300 Mr. stokes, ON THE FRICTION OP FLUIDS IN MOTION, 



motion of the fluid particles immediately about any imaginary surface dividing the fluid, the 

 tano-ential 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 indefinitely 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, 

 according 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 experiment. I then 

 tried Poisson's conditions in the case of a circular pipe, but with no better success. In fact, it 

 appears from experiment 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. 



First, suppose the solid at rest, and let L - be the equation to its surface. Let M' be a 

 point within the fluid, at an insensible distance h from M, Let w be the pressure which would 

 exi.^t about M if there were no motion of the particles in its neighbourhood, and let p^ be the 

 additional normal pressure, and t^ the tangential force, due to the relative velocities of the 

 particles, both with respect to one another and with respect to the surface of the solid. If the 

 motion is so slow that the starts take place independently of each other, on the hypothesis of starts, 

 or that the molecules are very nearly in their positions of relative equilibrium, and if we suppose 

 as before that the effects of different relative velocities are superimposed, it is easy to show that 

 p and t are linear functions of m, t>, w and their differential coefficients with respect to or, y, and *; 

 u, t), &.C. denoting here the velocities of the fluid about the point M', in the expressions for which 

 however the co-ordinates of M may be used for those of M', since h is neglected. Now the 



relative velocities about the points M and M' depending on — &c. are comparable with — A, 



while those depending on m, v and w are comparable with these quantities, and therefore in 

 considering the action of the fluid on the solid it is only necessary to consider the quantities 

 u, V and w. Now since, neglecting //, the velocity at M' is tangential to the surface at J/, 

 ««, I', and w are the components of a certain velocity V tangential to the surface. The pressure p^ 

 must be zero; for changing the signs of u, v, and ui the circumstances concerned in its production 

 remain the same, whereas its analytical expression changes sign. The tangential force at M will 

 be in the direction of V, and proportional to it, and consequently its components along the axes 

 of CO, y, X will be proportional to u, u, w. Reckoning the tangential force positive when, 

 /, m, and n being positive, the solid is urged in the positive directions of .r, j/, z, the resolved 

 parts of the tangential force will therefore be vu, vv, vw, where v must evidently be positive, 

 since the effect of the forces must be to check the relative motion of the fluid and solid. The normal 

 pressure of the fluid on the solid being equal to iir, its components will be evidently /isr, m-sr-t nnr- 



Suppose now the solid to be in motion, and let ti, v', w' be the resolved parts of the velocity 

 of the point M of the solid, and w, to', &>'" the angular velocities of the solid. By hypothesis, 

 the forces by which the pres>.ure at any point differs from the normal pressure due to the action of 

 the molecules supposed to be in a state of relative equilibrium about that point are independent of 

 any velocity of translation or rotation. Supposing then linear and angular velocities equal and 

 opposite to those of the solid impressed both on the solid and on the fluid, the former will be for 

 an instant at rest, and we have only to treat the resulting velocities of the fluid as in the first case. 



