8. SOME PROPERTIES OF IRROTATIONAL FLOW 



The following properties of the irrotational flow of incompressible fluids may be noted. 

 More rigorous proofs of some of them may be found in Milne-Thomson's Theoretical Hydro- 

 dynamics-^ or elsewhere. ^^ 



a. The distance between two given equipotential surfaces corresponding to slightly differ- 

 ent values of varies in inverse ratio to the magnitude of the velocity q. 



For, 8ch = - a 8 s; if 8(h is constant, Ss<>: — 



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b. The stream,lines are concave toward the side on which the magnitude of the velocity q 

 is larger. 



For on the concave side of a streamline neighboring equipotential surfaces, being per- 

 pendicular to the line, must converge, as at P in contrast to Q jn Figure 9a-, hence, by (a), q 

 is greater. 



c. The velocity q increases in the direction in which the streamlines converge, and hence 

 is greater near the concave side of an equipotential surface than near its convex side. 



For, if the streamlines converge in a certain direction, such as RS in Figure 9b, the 

 associated tube of flow diminishes in cross section in that direction; but the same volume of 

 an incompressible fluid must flow across every cross section of a tube; hence q increases in 

 this direction. 



d. In any given region, the maximum velocity must occur at a point on the boundary. The 

 same is true of the minimum velocity unless it is actually zero. 



For, suppose a maximum value of q occurred in the midst of the fluid, as at T in Figure 

 9c. Then q would decrease in all directions from this point. But then the tube of flow contain- 

 ing this point would have to flare in both directions from T by (c), and would also have to be 

 concave inward over the sides of the tube, by (b), which is impossible. The proof for a nonzero 

 minimum is similar. 



+ S(/. 



Figure 9 - Illustrating some geometrical properties of streamlines and equipotential surfaces. 



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