IX. On some Cases of Fluid Motion. By G. G. Stokes, B.A , Fellow of 



Pembroke College. 



[Read May 29, 1843.] 



Thk equations of Hydrostatics are founded on the principles that the mutual action of two adja- 

 cent elements of a fluid is normal to the surface which separates them, and that the pressure is equal 

 in all directions. The latter of these is a necessary consequence of the former, as has been shewn 

 by Mr. Airy*. An exactly similar proof may be employed in Hydrodynamics, by which it may 

 be shewn that, if the mutual action of two adjacent elements of a fluid in motion is normal to their 

 common surface, the pressure must be equal in all directions, in order that the accelerating force 

 which acts on the centre of gravity of an element may not become infinite, when we suppose the 

 dimensions of the element indefinitely diminished. In Hydrostatics, the accurate agreement of the 

 results of our calculations with experiments, (those phenomena which depend on capillary attraction 

 beinrr excepted), fully justifies our fundamental assumption. The same assumption is made in 

 Hydrodynamics, and from it are deduced the fundamental equations of fluid motion. But the 

 verification of our fundamental law in the case of a fluid at rest, does not at all prove it to be 

 true in the case of a fluid in motion, except in the very limited case of a fluid moving as if it were 

 solid. Thus, oil is sufficiently fluid to obey the laws of fluid equilibrium, (at least to a great extent), 

 yet no one would suppose that oil in motion ought to be considered a perfect fluid. It would 

 appear from the following consideration, that the fluidity of water and other such fluids is not 

 quite perfect. When a mass of water contained in a vessel of the form of a solid of revolution is 

 stirred round, and then left to itself, it presently comes to rest. This, no doubt, is owing to the 

 friction against the sides of the vessel. But if the fluidity of water were perfect, it does not 

 appear how the retardation due to this friction could be transmitted through the mass. It would 

 appear that in that case a thin film of fluid close to the sides of the vessel would remain at rest, the 

 remaining part of the fluid being unaffected by it. And in this respect, that part of Poisson's 

 solution of tile problem of an oscillating sphere, which relates to friction, appears to me in some 

 degree unsatisfactory. A term enters into the equation of motion of the sphere depending on the 

 friction of the fluid on the sphere, while no such term enters into the equations of motion of the 

 fluid, to express the equal and opposite friction of the sphere on the fluid. In fact, as long as we 

 regard the fluidity of the fluid as perfect, no such term can enter. The only way by which to 

 e.stiniate the extent to which the imperfect fluidity of fluids may modify the laws of their motion, 

 without making any hypothesis as to the molecular constitution of fluids, appears to be, to calculate 

 according to the hypothesis of perfect fluidity some cases of fluid motion, which are of such a 

 nature as to be capable of being accurately compared with experiment. The cases of that nature 

 which have hitherto been calculated, are by no means numerous. My object in the present paper 

 which I have the honour to lay before the Society, has been partly to calculate some such cases 

 which may be useful in determining how far we are justified in regarding fluids as perfectly fluid, 

 and partly to give examples of the methods by which the solution of problems depending on partial 

 differential equations may be eftected. 



In the first seven articles, I have mentioned and explained some general principles, which are 

 afterwards a])plied. Some of these are not new, but it was convenient to state them for the sake 

 of reference. Others are I believe new, at least in their develo))ement. In the remaining articles, I 

 have given different problems, of which I have succeeded in obtaining the solutions. As the pro- 



• .See also ProfeMsor MtUcr's Hydrostatics, page 2. 



Vol. VIII. I'.uir I. O 



