212 Prof. W. J. M. Rankine on the Thermal Energy 



§ 3. Mean Pressure due to Centrifugal Force. — Let an ele- 

 mentary circulating stream (that is, a circulating stream of in- 

 definitely small sectional area) be supposed to flow round and 

 round in an endless tube with the uniform velocity w ; let p de- 

 note the density of the stream, da the sectional area. Consider 

 two cross sections of the stream at which the directions of motion 

 of the particles are contrary ; and consider what resultant forces 

 are exerted by the stream on the two parts into which those two 

 cross sections divide the tube. The mass of matter which flows 

 through each cross section of the tube in a unit of time is 



pwda) 



and in each unit of time a mass of matter of that amount has its 

 velocity reversed. The force required in order to produce that 

 reversal of velocity is of the following amount in absolute units, 



2/3 uP da- ; 



and such is the amount of each of the pair of inward pressures 

 which the tube exerts on the stream, and of each of the pair of 

 equal and opposite outward pressures exerted by the stream on 

 the tube, tending to pull it to pieces. It may be called the cen- 

 trifugal tension of an elementary stream. 



The velocity of the particles flowing in the stream may undergo 

 periodical fluctuations, positive and negative alternately ; these 

 will cause periodical variations in the centrifugal tension ; but 

 the mean value of that tension will continue to be that given by 

 the formula. 



The mean intensity of the centrifugal tension, in a direction 

 tangential to the stream, is found by dividing the amount given 

 in the preceding expression by the collective area, 2da, of the two 

 cross sections, giving the following result, 



pw 2 . 



Suppose now that the stream is cut by an oblique sectional 

 plane, making the angle 6 with a transverse section. Then the 

 area of that oblique section is greater than that of a transverse 

 section in the ratio of 1 : cos 6 ; and the amount of the compo- 

 nent tension in a direction normal to the oblique section is less 

 than that of the total centrifugal tension in the ratio of cos# : 1; 

 whence it follows that the mean intensity of the component cen- 

 trifugal tension in a direction making an angle 6 with a tangent 

 to the stream is 



puJ 2 cos 2 6. 



Next, suppose a vessel of any invariable volume and figure to 

 be filled with vortices or circulating streams, the velocity of 

 steady circulation being w, and the mean density p. The cen- 



