﻿Laminar 
  Motion 
  of 
  an 
  Inviscid 
  Fluid. 
  1003 
  

  

  shows, 
  n 
  may 
  be 
  complex 
  and 
  the 
  disturbance 
  exponentially 
  

   unstable. 
  

  

  Two 
  special 
  cases 
  at 
  once 
  suggest 
  themselves. 
  If 
  the 
  

   motion 
  be 
  that 
  which 
  is 
  possible 
  to 
  a 
  viscous 
  fluid 
  moving 
  

   steadily 
  between 
  two 
  fixed 
  walls 
  under 
  external 
  pressure 
  or 
  

   impressed 
  force, 
  so 
  that 
  for 
  example 
  U==y 
  2 
  — 
  b 
  2 
  , 
  d 
  2 
  U/dy 
  2 
  

   is 
  a 
  finite 
  constant, 
  and 
  complex 
  values 
  of 
  n 
  are 
  clearly 
  

   excluded. 
  In 
  the 
  case 
  of 
  a 
  simple 
  shearing 
  motion, 
  ex- 
  

   emplified 
  by 
  U 
  = 
  ?/, 
  d 
  2 
  U/dy 
  2 
  =^0, 
  and 
  no 
  inference 
  can 
  be 
  

   drawn 
  from 
  (4). 
  But 
  referring 
  back 
  to 
  (3), 
  we 
  see 
  that 
  in 
  

   this 
  case 
  if 
  n 
  be 
  complex, 
  

  

  $-^' 
  = 
  ( 
  5 
  > 
  

  

  would 
  have 
  to 
  be 
  satisfied 
  over 
  the 
  whole 
  range 
  between 
  the 
  

   limits 
  where 
  i? 
  = 
  0. 
  Since 
  such 
  satisfaction 
  is 
  not 
  possible, 
  

   we 
  infer 
  that 
  here 
  too 
  a 
  complex 
  n 
  is 
  excluded. 
  

  

  It 
  may 
  appear 
  at 
  first 
  sight 
  as 
  if 
  real, 
  as 
  well 
  as 
  complex, 
  

   values 
  of 
  n 
  were 
  excluded 
  by 
  this 
  argument. 
  But 
  if 
  n 
  be 
  

   such 
  that 
  n/i-fU 
  vanishes 
  anywhere 
  within 
  the 
  range, 
  

   (5) 
  need 
  not 
  there 
  be 
  satisfied. 
  In 
  other 
  words, 
  the 
  

   arbitrary 
  constants 
  which 
  enter 
  into 
  the 
  solution 
  of 
  (5) 
  

   may 
  there 
  change 
  values, 
  subject 
  only 
  to 
  the 
  condition 
  

   of 
  making 
  v 
  continuous. 
  The 
  terminal 
  conditions 
  can 
  

   then 
  be 
  satisfied. 
  Thus 
  any 
  value 
  of 
  —n/k 
  is 
  admissible 
  

   which 
  coincides 
  with 
  a 
  value 
  of 
  U 
  to 
  be 
  found 
  within 
  the 
  

   range. 
  But 
  other 
  real 
  values 
  of 
  n 
  are 
  excluded. 
  

  

  Let 
  us 
  now 
  examine 
  how 
  far 
  the 
  above 
  argument 
  applies 
  

   to 
  real 
  values 
  of 
  n, 
  when 
  d' 
  2 
  U/dy 
  2 
  in 
  (3) 
  does 
  not 
  vanish 
  

   throughout. 
  It 
  is 
  easy 
  to 
  recognize 
  that 
  here 
  also 
  any 
  

   value 
  of 
  — 
  k~U 
  is 
  admissible, 
  and 
  for 
  the 
  same 
  reason 
  as 
  

   before, 
  viz., 
  that 
  when 
  w 
  + 
  HJ 
  — 
  0, 
  dvjdy 
  may 
  be 
  dis- 
  

   continuous. 
  Suppose, 
  for 
  example, 
  that 
  there 
  is 
  but 
  one 
  

   place 
  where 
  n 
  + 
  /;U 
  = 
  0. 
  We 
  may 
  start 
  from 
  either 
  wall 
  

   with 
  v 
  = 
  and 
  with 
  an 
  arbitrary 
  value 
  of 
  dvjdy 
  and 
  

   gradually 
  build 
  up 
  the 
  solutions 
  inwards 
  so 
  as 
  to 
  satisfy 
  (3)*. 
  

   The 
  process 
  is 
  to 
  be 
  continued 
  on 
  both 
  sides 
  until 
  we 
  come 
  

   to 
  the 
  place 
  where 
  n 
  + 
  yfcU 
  = 
  0. 
  The 
  two 
  values 
  there 
  found 
  

   for 
  v 
  and 
  for 
  dvjdy 
  will 
  presumably 
  disagree. 
  But 
  by 
  

   suitable 
  choice 
  of 
  the 
  relative 
  initial 
  values 
  of 
  dvjdy, 
  

   v 
  may 
  be 
  made 
  continuous, 
  and 
  (as 
  has 
  been 
  said) 
  a 
  dis- 
  

   continuity 
  in 
  dvjdy 
  does 
  not 
  interfere 
  with 
  the 
  satisfaction 
  

   of 
  (3). 
  If 
  there 
  are 
  other 
  places 
  where 
  U 
  has 
  the 
  same 
  

  

  * 
  Graphically, 
  the 
  equation 
  directs 
  us 
  with 
  what 
  curvature 
  to 
  proceed 
  

   at 
  anv 
  point 
  already 
  reached. 
  

  

  3 
  Y2 
  

  

  