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Spiral motions of viscous fluids

Material Information

Title:
Spiral motions of viscous fluids
Series Title:
NACA TM
Physical Description:
44 p. : ; 27 cm.
Language:
English
Creator:
Hamel, Georg, 1877-1954
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

Subjects

Subjects / Keywords:
Aerodynamics   ( lcsh )
Viscosity solutions   ( lcsh )
Genre:
federal government publication   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
Exact solutions of the steady incompressible viscous flow equations are obtained. The streamlines corresponding to such solutions are in general logarithmic spirals. The more specific cases of purely concentric and purely radial flows are fully investigated. Corresponding to the radial flows are the physically important cases of flow in radially convergent channels and in divergent channels. A second method is used to investigate exact steady and unsteady two-dimensional motions in free spirals. Neighborhood solution to the radial flow are also discussed.
Funding:
Statement of Responsibility:
by Georg Hamel.
General Note:
"Report date January 1953."
General Note:
"Translation of "Spiralförmige bewegungen zäher flüssigkeiten." Jahresber. d. deutschen Math. Ver. 25, 1917."

Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003778985
oclc - 86087418
sobekcm - AA00006166_00001
System ID:
AA00006166:00001

Full Text
kAwA-T0 13 4

,'O 7' / ,

TECHNICAL MEMORANDUM 1342

SPIRAL MOTIONS OF VISCOUS FLUIDS*

By Georg Hamel

INTRODUCTION

The equations for the plane
volume are, after elimination of
stream function by which the

v -=
x 3y

motion of viscous fluids of constant
the pressure and introduction of the
velocity components

v =-
y 6x

are expressed, reduced to the one equation

t + 6 a y 3 o A Ax
at ox 6y ay 6x

therein a
cific mass

indicates the ratio between viscosity coefficient and spe-
p, and A signifies the Laplace operator.

This equation is satisfied by all potential motions

L0 = 0

however, this fact is of little significance since viscous fluids adhere
to solid walls and, from well-known considerations of function theory,
there cannot exist a potential motion which would do so. Otherwise,
properly speaking, only Poiseuille's laminar motion is known as exact
solution of equation (I) and that solution does not even show the sig-
nificance of the quadratic terms because they identically disappear
there.

"Spiralf6rmige Bewegungen zaher Flissigkeiten." Jahresber. d.
deutschen Math. Ver. 25, 1917, pp. 34-60.

NACA TM 1342

Under these circumstances it seems perhaps useful to know a few
more exact solutions of equation (I) for which the quadratic terms do
not disappear; such solutions will be indicated below according to two
methods.

In both cases, one deals with motions in spiral-shaped streamlines
(which are observed frequently).

Third, we shall, in addition, investigate the neighborhood solu-

FIRST PART

We raise the question:

Are there solutions of equation (I) which are not potential motions
for which, however, the stream paths are the same as for a potential
motion whereas the velocity distribution is to be different?

We shall be able to indicate such solutions, in fact all of them:
the streamlines are logarithmic spirals (including concentric circles
and pure radial flow); for the velocity distribution, one arrives at an
elliptic functions. In the discussion, the influence of the quadratic
terms becomes manifest in a considerable difference between inflow and
outflow (see paragraphs 7, 8, and 9).

We require, therefore, solutions s of equation (I) for which

= f(,= )

and AQp = 0, but not A4 = 0. The latter condition excludes

f' '( ( ) = 0

We limit ourselves to steady motions = 0.
at

1. The calculation becomes clearer if first the auxiliary problem
has been solved:

NACA TM 1342

Transformation of equation (I) into isometric coordinates, that is,
such curvilinear coordinates c,X that

P + ix:' = w(x + iy) = w(z)

Let us thus assume

* = (Cp, x.)

&x =
6x 6y

If one denotes

aviation

2 )+ 2
692 6-/2

by A'*, there results first, with the abbre-

dw2
dz

AWi = a'

With the double integral extended over an arbitrary region, one has

6 A o A a A
J x 6y yx

fj (A') 5

JJ [_ oc 57/

.)dx dy = i dA dl

drP dX

a(Q A')
ox

Since, however,

xp 6x
ox oy

Sa ax
oy ox

dw =2
dzi

8x

ate a' )
a~

- -6Xdx dy

a5CPJx y

NACA TM 1342

there follows

A Y = Q2
by bx

3P 6ai

ca' O'ai'
6> 6c

+A(4 n n (6i
1nT

8 In Qa 64
6x P

2
dw
SIn = 2 R n 2R d- In dw = 2R dz2
2P p dz dw dz (dw\2
\dzJ

and

6 In q = 2 -1R In dw
6x 6x dz

S-2 3 J In dw = -2J d- In dW = -2J
8P dz dw dz

are valid. If one puts the analytic function of

2
dw

2 d a + bi

(dw\2
dz

one obtains

6 A 6t
6x 6y

6 A 3x
6y ax

- dw
dz

(C 6 6:

6x 6cp)JI

+ A'(a

6 A 6~
6x 5y

However,

d2
dz
(dw2
dz

- +
oX

NACA TM 1342

Finally, there results

A At = Q A'(Q A'') -= Q2' A'* + Q A'Q A'' + 2 Q A

= Q2A- A'* A-+ -- + 2 a p- a 'b)j
LA I ( 6x5\ ^

+ A'* A
ix dx/

In Q = In

dw 2
dz

is a harmonic function, thus

A' In Q = O

hence,

AL'Q = ( In Q 2 (+ In = 2 + b2
Q \ )p + x j

Thus one obtains as the result of the conversion
isometric coordinates C,X for steady motion

Am dx
0'p ax

- A'' 6* +A't a +b b -
ax YP ax W/]

of equation (I) to

= A' A'' +

A'+(a2 t b2)

+ 2( A, a
( fYP

-3 A' b)
X Y,

(II)

therein, a + bi is the analytic function

d2
dw
2 dz
(d/)2
\dz/

(w = v2 + iX, z = x + ly)

and A' denotes the operator

2
a3f

62
6C'-2

6 NACA TM 1342

2. We return to the question on page 2: must be a mere function
of 9>

4 = f(cP)

If derivatives, with respect to q, are denoted by primes, equation (II)
becomes

f"f'b = a f + f''(a2 + b2) + 2f'''a (III)

f may depend only on cP but must not depend on X.

This is certainly possible if a and b do not depend on X, thus,
since a + bi is an analytic function of CP + Xi, do not depend on 9
either, if a + bi is, therefore

d2w

a + bi = 2 dz C

dz

that is, constant. We shall see later (paragraph 3) that this is the
only possibility.

From a and b being constant, there follows

w =- 2 in (z o) + wo
a + bi

thus, after introduction of the polar coordinates

z z0 = redi

S= 2 2 (a In r + b)) + CP
a + b

NACA TM 1342

Thus, the streamlines q = const are identical with the logarithmic
spirals

a In r + bO = const

a = 0 signifies pure radial flow, b = 0 flow in concentric circles.
The velocity distribution, however, is given by equation (III): the

3l 2b 1
r 6- r 63 a2 + b2 r

the circular component

c P 2a f, r
Tr 3r a2 + b2 r
a tb r

consequently 2 f' I the magnitude of the
a2 + b r
fore, f' must disappear on solid walls.

velocity.

There-

Without restriction of the generality, one may presuppose left-
hand spirals so that r increases with 6, thus a and b have dif-
ferent signs; since, furthermore, -(4 + ix) is an analytical function
just as T + iK., and equation (III) is actually invariant with respect
to a simultaneous signI change of iP,a,b, one may presuppose

a 0

b 0

Therefore, positive velocity components 3 and -
r 6be r

f' >0

signify for

outflow,

in contrast for

f' <0

inflow.

iTranslator's note: The original says "time change," obviously a
misprint.

NACA TM 1342

Since 9' may be replaced by cT, one may in addition impose a con-
dition on the constants a and b.

3. We now want to conduct the proof that on the basis of our require-
ments a and b must be constant, that therefore the flows in loga-
rithmic spirals are the only ones the flow patterns of which correspond
to a potential motion without themselves being a potential motion.

If a and b were not constant, the analytical function a + bi
would produce a conformal transformation of the mP + iX-plane; by virtue
of equation (III) which with the abbreviations

A f--
f I

B f'
20

fIV
f ,

(f' = 0 is excluded) may also be written

a2 + b2 2A(C)a 2B(q)b + C(P) = 0

(III')

the circles (equation (III')) would correspond to the straight lines
( = const in this transformation.

These circles would therefore have to form an isometric curve
family.

However, if the family of curves

g(a,b,T) = 0

(III')

is to be an isometric one so that AP = 0, the function
the equation

g must satisfy

,gg.,2 23g (ga + ,b) pc a2 + 2) = 0

(IV)

a2 2
oa db /2

and this equation must either be identically satisfied, or be a conse-
quence of equation (III').

NACA TM 1342

One has

ga = 2(a A),

gb = 2(b B), Ag = 4, ga = -2A',

gqb = -2B'

gq = -2A'a 2B'b + C', g = -2A''a 2B''b + C"

ga2 gb2 = 4(a A)2 + 4(b B)2 = 4(A2 + B2 C)

Thus, equation (IV) is quadratic in a and b; an easy calculation shows
the result that the quadratic terms are automatically eliminated. There-
fore, the coefficients of the two terms must be zero whence follow three
conditions

0 A'' C' 2AA' 2BB' _B'' C' 2AA' 2BB'
A' 2C 9 B' 2 2
C A B2 C A B

C'' C' 2AA' 2BB'
C' C A2 B2

Hence, there follows further that A', B'
furthermore that

B'
C A2 B2

must be constant.

The final condition yields

C = alB + 0

, C', must be proportional and

A = y 1B 8

or with

l
2o

71
- --7
2o

fIV = af'f" + Bf''

f"' = yf'f'' + 8f"

and

NACA TM 1342

which, integrated, yields

f"l =1 af'2 + Of' + e
2

and

f"' = I yf.2 + 6f' +
2

Comparison of the two values for f''' results in

1 af'2 + Of' +
f"f = 2
yf' + b

which must be
son requires

identical with the preceding value of f'.

7 =0

a=O

The compari-

e = r7I

thus C =A2 = 2 constant and f" = 5f' + T.

The second condition

B B' const
A2 + B2 C B2

however would yield

S= const and this together with f' = 5f' +
f'2

f' = const

therewith, the proof has been produced.

4. We now turn to the determination of the velocity according to
differential equation (III) which may be integrated once and assumes,
after introduction of the quantity proportional to the velocity at unit
distance

u = f'(P)

0 = 62

NACA TM 1342

the form

u' + 2au' + u a2 + b2) b u2 + C = 0
2a

This equation is identical with a damped oscillation which takes place
under the influence of the potential

3 + a2 + b2)u2 + Cu
60 2

1. The streamlines are

concentric circles: b = 0.

Then

u -C- + e-am(A + BT)
a2

and, because of

S=- 2 In r
a

u = const + r2(A + B1 In r)

whereby, the velocity distribution

2 u
ar

is given. The exact solution of Conette's case is also contained therein:
the three constants here are determined from the two limiting values of
the velocity and from the fact that in case of a full turn around the
circular annulus, the pressure must revert to its initial value. An
easy calculation yields B1 = 0 and thus

v = r2 r12)

(More details on the determination of the pressure are seen in para-
graph 10.)

NACA TM 1342

2. The flow is purely radial: a = 0.

u" = b2u b u2 + C = 0
2o

u' = j- u 3 + 3obu2 + const u + const

= el u 2 u)e -u)

where the three e's are only subject to the one condition

el + e2 t e3 = 3ab

but otherwise are still at disposal.

Since, according to the remark on page 8 one relation between a,b
is still unused, it will be expedient to put

b = -2

so that one obtains, according to page 7

Cj = 5

Then the conditional equation for the e reads

el + e2 + e2 = -60

and one has

U' = el u)(e2 u)(e3 u)

NACA TM 1342

thus

u = -20 +P 0. O0) g2,;

where SOg2,g3 are the three integration constants. For the pressure
(see paragraph 10) there results the equation

m.p + 1 v2 = ff f 2 f 1 (1 f2 2ofT
6p\u 2 r ,2 r2 f' r2\2

its uniqueness is a priori ensured, thus does not determine here any of
the constants.

5. The condition

e + e + e = -60 (1)
1 2 j

requires at least one e to have a negatively real part, for instance

R(el) R(e2) ? R(e)

then

with the equality sign being
same real part.

Furthermore, since this

(a) for three real e's

-m

R(e3) -20

valid only when all three e's have the

part is real, there must apply

either

< u 5 e < -2o

e2 \$ u 5 el

14 NACA TM 1342

(b) for one real e

-om u e e

where, however, this e may be positive.

Furthermore, two possible types of flow must be distinguished:

1. Either there are no solid walls, thus a source or sink in an
unlimited fluid. Then u must be a periodic function of P, with a
period which is an integral part of 2n. u = -w is excluded, u = 0
need not occur. Therefore, this case can occur only for three real e's,
and

e2 5 u < el

must be valid.

2. Or there are two solid walls, for instance for b = 0
- = d1 (which may also be equal 2x); then at these walls u
u = 0.

and for
must be

(a) In case of three real e's there must be, additionally

e2 <

e >0

and either

e2 u < 0

0 < u < e
1

(b) In the case of one real e, this e

must be positive and

0 < u e

One remembers, furthermore, that according to page 7, paragraph 2,
u > O signifies outflow, u < 0 signifies inflow; so that one has
inflow in the case of 2(a)(a), and outflow in the case of 2(a)(0), and
2(b) above. For the case 1, both cases may occur.

NACA TM 1342

First Case: Free Flow

6. One must assume a = 0 for u = e2 and has therefore

e ) e2)(u e3)

Hence, there must

with n bein an
2

with n being an int

(e u)(u e2)(u

By the known substitution

u = e2 + (el e2) sin2

equation (2) becomes

2

e2 e3

d+

i + 2sin22

32 n
3a n

If one now introduces the mean velocity2

u = (e 1+ e2)

and the velocity fluctuation2

S= el e2

2At the distance

- e3)

V3a n

2 el e2
e2 e3

- 2

r = 1.

NACA TM 1342

there becomes because of equation (1)

e2 e = 6 +3u &>0
2 3 m 2

thus

60 + 3um 2 8

2 d2
1 t at
0 1 +X2sin2*

2
n

From this, one may draw several interesting conclusions.

One has

1+ X 2sin2 r

I
0

dJ +

\1 +X2sin2*

\Il i2(1 + cos 2*)
2j (

J
10 4

d1

1 + X2cos2

+

>2 4
0

d2

l+1

02
0

6a + 2u 2 5

\Fo

NACA T 1342

thus

2f o
It O0

1+ X2sin 2

where

0 < e 1

Thus the relation (2') between
of x2

Urmb,n,a

2 1
n 6o

6a + 3u (1 -)
m 2

1 n2
60 + 3um 1 25 = 60 -

with j being a proper fraction.

Since, furthermore

(2")

12 % ds >
O \1 + X2sin2

I2
0

X d*

I\f 1 X22

= sinh X-
2

thus becomes arbitrarily large with increasing X, one has lim E = 0,

thus,

liam = 1
U=mO

1
1 2
2

NACA TM 1342

From equation (2'') there follows

u > -2a )

which, with u = 1, gives as the minimum value

3

The mean inflow velocity is therefore considerably limited upward, the
more so, the easier movable the fluid.

However, this is the only restriction: If u
are selected so that

2
60 + 3um > 60 "
4

there exists, certainly, a pertaining 8.

For if I 8 increases from zero to the value
2

and the integer

6a + 3um,

1 12g
2

lies between zero and 60 + 3um (because for the second value 2
9 1 2
becomes infinite and, hence, n" = 1) so that certainly sometime 2 25
2
2
becomes equal to 6a + 3u 6a which is presupposed to be positive.
24

One sees, furthermore, that for a prescribed
fluctuation 8 and for a prescribed fluctuation
must increase to infinity with the mean velocity

period number n the
5 the period number n
um.

Second Case: Outflow Between Solid Walls

7. The cases 2(a)(p), and 2(b) may be summarized thus

du

\/(e u)(u2 + 2u +

u
o~~?

NACA TM 1342 19

e > 0 is the maximum velocity (at the distance r = 1); because of

2a = -e e = 60 + e

and p = e2e3 > 0, otherwise, however, arbitrary

u2 + 2au + 0

may for prescribed e assume all values from u2 + 2ou to o, so that

1 = 2 du
0 j(e u)(u2 + 2au + )

appears not at all restricted downward, but upward restricted by

e
1,max 21 _d
v 0 V(e u)u(u + e + 60)

since

e
dn

0 (e u)u

one has

3ma = 21n/ 3' (4)
l,max V2e(l e) + 120

where E signifies a positive proper fraction.

For the outflow, the width of the wall opening appears therefore
restricted, according to the preceding equation, by the maximum value e
of the velocity. For small velocity and large viscosity, the maximum
lies near n, otherwise, however, lower; with increasing e it drops
below all limits.

NACA TM 1342

If, therefore, an angle opening smaller than n is prescribed, it
permits an outflow only up to a certain maximum value. If a greater
outflow quantity is prescribed, the jet will, therefore, actually prob-
ably separate from the walls.

Also, there is, of course, for any prescribed angle 01 a flow
possible where partly inflow partly outflow occurs.

Third Case: Inflow Between Solid Walls

8. There remains the case 2(a)(a)

e u O0
2

all three roots e real,

e 3, e2 negative, el positive.

Here

l iP e2

-0

e2
c ^e0

du

u e)(u e)(el u)

du

- 2au )

where

2a =-(e + e3) = 60 + e

p = ee <
1 3

otherwise, however, arbitrary. Thus, the angle 01 may be made arbi-
trarily small for prescribed e2. On the other hand, however, it may
also be made arbitrarily large: one takes, for prescribed e2, the

- e2)(-u2

NACA TM 1342

negative value e3 sufficiently close to e2, as far as this is not made
impossible by el < 0. The sole relation between the e

el + e2 + e3 = -6a

however, results with el > 0 in

-e2 e > 60

If -e2 ? 3a, e3 may actually be assumed arbitrarily close to e2.

If the maximum inflow velocity is larger than 3a, any angle 61
is possible between the solid walls.

If, however, -e2 < 3c, say e3a, where E is a positive proper
fraction, only

-e3 = e + (6 3E)o = -e2

+ el + 6(1 c)o

is possible and

0
61 fe2

+ (6 3e)O +

e l (e u)

attains its highest value for el = 0

\j- r

1 ,max

(u + 3Eo)u + (6 -

3e)O](-u)

\ 3E(2 + rr 1 ( + TI)

I

u e2 u

NACA TM 1342

where I and 5 are positive proper fractions. Thus, the maximum of
lI is larger than n.

When the maximum inflow velocity is smaller than 3a, the angle
openings of the solid walls also may attain any magnitude up to n.

Flow in Spirals

9. Because of the damping 2au' (see paragraph 4, page 11), a
periodic solution, aside from u = const, is not possible.

A free motion in logarithmic spirals is always a potential motion.
In contrast, there exist other flows on logarithmic spirals between
solid walls.

In order
angle -, one
a manner that

to have, for r = const, the variable q agree with the
may furthermore prescribe for the constants a,b, in such
one obtains

2b -
2 b2
a + b

thus

b = -1 + \ a2

a must be a proper fraction, otherwise it remains arbitrary.

Equation III, once integrated, (see page 10) then reads

2
u' + 2au' + 02u + C = 0
4a

2 2.
where 0 = -2b = 2 T2 21 a < 4, but > a

NACA TM 1342

The velocity at unit distance is

2

\fa2 + b2

2 u
ia2
1+7 1 ia

u is, therefore, the velocity at the distance

r _2 _2

If one first omits the damping, one has exactly the same case one

\6o

is in front of the square root (see page 12). The relation for the e
remains the former one. Since P2 < 4, the angle opening is increased
by this influence 91.

The damping, however, takes effect in the same sense. Nevertheless,
the main result remains correct.

For outflow the admissible angle opening ia is restricted by the
maximum flow velocity in such a manner that it tends toward zero when
this velocity increases beyond all limits.

If one puts

u = ve-a

the above differential equation becomes

v" +(p2 a 2)v + ve + Cea =0
S' 4a

NACA TI 1342

If cp = 0 is assumed to be the location of the maximum v0 for v,
multiplication by 2v' and integration yields

v2 + (32 a)(v2 v02)

2 v
+ --

0

2 -amP i am
0

From the corresponding equation for u

u'2 + 4a u u' du +
u0

2 2 2) (u3 u03) + 2C(u uO) = 0

one can see that for equal u0 the u-curve will be the steeper, -l
therefore the smaller, the larger C. For value close to u0 this is
immediately clear from the differential equation, for in case of u' = 0,
u'' will be the smaller, the larger C, thus lu'l the larger; from
the preceding equation one may see, however, that for larger C

U
u,2 + 4a u
u"0

u u' du
Ju

has the higher value. Hence, follows directly for the
(u' > 0) that always Iu'I has the higher value when
value. For if u' would once reach for the initially

(C smaller) the value of the steeper curve,

ascending branch
C is the larger
flatter curve

u u' du would have to
u0

SuO
have for the former the smaller, thus u' du the higher value
u
which for u' > 0 immediately leads to contradiction since, up to then,
that is between u and uO, u' had been the smaller value. If, how-
ever, u' < 0, one has in case of a variation of the C by bC

u -u 4a 1 f u' du = 2 AC
UO -u u u

NACA TM 1342

or, since

A',2 = Alu'l(21u'l + Alu' I)

lu'l (21u'l + Alu') + UO Alu'l dn+2 AC
UO u u0 uJ

If there were at one point A u'I = 0, then at this point the first
term would, for fixed AC, decrease with decreasing u, that is, go over
from positive to negative values; the second term also would decrease

since the part of the integral I Au' du supervening with

decreasing u would be negative. This is impossible, however, since
the sum of both terms is supposed to be constant 2 AC.

Therewith, it has been generally proved that the angle opening al
decreases with increasing C (for fixed u0); since the maximum possible
9l is desired, the minimum admissible value for C may be assumed.

This value is determined from v'2 > 0

PO aP> LA2 2V 2 2\ 2 vYO 2 -am
2C e dv a 2v2- v2) v- ye dv
V V

whence, one may see that the minimum admissible value of C is zero or
negative3.

The above inequality must be valid for all v's between v0 and
zero and for the positive and negative m attained. One may write them

C -2 2) m )e 2 -a(2 + 9 O)
20 > -Q3 a) v0+ v)eaO v02 +o v )e + V
6a -o

According to page 22, 02 a2 > 0.

NACA TM 1342

where p0,cp2 are certain mean values. One must further note that, for
v = VO, I01 and IT2 I must be zero whereas they have maximum values
for v = 0. The severest restriction is due to the absolutely smaller
value of the right side, thus the minimum admissible C is given by

2C=_(2 a2 voe-aP O 12 -a(P 2'+c0')
20C p a2)ve 60 e

CPO'C 2' are positive and the maximum of c0 and 2 which occur for
v = 0.
Consequently, one has for the maximum possible -l

v'2 = (2 a2) v2 v2 e-acPO' T eacdv +

2a v 3 e

(P2 a2) 0v2 v) v 0 v)e- a '-P0

o2 3 v3 -aT 2 1 2 -a cP 2-af ( '- 10
2 3 3) 2 -v 1 v02o 0 v)e z Oj

Since CPQ' > cp and P2' > P2

v'2 >V v) v k a( + (v )e ]

NACA TM 1342

and since cP2 < 91

v > ( v)v 2 a2)+ (v + V)e-a

whence follows

23t
91 < 2

2 a2+ v0(1 + e)e-aB1

(Compare formula (4), page 19).

Hence follows that with increasing vo, thus also with increasing
uo, 1a must drop below all limits: a certain width of the spiral
permits only a limited outflow velocity.

SECOND PART

10. Since the only spiral motion, possible without walls, of the
type used so far, lead to be a potential motion, exact steady and non-
steady two-dimensional motions in free spirals will be investigated
according to another method.

In polar coordinates the differential equation (I) reads

SA 15A+ .\ = a A A
at r\ -r 6(P 6r)

where

lA 1 82
A = r + 2
r or or r2 cP2

28 NACA TM 1342

Obviously this equation permits solutions which are linear in I?

S= u + cpx

in order to make the velocity which has the components

v =
r r

and

v =-u
Cp 6r

- c
6r

unique and thus enable a free motion, C must
statement, the differential equation becomes

SAu
)t

be constant.

+X KAu = aA A
r or

with

u= 1 = ru
r 6r 6r

Here it is necessary also
in a motion about the singular
pressure becomes evident.

to investigate the pressure lest perhaps
point r = 0 a multivaluedness of the

Now one may write the equations of motion
the pressure in the following form

-v
2

atA -
= At d* + dx -
;by

without elimination of

xA -L

6x

or because of the invariance of the last term

3 aA -q
= A* d* + dr -
r 2P

By this

"(a A 6 *^
Ardr
rr

NACA TM 1342

Hence, follows

+ 1 v = At r -L a t = X Au ar 6 + r
bp 2 6/ cP r\ ot/ (r

By virtue of the differential equation for u the right side
stant; thus it must be zero to make the pressure in case of a
about r = 0 revert to its former value, so that one obtains

2
r Au r 2u Au = 0
6r a 6r 6t a

which by introduction of r 6u = v assumes the form
6r

- 1 +

62u
at 6r

is con-
revolution

a r _r
ar 6r

11. The solution independent of t is

2+2
u = clr

when '> -2, otherwise, when
a

+ c2 In r + c3
2 3

a= -
a

u = c1(in r)2 + c2 In r + c

30 NACA IM 1342

If one disregards the trivial case of potential motion, a spiral motion
the velocity of which disappears at infinity exists when

A + 1 < 0
a

that is, X < a, thus a sufficiently strong inflow takes place.

The spirals then have the form

+-+2
1P u = Clr

+ C2 In r + C
2 3

If 2 + 2 < 0, they approach at infinity the logarithmic spirals; near

the sink, in contrast, they converge considerably less pronouncedly
toward the sink point, and the vortex velocity is considerably higher
than in case of potential flow in logarithmic spirals.

12. If one uses the formulation

v = entXn(r)

one obtains from equation (V)

X 1 ++ X '- X = 0
1n r oan n a X =

thus, with the abbreviation X = 1 + 2
2o

n +=r (J r

NACA TM 1342

where the J are the Bessel functions

--r)

= const rX + n +
a 4 1(1 + x)

n2 4
a )

1
2'(l + X)(2 + x)

If X does not happen to be an integer, rXJ and rXJ_
regarded as independent solutions.

may be

13. Similarly to the case of the heat conduction equation there
exist also of equation (V) integrals which show for r = 0 and t = 0
an indeterminate point.

Since the differential equation (V) remains unchanged if v is
multiplied by an arbitrary factor, r by a similar factor, t by its
square, there must exist solutions of the form

v = ract Ow = rat w(z)

After substitution, one obtains for w the, differential equation

w" + w1 + 1 w 2 +

(VI)

= 0

When does this equation permit a solution of the form

w = epz

A simple calculation yields

P = -1

NACA !M 1342

and then either

a =0

or

a = 2x

3 = -x 1

One thus has two simple integrals of the required type

1 r

v = tX-le

and

v = r2kt-l-ke

1 r2
4a t

for X = 0, both are transformed into the known integral of the heat
conduction equation.

Let us continue the discussion of the differential equation (VI).

The singular point z = 0 is a determinate point. The determining

p2 + p(a ) + a2 2xa 0
4

and has the roots

P = p a =-
1 2 2 2

NACA TM 1342

so that generally there exist developments of the form

XaM
w = z 2l + c z + c2 z

aC
w = z 2(1

thus

v = r2kt
1

2 1

+c1, z c z2 +
+ c 'z + c2'Z2 + .

2 4
+ c r + c r +
14wt 2 2
(4ht)

and

B+ 2 4
v = t 2 + c, r + C, r +
2 1 4at 2 (42t)

with the power series continuously converging since
singular point of the differential equation.

z = 0 is the only

If one assumes p = 0, that is, if one desires solution of equa-
tion (V) of the form

ral I r2

an integration by definite integrals is possible.

The differential equation (VI) reads after introduction of the
roots p ,p
12

z2 d2 + z dl p + z + wpp = 0
dz2 dz\- 1l 2 12
dz

(VI')

34 NACA TM 1342

The connection with Gauss' equation for the hypergeometric function can
be easily recognized. If one makes the Euler transformation

W = e-3(1 ) y(s) ds

with the integral extended over a suitable closed path, one finds for y
a differential equation which may be satisfied for

n = -pl

by y = +p2
by y=s

and for

n = -p

-by y l+
by y = s

Therefore

w = e3fl -

v = e3(l
'' M1 -~(

I-P1 -1+p
Ss 2ds

S-P -1+p
)s Ids
z/

are integral of equation (VI'). The integrals are extended best over
a path which leads from R(s) = +o around the points s = 0 and a = z
back to R(s) = +0.

Since

fe-3(z )Pls-l+Pd

and

NACA TM 1342

is analytically regular in the neighborhood of z = 0, there is

wl = C1 e-31

2 = C2 e(

-P1 -l+p
- s)-P s -+p2ds

S -P2 s-l+pds

One can show that

v=oo

raC 1w 1- + C2 w 2
[ lklWV c tw

are the general solutions of equation (V) and likewise are represented
by definite integrals in closed form. I shall perhaps refer back to
this and to the connection with the representation and the development
in terms of Bessel functions elsewhere.

THIRD PART

14. We shall first look for steady neighborhood solutions to the
radial flow (pages 12 and 13) by putting

= = f(m) + p(I,r)

where p is assumed to be a small quantity, the square of which is
neglected.

We then obtain for f the former equation

f(CV) + 4f' + 2 f'f = 0

NACA TM 1342

with f' = u and integrating once

'' + 4u + 1u2 + C = 0
o

For p we obtain

SAp + u Ap 2u' 6P
at r Sr 4 kp
r

u" ap
a A Ap
r3 or

a I8 r
r r -r
r 6r 6r

Since the differential equation in
when r and p each are multiplied by
exist solutions of the form

an arbitrary factor, there must

p = r'w(')

One obtains for w the differential equation

wIV + w' 2X2 4 + 4 + Zu u) + 2u'w' +
a a a

w( k43 + 2X2 +22 u + u" = 0
o a

We are particularly interested in free flows and thus in
tions in ''.

periodic solu-

As concerns the uniqueness of the pressure (see paragraph 10,
page 28), one obtains by a simple calculation, the condition

2 02 w[u- ( 2) ]dCP = 0

where

+ 1 2
r2 acp2

(VII)

NACA TM 1342

On the other hand, there follows from the above differential equation
itself, by integration over the interval from 0 to 2x with assump-
tion of periodicity

.r2x
X2(A 2) w[u (A 2)o dc = 0
-'0

so that in general the uniqueness of the pressure follows from the peri-
odicity except for the case when X = 2.

For'free flow, u
is an integral w part
have the same period as
part of 2n.

itself is a periodic function of q'; the period
of 2A. However, it is not necessary that w
u; but this period must likewise be an integral

Since

u"' = -C 4u -1 u2
a

as well as

u'2 = -2Cu 4u2 2 u3 + D = e -u)u e2u e3)

may be rationally expressed by u, it will be useful to introduce u
instead of 'r as independent variable in equation (VII). Because of

w' = dw u
du

w' = d-w u'2 + d-- u
du2 du

wi" = d3w u'3 + 3
du3

IV = d4w u'4 + 6 d3w u'2u" + 3
du du3

d2w u'ui + dw u',
du2 du

d2w u,2 + 4 d2w u'u''' + dw uV
S2 2 du
du du

38 NACA TM 1342

and because of

urV = 4 u)u 2 u,2 u'u' = (-4 2 u'2
Sa \ a

all coefficients of the new equation are integral and rational in u;
indicating the degree, one writes them

R6 d + R5 d3w+ Rdw +R dw + R2w = 0 (VII')
du du du2

with

R 6= 4 = 2 (e u)2(u e2)2(u e )2
9a

From the form (VII) one can see that w possesses singularities
only where they occur for u, thus certainly not in the real part of 9
(which is of interest); equation (VII') shows that, as a function of u,
w becomes singular only at the branch points el,e2,e .

Since R5 = 6u'2u is divisible by (el u)(u e2)(u e3),
the points el,e2,e3 are determinate points, and since the degree of
the coefficients decreases steadily by 1 with the order of the deriva-
tives, u = m also is a determinate point; the differential equa-
tion (VII') belongs to the Fuchs class.

A well-known calculation yields as the four roots of the determining
equation for the points e the values

p = 0 p =1 p =1 p
1 2 3 2 4 2

Although, therefore, two root differences here are integral, no loga-
rithmics appear in the developments: For from the form (VII) there fol-
lows that at the points P for which u becomes = e, where, therefore,
u and u' are regular functions of P, w also must be such a func-
tion, whereas In(u e) does not possess this regular character.

NACA TM 1342

Therefore, the solutions of equation (VII') have at every point e the
form

u = Pl(u e) + /u eP2(u e)

other singularities do not exist in a finite domain.
For u = = there results the determining equation

(2p2 + 3) (2 + j + ) =0

which has the roots

1 =1 =-_2

which are independent of X, and the roots

} =- i W-i
41 4- 4

which are dependent on X.

Continuation
15. Solutions with the real period 2n (this period must be present
at least in case of free flow) will exist only for certain X. In
analogy with Hermite's method for Laine's differential equation, one can
proceed as follows:
If wlw 2W3,W4 are a fundamental system of equation (VII), the
w(cp + 2x) are expressed homogeneously linearly by the w

4
w (C + 2x) = Z ,w() (v = 1,2,3,4)
u=1

40 NACA TM 1342

There certainly exist periodic functions of the second kind, that
is, there exist solutions w for which

w(cP + 2) = aw(c)

This a is a root of the equation of the fourth degree

- a a12

13

a22 a a23

a32

a33

a43

a24

a a34

a44 a

If a periodic solution is to
obtains for X the equation

exist, a = 1 must be a root, and one

D(1,X) = 0

The characteristic exponents which were calculated suggest the
attempts of putting

w = u + const, w = \u- e

and w= ea u)( e

Elementary calculation yields the following particular solutions:

1. The trivial possibility w = u for k = 0

2. w = u for k = 2, that is

p = ar2u

I = f(cp) + a~r2f'(cp)

all

a21

a31

D(a; X) =

= 0

NACA TM 1342

where a must be small and therefore with the same approximation

= f(p + ar2)

so that the streamlines are approximately the spirals

Cp = 'T ar2

Remains the same elliptic function discussed before in the case of
f' remains the same elliptic function discussed before in the case of

radial flow. It is true that this flow
since this is precisely the exceptional
the condition for the uniqueness of the
satisfied for w = u.

now cannot exist as free flow,
case X = 2 (see page 37); and
pressure can certainly not be

3. w = u t 30, when X = 1 and C = 30 whence for el e2 < CJ3

4. w = u e2 for

X = -1 and e2 = 0. This solution has a

period twice that of u; likewise, w = el u for X = -1 and

5. w = e u)(u e3) or w
and e2 = 0 or el = 0. This solution

(u e2)(u e) when X = 1
too has a period twice that of u.

The large X may be easily calculated approximately from equa-
tion (VII). For such large X there is in first approximation

wIV + 2X2w,, + X4w = 0

that is, w = e1ki' (we restrict ourselves to the periodic solutions),

so that is the period. The large set-apart k-values are therefore

approximately integral.

Finally, one case may be calculated quite elementarily: the case
when u is constant, the basic flow therefore an all around uniformly
distributed flow.

e = 0.

42 NACA TM 1342

This case is also of significance for the more general one since
4
according to a well-known theorem by Cauchy and Boltzmann the period
of w in first approximation is obtained if the constant mean value is
inserted for the periodic u, under the presupposition that the larger
fluctuation el e2 be sufficiently small.

For constant u there follows from equation (VII), page 36

wV + W" (22 4 + 2 u) + 4 + 3 + 4X2 + 2 2 u)

= 0

thus with the formulation

v = euip

Sa a
4- 2(2X2 2K + 4 + 2- u) + x4 4X3 + 4X2 + 2 22- K u =0

This equation has four roots

P = X p2 = (A 2)2 2 u
a

so that all integral positive and negative K are possible (potential
motions) as well as all K which are calculated from

K = 2 + +
4 -

2 2
Fta)

with integral w.

For the case u = const, that is:

16. For the radial flow which is uniform all around, the unsteady
neighborhood solutions also can be given.

Boltzmann, Ges. Abh., Bd. 1, S. 43.

NACA TM 1342

The differential equation now reads (see page 36)

8Ap u6p=
+ o Ap
at r 6r

One may integrate it either by means of the formulation

Ap = e kt+ni w(r)

(n integral) and thus arrives at the differential equation

2
d2w
dr2

u
1 H
G dw
+ d
r dr

which may be solved
(compare page 32)

by Bessel functions, or by means of the formulation

Ap = enim r2)
aot

= eniQrmw(z)

whereby one obtains for w(z) the differential equation

z2'' + zw'(m + 1 -

For z = 0 this equation has
tion has the real roots

u- + z + w -2 =
20 4 h

a determinate point, the determining equa-

p mui +1 2 lu2
2 4a 2 4 2
aT

n+ w = 0
r2 ci

NACA 'M 1342

By introduction of the roots pl and p2 the differential equation
assumes the form

zw' + zw'l P+ z) + PlP2 =0

This is, however, exactly the differential equation (VI') of page 33 so
that everything said about it there is also valid here.

Translated by Mary L. Mahler
for Aeronautics

NACA-Langley 1-16-53 1000

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