516 



VISCOSITY. 



[CHAP, xi 



The equations (4) admit of an interesting interpretation. The first of 

 them, for example, may be written 



Du 1 dt 



The first two terms on the right hand express the rate of variation of u in 

 consequence of the external forces and of the instantaneous distribution of 

 pressure, and have the same forms as in the case of a frictionless liquid. The 

 remaining term vv 2 u, due to viscosity, gives an additional variation following 

 the same law as that of temperature in Thermal Conduction, or of density in 

 the theory of Diffusion. This variation is in fact proportional to the (positive 

 or negative) excess of the mean value of u through a small sphere of given 

 radius surrounding the point (#, y, z) over its value at that point*. In 

 connection with this analogy it is interesting to note that the value of v for 

 water is of the same order of magnitude as that (-01249) found by Dr Everett 

 for the thermometric conductivity of the Greenwich gravel. 



When the forces X, F, Z have a potential Q, the equations (4) may be 

 written 



dy 



where 



q denoting the resultant velocity, and , 77, the components of the angular 

 velocity of the fluid. If we eliminate x by cross-differentiation, we find, 



du 



f& 



dv dv . dv 



+ "^ + f S 



dw dw ..dw 



(iv). 



The first three terms on the right hand of each of these equations express, as 

 in Art. 143, the rates at which , ?/, vary for a particle, when the vortex-lines 

 move with the fluid, and the strengths of the vortices remain constant. The 

 additional variation of these quantities, due to viscosity, is given by the last 

 terms, and follows the law of conduction of heat. It is evident from this 

 analogy that vortex-motion cannot originate in the interior of a viscous liquid, 

 but must be diffused inwards from the boundary. 



* Maxwell, Proc. Lond. Math. Soc., t. iii., p. 230; Electricity and Magnetism, 

 Art. 26. 



