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HYDROSTATICS]
117
HYDROMECHANICS


According to Maxwell (Theory of Heat) “When a continuous alteration of form is produced only by a stress exceeding a certain value, the substance is called a solid, however soft and plastic it may be. But when the smallest stress, if only continued long enough, will cause a perceptible and increasing change of form, the substance must be regarded as a viscous fluid, however hard it may be.” Maxwell illustrates the difference between a soft solid and a hard liquid by a jelly and a block of pitch; also by the experiment of supporting a candle and a stick of sealing-wax; after a considerable time the sealing-wax will be found bent and so is a fluid, but the candle remains straight as a solid.

4. Definition of a Fluid.—A fluid is a substance which yields continually to the slightest tangential stress in its interior; that is, it can be divided very easily along any plane (given plenty of time if the fluid is viscous). It follows that when the fluid has come to rest, the tangential stress in any plane in its interior must vanish, and the stress must be entirely normal to the plane. This mechanical axiom of the normality of fluid pressure is the foundation of the mathematical theory of hydrostatics.

The theorems of hydrostatics are thus true for all stationary fluids, however viscous they may be; it is only when we come to hydrodynamics, the science of the motion of a fluid, that viscosity will make itself felt and modify the theory; unless we begin by postulating the perfect fluid, devoid of viscosity, so that the principle of the normality of fluid pressure is taken to hold when the fluid is in movement.

5. The Measurement of Fluid Pressure.—The pressure at any point of a plane in the interior of a fluid is the intensity of the normal thrust estimated per unit area of the plane.

Thus, if a thrust of P ℔ is distributed uniformly over a plane area of A sq. ft., as on the horizontal bottom of the sea or any reservoir, the pressure at any point of the plane is P/A ℔ per sq. ft., or P/144A ℔ per sq. in. (℔/ft.2 and ℔/in.2, in the Hospitalier notation, to be employed in the sequel). If the distribution of the thrust is not uniform, as, for instance, on a vertical or inclined face or wall of a reservoir, then P/A represents the average pressure over the area; and the actual pressure at any point is the average pressure over a small area enclosing the point. Thus, if a thrust ΔP ℔ acts on a small plane area ΔA ft.2 enclosing a point B, the pressure p at B is the limit of ΔP/ΔA; and

p = lt (ΔP/ΔA) = dP/dA,
(1)

in the notation of the differential calculus.

6. The Equality of Fluid Pressure in all Directions.—This fundamental principle of hydrostatics follows at once from the principle of the normality of fluid pressure implied in the definition of a fluid in § 4. Take any two arbitrary directions in the plane of the paper, and draw a small isosceles triangle abc, whose sides are perpendicular to the two directions, and consider the equilibrium of a small triangular prism of fluid, of which the triangle is the cross section. Let P, Q denote the normal thrust across the sides bc, ca, and R the normal thrust across the base ab. Then, since these three forces maintain equilibrium, and R makes equal angles with P and Q, therefore P and Q must be equal. But the faces bc, ca, over which P and Q act, are also equal, so that the pressure on each face is equal. A scalene triangle abc might also be employed, or a tetrahedron.

Fig. 1a.

It follows that the pressure of a fluid requires to be calculated in one direction only, chosen as the simplest direction for convenience.

7. The Transmissibility of Fluid Pressure.—Any additional pressure applied to the fluid will be transmitted equally to every point in the case of a liquid; this principle of the transmissibility of pressure was enunciated by Pascal, 1653, and applied by him to the invention of the hydraulic press.

This machine consists essentially of two communicating cylinders (fig. 1a), filled with liquid and closed by pistons. If a thrust P ℔ is applied to one piston of area A ft.2, it will be balanced by a thrust W ℔ applied to the other piston of area B ft.2, where

p = P/A = W/B,
(1)

the pressure p of the liquid being supposed uniform; and, by making the ratio B/A sufficiently large, the mechanical advantage can be increased to any desired amount, and in the simplest manner possible, without the intervention of levers and machinery.

Fig. 1b shows also a modern form of the hydraulic press, applied to the operation of covering an electric cable with a lead coating.

8. Theorem.—In a fluid at rest under gravity the pressure is the same at any two points in the same horizontal plane; in other words, a surface of equal pressure is a horizontal plane.

This is proved by taking any two points A and B at the same level, and considering the equilibrium of a thin prism of liquid AB, bounded by planes at A and B perpendicular to AB. As gravity and the fluid pressure on the sides of the prism act at right angles to AB, the equilibrium requires the equality of thrust on the ends A and B; and as the areas are equal, the pressure must be equal at A and B; and so the pressure is the same at all points in the same horizontal plane. If the fluid is a liquid, it can have a free surface without diffusing itself, as a gas would; and this free surface, being a surface of zero pressure, or more generally of uniform atmospheric pressure, will also be a surface of equal pressure, and therefore a horizontal plane.

Fig. 1b.

Hence the theorem.—The free surface of a liquid at rest under gravity is a horizontal plane. This is the characteristic distinguishing between a solid and a liquid; as, for instance, between land and water. The land has hills and valleys, but the surface of water at rest is a horizontal plane; and if disturbed the surface moves in waves.

9. Theorem.—In a homogeneous liquid at rest under gravity the pressure increases uniformly with the depth.

This is proved by taking the two points A and B in the same vertical line, and considering the equilibrium of the prism by resolving vertically. In this case the thrust at the lower end B must exceed the thrust at A, the upper end, by the weight of the prism of liquid; so that, denoting the cross section of the prism by α ft.2, the pressure at A and By by p0 and p ℔/ft.2, and by w the density of the liquid estimated in ℔/ft.3,

pαp0α = wα·AB,
(1)
p = w·AB + p0.
(2)

Thus in water, where w = 62.4℔/ft.3, the pressure increases 62.4 ℔/ft.2, or 62.4 ÷ 144 = 0.433 ℔/in.2 for every additional foot of depth.

10. Theorem.—If two liquids of different density are resting in vessels in communication, the height of the free surface of such liquid above the surface of separation is inversely as the density.

For if the liquid of density σ rises to the height h and of density ρ to the height k, and p0 denotes the atmospheric pressure, the pressure in the liquid at the level of the surface of separation will be σh + p0 and ρk + p0, and these being equal we have

σh = ρk.
(1)

The principle is illustrated in the article Barometer, where a column of mercury of density σ and height h, rising in the tube to the Torricellian vacuum, is balanced by a column of air of density ρ, which may be supposed to rise as a homogeneous fluid to a height k, called the height of the homogeneous atmosphere. Thus water being about 800 times denser than air and mercury 13.6 times denser than water,

k/h = σ/ρ = 800 × 13.6 = 10,880;
(2)

and with an average barometer height of 30 in. this makes k 27,200 ft., about 8300 metres.

11. The Head of Water or a Liquid.—The pressure σh at a depth h ft. in liquid of density σ is called the pressure due to a head of h ft. of the liquid. The atmospheric pressure is thus due to an average head of 30 in. of mercury, or 30 × 13.6 ÷ 12 = 34 ft. of water, or 27,200 ft. of air. The pressure of the air is a convenient unit to employ in practical work, where it is called an “atmosphere”; it is made the equivalent of a pressure of one kg/cm2; and one ton/inch2, employed as the unit with high pressure as in artillery, may be taken as 150 atmospheres.

12. Theorem.—A body immersed in a fluid is buoyed up by a force equal to the weight of the liquid displaced, acting vertically upward through the centre of gravity of the displaced liquid.

For if the body is removed, and replaced by the fluid as at first, this fluid is in equilibrium under its own weight and the thrust of the surrounding fluid, which must be equal and opposite, and the surrounding fluid acts in the same manner when the body replaces the displaced fluid again; so that the resultant thrust of the fluid acts vertically upward through the centre of gravity of the fluid displaced, and is equal to the weight.

When the body is floating freely like a ship, the equilibrium of this liquid thrust with the weight of the ship requires that the weight of water displaced is equal to the weight of the ship and the two centres of gravity are in the same vertical line. So also a balloon begins to rise when the weight of air displaced is greater than the weight of the balloon, and it is in equilibrium when the weights are equal. This theorem is called generally the principle of Archimedes.

It is used to determine the density of a body experimentally; for if W is the weight of a body weighed in a balance in air (strictly in vacuo), and if W′ is the weight required to balance when the body is suspended in water, then the upward thrust of the liquid