Desai and Lieber 



compatible with the equations of motion? Third, did Stokes in fact answer these 

 questions ? Below we quote two relevant paragraphs from Stokes' [15, 16] papers, 

 the first from the paper of 1845 and the second from the paper of 1851. 



The next case to consider is that of a fluid in contact with a solid. 

 The condition which first occurred to me to assume for this case 

 was, that the film of fluid immediately in contact with the solid 

 did not move relatively to the surface of the solid. 1 was led to 

 try this condition from the following considerations. According to 

 the hypotheses adopted, if there was a very large relative motion 

 of the fluid particles immediately about any imaginary surface 

 dividing the fluid, the tangential forces called into action would be 

 very large, so that the amount of relative motion would be rapidly 

 diminished. Passing to the limit, we might suppose that if at any 

 instant the velocities altered discontinuously in passing across 

 any imaginary surface, the tangential force called into action would 

 immediately destroy the finite relative motion of particles indefi- 

 nitely close to each other, so as to render the motion continuous; 

 and from analogy the same might be supposed to be true for the 

 surface of junction of a fluid and solid. But having calculated, ac- 

 cording to the conditions which I have mentioned, the discharge of 

 long straight circular pipes and rectangular canals, and compared 

 the resulting formulae with some of the experiments of Bossut and 

 Dubuat, I found that the formulae did not at all agree with experi- 

 ment. I then tried Poisson's conditions in the case of a circular 

 pipe, but with no better success. In fact, it appears from experi- 

 ment that the tangential force varies nearly as the square of the 

 velocity with which the fluid flows past the surface of a solid, at 

 least when the velocity is not very small. It appears however from 

 experiments on pendulums that the total friction varies as the first 

 power of the velocity, and consequently we may suppose that 

 Poisson's conditions, which include as a particular case those 

 which I first tried, hold good for very small velocities. I proceed 

 therefore to deduce these conditions in a manner conformable 

 with the views explained in this paper. 



For the purposes of the present paper there will be no occasion to 

 consider the case of a free surface, but only that of the common 

 surface of the fluid and a solid. Now, if the fluid immediately in 

 contact with a solid could flow past it with a finite velocity, it 

 would follow that the solid was infinitely smoother with respect 

 to its action on the fluid than the fluid with respect to its action 

 on itself. For, conceive the elementary layer of fluid comprised 

 between the surface of the solid and a parallel surface at a dis- 

 tance h, and then regard only so much of this layer as corre- 

 sponds to an elementary portion ds of the surface of the solid. 

 The impressed forces acting on the fluid element must be in equi- 

 librium with the effective forces reversed. Now conceive h to 

 vanish compared with the linear dimensions of ds, and lastly let 

 ds vanish. It is evident that the conditions of equilibrium will 

 ultimately reduce themselves to this, that the oblique pressure 



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