The components of acceleration of a particle of fluid are consequently
| Du | = | du | + u | du | + v | du | + w | du | , |
| dt | dt | dx | dy | dz |
| Dv | = | dv | + u | dv | + v | dv | + w | dv | , |
| dt | dt | dx | dy | dz |
| Dw | = | dw | + u | dw | + v | dw | + w | dw | , |
| dt | dt | dx | dy | dz |
leading to the equations of motion above.
If F (x, y, z, t) = 0 represents the equation of a surface containing always the same particles of fluid,
| DF | = 0, or | dF | + u | dF | + v | dF | + w | dF | = 0, |
| dt | dt | dx | dy | dz |
which is called the differential equation of the bounding surface.
A bounding surface is such that there is no flow of fluid across it,
as expressed by equation (6). The surface always contains the same
fluid inside it, and condition (6) is satisfied over the complete surface,
as well as any part of it.
But turbulence in the motion will vitiate the principle that a bounding surface will always consist of the same fluid particles, as we see on the surface of turbulent water.
24. To integrate the equations of motion, suppose the impressed force is due to a potential V, such that the force in any direction is the rate of diminution of V, or its downward gradient; and then
and putting
| dw | − | dv | = 2ξ, | du | − | dw | = 2η, | dv | − | du | = 2ζ, |
| dy | dz | dz | dx | dx | dy |
| dξ | + | dη | + | dζ | = 0, |
| dx | dy | dz |
the equations of motion may be written
| du | − 2vζ + 2wη + | dH | = 0, |
| dt | dx |
| dv | − 2wξ + 2uζ + | dH | = 0, |
| dt | dy |
| dw | − 2uη + 2wξ + | dH | = 0, |
| dt | dz |
where
and the three terms in H may be called the pressure head, potential head, and head of velocity, when the gravitation unit is employed and 12q2 is replaced by 12q2/g.
Eliminating H between (5) and (6)
| Dξ | − ξ | du | − η | dw | − ζ | dv | + ξ ( | du | + | dv | + | dw | ) = 0, |
| dt | dx | dx | dx | dx | dy | dz |
and combining this with the equation of continuity
| 1 | Dρ | + | du | + | dv | + | dw | = 0, | |
| ρ | dt | dx | dy | dz |
we have
| D | ( | ξ | ) − | ξ | du | − | η | dv | − | ζ | dw | = 0, | |||
| dt | ρ | ρ | dx | ρ | dx | ρ | dx |
with two similar equations.
Putting
a vortex line is defined to be such that the tangent is in the direction of ω, the resultant of ξ, η, ζ, called the components of molecular rotation. A small sphere of the fluid, if frozen suddenly, would retain this angular velocity.
If ω vanishes throughout the fluid at any instant, equation (11) shows that it will always be zero, and the fluid motion is then called irrotational; and a function φ exists, called the velocity function, such that
and then the velocity in any direction is the space-decrease or downward gradient of φ.
25. But in the most general case it is possible to have three functions φ, ψ, m of x, y, z, such that
as A. Clebsch has shown, from purely analytical considerations (Crelle, lvi.); and then
| ξ = 12 | d(ψ, m) | , η = 12 | d(ψ, m) | , ζ = 12 | d(ψ, m) | , |
| d(y, z) | d(z, x) | d(x, y) |
and
| ξ | dψ | + η | dψ | + ζ | dψ | = 0, ξ | dm | + η | dm | + ζ | dm | = 0, |
| dx | dy | dz | dx | dy | dz |
so that, at any instant, the surfaces over which ψ and m are constant intersect in the vortex lines.
Putting
| H − | dφ | − m | dψ | = K, |
| dt | dt |
the equations of motion (4), (5), (6) § 24 can be written
| dK | − 2uζ + 2wη − | d(ψ,m) | = 0, . . . , . . . ; |
| dx | d(x,t) |
and therefore
| ξ | dK | + η | dK | + ζ | dK | = 0. |
| dx | dy | dz |
Equation (5) becomes, by a rearrangement,
| dK | − | dψ | ( | dm | + u | dm | + v | dm | + w | dm | ) |
| dx | dx | dt | dx | dy | dz |
| + | dm | ( | dψ | + u | dψ | + v | dψ | + w | dψ | ) = 0, . . . , . . . , |
| dx | dt | dx | dy | dz |
| dK | − | dψ | Dm | + | dm | Dψ | = 0, . . . , . . . , | ||
| dx | dx | dt | dx | dt |
and as we prove subsequently (§ 37) that the vortex lines are composed of the same fluid particles throughout the motion, the surface m and ψ satisfies the condition of (6) § 23; so that K is uniform throughout the fluid at any instant, and changes with the time only, and so may be replaced by F(t).
26. When the motion is steady, that is, when the velocity at any point of space does not change with the time,
| dK | − 2vζ + 2wη = 0, . . ., . . . |
| dx |
| ξ | dK | + η | dK | + ζ | dK | = 0, u | dK | + v | dK | + w | dK | = 0, |
| dx | dy | dz | dx | dy | dz |
and
is constant along a vortex line, and a stream line, the path of a fluid particle, so that the fluid is traversed by a series of H surfaces, each covered by a network of stream lines and vortex lines; and if the motion is irrotational H is a constant throughout the fluid.
Taking the axis of x for an instant in the normal through a point on the surface H = constant, this makes u = 0, ξ = 0; and in steady motion the equations reduce to
where θ is the angle between the stream line and vortex line; and this holds for their projection on any plane to which dν is drawn perpendicular.
In plane motion (4) reduces to
| dH | = 2qζ = q ( | dQ | + | q | ), |
| dν | dv | r |
if r denotes the radius of curvature of the stream line, so that
| 1 | dp | + | dV | = | dH | − | d 12q2 | = | q2 | , | |
| ρ | dν | dν | dν | dν | r |
the normal acceleration.
The osculating plane of a stream line in steady motion contains the resultant acceleration, the direction ratios of which are
| u | du | + v | du | + w | du | = | d 12q2 | − 2vζ + 2wη = | d 12q2 | − | dH | , . . . , |
| dx | dy | dz | dx | dx | dx |
and when q is stationary, the acceleration is normal to the surface H
= constant, and the stream line is a geodesic.
Calling the sum of the pressure and potential head the statical head, surfaces of constant statical and dynamical head intersect in lines on H, and the three surfaces touch where the velocity is stationary.
Equation (3) is called Bernoulli’s equation, and may be interpreted as the balance-sheet of the energy which enters and leaves a given tube of flow.
If homogeneous liquid is drawn off from a vessel so large that the motion at the free surface at a distance may be neglected, then Bernoulli’s equation may be written
where P denotes the atmospheric pressure and h the height of the free surface, a fundamental equation in hydraulics; a return has been made here to the gravitation unit of hydrostatics, and Oz is taken vertically upward.
In particular, for a jet issuing into the atmosphere, where p = P,
or the velocity of the jet is due to the head k − z of the still free surface above the orifice; this is Torricelli’s theorem (1643), the foundation of the science of hydrodynamics.
27. Uniplanar Motion.—In the uniplanar motion of a homogeneous liquid the equation of continuity reduces to
| du | + | dv | = 0, |
| dx | dy |
so that we can put
