﻿Motion 
  of 
  a 
  Viscous 
  Fluid. 
  785 
  

  

  When 
  fi 
  is 
  small 
  and 
  there 
  is 
  no 
  special 
  limitation 
  upon 
  the 
  

   disturbance, 
  instability 
  probably 
  prevails. 
  The 
  question 
  

   whether 
  //, 
  is 
  to 
  be 
  considered 
  great 
  or 
  small 
  depends 
  of 
  

   course 
  upon 
  the 
  other 
  data 
  of 
  the 
  problem. 
  If 
  D 
  be 
  the 
  

   distance 
  between 
  the 
  planes, 
  we 
  have 
  to 
  deal 
  with 
  BD 
  2 
  /v 
  

   (Reynolds) 
  . 
  

  

  In 
  an 
  important 
  paper 
  * 
  Orr, 
  starting 
  from 
  equation 
  (34), 
  

   has 
  shown 
  that 
  if 
  BD 
  2 
  /i>is 
  less 
  than 
  177 
  " 
  every 
  disturbance 
  

   must 
  automatically 
  decrease, 
  and 
  that 
  (for 
  a 
  higher 
  value 
  

   than 
  177) 
  it 
  is 
  possible 
  to 
  prescribe 
  a 
  disturbance 
  which 
  

   will 
  increase 
  for 
  a 
  time/' 
  We 
  must 
  not 
  infer 
  that 
  when 
  

   BD 
  2 
  /j/>177 
  the 
  regular 
  motion 
  is 
  necessarily 
  unstable. 
  As 
  

   the 
  fluid 
  moves 
  under 
  the 
  laws 
  of 
  dynamics, 
  the 
  initial 
  

   increase 
  of 
  certain 
  disturbances 
  may 
  after 
  a 
  time 
  be 
  ex- 
  

   changed 
  for 
  a 
  decrease, 
  and 
  this 
  decrease 
  may 
  be 
  without 
  

   limit. 
  

  

  At 
  the 
  other 
  extreme 
  when 
  v 
  is 
  very 
  small, 
  observation 
  

   shows 
  that 
  the 
  tangential 
  traction 
  on 
  the 
  walls, 
  moving 
  (say) 
  

   with 
  velocities 
  ±U, 
  tends 
  to 
  a 
  statistical 
  uniformity 
  and 
  to 
  

   become 
  proportional, 
  no 
  longer 
  to 
  U, 
  but 
  to 
  IP. 
  If 
  we 
  

   assume 
  this 
  law 
  to 
  be 
  absolute 
  in 
  the 
  region 
  of 
  high 
  velocity, 
  

   the 
  principle 
  of 
  dynamical 
  similarity 
  leads 
  to 
  rather 
  re- 
  

   markable 
  conclusions. 
  For 
  the 
  tangential 
  traction, 
  having 
  

   the 
  dimensions 
  of 
  a 
  pressure, 
  must 
  in 
  general 
  be 
  of 
  the 
  form 
  

  

  ^•/(ot)- 
  ( 
  37 
  > 
  

  

  D 
  being 
  the 
  distance 
  between 
  the 
  walls, 
  and 
  / 
  an 
  arbitrary 
  

   function. 
  In 
  the 
  regular 
  motion 
  (z 
  large) 
  f 
  (z) 
  = 
  %z, 
  and 
  

   (37) 
  is 
  proportional 
  to 
  U. 
  If 
  (37) 
  is 
  proportional 
  to 
  IP, 
  

   /must 
  be 
  a 
  constant 
  and 
  the 
  traction 
  becomes 
  independent 
  

   not 
  only 
  of 
  //,, 
  but 
  also 
  of 
  D. 
  

  

  If 
  the 
  velocity 
  be 
  not 
  quite 
  so 
  great 
  as 
  to 
  reduce 
  / 
  to 
  

   constancy, 
  we 
  may 
  take 
  

  

  ■ 
  f-(z)-a 
  + 
  bz, 
  

  

  where 
  a 
  and 
  b 
  are 
  numerical 
  constants, 
  so 
  that 
  (37) 
  becomes 
  

  

  apIF 
  + 
  fyaU/D 
  (38) 
  

  

  It 
  could 
  not 
  be 
  assumed 
  without 
  further 
  proof 
  that 
  b 
  

   has 
  the 
  value 
  (2) 
  appropriate 
  to 
  a 
  large 
  z 
  ; 
  nevertheless, 
  

  

  * 
  Proc. 
  Rov. 
  Irish 
  Acad. 
  1907. 
  

  

  