293-296] General Theory of Viscosity 243 



There is also an entirely different method of attacking the problem, due 

 to Maxwell, which will be presented in Chapter XV. This method is an 

 exact method, but unfortunately only leads to a solution in very special 

 cases. We shall give the solution obtained by Maxwell on the supposition 

 that molecules are point-centres of force repelling one another with a force 

 proportional to the inverse fifth power of the distance. This exact solution 

 will be of interest in that it will enable us to estimate the amount of in- 

 accuracy in the results obtained by the general method of the present 

 chapter. 



General Equations of Viscosity. 



295. We shall now discuss the motion in a gas in which the mass- 

 velocity varies from point to point. At the particular point considered, let 

 us choose our axes so that the mass- velocity is parallel to the axis of x, 

 while the surfaces of equal velocity are parallel to the plane of aey, so that 

 in the neighbourhood of the point in question the mass-velocity is a function 

 of z only. 



Let us write p, for mu, the momentum of any single molecule in the 

 direction of the #-axis, so that the total momentum of any element of the 

 gas is obtained by summing p, over all the molecules inside the element. 

 The mean value of p, at any point will be denoted by JL, and of course /Z 

 varies from point to point in the gas. At the particular point considered we 

 have chosen the direction of our axes so that the gas is arranged, as regards 

 the distribution of /I, in a series of layers parallel to the plane of xy. We 

 proceed to attempt to calculate the amount of p, which is transferred by the 

 molecular motion across any one of the planes z = constant. 



296. The physical principle underlying the calculation can easily be 

 explained. Let us, to fix our ideas, suppose that the average value of p, 

 increases as z increases, that the planes z = constant are horizontal and 



' that z increases as we move upwards. The molecules will cross the planes 

 z = constant in both directions. Those which cross any plane, say z = z , in 

 the downward direction will, however, be coming from regions in which the 

 average value of p, per molecule is greater than it is over the plane z = z , 

 and will therefore, on the average, possess a value of p, in excess of that 

 appropriate to the plane z = z . In the same way, those molecules which 

 cross this plane in the upward direction will, on the average, possess a value 

 of p, smaller than that appropriate to the plane z = z . Since, however, there 

 is no mass motion parallel to the axis of z, the number of molecules which 

 cross the plane z = z in one direction is exactly equal to the number which 

 cross it in opposite directions. There is, therefore, more momentum carried 

 through the plane z = z in the downward direction than in the upward 

 direction. In other words, there is a downward transfer of momentum. 



162 



