222 VISCOSITY. [CHAP. ix. 



tions of the kinetic theory, point to the validity of the formulae (11) 

 within very wide limits as to the values of a, 6, c, &c. Since the 

 postulates of this theory are in the main highly probable, we are 

 warranted in regarding the equations (14) as sensibly accurate for 

 all ordinary cases of motion of gases. 



181. We have still to consider the conditions to be satisfied 

 at a boundary ; at the common surface, for instance, of the fluid 

 and of a solid with which it is in contact. It is found that in 

 many cases* the particles of fluid in contact with the solid move 

 with the latter, so that if u, v, w and u, v , w denote the component 

 velocities of contiguous elements of the fluid and the solid respec 

 tively, we have 



11= u, v = v y w = w (16). 



In other cases it appears that there is a finite slipping of the 

 fluid past the surface of the solid. The boundary-condition may 

 then be obtained as follows. Considering the motion of a small 

 film of fluid, of thickness infinitely small compared with its lateral 

 dimensions, in contact with the solid, we see that the tangential 

 stress on its inner surface must ultimately balance the force 

 exerted on its outer surface by the solid. The former stress may 

 be calculated from (9) ; the latter may be taken as directly oppos 

 ing the velocity of the fluid relatively to the solid, and as approxi 

 mately proportional to this relative velocity, provided it be small. 

 The constant (/3, say,) which expresses the ratio of the tangential 

 force to the relative velocity may be called the coefficient of 

 sliding friction. 



The conditions to be satisfied at the common surface of two 

 different fluids, or of two portions of the same fluid separated by a 

 surface of discontinuity, may be obtained in the same way. 



The boundary-conditions here given are dynamical; the purely 

 kinematical condition of Art. 10 of course obtains here as always. 



* Stokes (Camb. Trans. Vol. ix.) and Maxwell (Phil. Trans. 1866) find this to be 

 true of air in contact with slowly vibrating bodies. Helmholtz and Piotrowski infer 

 from their experiments (Wiener Sitz. B. t. 40, 1860) that there is little or no slipping 

 for the cases of ether and alcohol in contact with a polished and gilt metal surface. 

 On the other hand, they found that in the case of water, and of most other liquids 

 which they experimented upon, an appreciable amount of slipping took place. 



