)l.

w

)

Therefore

~s'"t ~À/l~ol\.+t; à~l+"~ÓÖL

'

~y

"L

t-

~ ~e~

+

-c

1./-1

~

1

@.&«J'-'=

I

= ;

-(50) (5l)

Hence in spherical polar coordtnates

-à

_-'.a--

_'t-~

_J....

lr-7

::=

-R..

~

-4-"W'-

~

(52)

""""' '"e-

f'(~~é (see equa t i on (15))

V

.?<.

.e

::..

0

·

V x

~ -'----~

o.

\ "

:: .r

0-

C1SO$)

c:

(f.)~~)

(53) Vx~

-.y

:"" ~ ~

'"ÎI)~

@

-

-.0. ~

-

_~~

_~ (f r om equations (18» 'ÇJ I

-t

(2

-r

l>~

éJ

)

-1.. ::-

-1. •

S

.,....

"'( f>

Vv-= ,

I

.

€ (....-

""",qS)

I

. 'l'~::;;'"

-

_-(~

(54) TA"""

I

_{(0 )}

V·

V\

_{~ -r7~i.Y.~}

-

-

0

(40)

.(from equations (19)) ' .

'V.~ :

-.,.~;..~

ft(Qtv\..

e

)+~(a.."'A~~)

..

t/

~~-r-)]

₍₅₅₎

In the same way we may evaluate the remaining expressions in section 26 above,

30. Cylindrical Coordinates

)

For cylindrical coordinates (see figure 19) (>(, :::::. ::>t

f

_

.

...,

\-

-:=:. W ~.x

_-

O~( -.

₀

0:)(

-

·0

J:-::- .,(

-~( :=. (j-y-.- ") ~u.J

-,

\

'i

"=y- C't/)

liJ

~

d~

-

0-

~

_':

~~W

~

_'w

)

-è.)(. - ) ) lI..)

=

~ A.~ •

h.:

0

- ó"2r

'

w

è~

M

'0

z:

-c

IJ\.v..Lû

ï

_-.-P"'"

_-

_{dLû ::}

_-r

r Cl>

O.)c.

)

_{dv- -}

) Equations (11) give

.

h..

_{1 ':. ' )}

J... 1 :::

I

so that for cylindrical coordinates

~

"- : S.){

'L.

-+

~

1.

-+ .,...

1.

&

LU 1-In cylindrical coordinates, therefore

~

~-2-V

=

:e

~x-

'i-

~ ~

-t-

"Y'-

rów

(56) • (57) (58) (59) (60)

The remaining expressions in section 26 above may be eviiluated in terms of cylindrical coordinates by substituting the appropriate values tör hj , h 2, h3•

•.,..1

~.

...

(41)

IV.' BASIC EQUATIONS OF FLUID DYNAMICS 31. Taylor's Theorem in Vector Form

The various forms of Taylor's theorem, depending on the number of variables, are as follows:

)

~

0(.. )

.

~l..

ó

"'L{6-)

, ' f (r + & .:

~(~) ~-e.

.

~

-+

2T

cl.)l&"

+ -

-

-

(1)

tc

:.+

-t)

'0+

m)

=

~ (~,V

+

r

~

~~

~(J2

-I- - -

à

f~~

'ä)l

+;;

ç

lP

;)1f("})

t-2~'l>-.

à

\.

ft~

+»\1.

~J

?

• (

~

)(

c\>t

'6

ab

1.

j

-l-

-

- _

(2) Also

where f on the right hand side denotes f(x, y, z}, This equation mayalso be

writte n

l