Figure 1 - Two successive positions of 



several streamlines and, shown by heavy 



curves, the paths of two particles. 



(l.m,n) 



Figure 2 — Relation between the velocity and 

 its components. 



In addition to streamlines, the concept of tubes of flow is sometimes useful. A tube of 

 flow is a slender filament of fluid whose bounding surface is composed of streamlines. 



The particle velocity is a vector quantity. Its magnitude will be denoted by q; its com- 

 ponents in the directions of the x-, y-, 2-axes of a rectangular Cartesian coordinate system will 

 be denoted by u, v, w. Thus 



U + V" + w 



[la] 



The component q of the velocity in a direction whose direction cosines are I, m, n can then 

 be written 



lu + mv + nw 



[lb] 



as is evident from Figure 2, in which the component in the direction OP is represented by the 

 projection on OP of either the vector OQ representing the velocity or of the broken line ORSQ, 

 whose segments represent u, v, and w. 



Since q, u, v, and w may vary from point to point, and also with the time, they may be 

 regarded as functions of the four variables x, y, z, and t. In steady motion, however, every- 

 thing is a function of x, y, z only. 



2. THE EQUATION OF CONTINUITY 



A relation must exist between the motion of a fluid and changes in its density. If, for 

 example, more fluid enters a given volume than leaves it, the density of the fluid in the 

 volume must increase. 



Consider a small cubical element of sides 5a;, Sy, Ss whose center is situated at the 

 point (a;, y, z), as in Figure 3; let it be fixed in size as well as in position in space. Fixing 

 attention first on the increase in mass due to flow through the two faces perpendicular to the 



