514 VISCOSITY. [CHAP, xi 



for the temperature C. For air at atmospheric pressure, assuming p = 00129 

 this gives 



The value of v varies inversely as the pressure*. 



285. We have still to inquire into the dynamical conditions 

 to be satisfied at the boundaries. 



At a free surface, or at the surface of contact of two dissimilar 

 fluids, the three components of stress across the surface must be 

 continuous )-. The resulting conditions can easily be written down 

 with the help of Art. 283 (4). 



A more difficult question arises as to the state of things at the 

 surface of contact of a fluid with a solid. It appears probable that 

 in all ordinary cases there is no motion, relative to the solid, of the 

 fluid immediately in contact with it. The contrary supposition 

 would imply an infinitely greater resistance to the sliding of one 

 portion of the fluid past another than to the sliding of the fluid 

 over a solid . 



If however we wish, temporarily, to leave this point open, the most 

 natural supposition to make is that the slipping is resisted by a tangential 

 force proportional to the relative velocity. If we consider the motion of a 

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

 dimensions, in contact with the solid, it is evident that the tangential traction 

 on its inner surface must ultimately balance the force exerted on its outer 

 surface by the solid. The former force may be calculated from Art. 283 (4) ; 

 the latter is in a direction opposite to the relative velocity, and proportional 

 to it. The constant (/3, say) which expresses the ratio of the tangential force 

 to the relative velocity may be called the coefficient of sliding friction. 



286. The equations of motion of a viscous fluid are obtained 

 by considering, as in Art. 6, a rectangular element Sx&ySz having 

 its centre at P. Taking, for instance, the resolution parallel 

 to #, the difference of the normal tractions on the two yz- faces 

 gives (dp xx jdx) x . Sy Sz. The tangential tractions on the two 

 #-faces contribute (dp yx /dy) Sy . 8z$x, and the two xy-f&ces give 



* A very full account of the results obtained by various experimenters is 

 given in Wmkelmann s Handbuch der Physik, t. i., Art. Keibung. 



t This statement requires an obvious modification when capillarity is taken into 

 account. Cf. Art. 302. 



Stokes, 1. c. p. 518. 



