e. Within a single connected region the streamlines must begin and end on the boundary. 

 Hence, if the boundary is entirely stationary, so is the fluid. 



For, otherwise, closed streamlines would necessarily occur, and this is impossible 

 under the conditions assumed; since, in going around a closed streamline in the direction of 

 the flow, the velocity potential would decrease continually, and hence it could not return to 

 its initial value upon returning to the starting point. Furthermore, no streamline can end on a 

 stationary boundary, since there the normal component of the velocity must vanish. 



f. The flow within any region is uniquely determined if the velocity is given, in magnitude 

 and in direction, at all points on the boundary of the region. 



For, if two different distributions of velocity satisfying the given conditions were 

 possible, a third one would also be possible in which the velocity is the vector difference of 

 the velocities in the given distributions. In this latter distribution the velocity would be zero 

 at all points on the boundary; hence, by (d), the velocity must vanish everywhere. It follows 

 that the two original distributions of velocity must be identical. 



g. In a singly connected finite region, the flow is uniquely determined if the normal com- 

 ponent of the velocity is assigned at ail points of the boundary. 



The proof is similar to that of (f). The difference between two types of flow satisfying 

 the given condition would be another in which the normal component of the velocity vanishes 

 over the boundary, so that no streamlines could begin or end there; hence, as explained under 

 (e), there can be no streamlines, and the two assumed distributions of velocity must be identi- 

 cal. 



In all cases, the word boundary may refer either to a physical boundary or merely to a 

 geometrical surface drawn through the fluid. Furthermore, except where the contrary is 

 specified, the boundary may lie partly or wholly at infinity. 



It may also be noted that differentiation of Equations [6b, c, d] leads to the equations 



il_^=0, ^-i« = 0, iM.dv^o, [8a, b, c] 



dz dy dx dz dy dx 



These differential equations may be regarded as an alternative characterization of irrotational 

 motion; for it can be shown by means of a theorem known as Stokes' theorem that the circula- 

 tion vanishes around any closed curve drawn in a singly connected region in which Equations 

 [8a, b, c] are satisfied. The three left-hand members of the equations are the components of 

 a vector, which is called in vector analysis the curl of the particle velocity and in hydro- 

 dynamics is often called the vorticity. 



From Equations [8a, b, c] it can be shown, furthermore, that in irrotational flow the 

 motion of any particle is compounded of a motion of translation and one of pure strain. In a 

 pure straining motion there are three mutually perpendicular lines through any particle of the 

 material which do not change their directions; these lines are the strain axes. In rigid- 

 rotational motion, on the other hand, only one line through a given particle retains its 



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