31 



45 



may be a good approximation; for the paraboloidal motion, 0.8 is replaced by 

 0.78, and for the piston motion, by 0.8U. 

 A third type of some Interest is 



2\-? 



/(x,v) = (i-y ^ ^=^c(i-ji) 



for which, as in Equation [152] of the Appendix, 



\t 2 3 



Ml = n pa 

 and, if Pi is uniform, Equation [31a] gives 



r^ 2 - — 



F, = 2np, / (1 - fr) ^ rdr = 27ra'pi 



[^+53, b] 



[^6] 



[4?] 



This form of /(x,j/) represents the 

 cording to non-compressive theory, 

 lar hole because of a sudden 

 application of pressure; see 

 the Appendix, and Reference 

 (1 )• Here the liquid surface 

 Is assumed to be plane initial- 

 ly. The average velocity is 

 2d,zjdt. As the motion con- 

 tinues, however, second-order 

 effects become appreciable and 

 the usual vena contracta de- 

 velops; at the edge it will 

 begin forming Immediately. 



The distribution of 

 velocity over the plate is Il- 

 lustrated for the three types 

 of motion in Figure 19- 



In all three cases 

 a rigid baffle beyond the plate 

 or hole has been assumed. If 

 the plate merely forms one side 

 of an alr-fllled caisson or box, 

 the estimation of Af, is more 

 difficult. From the consideration 

 pears that the absence of a baffle 

 phragm by a factor of about 2, and 

 factor nearer 3. 



distribution of velocities with which, ac- 

 liquid should begin to issue from a circu- 



2.0 



1.8 

 i.e 

 1.4 

 1.2 



di 10 



dt 



0.6 

 0.6 

 0.4 

 0.2 



0.1 0.2 0.3 0.4 0.5 Q6 0.7 0.8 0.9 1.0 



o 



Figure 19 - Distribution of Velocities In 



Three Types of Proportional Motion, 



for a Circular Diaphragm 



dt/dt is the velocity perpendicular to the initial plane 



at a distance r from the center of the diaphragm whoso- 



radius is a. The velocity is shown in each case on an 

 arbitrary scale. 



of a solvable case in the Appendix it ap- 

 mlght reduce Af^ for the paraboloidal dia- 

 for a diaphragm moving like a piston by a 



