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Encyclopaedia Britannica, 11th Edition, "Hydromechanics" to "Ichnography": Volume 14, Slice 2Various

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Encyclopaedia Britannica, 11th Edition, "Hydromechanics" to "Ichnography": Volume 14, Slice 2

Name: Encyclopaedia Britannica, 11th Edition, "Hydromechanics" to "Ichnography": Volume 14, Slice 2
Author: Various

Various

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Taking the fixed direction parallel to the axis of x, the time-rate of
  increase of momentum, due to the fluid which crosses the surface, is
       _ _                            _ _
      / /                            / /
    - | | [rho]uq cos [theta] dS = - | | (l[rho]u² + m[rho]uv + n[rho]uw) dS,  (1)
     _/_/                           _/_/

  which by Green's transformation is
       _ _ _
      / / /  /d([rho]u²)   d([rho]uv)   d([rho]uw)\
    - | | | (---------- + ---------- + ----------  ) dx dy dz.   (2)
     _/_/_/  \    dx           dy           dz    /

  The rate of generation of momentum in the interior of S by the
  component of force, X per unit mass, is
      _ _ _
     / / /
     | | | [rho]X dx dy dz,   (3)
    _/_/_/

  and by the pressure at the surface S is
       _ _           _ _ _
      / /           / / / dp
    - | | lp dS = - | | | -- dx dy dz,   (4)
     _/_/          _/_/_/ dx

  by Green's transformation.

  The time rate of increase of momentum of the fluid inside S is
      _ _ _
     / / / d([rho]u)
     | | | --------- dx dy dz;   (5)
    _/_/_/    dt

  and (5) is the sum of (1), (2), (3), (4), so that
      _ _ _
     / / /  /d[rho]u   d[rho]u²   d[rho]uv   d[rho]uw            dp \
     | | | ( ------- + -------- + -------- + -------- - [rho]X + --  ) dx dy dz = 0,   (6)
    _/_/_/  \  dt         dx         dy         dz               dx /

  leading to the differential equation of motion

    d[rho]u   d[rho]u²   d[rho]uv,  d[rho]uw            dp
    ------- + -------- + -------- + -------- = [rho]X - --,   (7)
      dt         dx         dy         dz               dx

  with two similar equations.

  The absolute unit of force is employed here, and not the gravitation
  unit of hydrostatics; in a numerical application it is assumed that
  C.G.S. units are intended.

  These equations may be simplified slightly, using the equation of
  continuity (5) § 21; for

    d[rho]u   d[rho]u²   d[rho]uv   d[rho]uw
    ------- + -------- + -------- + --------
      dt         dx         dy         dz

             /du    du    du    du \
    = [rho] ( -- + u-- + v-- + w--  )
             \dt    dx    dy    dz /

         /d[rho]   d[rho]u   d[rho]v   d[rho]w \
    + u ( ------ + ------- + ------- + -------  ),   (8)
         \  dt        dx        dy        dz   /

  reducing to the first line, the second line vanishing in consequence
  of the equation of continuity; and so the equation of motion may be
  written in the more usual form

    du    du    du    du         1   dp
    -- + u-- + v-- + w-- = X - ----- --,   (9)
    dt    dx    dy    dz       [rho] dx

  with the two others

    dv    dv    dv    dv         1   dp
    -- + u-- + v-- + w-- = Y - ----- --,   (10)
    dt    dx    dy    dz       [rho] dy

    dw    dw    dw    dw         1   dp
    -- + u-- + v-- + w-- = Z - ----- --.   (11)
    dt    dx    dy    dz       [rho] dz

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Encyclopaedia Britannica, 11th Edition, "Hydromechanics" to "Ichnography": Volume 14, Slice 2

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