CHAPTER I 

 FUNDAMENTALS OF THEIRROTATIONAL FLOW OF FRICTIONLESS FLUIDS 



In this chapter the nature and properties of the irrotational or potential flow of friction- 

 less fluids will be discussed to the extent that is desirable for the understanding and use of 

 the material that forms the body of the report. This chapter may be regarded as an introduction 

 to the subject, but it does not aim at a complete exposition of the mathematical theory of the 

 potential. Further information on the mathematical side may be found in the textbooks of 

 Lamb^ and Milne-Thomson, in MacMillan's or Kellog's "Theory of the Potential," ^-^ or in the 

 periodical literature. 



1. PARTICLE VELOCITY AND STREAM LINES 



The velocity of the particles in a fluid may vary from point to point in a complicated 

 manner. By a particle of the fluid is meant a portion so small that both its linear dimensions 

 and differences in the motion of its parts may be neglected. The motion of the fluid at any 

 instant can be described completely by specifying the particle velocity at each point. 



At any given instant, a set of curves can be drawn such that at every point on a curve 

 its tangent has the direction of the particle velocity at that point. These curves are called 

 streamlines; the aggregate of them is sometimes called a flow pattern. Thus at any given 

 instant the particles are all moving along the streamlines as they exist at that instant. 



If the streamlines remain fixed in position, the particles will continue to follow them, 

 and the streamlines will then represent the actual paths of the particles. If the motion under- 

 goes changes, however, the actual paths of the particles may be quite different from any of the 

 instantaneous streamlines. Thus in Figure 1, curves a, b, c may represent streamlines at a 

 time t, and curves a', b', c'may be the streamlines at a later time t\ whereas the actual paths 

 pursued by particles P., P from time t to t' are as shown by the heavy curves. 



An important case in which the paths of the particles coincide permanently with the 

 streamlines is the case of steady motion. The motion of a fluid is called steady when the 

 particle velocity at each point in space remains constant. The velocity of a given particle 

 may vary, however, as it moves from point to point. Motion may be steady when referred to 

 one frame of reference and variable when referred to another. Thus the motion of the air 

 around an airplane in steady flight is a steady motion as seen by the pilot, whereas at a 

 fixed point above the ground the velocity of the air changes as the airplane goes past. 



References are listed on page 396. 



