276 Sir W. Thomson on a Broad Rivet 



the fluid between its boundaries. But the infinity corre- 

 sponding to 2/ = ? or y=^ will vitiate this solution if co/?n is 

 equal to the value of U for some one plane of the fluid or 

 for two planes of the fluid ; and the true solution will involve 

 the " cat's-eye pattern " of stream-lines, and the enclosed 

 elliptic whirls*, at this plane or these planes. 



48. Now let the fluid be given moving with the steady 

 laminar flow between two parallel boundary planes, expressed 

 by (57), which would be a condition of kinetic equilibrium 

 (proved stable in § 43) under the influence of gravity and 

 viscosity ; and let both gravity and viscosity be suddenly 

 annulled. The fluid is still in kinetic equilibrium ; but is the 

 equilibrium stable ? To answer this question, let one or both 

 bounding-surfaces be infinitesimally dimpled in any place and 

 made plane again. The Fourier synthesis of this surface- 

 operation is 



/^oo < oo f*aa 



111 da)dmdqf((o) F(m) %(q) cos cot cos mx cos qz (78), 

 Jo Jo *^o 



or 



/»oo /too /"*co 



2 I j j dcodmdqf(co) F l (m)$(q) {cos (cot — mx) 

 oJq Jo Jo 



— cos (cot + mx)} cos qz . (79), 



which implies harmonic surface-undulations travelling in 

 opposite ^-directions, with all values from to oo of (w/m), 

 the ±x of wave-velocity. Hence (§ 47) the interior dis- 

 turbance essentially involves elliptic whirls. Thus we see 

 that the given steady laminar motion is thoroughly unstable, 

 being ready to break up into eddies in every place, on the 

 occasion of the slightest shock or bump on either plastic 

 plane boundary. The slightest degree of viscosity, as 

 we have seen, makes the laminar motion stable ; but the 

 smaller the viscosity with a given value of g sin I, or the 

 greater the value of g sin I with the same viscosity, the nar- 

 rower are the limits of this stability. Thus we have been led 

 by purely mathematical investigation to a state of motion 

 agreeing perfectly with the following remarkable descriptions 

 of observed results by Osborne Reynolds (Phil. Trans. March 

 15, 1883, pp. 955, 956):— 



" The fact that the steady motion breaks down suddenly, 

 shows that the fluid is in a state of instability for disturbances 

 of the magnitude which cause it to break down. But the 

 fact that in some conditions it will break down for a large 

 disturbance, while it is stable for a smaller disturbance, shows 

 * See my former paper on the u Disturbing Infinity " already referred to. 



