﻿786 
  On 
  the 
  Motion 
  of 
  a 
  Viscous 
  Fluid. 
  

  

  Korteweg's 
  equation 
  (6) 
  suggests 
  that 
  such 
  m&y 
  be 
  the 
  

   case. 
  

  

  From 
  data 
  given 
  by 
  Couette 
  I 
  calculate 
  that 
  in 
  C.Gr.S. 
  

   measure 
  

  

  a 
  = 
  '000027. 
  

  

  The 
  tangential 
  traction 
  is 
  thus 
  about 
  a 
  twenty 
  thousandth 
  

   part 
  of 
  the 
  pressure 
  O2/0U 
  2 
  ) 
  due 
  to 
  to 
  the 
  normal 
  impact 
  of 
  

   the 
  fluid 
  moving 
  with 
  velocity 
  U. 
  

  

  Even 
  in 
  cases 
  where 
  the 
  steady 
  motion 
  of 
  a 
  viscous 
  fluid 
  

   satisfying 
  the 
  dynamical 
  equations 
  is 
  certainly 
  unstable, 
  

   there 
  is 
  a 
  distinction 
  to 
  be 
  attended 
  to 
  which 
  is 
  not 
  without 
  

   importance. 
  It 
  may 
  be 
  a 
  question 
  of 
  the 
  time 
  during 
  which 
  

   the 
  fluid 
  remains 
  in 
  an 
  unstable 
  condition. 
  When 
  fluid 
  

   moves 
  between 
  two 
  coaxal 
  cylinders, 
  the 
  instability 
  has 
  an 
  

   indefinite 
  time 
  in 
  which 
  to 
  develop 
  itself. 
  But 
  it 
  is 
  other- 
  

   wise 
  in 
  many 
  important 
  problems. 
  Suppose 
  that 
  fluid 
  has 
  

   to 
  move 
  through 
  a 
  narrow 
  place, 
  being 
  guided 
  for 
  example 
  

   by 
  hyperbolic 
  surfaces, 
  either 
  in 
  two 
  dimensions, 
  or 
  in 
  three 
  

   with 
  symmetry 
  about 
  an 
  axis. 
  If 
  the 
  walls 
  have 
  suitable 
  

   tangential 
  velocities, 
  the 
  motion 
  may 
  be 
  irrotational. 
  This 
  

   irrotational 
  motion 
  is 
  that 
  which 
  would 
  be 
  initiated 
  from 
  

   rest 
  by 
  propellent 
  impulses 
  acting 
  at 
  a 
  distance. 
  If 
  the 
  

   viscosity 
  were 
  great, 
  the 
  motion 
  would 
  be 
  steady 
  and 
  stable 
  ; 
  

   if 
  the 
  viscosity 
  is 
  less, 
  it 
  still 
  satisfies 
  the 
  dynamical 
  equa- 
  

   tions, 
  but 
  is 
  (presumably) 
  unstable. 
  But 
  the 
  instability, 
  as 
  

   it 
  affects 
  any 
  given 
  portion 
  of 
  the 
  fluid, 
  has 
  a 
  very 
  short 
  

   duration. 
  Only 
  as 
  it 
  approaches 
  the 
  narrows 
  has 
  the 
  fluid 
  

   any 
  considerable 
  velocity, 
  and 
  as 
  soon 
  as 
  the 
  narrows 
  are 
  

   passed 
  the 
  velocity 
  falls 
  off 
  again. 
  Under 
  these 
  circum- 
  

   stances 
  it 
  would 
  seem 
  probable 
  that 
  the 
  instability 
  in 
  the 
  

   narrows 
  would 
  be 
  of 
  little 
  consequence, 
  and 
  that 
  the 
  irro- 
  

   tational 
  motion 
  would 
  practically 
  hold 
  its 
  own. 
  If 
  this 
  be 
  

   so, 
  the 
  tangential 
  movement 
  of 
  the 
  walls 
  exercises 
  a 
  profound 
  

   influence, 
  causing 
  the 
  fluid 
  to 
  follow 
  the 
  walls 
  on 
  the 
  down 
  

   stream 
  side, 
  instead 
  of 
  shooting 
  onwards 
  as 
  a 
  jet 
  — 
  the 
  be- 
  

   haviour 
  usually 
  observed 
  when 
  fluid 
  is 
  invited 
  to 
  follow 
  

   fixed 
  divergent 
  walls, 
  unless 
  indeed 
  the 
  expansion 
  is 
  very 
  

   gradual. 
  

  

  July 
  1913. 
  

  

  