585 



In order to simplify the mathematical treatment it lias been 

 assumed that the motion is confined to a plane. 



Finally in paragraph 7 some results are given for the flow between 

 two fixed parallel walls. 



§ 2. The principal equation. 



In the following lines the mean or principal motion of the fluid 

 will be denoted by U. It is a function of the variable y only; at 

 the wall // =^ it is equal to 0, at the wall y = I it takes the 

 value V. The components of the velocity of the relative motion are 

 written ?< and v; the vorticity of the relative motion is written: 



ox ay 



These latter quantities are functions of the variables x, y and t. 

 The velocities u and v are subjected to the boundary conditions: 



u — 0,1=0 for ;(/ = and \ox y = 1 . . . . (6) 

 and to the equation of continuity : 



du dv 

 dx dy 



Now both Reynolds and Lorkntz have shown that the peculiar 

 character of turbulent motion is caused by the action of an apparent 

 frictional force, influencing the principal motion, and due to the 

 existence of the relative motions. This is expressed by the formula: 



dU — 

 fi ^ uv = S (8) 



dy 



The bar over iiv indicates that the mean value of this quantity is 

 meant, taken at a certain point during a certain lapse of time, or 

 taken at a certain moment along.a line parallel to the axis of ,c. This 

 mean value is a function of the variable y only (the same remark 

 applies to $' in formula (9)). The quantity uv is negative, and 



The relative motions, however, are not independent of the mean 

 motion. In order that the relative motions may always retain the 

 same energy, it is necessary that the following equation is fulfilled: 



r -dU r - 



- I dyQiiv — = j dynP (9) 







