﻿Motion 
  of 
  a 
  Viscous 
  Fluid, 
  779 
  

  

  so 
  that 
  (13) 
  reduces 
  to 
  

  

  <^-Tf)W-^ 
  d 
  * 
  dyd 
  >- 
  

  

  Since 
  the 
  sign 
  of 
  C 
  is 
  at 
  disposal, 
  this 
  may 
  be 
  made 
  

   positive 
  or 
  negative 
  at 
  pleasure. 
  Also 
  F' 
  in 
  (2) 
  may 
  be 
  

   neglected 
  as 
  of 
  the 
  second 
  order 
  when 
  u', 
  v\ 
  w' 
  are 
  small 
  

   enough. 
  It 
  follows 
  that 
  F 
  is 
  not 
  an 
  absolute 
  minimum 
  for 
  

   z/ 
  , 
  v 
  , 
  iv 
  , 
  unless 
  the 
  conditions 
  (9) 
  are 
  satisfied. 
  

  

  Korteweg 
  has 
  also 
  shown 
  that 
  the 
  slow 
  motion 
  of 
  a 
  viscous 
  

   fluid 
  denoted 
  by 
  M 
  is 
  stable. 
  " 
  When 
  in 
  a 
  given 
  region 
  

   occupied 
  by 
  viscous 
  incompressible 
  fluid 
  there 
  exists 
  at 
  a 
  

   certain 
  moment 
  a 
  mode 
  of 
  motion 
  M 
  which 
  does 
  not 
  satisfy 
  

   equation 
  (-4), 
  then, 
  the 
  velocities 
  along 
  the 
  boundary 
  being 
  

   maintained 
  constant, 
  the 
  change 
  which 
  must 
  occur 
  in 
  the 
  

   mode 
  of 
  motion 
  will 
  be 
  such 
  (neglecting 
  squares 
  and 
  pro- 
  

   ducts 
  of 
  velocities) 
  that 
  the 
  dissipation 
  of 
  energy 
  by 
  internal 
  

   friction 
  is 
  constantly 
  decreasing 
  till 
  it 
  reaches 
  the 
  value 
  F 
  

   and 
  the 
  mode 
  of 
  motion 
  becomes 
  identical 
  with 
  M 
  ." 
  

  

  This 
  theorem 
  admits 
  of 
  instantaneous 
  proof. 
  If 
  the 
  terms 
  

   of 
  the 
  second 
  order 
  are 
  omitted, 
  the 
  equations 
  of 
  motion, 
  

   such 
  as 
  (7), 
  are 
  linear, 
  and 
  any 
  two 
  solutions 
  may 
  be 
  

   superposed. 
  Consider 
  two 
  solutions, 
  both 
  giving 
  the 
  same 
  

   velocities 
  at 
  the 
  boundary. 
  Then 
  the 
  difference 
  of 
  these 
  is 
  

   also 
  a 
  solution 
  representing 
  a 
  possible 
  motion 
  with 
  zero 
  

   velocities 
  at 
  the 
  boundary. 
  But 
  such 
  a 
  motion 
  necessarily 
  

   comes 
  to 
  rest. 
  Hence 
  with 
  flux 
  of 
  time 
  the 
  two 
  original 
  

   motions 
  tend 
  to 
  become 
  and 
  to 
  remain 
  identical. 
  If 
  one 
  of 
  

   these 
  is 
  the 
  steady 
  motion, 
  the 
  other 
  must 
  tend 
  to 
  become 
  

   coincident 
  with 
  it. 
  

  

  The 
  stability 
  of 
  the 
  sloio 
  steady 
  motion 
  of 
  a 
  viscous 
  fluid, 
  

   or 
  (as 
  we 
  may 
  also 
  put 
  it) 
  the 
  steady 
  motion 
  of 
  a 
  very 
  

   viscous 
  fluid, 
  is 
  thus 
  ensured. 
  When 
  the 
  circumstances 
  are 
  

   such 
  that 
  the 
  terms 
  of 
  the 
  second 
  order 
  must 
  be 
  retained, 
  

   there 
  is 
  but 
  little 
  definite 
  knowledge 
  as 
  to 
  the 
  character 
  

   of 
  the 
  motion 
  in 
  respect 
  of 
  stability. 
  Viscous 
  fluid, 
  con- 
  

   tained 
  in 
  a 
  vessel 
  which 
  rotates 
  with 
  uniform 
  velocity, 
  would 
  

   be 
  expected 
  to 
  acquire 
  the 
  same 
  rotation 
  and 
  ultimately 
  to 
  

   revolve 
  as 
  a 
  solid 
  body, 
  but 
  the 
  expectation 
  is 
  perhaps 
  

   founded 
  rather 
  upon 
  observation 
  than 
  upon 
  theory. 
  We 
  

   might, 
  however, 
  argue 
  that 
  any 
  other 
  event 
  would 
  involve 
  

   perpetual 
  dissipation 
  which 
  could 
  only 
  be 
  met 
  by 
  a 
  driving 
  

   force 
  applied 
  to 
  the 
  vessel, 
  since 
  the 
  kinetic 
  energy 
  of 
  the 
  

   motion 
  could 
  not 
  for 
  ever 
  diminish. 
  And 
  such 
  a 
  maintained 
  

   driving 
  couple 
  would 
  generate 
  angular 
  momentum 
  without 
  

  

  