XXII. On the Theories of the Internal Friction of Fluids in Motion, and of the 

 Equilibrium and Motion of Elastic Solids. By G. G. Stokes, M.A., Fel- 

 Iniv of Pembroke College. 



[Read April 14, 1845.] 



The equations of Fluid Motion commonly employed depend upon the fundamental hypothesis 

 that the mutual action of two adjacent elements of the fluid is normal to the surface which 

 separates them. From this assumption the equality of pressure in all directions is easily deduced, 

 and then the equations of motion are formed according to D'Alembert's principle. This appears 

 to me the most natural light in which to view the subject ; for the two principles of the absence 

 of tangential action, and of the equality of pressure in all directions ought not to be assumed 

 as independent hypotheses, as is sometimes done, inasmuch as the latter is a necessary consequence 

 of the former*. The equations of motion so formed are very complicated, but yet they admit 

 of solution in some instances, especially in the case of small oscillations. The results of the theory 

 agree on the whole with observation, so far as the time of oscillation is concerned. But there 

 is a whole class of motions of which the common theory takes no cognizance whatever, namely, those 

 which depend on the tangential action called into play by the sliding of one portion of a fluid along 

 another, or of a fluid along the surface of a solid, or of a different fluid, that action in fact which 

 performs the same part with fluids that friction does with solids. 



Thus, when a ball pendulum oscillates in an indefinitely extended fluid, the common theory 

 gives the arc of oscillation constant. Observation however shows that it diminishes very rapidly 

 in the case of a liquid, and diminishes, but less rapidly, in the case of an elastic fluid. It has 

 indeed been attempted to explain this diminution by supposing a friction to act on the ball, 

 and this hypothesis may be approximately true, but the imperfection of the theory is shown 

 from the circumstance that no account is taken of the equal and opposite friction of the ball on 

 the fluid. 



Again, suppose that water is flowing down a straight aqueduct of uniform slope, what will be 

 the discharge corresponding to a given slope, and a given form of the bed.'' Of what magnitude 

 must an aqueduct be, in order to supply a given place with a given quantity of water? Of what 

 form must it be, in order to ensure a given supply of water with the least expense of materials 

 in the construction .' These, and similar questions are wholly out of the reach of the common 

 theory of Fluid Motion, since they entirely depend on the laws of the transmission of that 

 tangential action which in it is wholly neglected. In fact, according to the common theory 

 the water ought to flow on with uniformly accelerated velocity ; for even the supposition of 

 a certain friction against the bed would be of no avail, for such friction could not be transmitted 

 through the mass. The practical importance of such questions as those above mentioned lias 

 made them the object of numerous experiments, from which empirical formulae have been con- 

 structed. But such formulae, although fulfilling well enough the purposes for which they were 



• Thin may be eauily shown by ihc coniideration of s tetrahedron of the fluid, as in An. 4. 



