128

THE INERTIA-COEFFICIENTS OF AN ELLIPSOID MOVING IN FLUID.

By HORACE LAMB, F.R,S.

Reports and Memoranda, No. 623. October, 1918.

When a body moves t h rough a fr ic t ionless fluid i ts iner t ia is a p p a r e n t l y modif ied in var ious ways. I n the case of rect i- l inear mo t ion the effect is equiva len t to increasing the mass of the body b y k t imes the mass of the fluid which i t displaces, where k is a cer ta in cons tan t depending on the di rect ion of mot ion re la t ive to the body. For a sphere k = ½ ; for a cy l inder moving broads ide on k = 1.

Fo r a p ro la te ell ipsoid of axes a, a, c, moving end-on, the coefficient is

_ Y k l 2 - - "~

2 ( 1 - - e ~ ) ( 1 1 1-~ - e ) where y = ea 2- log l ~ e - - e ,

where e is the eccent r ic i ty of the longi tudinal section, viz. :

4[ e : 1 - - C~ .

I f we pu t e = sin ¢, a = c cos ¢, we h a v e

( ' ] ) Y __ 2sin ac°s 2¢5 log t a n • ~- ~ ~ - - sin

A formula which is convenient, when hyperbol ic tables are a t hand is got b y p u t t i n g e ----- t a n h u, c = a cosh u. I t is

2 y ---- s ~ 2 u (u coth u - - 1).

I n mak ing the following tab le the values of u were selected so as to give values of the ra t io (c/a) of length to b read th , which should be as near ly equal to whole number s as possible, w i thou t in te rpola t ion .

c/a. k. c/a. ki.

1 1-50 2-00 2.51 2.99 3.99

0.5 4.99 0.305 6.01 0-209 6.97 0.156 8.01 0.122 9.02 0.082 9.97

oo

0 . 0 5 9 0 . 0 4 5 0 . 0 3 6 0 - 0 2 9 0 . 0 2 4 0 . 0 2 1

0

REPORT Noaz3.

7f/E /NERTI,4 -COEFFICIENT Of'AN O_LIPSOID •IOVING IN FLUID~

/-0

0 " 9

~7

#'6 /

O'.5 / ° ° \ /

~3 _i! 0"/ 7 -

..~...~ f I - ' - - " - -

J Y /

!& I K

6 7 8 9 /o

L , ~ - ~ . , . ~ m ~ . ~ ® ~ . , ~ , ~ . ~ . . . . . . . i ..................... i¸..¸.--..,~.,~.e~mm~ .................... JlllJl~m~

129

For a prola te ellipsoid moving broadside-on, the coefficient is

k2 = 2 - - ~

where 1 1 - - e ~ l + e

---- e 2 2 e ~ l o g 1 - - e"

With the same meanings of ¢ and u as before

e - - s i n 2 ¢ 1 s -~¢ log t an ~ 7 : + ~ ¢

or 1 1 2U

= ~ [ s i n h 2 u l"

T h i s g i v e s the following resul ts :

via. ~.

1 1.50 2.00 2.51 2.99 3"99

]¢. C ]q.

4.99 0.621 6.01 0.702 6.97 0.763 8.01 0.803 9.02 0.860 9.97

0.895 0.918 0.933 0.945 0.954 0.960 1

There is a corresponding correct ion to the momen t of iner t ia for rotation abou t a t ransverse diameter . I f k' he the ra t io of the appa ren t increase of the momen t to the m o m en t of iner t ia of the displaced fluid, the formula is

e4 (~ _ ~,)

(2 - e2 ) ( 2 e2 _ ( 2 - e2) (~ - y ) }

where g, y have the same meanings as before. The following ~able is calculated from this :

c/a. k'.

1 1 '50 2.00 2.51 2.99 3.99

0 4.99 0.094. 6.01 0.240 6.97 0.367 8.01 0.465 9.02 0.608 9.97

0.701 0.764 0.805 0.840 0.865 0.883 1