299-302] General Theory of Viscosity 247 



301. If F is the aggregate transfer of momentum per unit of time across 

 a unit area of a plane parallel to the plane of xy, we have shewn that 



r = i^ (569). 



Across the plane z + dz the similar transfer is 



hence the gain of momentum to the layer between the planes z and z + dz is 



Also if we have a viscous fluid of coefficient of viscosity K moving with 

 the mass velocity of the gas, of which the components are u , 0, 0, the force 

 per unit area of the z plane in the direction of the axes of x, acting upon the 

 layer of fluid enclosed by the planes z and z + dz is 



Similarly that on the plane z + dz, acting in the other direction, is 



fdu 3 2 w \ 

 K ( - 4- dz 3 . 

 \oz dz* )' 



Hence, acting upon this layer per unit area there is a resultant force 



3X -, 

 K ^ dz ' 



This force increases the momentum per unit area of the layer per unit 

 time at a rate 



In expression (570), we can replace /Z by mw , where u is the mass- 

 velocity of the gas, and expression (571) now becomes identical with (570) if 



tc = %vclm ....................... . ...... (572). 



We have therefore found that our gas will behave exactly like a viscous 

 fluid, of which the coefficient of viscosity is given by equation (572). If we 

 replace mv by p, this takes the simple form 



K = $pcl. ................................. (573). 



302. From the results of our analysis we can now obtain an insight into 

 the molecular mechanics of viscosity in the case of a gas. Let us imagine 

 two molecules, with velocities u, v, w and u,v,w penetrating from a layer 

 at which the mass velocity is 0, 0, to one at which it is u, 0, 0. By the 



