20. In considering the motion of a fluid we shall suppose it
non-viscous, so that whatever the state of motion the stress across any
section is normal, and the principle of the normality and thence of the
equality of fluid pressure can be employed, as in hydrostatics. The
practical problems of fluid motion, which are amenable to mathematical
analysis when viscosity is taken into account, are excluded from
treatment here, as constituting a separate branch called "hydraulics"
(q.v.). Two methods are employed in hydrodynamics, called the Eulerian
and Lagrangian, although both are due originally to Leonhard Euler. In
the Eulerian method the attention is fixed on a particular point of
space, and the change is observed there of pressure, density and
velocity, which takes place during the motion; but in the Lagrangian
method we follow up a particle of fluid and observe how it changes. The
first may be called the statistical method, and the second the
historical, according to J. C. Maxwell. The Lagrangian method being
employed rarely, we shall confine ourselves to the Eulerian treatment.
_The Eulerian Form of the Equations of Motion._
21. The first equation to be established is the _equation of
continuity_, which expresses the fact that the increase of matter within
a fixed surface is due to the flow of fluid across the surface into its
interior.
In a straight uniform current of fluid of density [rho], flowing with
velocity q, the flow in units of mass per second across a plane area
A, placed in the current with the normal of the plane making an angle
[theta] with the velocity, is [rho]Aq cos [theta], the product of the
density [rho], the area A, and q cos [theta] the component velocity
normal to the plane.
Generally if S denotes any closed surface, fixed in the fluid, M the
mass of the fluid inside it at any time t, and [theta] the angle which
the outward-drawn normal makes with the velocity q at that point,
dM/dt = rate of increase of fluid inside the surface, (1)
= flux across the surface into the interior
_ _
/ /
= - | | [rho]q cos [theta] dS,
_/_/
the integral equation of continuity.
In the Eulerian notation u, v, w denote the components of the velocity
q parallel to the coordinate axes at any point (x, y, z) at the time
t; u, v, w are functions of x, y, z, t, the independent variables; and
d is used here to denote partial differentiation with respect to any
one of these four independent variables, all capable of varying one at
a time.
To transfer the integral equation into the differential equation of
continuity, Green's transformation is required again, namely,
_ _ _ _ _
/ / / /d[xi] d[eta] d[zeta] \ / /
| | | ( ----- + ------ + ------- )dx dy dz = | | (l[xi] + m[eta] + n[zeta]) dS, (2)
_/_/_/ \ dx dy dz / _/_/
or individually
_ _ _ _ _
/ / / d[xi] / /
| | | ----- dx dy dz = | | l[xi] dS,..., (3)
_/_/_/ dx _/_/
where the integrations extend throughout the volume and over the
surface of a closed space S; l, m, n denoting the direction cosines of
the outward-drawn normal at the surface element dS, and [xi], [eta],
[zeta] any continuous functions of x, y, z.
The integral equation of continuity (1) may now be written
_ _ _ _ _
/ / / d[rho] / /
| | | ----- dx dy dz = | | (l[rho]u + m[rho]v + n[rho]w) dS = 0, (4)
_/_/_/ dt _/_/
which becomes by Green's transformation
_ _ _
/ / / /d[rho] d([rho]u) d([rho]v) d([rho]w)\
| | | ( ------ + --------- + --------- + -------- ) dx dy dz = 0, (5)
_/_/_/ \ dt dx dy dz /
leading to the differential equation of continuity when the
integration is removed.
22. The equations of motion can be established in a similar way by
considering the rate of increase of momentum in a fixed direction of the
fluid inside the surface, and equating it to the momentum generated by
the force acting throughout the space S, and by the pressure acting over
the surface S.
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
23. As a rule these equations are established immediately by determining
the component acceleration of the fluid particle which is passing
through (x, y, z) at the instant t of time considered, and saying that
the reversed acceleration or kinetic reaction, combined with the
impressed force per unit of mass and pressure-gradient, will according
to d'Alembert's principle form a system in equilibrium.
To determine the component acceleration of a particle, suppose F to
denote any function of x, y, z, t, and investigate the time rate of F
for a moving particle; denoting the change by DF/dt,
DF F(x + u[delta]t, y + v[delta]t, z + w[delta]t, t + [delta]t) - F(x, y, z, t)
-- = lt·----------------------------------------------------------------------------
dt [delta]t
dF dF dF dF
= -- + u-- + v-- + w--; (1)
dt dx dy dz
and D/dt is called particle differentiation, because it follows the
rate of change of a particle as it leaves the point x, y, z; but
dF/dt, dF/dx, dF/dy, dF/dz (2)
represent the rate of change of F at the time t, at the point, x, y,
z, fixed in space.