Vortices between parallel walls: Jaff^^'' and Masotti;^^^ trains of vortices, Glauert/^^ 



Rosenhead,^^'* Tomotika,^^^ Imai/^^ and Schwarz.^^'' 

 Two vortices and two parallel laminas: Riabouchinsky^^^ and Villat.^^^ 

 Source and vortex near a plane lamina: Cisotti and Agostinelli. ° By using two 



out of the three elements-source, vortex, and circulation the velocity can be made 



finite on both edges of the lamina. 



ROTATING BOUNDARIES 



99. MOVING BOUNDARIES 



From the properties of the stream function xp, the required condition at a moving 

 boundary is readily seen to take the form 





= 1n- 



[99a] 



Here dyjj/ds is the space rate of change of i// along the boundary in a chosen positive direc- 

 tion; q is the component of the velocity of the boundary in the direr-Uon of its normal, taken 

 as positive when directed toward the side that lies on the left as the boundary is traced in 

 the positive direction; compare Figure 166a. If the boundary is at rest, §■ = 0, hence 

 di/j/ds = and, as hitherto assumed, tp is constant. 







A 



V/ 



/h 







/ 



7/ 



/ Ts 







/^^ 



1/ 





-X- 



"u 



^ 



A 





Figure 166a Figure 166b 



Figure 166 - Relations at a moving boundary. See Section 99. 



In any given field of irrotational flow, a physical boundary may be supposed inserted 

 along any chosen curve provided the boundary is assumed to move at every point as is re- 

 quired by Equation [99a]; the flow will then be undisturbed by the insertion of the boundary. 

 In this way the flow around moving boundaries of many forms can be found. 



If the boundary moves in translation, an alternative procedure is the familiar one of 

 first solving the problem with the boundary at rest and then imposing an additional uniform 

 velocity upon everything. A more useful application of Equation [99a] is to rotating 

 cylinders. 



245 



