﻿Problems 
  connected 
  with 
  the 
  Circular 
  Arc. 
  403 
  

  

  Let 
  the 
  total 
  thrust, 
  which 
  evidently 
  must 
  act 
  through 
  the 
  

   centre, 
  have 
  components 
  P^, 
  P 
  y 
  , 
  then 
  

  

  P-p 
  = 
  I 
  Ap 
  sin 
  0d0 
  = 
  7rp 
  sin 
  2 
  a 
  cos 
  a 
  sin 
  ft 
  cos 
  ft, 
  , 
  . 
  . 
  . 
  

  

  Y 
  y 
  = 
  — 
  I 
  Ap 
  cos 
  6 
  dd—— 
  np 
  sin 
  2 
  a 
  ( 
  cos 
  2 
  /3 
  sin 
  2 
  -= 
  -f 
  sin 
  2 
  /3 
  cos 
  

  

  These 
  results 
  are 
  apparently 
  in 
  contradiction 
  with 
  the 
  

   known 
  fact 
  that 
  the 
  resultant 
  of 
  the 
  pressures 
  should 
  reduce 
  

   to 
  a 
  couple. 
  The 
  explanation 
  is 
  that 
  the 
  velocity 
  at 
  the 
  edge 
  

   becomes 
  infinite, 
  and 
  consequently 
  the 
  pressure 
  also, 
  and 
  

   though 
  the 
  thickness 
  of 
  the 
  lamina 
  is 
  assumed 
  to 
  be 
  infinitely 
  

   small, 
  yet 
  considered 
  as 
  the 
  limiting 
  case 
  of 
  a 
  thin 
  lamina, 
  

   with 
  finite 
  velocities 
  and 
  pressures, 
  the 
  resultant 
  pressure 
  at 
  

   the 
  end 
  tends 
  to 
  a 
  finite 
  limit 
  as 
  the 
  thickness 
  decreases. 
  

   To 
  see 
  this, 
  consider 
  the 
  case 
  of 
  the 
  flow 
  round 
  a 
  semi-infinite 
  

   plane, 
  given 
  by 
  w 
  — 
  Az*. 
  Suppose 
  the 
  fluid 
  inside 
  one 
  of 
  

   the 
  parabolic 
  stream 
  lines 
  is 
  solidified. 
  The 
  motion 
  of 
  the 
  

   remaining 
  fluid 
  is 
  unaffected, 
  and 
  when 
  we 
  calculate 
  the 
  

   resultant 
  thrust 
  exerted 
  by 
  it 
  on 
  the 
  solid, 
  we 
  obtain 
  a 
  force 
  

  

  j/oA 
  2 
  parallel 
  to 
  the 
  axis, 
  tending 
  to 
  drag 
  the 
  solid 
  further 
  

  

  into 
  the 
  liquid. 
  The 
  magnitude 
  of 
  this 
  force 
  is 
  independent 
  

   of 
  the 
  particular 
  stream-line 
  selected. 
  Turning 
  now 
  to 
  the 
  

   problem 
  in 
  band, 
  the 
  flow 
  in 
  the 
  vicinity 
  of 
  the 
  edge 
  is 
  of 
  

   the 
  type 
  just 
  mentioned, 
  for 
  on 
  substituting 
  e 
  ta 
  { 
  — 
  t 
  + 
  f) 
  for 
  z, 
  

   where 
  f 
  is 
  small, 
  we 
  obtain 
  from 
  (14) 
  

  

  io 
  = 
  const— 
  \/2 
  sin 
  a 
  . 
  sinf- 
  — 
  ft 
  J 
  fa-f- 
  higher 
  powers 
  of 
  f. 
  (23) 
  

  

  Hence 
  there 
  is 
  a 
  force 
  along 
  the 
  tangent 
  at 
  the 
  end, 
  of 
  

  

  magnitude 
  ^/>sinasin 
  2 
  |^ 
  — 
  ft\, 
  and 
  consequently 
  at 
  the 
  

  

  other 
  end, 
  one 
  of 
  magnitude 
  - 
  psinasinM 
  - 
  +ft). 
  The 
  

   resultant 
  of 
  these 
  has 
  components 
  

  

  P 
  x 
  '=-7rpsina 
  -j 
  sin 
  2 
  /"— 
  ft)— 
  sinM* 
  +ft) 
  V 
  cos 
  a, 
  

  

  = 
  —7rp 
  sin 
  2 
  a 
  cos 
  a 
  sin 
  ft 
  cos 
  ft 
  ; 
  

  

  P 
  y 
  '=| 
  W/)S 
  in«| 
  S 
  in 
  2 
  (|-^) 
  + 
  siting 
  +/3j\ 
  sin«, 
  r 
  

  

  = 
  777) 
  sin 
  2 
  a 
  < 
  cos 
  2 
  ,8 
  sin 
  2 
  -- 
  + 
  sin 
  2 
  /3 
  cos 
  5 
  g 
  V 
  ; 
  

  

  (22) 
  

  

  