34 
PROFESSOR G. H. DARWIN ON THE MECHANICAL CONDITIONS OF 
§ 11. On the Diffusion of Kinetic Energy and on the Viscosity. 
Ill order to discuss these questions, it will be well to begin with a simple case of 
fluid motion. 
Consider two-dimensional motion, in which there are a number of streams of equal 
breadth moving parallel to y with velocity V, and, interpolated between them, let 
there be strata of quiescent fluid; suppose then that we wish to find the motion at 
any time after this initial state. Let the boundaries of the streams V be from x = ml 
to ^ (2711 + 1) /. Then, if u be the velocity at x, parallel to y at time t, and v the 
kinetic modulus of viscosity, the equation of motion is 
dto d~to 
dt dx~ 
The solution of this being of the form cos jiai, the complete solution satisfying the 
initial condition is— 
2V 
e cos 
_ I cog + A 
I I 
blTX 
cos 
Now, if we refer time to a period r, where r = then after a time 6 t, which is 
greater than r, the solution is sensibly 
1 T/ I 1 I ^ 
w = ^ K 1 + — COS — 
Tree I 
It is clear that the maximum of u occurs when a: = 0, and the minimum when x = /, 
and that they are 
I 
1 
Hence, the difference betwmen the maximum and minimum is 4F/7re^ Therefore, 
the ratio of the greatest difierence of velocities after time 6t to the initial difference of 
velocities is 4/7re^ When 6 is 1, 2, 3, tins ratio assumes the values 1/2T35, l/5'804, 
1/15'73 respectively. Thus, after three times the inteiwal r, the difference of velocities 
is small. The time t may be therefore taken as a convenient measure of viscosity. 
In our problem the streams must be taken of a width comparable with the linear 
dimensions of the solar system. I therefore take I, the width of the streams, as equal 
to ttg, the Earth’s distance from the Sun, and we have 
7r~v 
Now’, according to the kinetic theory of gases, the kinetic modulus of viscosity is 
I/tt into the mean free path multiplied by the mean velocity. Hence, 
