The Momentum Balance for Three-Dimensional Flow

In the preceding material of this chapter, the flow considered was onedimensional even in Example 7.5, all the fluid leaving the system had the same velocity and direction. Most practical fluid mechanics problems (flow in pipes, channels, jets, etc.) are one-dimensional. However, there are some important two- or three-dim ensional flows, particularly in flows of very viscous materials and of fluids around airplanes and ships. In such cases it is useful to have the momentum balanjce set up for a three-dimensional flow. The notation to be used is shown in Fig. 7.18. 

Our system is the small cube shown. We may think of it as being a wire frame with flow iii or put through all six faces. The frame itself is fixed in space 

Previously we have written the first term as didv, here we write it as 3/d/ because is a function of both time and position that is, V = V ix, y, z, /). Thus, our momentum balance is a partial differential equation, and diBt implies didt at some fixed location. 

It is common practice in fluid mechanics not to include in the partial differentials the subscripts indicating which variables are being held constant. This is so because with very few exceptions the independent variables are x, y, z, and t or r, 0, z, and i in polar coordinates. Thus, the symbol d pV ) dx really means z,/- Throughout this text any partial derivative that 

The mass of fluid in the system is equal to the system volume times the density of the fluid in the system so we may write 

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