﻿[ 
  7 
  ™ 
  ] 
  

  

  LXV. 
  On 
  the 
  Motion 
  of 
  a 
  Viscous 
  Fluid. 
  

   By 
  Lord 
  Rayleigh, 
  031., 
  F.B.S.* 
  

  

  IT 
  has 
  been 
  proved 
  by 
  Helmholtz 
  f 
  and 
  Korteweg 
  t 
  that 
  

   when 
  the 
  velocities 
  at 
  the 
  boundary 
  are 
  given, 
  the 
  slow- 
  

   steady 
  motion 
  of 
  an 
  incompressible 
  viscous 
  liquid 
  satisfies 
  

   the 
  condition 
  of 
  making 
  F, 
  the 
  dissipation, 
  an 
  absolute 
  

   minimum. 
  If 
  u 
  , 
  v 
  , 
  w 
  be 
  the 
  velocities 
  in 
  one 
  motion 
  M 
  , 
  

   and 
  a, 
  v, 
  w 
  those 
  of 
  another 
  motion 
  M 
  satisfying 
  the 
  same 
  

   boundary 
  conditions, 
  the 
  difference 
  of 
  the 
  two 
  u\ 
  v', 
  w\ 
  

   where 
  

  

  u=u—u 
  , 
  v'=v—v 
  , 
  iv'=iv 
  — 
  iv 
  , 
  . 
  . 
  (1) 
  

  

  will 
  constitute 
  a 
  motion 
  M' 
  such 
  that 
  the 
  boundary 
  velocities 
  

   vanish. 
  If 
  F 
  , 
  F, 
  F' 
  denote 
  the 
  dissipation-functions 
  for 
  the 
  

   three 
  motions 
  M 
  , 
  M, 
  M' 
  respectively, 
  all 
  being 
  of 
  necessity 
  

   positive, 
  it 
  is 
  shown 
  that 
  

  

  F 
  = 
  F 
  +F— 
  2ji\(u'S/ 
  2 
  u 
  +v'V\+w'V 
  2 
  iv 
  )divd'i/dz, 
  . 
  (2) 
  

  

  the 
  integration 
  being 
  over 
  the 
  whole 
  volume. 
  Also 
  

  

  F= 
  -//, 
  j 
  (u'VV 
  + 
  fl'VV 
  + 
  w'W) 
  dx 
  dy 
  dz 
  

  

  Crfdw' 
  dv 
  ! 
  \ 
  2 
  , 
  tdu' 
  dw'\ 
  2 
  

   = 
  ^l\-^--dz) 
  + 
  Kdz-J^) 
  

  

  + 
  (£-$>** 
  w 
  

  

  These 
  equations 
  are 
  purely 
  kinematical, 
  if 
  we 
  include 
  

   under 
  that 
  head 
  the 
  incompressibility 
  of 
  the 
  fluid. 
  In 
  the 
  

   application 
  of 
  them 
  by 
  Helmholtz 
  and 
  Korteweg 
  the 
  motion 
  

   M 
  is 
  supposed 
  to 
  be 
  that 
  which 
  would 
  be 
  steady 
  if 
  small 
  

   enough 
  to 
  allow 
  the 
  neglect 
  of 
  the 
  terms 
  involving 
  the 
  

   second 
  powers 
  of 
  the 
  velocities 
  in 
  the 
  dynamical 
  equations. 
  

   We 
  then 
  have 
  

  

  fiV%u 
  , 
  v 
  , 
  w 
  ) 
  = 
  ^, 
  ^, 
  faYVfi+Po)* 
  • 
  ( 
  4 
  ) 
  

  

  where 
  V 
  is 
  the 
  potential 
  of 
  impressed 
  forces. 
  In 
  virtue 
  

   of 
  (4) 
  

  

  \(u'T7 
  2 
  u 
  + 
  v'V 
  2 
  v 
  + 
  io'T7 
  a 
  wo)da: 
  dy 
  dz 
  = 
  0, 
  . 
  (5) 
  

  

  * 
  Communicated 
  by 
  the 
  Author. 
  

  

  f 
  ' 
  Collected 
  Works/ 
  vol. 
  i. 
  p. 
  223 
  (1869). 
  

  

  \ 
  Phil. 
  Mag. 
  vol. 
  xvi. 
  p. 
  112 
  (1883). 
  

  

  