﻿Laminar 
  Motion 
  of 
  an 
  Inviscid 
  Fluid. 
  

  

  1009 
  

  

  velocity 
  U 
  in 
  the 
  laminar 
  motion 
  satisfies 
  generally 
  an 
  equation 
  

   of 
  the 
  form 
  

  

  <™-% 
  (19) 
  

  

  dt 
  

  

  ay- 
  

  

  vrith 
  the 
  conditions 
  that 
  initially 
  (t 
  = 
  Q) 
  U 
  = 
  0,and 
  that 
  from 
  

   t 
  = 
  onwards 
  U 
  = 
  l 
  when 
  y=0, 
  and 
  (if 
  we 
  please) 
  U 
  = 
  

   when 
  */ 
  = 
  /. 
  We 
  might 
  employ 
  Fourier's 
  solution, 
  but 
  all 
  

   that 
  we 
  require 
  follows 
  at 
  once 
  from 
  the 
  differential 
  equation 
  

   itself. 
  It 
  is 
  evident 
  that 
  dJJ/dt, 
  and 
  therefore 
  d 
  2 
  \Jjdy 
  2 
  , 
  is 
  

   everywhere 
  positive 
  and 
  accordingly 
  that 
  a 
  non-viscous 
  liquid, 
  

   moving 
  laminarly 
  as 
  the 
  viscous 
  fluid 
  moves 
  in 
  any 
  of 
  these 
  

   stages, 
  is 
  stable. 
  It 
  would 
  appear 
  then 
  that 
  no 
  explanation 
  

   is 
  to 
  be 
  found 
  in 
  this 
  direction. 
  

  

  Hitherto 
  we 
  have 
  supposed 
  that 
  the 
  disturbance 
  is 
  periodic 
  

   as 
  regards 
  x, 
  but 
  a 
  simple 
  example, 
  not 
  coining 
  under 
  this 
  

   head, 
  may 
  be 
  worthy 
  of 
  notice. 
  It 
  is 
  that 
  of 
  the 
  disturbance 
  

   due 
  to 
  a 
  single 
  vortex 
  filament 
  in 
  which 
  the 
  vorticity 
  differs 
  

   from 
  the 
  otherwise 
  uniform 
  vorticity 
  of 
  the 
  neighbouring- 
  

   fluid. 
  In 
  the 
  figure 
  the 
  lines 
  AA, 
  BB 
  represent 
  the 
  situation 
  

  

  of 
  the 
  walls 
  and 
  AM 
  the 
  velocity-curve 
  of 
  the 
  original 
  

   shearing 
  motion 
  rising 
  from 
  zero 
  at 
  A 
  to 
  a 
  finite 
  value 
  at 
  M. 
  

   For 
  the 
  present 
  purpose, 
  however, 
  we 
  suppose 
  material 
  walls 
  

   to 
  be 
  absent, 
  but 
  that 
  the 
  same 
  effect 
  (of 
  prohibiting 
  normal 
  

   motion) 
  is 
  arrived 
  at 
  by 
  suitable 
  suppositions 
  as 
  to 
  the 
  fluid 
  

   lying 
  outside 
  and 
  now 
  imagined 
  infinite. 
  It 
  is 
  only 
  necessary 
  

   to 
  continue 
  the 
  velocity-curve 
  in 
  the 
  manner 
  shown 
  AMCN 
  . 
  . 
  ., 
  

   the 
  vorticity 
  in 
  the 
  alternate 
  layers 
  of 
  equal 
  width 
  being 
  

   equal 
  and 
  opposite. 
  Symmetry 
  then 
  shows 
  that 
  under 
  the 
  

   operation 
  of 
  these 
  vorticities 
  the 
  fluid 
  moves 
  as 
  if 
  AA, 
  BB, 
  &c. 
  

   were 
  material 
  walls. 
  

  

  We 
  have 
  now 
  to 
  trace 
  the 
  effect 
  of 
  an 
  additional 
  vorticity, 
  

   supposed 
  positive, 
  at 
  a 
  point 
  P. 
  If 
  the 
  wall 
  AA 
  were 
  alone 
  

   concerned, 
  its 
  effect 
  would 
  be 
  imitated 
  by 
  the 
  introduction 
  

   of 
  an 
  opposite 
  vorticity 
  at 
  the 
  point 
  Q 
  which 
  is 
  the 
  image 
  

  

  