﻿960 
  Prof. 
  A. 
  H. 
  Gibson 
  on 
  the 
  Motion 
  of 
  

  

  The 
  ratio, 
  k, 
  of 
  the 
  true 
  to 
  the 
  apparent 
  kinetic 
  energy 
  

  

  Z 
  IT 
  

  

  depends 
  on 
  the 
  value 
  of 
  p. 
  It 
  becomes 
  ~- 
  when 
  this 
  

  

  fraction 
  is 
  very 
  small, 
  and 
  has 
  a 
  value 
  of 
  1*58 
  when 
  

  

  z 
  z 
  

  

  :- 
  =-05, 
  and 
  of 
  1*61 
  when 
  ^ 
  = 
  -15. 
  Over 
  the 
  range 
  of 
  

  

  values 
  of 
  ^ 
  found 
  in 
  the 
  experiments, 
  its 
  value 
  may 
  be 
  

  

  taken 
  as 
  1*60, 
  with 
  an 
  error 
  not 
  exceeding 
  1 
  per 
  cent. 
  

  

  Experiments* 
  show 
  that 
  where 
  the 
  velocity 
  of 
  a 
  stream 
  of 
  

   water 
  is 
  reduced 
  by 
  a 
  gradual 
  enlargement 
  of 
  its 
  cross- 
  

   section, 
  the 
  loss 
  of 
  energy 
  accompanying 
  the 
  change 
  from 
  

  

  kinetic 
  to 
  pressure 
  energy 
  is 
  given 
  by 
  k' 
  — 
  l 
  — 
  — 
  — 
  , 
  where 
  v 
  t 
  

  

  is 
  the 
  initial 
  and 
  v 
  2 
  the 
  final 
  velocity, 
  and 
  that 
  if 
  the 
  

   angle 
  of 
  divergence 
  of 
  the 
  boundaries 
  is 
  greater 
  than 
  about 
  

   40°, 
  k' 
  is 
  greater 
  than 
  unity. 
  An 
  examination 
  of 
  photo- 
  

   graphs 
  of 
  long 
  bubbles 
  shows 
  that 
  the 
  effective 
  angle 
  of 
  

   divergence 
  of 
  the 
  streams 
  leaving 
  the 
  annulus 
  is 
  in 
  the 
  

   neighbourhood 
  of 
  60°, 
  for 
  which 
  angle 
  k' 
  is 
  about 
  1*25, 
  so 
  

   that 
  in 
  this 
  case 
  the 
  total 
  loss 
  due 
  to 
  the 
  production 
  of 
  

  

  kinetic 
  energy 
  is 
  given 
  by 
  / 
  kk' 
  »- 
  V 
  where 
  kk' 
  or 
  k" 
  will 
  

  

  have 
  a 
  value 
  somewhat 
  greater 
  than 
  1*60 
  X 
  1*25 
  or 
  2*0. 
  

  

  The 
  exact 
  value 
  of 
  this 
  constant 
  cannot 
  be 
  predetermined 
  

   with 
  any 
  great 
  degree 
  of 
  accuracy. 
  Under 
  the 
  conditions 
  

   outlined 
  it 
  cannot 
  differ 
  greatly 
  from 
  2*0, 
  and 
  in 
  the 
  follow- 
  

   ing 
  calculations 
  has 
  been 
  taken 
  as 
  having 
  this 
  value. 
  

  

  The 
  viscous 
  loss 
  in 
  the 
  annulus 
  is, 
  except 
  in 
  very 
  long 
  

   bubbles, 
  a 
  small 
  fraction 
  of 
  that 
  due 
  to 
  eddy 
  formation 
  in 
  

   the 
  rear 
  of 
  the 
  bubble. 
  An 
  approximation 
  to 
  its 
  value 
  may 
  

   be 
  obtained 
  by 
  considering 
  that 
  since 
  the 
  mean 
  velocity 
  in 
  

   the 
  annulus 
  at 
  the 
  minimum 
  section 
  is 
  given 
  by 
  

  

  

  :)udz 
  

  

  7T(R 
  2 
  -R 
  /2 
  ) 
  7 
  

  

  we 
  have 
  from 
  (9) 
  

  

  dp 
  v 
  . 
  yu-R 
  

  

  \ 
  {U-z)zdz 
  

  

  Jo 
  

  

  If 
  A 
  is 
  the 
  cross-section 
  of 
  the 
  tube, 
  and 
  a' 
  that 
  of 
  the 
  

   * 
  Trans. 
  Roy. 
  Soc. 
  Edin. 
  vol. 
  xlviii. 
  pt. 
  1, 
  p. 
  97 
  (1911). 
  

  

  