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118
[HYDROSTATICS
HYDOROMECHANICS


or weight of liquid displaced is W − W′, so that the specific gravity (S.G.), defined as the ratio of the weight of a body to the weight of an equal volume of water, is W/(W − W′).

As stated first by Archimedes, the principle asserts the obvious fact that a body displaces its own volume of water; and he utilized it in the problem of the determination of the adulteration of the crown of Hiero. He weighed out a lump of gold and of silver of the same weight as the crown; and, immersing the three in succession in water, he found they spilt over measures of water in the ratio 1/14 : 4/77 : 2/21 or 33 : 24 : 44; thence it follows that the gold : silver alloy of the crown was as 11 : 9 by weight.

13. Theorem.—The resultant vertical thrust on any portion of a curved surface exposed to the pressure of a fluid at rest under gravity is the weight of fluid cut out by vertical lines drawn round the boundary of the curved surface.

Theorem.—The resultant horizontal thrust in any direction is obtained by drawing parallel horizontal lines round the boundary, and intersecting a plane perpendicular to their direction in a plane curve; and then investigating the thrust on this plane area, which will be the same as on the curved surface.

The proof of these theorems proceeds as before, employing the normality principle; they are required, for instance, in the determination of the liquid thrust on any portion of the bottom of a ship.

In casting a thin hollow object like a bell, it will be seen that the resultant upward thrust on the mould may be many times greater than the weight of metal; many a curious experiment has been devised to illustrate this property and classed as a hydrostatic paradox (Boyle, Hydrostatical Paradoxes, 1666).

Fig. 2.

Consider, for instance, the operation of casting a hemispherical bell, in fig. 2. As the molten metal is run in, the upward thrust on the outside mould, when the level has reached PP′, is the weight of metal in the volume generated by the revolution of APQ; and this, by a theorem of Archimedes, has the same volume as the cone ORR′, or 1/3πy3, where y is the depth of metal, the horizontal sections being equal so long as y is less than the radius of the outside hemisphere. Afterwards, when the metal has risen above B, to the level KK′, the additional thrust is the weight of the cylinder of diameter KK′ and height BH. The upward thrust is the same, however thin the metal may be in the interspace between the outer mould and the core inside; and this was formerly considered paradoxical.

Analytical Equations of Equilibrium of a Fluid at rest under any
System of Force.

14. Referred to three fixed coordinate axes, a fluid, in which the pressure is p, the density ρ, and X, Y, Z the components of impressed force per unit mass, requires for the equilibrium of the part filling a fixed surface S, on resolving parallel to Ox,

∫∫ lpdS = ∫∫∫ρX dx dy dz,
(1)

where l, m, n denote the direction cosines of the normal drawn outward of the surface S.

But by Green’s transformation

∫∫ lp dS = ∫∫∫ dp dx dy dz,
dx
(2)

thus leading to the differential relation at every point

dp = ρX,   dp = ρY,   dp = ρZ.
dx dy dz
(3)

The three equations of equilibrium obtained by taking moments round the axes are then found to be satisfied identically.

Hence the space variation of the pressure in any direction, or the pressure-gradient, is the resolved force per unit volume in that direction. The resultant force is therefore in the direction of the steepest pressure-gradient, and this is normal to the surface of equal pressure; for equilibrium to exist in a fluid the lines of force must therefore be capable of being cut orthogonally by a system of surfaces, which will be surfaces of equal pressure.

Ignoring temperature effect, and taking the density as a function of the pressure, surfaces of equal pressure are also of equal density, and the fluid is stratified by surfaces orthogonal to the lines of force;

1   dp ,   1   dp ,   1   dp , or X, Y, Z
ρ dx ρ dy ρ dz
(4)

are the partial differential coefficients of some function P, = ∫ dp/ρ, of x, y, z; so that X, Y, Z must be the partial differential coefficients of a potential −V, such that the force in any direction is the downward gradient of V; and then

dP + dV = 0, or P + V = constant,
dx dx
(5)

in which P may be called the hydrostatic head and V the head of potential.

With variation of temperature, the surfaces of equal pressure and density need not coincide; but, taking the pressure, density and temperature as connected by some relation, such as the gas-equation, the surfaces of equal density and temperature must intersect in lines lying on a surface of equal pressure.

15. As an example of the general equations, take the simplest case of a uniform field of gravity, with Oz directed vertically downward; employing the gravitation unit of force,

1   dp = 0,   1   dp = 0,   1   dp = 1,
ρ dx ρ dy ρ dz
(1)
P = dp/ρ = z + a constant.
(2)

When the density ρ is uniform, this becomes, as before in (2) § 9

p = ρz + p0.
(3)

Suppose the density ρ varies as some nth power of the depth below O, then

dp/dz = ρ = μzn
(4)
p = μ zn+1 = ρz = ρ ( ρ ) 1/n ,
n + 1 n + 1 n + 1 μ  
(5)

supposing p and ρ to vanish together.

These equations can be made to represent the state of convective equilibrium of the atmosphere, depending on the gas-equation

p = ρk = R ρθ,
(6)

where θ denotes the absolute temperature; and then

R dθ = d ( p ) = 1 ,
dz dz ρ n + 1
(7)

so that the temperature-gradient dθ/dz is constant, as in convective equilibrium in (11).

From the gas-equation in general, in the atmosphere

1   dp = 1   dp 1   dθ = ρ 1   dθ = 1 1   dθ ,
ρ dz p dz θ dz p θ dz k θ dz
(8)

which is positive, and the density ρ diminishes with the ascent, provided the temperature-gradient dθ/dz does not exceed θ/k.

With uniform temperature, taking k constant in the gas-equation,

dp/dz = ρ = p/k,   p = p0ez/k,
(9)

so that in ascending in the atmosphere of thermal equilibrium the pressure and density diminish at compound discount, and for pressures p1 and p2 at heights z1 and z2

(z1z2)/k = loge (p2/p1) = 2.3 log10 (p2/p1).
(10)

In the convective equilibrium of the atmosphere, the air is supposed to change in density and pressure without exchange of heat by conduction; and then

ρ/ρ0 = (θ/θ0)n, p/p0 = (θ/θ0)n + 1,
(11)
dz = 1   dp = (n + 1) p = (n + 1) R, γ = 1 + 1 ,
dθ ρ dθ ρθ n

where γ is the ratio of the specific heat at constant pressure and constant volume.

In the more general case of the convective equilibrium of a spherical atmosphere surrounding the earth, of radius a,

dp = (n + 1) p0   dθ = − a2 dr,
ρ ρ0 θ0 r2
(12)

gravity varying inversely as the square of the distance r from the centre; so that, k = p0/ρ0, denoting the height of the homogeneous atmosphere at the surface, θ is given by

(n + 1) k (1 − θ/θ0) = a(1 − a/r),
(13)

or if c denotes the distance where θ = 0,

θ = a · cr .
θ0 r ca
(14)

When the compressibility of water is taken into account in a deep ocean, an experimental law must be employed, such as

pp0 = k (ρρ0), or ρ/ρ0 = 1 + (pp0)/λ, λ = kρ0,
(15)

so that λ is the pressure due to a head k of the liquid at density ρ0 under atmospheric pressure p0; and it is the gauge pressure required on this law to double the density. Then

dp/dz = kdρ/dz = ρ,   ρ = ρ0ez/k,   pp0 = kρ0 (ez/k − 1);
(16)

and if the liquid was incompressible, the depth at pressure p would be (pp0)/p0, so that the lowering of the surface due to compression is

kez/kkz = 1/2z2/k, when k is large.
(17)

For sea water, λ is about 25,000 atmospheres, and k is then 25,000 times the height of the water barometer, about 250,000 metres, so that in an ocean 10 kilometres deep the level is lowered about 200 metres by the compressibility of the water; and the density at the bottom is increased 4%.

On another physical assumption of constant cubical elasticity λ,

dp = λdρ/ρ,   (pp0)/λ = log (ρ/ρ0),
(18)
dp = λ   dρ = ρ,   λ ( 1 1 ) = z,   1 − ρ0 = z ,   λ = kρ0,
zd ρ dz ρ0 ρ ρ k
(19)