The inertia coefficients of an ellipsoid moving in fluid

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Lab.

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A.1.a. Dynamical similarity, etc., 33

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V APR. t97

ARCHEF

AJWISORY COMMITT EE FOR

Ali RONAUTICS.

REPORTS AND MEMORANDA,

No. 623.

OCTOBER, 1918.

L O D O Ni

PUIJT.ISUED 11V IllS MAJESTYS STATIONLRV oI'FICE.

1)1).

Price 2d. Net.

o

THE INERTIA-COEFFICIENTS OF AN

_ELLIPSOID

MOVING IN FLUID.By HORACE LAMB, F.R.S.

Technisch Hoqeschool

To be purcbased through any Bookseller or direcUy (mm N.M. STATIONERY OFFICE at the following adtireses:

IPWAL Ifousa. KINOSWAV, LopoN, W.C.2, and 28. ABINGDON STRI:IT, LosmON, S.W.1_{37, Pr}

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21, FORTH SrRET, 1.bLNIRii

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7ÑE /NrRT/A

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Corrr/c/Lwr o ,w ri i /P5c/D

34O V/MG IN rLUID.

I-o 09 08 06

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THE INE WI'1A-COEF FICI ENTh O F AN EL Li S( )ID

MOVING IN FLUID.

By H0nACE LAMB, F.R.S.

Reports and Memoranda, No. 63. October, 1918.

When a body moves through a frictionless fluid its inertia is apparently modified in various ways. In the case of recti-linear motion the effect is equivalent to increasing the mass of

the body by k times the mass of the fluid which it displaces, where k is a certain constant depending on the direction of

motion relative to the body. For a s1)here k = ; for a cylinder

moving broadside on k = I.

For a prolate ellipsoid of axes a, a, c, moving end-on, the

coefficient is where

k -

y 2(1 e2)

I'

1

1 +e

y=

og-where e is the eccentricity of the longitudinal section, viz.:

A formula which is convenient when hyperbolic tables are at hand is got by putting e = tanh u, c = a cosh u. It is

2

- (ucothu 1).

sinh- u

In making the following table the values of u were selected so as to give values of the ratio (c Ja) of length tobreadth, which should be as nearly equal to whole numbers as possible, without interpolation. (B37%) Wt. &iO4. 1225. 11(19. Op. 32. ej o !a. k. e/a. k. i 05 4.99 O059 i 60 O-305 6-01 O-045 2-00 O-209 697 O-036 2-5! O-156 8-01 O-029 299 O-122 9-02 O-024 3.99 00 9.97 0-021

If we put e = sin , a c cos , wehave

2cos2Í

tan11 1 1 .

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cfa. k. i 2-00 2-51 2-99 3.99 i 0-621 0-702 0-763 0-803 O-860 4

For a prolate ellipsoid moving broadside-on. the coefficient is

"2 =

₂

-where

I

1e2

l+e

=

e2 log1

e

With the sanie meanings of andu as before

= r-- --

1 {

i - -

cos2 log tan sin

or

i r

H

tanh2 u sinh 2 74

This gives the following results

cc

j

Prmt.4 uuder the authority ot His MAEsTv's STArioIItv O,rlcK By Sir Joseph Custon & Sons, Limited, 9, Egstclieip, London, E.C. 3.

There is a corresponding correction to the moment of inertia

for rotation about a transverse diameter. If L' be the ratio of the apparent increase of the moment to the moment of inertia

of the displaced fluid, the formula is

e4 ( - y)

(2 - e2){ 2 e - (2 - e2) ( _{- -) }}

where , y have the same meanings as before. The following table is calculated from this

i c/a. ¡a. i O 4.99 O-701 1-50 O-094 45-0 I O-764 2-00 O-240 6-97 O-805 2-51 O-367 8-QL O-840 2-99 O-465 9-02 0-865 3.99 O-608 9.97 O-883 cc i k,. r /r7. 4.99 6-01 6-97 8-01 9-02 9.97 0-895 0-918 0-933 O-945 0-954 O-960