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120
[HYDRODYNAMICS
HYDROMECHANICS


the movement of a weight P through a distance c will cause the ship to heel through an angle θ about an axis FF′ through F, which is conjugate to the direction of the movement of P with respect to an ellipse, not the momental ellipse of the water-line area A, but a confocal to it, of squared semi-axes

a2 − hV/A, b2 − hV/A,
(1)

h denoting the vertical height BG between C.G. and centre of buoyancy. The varying direction of the inclining couple Pc may be realized by swinging the weight P from a crane on the ship, in a circle of radius c. But if the weight P was lowered on the ship from a crane on shore, the vessel would sink bodily a distance P/wA if P was deposited over F; but deposited anywhere else, say over Q on the water-line area, the ship would turn about a line the antipolar of Q with respect to the confocal ellipse, parallel to FF′, at a distance FK from F

FK = (k2hV/A)/FQ sin QFF′
(2)

through an angle θ or a slope of one in m, given by

sin θ = 1 = P = P · V FQ sin QFF′
m wA·FK W Ak2 − hV
(3)

where k denotes the radius of gyration about FF′ of the water-line area. Burning the coal on a voyage has the reverse effect on a steamer.

Hydrodynamics

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 ρ, 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 θ with the velocity, is ρAq cos θ, the product of the density ρ, the area A, and q cos θ 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 θ 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,

= flux across the surface into the interior

= − ∫∫ ρq cos θ dS,

(1)

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,

∫∫∫ ( dξ + dη + dζ ) dx dy dz = ∫∫ (lξ + mη + nζ) dS,
dx dy dz
(2)

or individually

∫∫∫ dξ dx dy dz = ∫∫ lξ dS, ...,
dx
(3)

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 ξ, η, ζ any continuous functions of x, y, z.

The integral equation of continuity (1) may now be written

∫∫∫ dρ dx dy dz = ∫∫ (lρu + mρv + nρw) dS = 0,
dt
(4)

which becomes by Green’s transformation

∫∫∫ ( dρ + d(ρu) + d(ρv) + d(ρw) ) dx dy dz = 0,
dt dx dy dz
(5)

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

− ∫∫ ρuq cos θ dS = − ∫∫ (lρu2 + mρuv + nρuw) dS,
(1)

which by Green’s transformation is

∫∫∫ ( d(ρu2) + d(ρuv) + d(ρuw) ) dx dy dz.
dx dy dz
(2)

The rate of generation of momentum in the interior of S by the component of force, X per unit mass, is

∫∫∫ ρX dx dy dz,
(3)

and by the pressure at the surface S is

− ∫∫ lp dS = − ∫∫∫ dp dx dy dz,
dx
(4)

by Green’s transformation.

The time rate of increase of momentum of the fluid inside S is

∫∫∫ d(ρu) dx dy dz;
dt
(5)

and (5) is the sum of (1), (2), (3), (4), so that

∫∫∫ ( dρu + dρu2 + dρuv + dρuw ρX + dp ) dx dy dz = 0,
dt dx dy dz dx
(6)

leading to the differential equation of motion

dρu + dρu2 + dρuv + dρuw = ρX − dp ,
dt dx dy dz dx
(7)

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

dρu + dρu2 + dρuv + dρuw
dt dx dy dz
= ρ ( du + u du + v du + w du )
dt dx dy dz
+ u ( dρ + dρu + dρv + dρw ),
dt dx dy dz
(8)

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 + u du + v du + w du = X − 1   dp ,
dt dx dy dz ρ dx
(9)

with the two others

dv + u dv + v dv + w dv = Y − 1   dp ,
dt dx dy dz ρ dy
(10)
dw + u dw + v dw + w dw = Z − 1   dp .
dt dx dy dz ρ dz
(11)

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 = lt· F(x + uδt, y + vδt, z + wδt, t + δt) − F(x, y, z, t)
dt δt
= dF + u dF + v dF + w dF ;
dt dx dy dz
(1)

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.