16

THE EQUALITY OF FLUID PRESSURE

11. The Equality of Fluid Pressure in all directions. We may now repeat the Definition of a Fluid given in Maxwell's Theory of Heat, chap. V.;

Definition of a Fluid.

'A fluid is a body the contiguous parts of which, when at rest, act on one another with a pressure which is perpendicular to the plane interface which separates those parts."

From the definition of a Fluid we deduce the important THEOREM. "The pressures in any two directions at a point of a fluid are equal."

Let the plane of the paper be that of the two given directions, and draw an isosceles triangle whose sides are perpendicular to the two given directions respectively, and consider the equilibrium of a small triangular prism of fluid, of which the triangle is the cross section (fig 6).

Let P, Q be the thrusts perpendicular to the sides and R that perpendicular to the base. Then since these three forces are in equilibrium, and since R makes equal angles with P and Q, therefore P and Q must be equal.

But the faces on which P and Q act are also equal; therefore the pressures, or thrusts per unit area, on these faces are equal, which was to be proved.

Generally for any scalene triangle abc, the thrusts or forces P, Q, R acting through the middle points of the sides and perpendicular to the sides are in equilibrium if proportional to their respective sides, so that the pressure is the same on each face; and a similar proof will hold if a tetrahedron or polyhedron of fluid is taken.

If we consider the equilibrium of any portion of the fluid enclosed in a polyhedron when the pressure of the fluid is uniform, we are led to the theorem in Statics that

IN ALL DIRECTIONS.

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"Forces acting all inwards or all outwards through the centres of gravity of the faces of a polyhedron, each proportional to and perpendicular to the face on which it acts, are in equilibrium."

12. The Transmissibility of Fluid Pressure. Hydraulic Press.

The

Any additional pressure applied to the fluid will, if the fluid is an incompressible liquid, be transmitted equally to every point of the liquid: this principle of the "Transmissibility of Pressure was enunciated by Pascal (Equilibre des liqueurs, 1653), and applied by him to the invention of

The Hydraulic Press.

This machine consists essentially of two communicating cylinders, filled with liquid, and closed by pistons (fig. 7); then if a thrust P lb is applied to one piston, of area B square feet, it will be balanced by the thrust W lb applied to the other piston of area A square feet such that

P/B = W/A,

the pressure of the liquid being supposed uniform and equal to P/B or W/A, lb/ft2; and by making the ratio of A/B sufficiently large, the mechanical advantage W/P can be increased to any desired amount.

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THE HYDRAULIC PRESS.

The difficulty of keeping the pistons tight against the leakage of the liquid prevented the practical application of Pascal's invention, until Bramah (in 1796) replaced the pistons by plungers (fig. 8) and made a water-tight joint by his invention of the cupped collar CC, pressed into U shape in cross section from an annular sheet of leather, which effectually prevents the escape of the water.

The applied thrust P can be applied, directly or by a lever, to the plunger of a force pump, provided with a stuffing box, the invention of Sir Samuel Morland, 1675; and then repeated strokes of the pump will cause the thrust W exerted by the head of the ram to act through any required distance.

In some portable forms, required for instance for punching or rail bending, the pressure is produced and kept up by a plunger P which advances on a screw thread.

For testing gauges Messrs. Schaffer and Budenberg employ an instrument consisting of a small ram working in a horizontal barrel full of water, the traverse of the ram being effected by its revolution in a screw. The gauge to be tested or graduated and the standard gauge are attached to the barrel and each registers the pressure of the water. The machine can even be used for testing vacuum gauges by turning the ram the reverse way, so as to diminish the pressure of the water below the atmospheric pressure.

The Hydrostatic Bellows was devised by Pascal as a mere lecture experiment to illustrate his Principle of the Transmissibility of Pressure; the large cylinder in fig. 7 is replaced by leather fastened to W, as in bellows, while the small cylinder is prolonged upwards by a pipe to a certain vertical height; and the thrust P is produced by

THE HYDRAULIC PRESS.

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the head of water poured in at the top of the pipe by a man standing on a ladder. In this way a small quantity of water poured in the pipe is shown lifting a considerable weight W supported by the bellows, and leakage is

C

Fig. 8.

avoided. For a diagram consult Ganot's Physique; the instrument is of no practical use, except for Nasmyth's attempt to replace the Hydraulic Press by his patent Steel Mattress (Engineer, 23 May, 1890, p. 426).

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ENERGY DUE TO PRESSURE.

13. The Principle of Virtual Velocities.

Pascal's Principle of the Transmissibility of Pressure was applied by him to verify the Principle of Virtual Velocities in the case of an incompressible liquid, thus showing that a liquid can be made to take the place of a complicated system of levers, in transmitting and multiplying a thrust.

For taking a closed vessel, filled with incompressible liquid, and fitted with cylindrical openings closed by pistons, of areas A, B, C,... ft2; then if the pistons move inwards through distances a, b, c, feet respectively, the condition that the volume of liquid is unchanged requires that Aa+Bb+Cc+...=0,

...

some of the quantities a, b, c,... being positive and some negative.

But if P, Q, R,... denote the thrust in lb on the pistons, then P/A=Q/B=R/C=...

= the uniform pressure in lb/ft2 of the liquid, and therefore

Pa+Qb+Re+...=0,

a verification of the Principle of Virtual Velocities.

14. The Energy of Liquid due to Pressure.

We have supposed the fluid employed to be incompressible liquid: for if a compressible gas had been used to transmit power, part of the energy would be used up in compressing the gas, if used to transmit power; so that a gas would behave like a machine composed of elastic levers.

But with an incompressible liquid the energy is entirely due to the pressure; and if the pressure is p lb/ft2, the energy of the liquid is p ft-lb per cubic foot (or p ft-lb/ft3).