﻿Long 
  Air-Bubbles 
  in 
  a 
  Vertical 
  Tube, 
  

  

  963 
  

  

  and 
  on 
  introducing 
  this 
  value 
  in 
  equation 
  (4), 
  or, 
  if 
  necessary, 
  

   in 
  (4) 
  modified 
  for 
  the 
  effect 
  of 
  restricted 
  depth, 
  the 
  wave- 
  

   length 
  I 
  is 
  determined. 
  

  

  When 
  examining 
  any 
  particular 
  tube 
  it 
  is 
  simpler 
  to 
  write 
  

   V 
  = 
  mA, 
  and 
  to 
  obtain 
  m 
  from 
  the 
  values 
  of 
  V 
  given 
  in 
  

   Table 
  I., 
  or 
  from 
  formula 
  (18). 
  Then 
  (17) 
  becomes 
  

  

  a[ 
  _ 
  gL(Km 
  + 
  2cc) 
  / 
  ( 
  gL(Km-h2u) 
  ) 
  2 
  a 
  ^£, 
  

  

  A~ 
  *2(V 
  2 
  -agL) 
  ±V 
  \ 
  2(V 
  2 
  -*gL) 
  J 
  + 
  W^ot< 
  

  

  gjj 
  

  

  (19) 
  

  

  e.g. 
  

  

  Tube 
  1*680 
  cm. 
  diam.: 
  Let 
  L 
  = 
  5 
  cm. 
  

   Y= 
  15-05 
  ; 
  m 
  (calculated) 
  = 
  6*80 
  ; 
  

  

  - 
  2 
  x 
  -0738 
  ,_„ 
  

   a 
  = 
  1 
  "^840^ 
  =9653 
  ' 
  

  

  Km 
  = 
  12x 
  -00001333 
  x6*80 
  = 
  *00109. 
  

  

  a' 
  

   On 
  substituting 
  these 
  values 
  in 
  (19) 
  we 
  get 
  -r- 
  = 
  *819, 
  

  

  XX 
  

  

  A 
  

  

  1 
  

  

  = 
  5*52, 
  

  

  A-a' 
  ~ 
  -181 
  

   *; 
  = 
  15-05x 
  5*52 
  = 
  82*9 
  cm.sec, 
  

   r' 
  = 
  -840x 
  V*819 
  = 
  -750cm. 
  

  

  Similarly 
  if 
  L 
  = 
  2 
  cm.; 
  ^ 
  =-7377; 
  v= 
  57*2; 
  R' 
  = 
  *720. 
  

  

  L 
  = 
  6 
  

   L 
  = 
  8 
  

  

  = 
  •8324; 
  v= 
  89*9; 
  K/ 
  = 
  *766. 
  

   = 
  •8500; 
  r=100-2; 
  R' 
  = 
  -775. 
  

  

  In 
  the 
  limit 
  when 
  L 
  is 
  large, 
  -r- 
  approximates 
  to 
  *967, 
  and 
  

  

  v 
  to 
  455 
  cm.sec. 
  Before 
  this 
  velocity 
  is 
  attained, 
  however, 
  

   the 
  motion 
  breaks 
  down 
  and, 
  as 
  is 
  evident 
  from 
  the 
  ap- 
  

   pearance 
  of 
  the 
  walls 
  of 
  the 
  bubble, 
  eddy 
  formation 
  takes 
  

   place 
  in 
  the 
  annulus. 
  This 
  point 
  will 
  receive 
  notice 
  at 
  a 
  

   later 
  stage 
  of 
  the 
  paper. 
  

  

  Values 
  of 
  v 
  and 
  of 
  B/, 
  and 
  hence 
  of 
  A, 
  the 
  thickness 
  of 
  the 
  

   film 
  of 
  water 
  in 
  which 
  ripple 
  formation 
  takes 
  place, 
  have 
  

   been 
  calculated 
  as 
  above 
  for 
  the 
  tubes 
  of 
  1*680, 
  2*695, 
  and 
  

   5*09 
  cm. 
  diam. 
  In 
  every 
  case 
  it 
  appears 
  that 
  the 
  ratio 
  of 
  

   li-^l 
  is 
  greater 
  than 
  *75, 
  and 
  consequently 
  formula 
  (4) 
  has 
  

  

  