On Certain Possible Cases of Steady Motion in a Viscous Fluid. [Continued]

On Certain Possible Cases of Steady Motion in a Viscous Fluid. [Continued]Author(s): Thomas CraigSource: American Journal of Mathematics, Vol. 3, No. 4 (Dec., 1880), pp. 289-293Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2369255 .

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Steady Motion in a Viscous Fluid. 289

(a4 + b4 + c4)

at each point of the ellipsoidal shell we can write

where de' is an element of area of the surface, and

r = (x-x)2 + (y y)2 + (Z- z)2.

If we denote byp1,P2,P3 the pressures on unit of area, and by dE the elernent

of area of an infinitely great sphere containing the fluid, the followinig equations are known to be satisfied:

ffpdE +f?ffpdE' O,

ffP2dE +?ff2dE' = ,

ffp3dE + ffp3d' =-0;

in our case these reduce to the first equation only since A2 -P3 0-

Making r 2 = X2 + y2 + z2, we have

1 _ 1 _ 1

/Vr' +r22 + r2rr r

for the points at infinity, i. e. the points on the sphere it will be sufficient to take

1 1 r' r

and consequently IT r;ffwdE'.

For brevity, write N2 V(a2? +I2) (b2 + #2) (C2 + 42),

N3A V(a2 + s) (b2++ 48) (C2+4f3).

The quantity wdEg is now given by (Ferrer's Spherical Harmonics, page 129)

iabc (i13 - k2) code'.= N N3 dt/2d43,

and consequently iabcf ('6J3 -4'2)

dkd3 -4J jJ N2N3 d'2d4'3,

or, since V'2 is between- c2 and - b2 and *3 between - b2 and - a2, - -2 b2 C3

~~~ab qldU. "4 d2 _d~s 412dql2 . ffWde' ~ _i --f:fa_N 4 N N f



290 CRAIG: On certain Possible Cases of

For N2 and N3 we can readily find values similar to that already found for N1. In fact we have

a~ + a1si2a _2 b2 + q = (a - C) dn 2 o c2 + +1=(a2- C2) ell2 ol sa2 c2 Sdn2 S1 2

__1_

giving 191 a2 C)dn 0, cli 01

N1 = (a2 - 2)

Also

a2 + q12 (a2-C2) dn2 02, b2 + 12 (b2 -C2) cn2 02, c2 + 2= -(b2-C2) sn20,

and

a2+ = (a2- c2)sn 20, b2?+ 3- (a2 C2)lj2cn2 03, C2?+ 3=- (a2-c2)dn2 0.

From these we get

41- 4 (2a2 -2) l so - dll2

41 - 3 =(a2 - )sn02 - /k2 sn203

Substituting in N2 and N3 the values now found for a2 + 412, etc., these become

A2= i(b2- c2) 2- csn 02 cn 02 dn 02,

N3 = (a2 - b2) Va2 - C2 sn 03 cn 03 dn 03;

the above integral now becomes

k_ 2 acf(dn2 02-hk sin2 03) d02d03, 0 0

since d

sn 0 cn 0 dn 0,

d tn 0 k2 sn 0 cn 0.

Now

fdn2 02d2= K 02 + Z02, o~~

where Z02 is the periodic part of the integral, and

-k2f sn2 03 = Z03 + (- 1)03. 0

Therefore, finally,

ff )d,' = - f d3 [E + Zo2] +f do2 [z3+ (K--1) 03] (



Steady Motion in a VTiscous Fluid. 291

the integration to be extended over the entire surface of the ellipsoid. We will denote this integral by J, and then shall have for the surface potential

J

P--.

For the determination of the total pressure upon the body we have now

Spide - -Spd'

the right-hand member of this equation, which we will denote by - X, can thus be computed by merely integrating pi over the surface of the infinite sphere. Suppose, first, the case of very slow motions; then, as we are only concerned with the value of p at infinity, we neglect the squares and products of the velocities, also the terms containing },

dp 2 apj dx,

p 2 a dk.

Now (Lamb's Fluid Motion, page 217),

Pi IPxx + rpXY + npxz, etc.,

wlhere 1, m, n are the direction-cosines of the normal to the bounding surface; in this case

zx y z - r- r

respectively. Also du P - + 2[t dx'

(du+ dv\ pxy iK + dx)

(dit dw> PXZ K t ), + d) etc.

The other relations are well known. We have now du d21.,

ax dx2'

du dv d___

dy d 2 aX dxdy'

du dv CP4 dz ? d -2 ax dxdz

We have then a d' x x d d@)



292 CRAIG: 02 certain Possible Cases of

x (d4 d d4) or p-2 a# r d- r dx dr|

Jx2 -x-6 ak JX-

Write - cos X, multiply by 2 irr2 sin XdX and integrate from X=0 to X-2 w;

ff _p,dE- - 8 ,walJ. Therefore we have

X = 8 rafLJ.

Introducing the found values of a and J, this is

4 _____ (a2_ fd03 [K 02 + Z02] +fd2 [Z03 + (K- ) 08]

k ~ 2 a2f0 sn2 01 d01-2 - (a2 _ c2) K

or fdE

x =k2 4 A (a2-c2)0 f 2 + Z02] +fdo2 [z03 + - i) 03]

a2 (1-_ 01 - Zo1] -2 7r (a2 - b2) K

The determination of this quantity in the general case will be extremely difficult, probably impossible. It will first be necessary to find the pressure of p froum either of the formulas, no external forces acting,

= G - dL + dM + dNzdlryd' p r where

L-vC - w- M -u, X= ur, or from

= = G-e q'r sinG siq+ dnady d P T7JJr d 2+ jj r2 xyz

The stream and vortex lines do not cut constantly at right angles, so no simpli- fication of this last is possible. A further investigation of this problem would require as a first step a thorough examination of tlle elliptic vortex rings formed in the fluid.*

We have for the angular velocities = 0,

d'1 a

d* a n j

* On this point see an article " On Circular Vortex Rings " by C. V. Coates, Quarterly Jourlnal, Vol. XVI. p. 170.



Steady Motion in a Viscous Fluid. 293

These differential coefficients are known to be given by

(a2 + V1) (C2 + V/1)

d4' 2 ( dy y (VI 1-2) (ql 1-3)

d4, (a 2+ qfl) (b 2 +). dz 2Z(ql _ q12) ( q1-)7

we have then for the differential equation of the vortex lines

ydy zdz b2+ qf + C2 += ?;

for 4,' 0, or all points on the surface of the ellipsoid, this gives

y 2 +z2

2 + 2consf.

Introducing elliptic functions,

dii 2 /a2 01 cn ol dnO cn 02 cn0

riy sn 011 - sn2 01 (dn2 02 + k2 Sn2 03) + k2 sn 2 0, dn2 02 Sn2 03

d4' 2 /a2- c2 dn2 01 Cn01 sn 02 dnO3 dz sn 01 1 - sn2 01 (dn2 02 + k2 sn2 03) + k2 sn201 dn202 sn2Oar

WASHINGTON, D. C., Nov. 22, 1880.



On Certain Possible Cases of Steady Motion in a Viscous Fluid. [Continued]

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