322 



PART I ■ 



Summary of Approximation Formulae 



The momentum constant m m the presence of an infi nit a ri,^td surface ► 



If the rigid plane is at a distance d below the DuDole and is horizontal, the momentjm 

 constant m (positive upwards) is 



P. 122 0.0K7 {i - l)^'3 

 m = — TTTt - 1 — TT-n (non-dimensional) (2) 



This equation is valid rijht up to tn^j point whore the DubDle touches the plane at its 

 TOxiiTium radius; the equation dift«rs from the results of full numerical intejration Dy, at most, 

 5» of the larger valuss of m, ovssr th.-. range i^ = t.S to 16. 



0. C107 (z - l)^'5 



If the plane is not horizontal the momentum 'i" lifb oirected towards the 



d z 

 plane, is to Pe added vectorially to the momentum 'h^ ^ upwards ouc to jravity. 



^0 



The Momentum Constant m m the presence of other rt ^ id surfaces . 

 m = (attraction coefficient of surface) (l.ll a^^ z^) 



(non-dimensional) (l) 



or m = (attraction coefficient of surface) — ^ 



(non— dimensional) ("Jal 



The attraction coefficient is a geometrical f.^ctcr t?,Buldted in Report B for a number 

 of surfaces and plotted in Fijuri 2 for thi cylinder, disc md sphere. 



Equations (*) and ("ia) are only valid when the maximum bubole radius is small compared 

 to the distance between bubble centra and sjrfic-?. 



Relation betueei momentum constatit m and di sp lac emnnt of bubble at end of 

 fi rst SCI 1 1 ation . 



Displacement h and momentum constant m have the same direction and are related numerically 

 by 



n^ = 3.57 (m- - 0.008) (non-dimensional) (3) 



or more accurately by th- curve in "ijur.; 1. In quation (3) h and m are the numerical values 

 without rejard to sign. 



Minimum radius a when no sur-^aces arc prp.s-'vt. 



aj = 0.U46 Zp"^^" + 0.198 c (non-<limensional) (5) 



This equation agrees with 3 or U pvr cent with the values given in rjgjre 3 oT report A. 

 If surfaces are present tn^n cqu-.tion (7) of th.-it paper must be us;-0. 



Peak Press u re in Pulr.e emitted by Coltapsm? Subble. 

 khen no surfaces are present 



^^m = "^ «"" t 'o - ^o"'"l f«) 



where ....> 



