﻿Incompressible 
  Viscous 
  Fluid 
  tlirougli 
  a 
  Circular 
  Tube. 
  37 
  

  

  that 
  without 
  sensible 
  error 
  r 
  = 
  x6^ 
  on 
  converting 
  to 
  polar 
  

   coordinates 
  x 
  and 
  6 
  the 
  equation 
  becomes 
  

  

  Since 
  u 
  is 
  symmetrical 
  with 
  respect 
  to 
  6, 
  putting 
  

  

  where 
  2a 
  is 
  the 
  vertical 
  angle 
  of 
  the 
  conical 
  pipe, 
  and 
  there- 
  

   fore 
  assuming 
  no 
  slip 
  at 
  the 
  boundaries, 
  we 
  get, 
  on 
  

   integration, 
  

  

  -P=-H^[ 
  \. 
  ' 
  + 
  3:^+-3^j+-27+^ 
  

  

  (2?i 
  AiU 
  1 
  ~| 
  tirn^ 
  -r> 
  

  

  When 
  ]) 
  =]:>(), 
  u 
  = 
  ito 
  and 
  .i' 
  = 
  a:'Q 
  ; 
  

  

  . 
  E 
  - 
  _ 
  >, 
  _ 
  ^ 
  / 
  2"o 
  4»o 
  ^ 
  "I 
  _ 
  ^'^o' 
  

  

  'ut'^ 
  2g~ 
  ^'^ 
  2g 
  ^ 
  L" 
  ;X«'-^')/ 
  L 
  •'^'o 
  «^J 
  " 
  ^' 
  "^ 
  

   Again 
  since 
  the 
  volume 
  discharged 
  per 
  second 
  is 
  given 
  by 
  

  

  Q 
  = 
  ^^2'iTx''eudd 
  

  

  ^2'TTK\\uW'-d'')dd 
  

   *^0 
  

  

  

  TrAa^ 
  

  

  

  ~ 
  2 
  ' 
  

  

  •*• 
  

  

  TTa"^' 
  

  

  * 
  

  

  

  '^-u 
  

  

  d'-^" 
  

  

  " 
  d'- 
  ■ 
  d'- 
  dy 
  U 
  ■ 
  -del 
  ' 
  'dr 
  

  

  

  _ 
  1 
  "d-ii 
  1 
  1 
  d-'* 
  

  

  

  ~ 
  a- 
  'da'- 
  ' 
  •■^ 
  ~ 
  ^'- 
  "do- 
  

  

  