﻿Long 
  Air-Bubbles 
  in 
  a 
  Vertical 
  Tube. 
  959 
  

  

  of 
  its 
  wails 
  shows 
  that 
  this 
  flow 
  is 
  steady, 
  i. 
  e. 
  unaccompanied 
  

   by 
  eddy 
  formation, 
  and 
  under 
  these 
  circumstances 
  the 
  

   velocity 
  at 
  any 
  radius 
  x 
  may 
  readily 
  be 
  shown 
  to 
  be 
  given 
  

  

  J. 
  dp 
  

  

  JB 
  2 
  -^ 
  + 
  Blog^ 
  + 
  CJ, 
  > 
  m 
  (7) 
  

  

  where 
  B 
  is 
  the 
  radius 
  of 
  the 
  tube, 
  

  

  " 
  TtT 
  iS 
  ^ 
  e 
  P 
  ressure 
  drop 
  per 
  unit 
  length 
  of 
  the 
  annulus, 
  

   aLl 
  necessary 
  to 
  maintain 
  steady 
  flow 
  against 
  viscous 
  

   resistance. 
  

   Determining 
  the 
  constants 
  B 
  and 
  C 
  to 
  satisfy 
  the 
  conditions 
  

  

  that 
  u 
  = 
  when 
  #=R, 
  and 
  -v- 
  =0 
  when 
  ,v 
  = 
  H\ 
  i. 
  e. 
  at 
  the 
  

  

  dx 
  

  

  surface 
  of 
  the 
  bubble, 
  this 
  becomes 
  

  

  If 
  x 
  is 
  nearly 
  =B, 
  say 
  x=H 
  — 
  z 
  where 
  -^ 
  is 
  small, 
  this 
  

   may 
  be 
  written 
  

  

  "-kH^-^Hi+A* 
  ■■■)} 
  

  

  = 
  ii,^- 
  w)-(^+ 
  ■■■}■■ 
  «• 
  

  

  and 
  as 
  in 
  these 
  experiments, 
  as 
  will 
  be 
  shown 
  later, 
  the 
  

   value 
  of 
  -^ 
  ranges 
  from 
  about 
  *07 
  to 
  a 
  maximum 
  of 
  about 
  

  

  •16, 
  expression 
  (9) 
  affords 
  a 
  sufficiently 
  close 
  approximation. 
  

   If 
  B 
  / 
  be 
  the 
  radius 
  of 
  the 
  bubble 
  at 
  its 
  maximum 
  cross- 
  

   section, 
  the 
  kinetic 
  energy 
  of 
  flow 
  past 
  the 
  minimum 
  section 
  

   of 
  the 
  annulus 
  is 
  given 
  by 
  

  

  /»E-R' 
  

  

  — 
  . 
  2tt 
  i 
  (B 
  — 
  z)u 
  z 
  dz 
  per 
  second, 
  

   *9 
  Jo 
  

  

  while 
  if 
  i* 
  be 
  the 
  mean 
  velocity 
  of 
  flow 
  past 
  this 
  cross-section, 
  

   the 
  apparent 
  kinetic 
  energy 
  is 
  measured 
  by 
  

  

  £ 
  77-(B 
  2 
  -B'> 
  3 
  = 
  Z 
  v 
  2 
  per 
  c.c. 
  

  

  * 
  For 
  the 
  suggestion 
  embodied 
  in 
  expressions 
  (8) 
  and 
  (9), 
  as 
  for 
  many 
  

   other 
  helpful 
  suggestions 
  in 
  the 
  mathematical 
  portion 
  of 
  the 
  paper, 
  the 
  

   author 
  would 
  express 
  his 
  indebtedness 
  to 
  Professor 
  Horace 
  Lamb. 
  

  

  