Plasticity

of solids.

Perfect and

unlimited

plasticity

unopposed

by internal

character

istic of the ideal perfect

dynamics.

of shape, or gradually very great change at a diminish: (asymptotic) rate through infinite time; and as the use of t term plasticity implies no more than does viscosity, any phys theory or explanation of the property, the word viscosity without inconvenience left available for the definition we hav given of it above.

742. A perfect fluid, or (as we shall call it) a fluid, is a: unrealizable conception, like a rigid, or a smooth, body: it friction, the defined as a body incapable of resisting a change of shape: . therefore incapable of experiencing distorting or tangenti fluid, of abs stress (§ 669). Hence its pressure on any surface, wheth tract hydro- of a solid or of a contiguous portion of the fluid, is at every point perpendicular to the surface. In equilibrium, all conn = liquids and gaseous fluids fulfil the definition. But there finite resistance, of the nature of friction, opposing change t shape at a finite rate; and, therefore, while a fluid is chan shape, it exerts tangential force on every surface other the normal planes of the stress (§ 664) required to keep this cha of shape going on. Hence; although the hydrostatical results to which we immediately proceed, are verified in practice; treating of hydrokinetics, in a subsequent chapter, we shali « obliged to introduce the consideration of fluid friction, exsp in cases where the circumstances are such as to render effects insensible.

Fluid pressure.

743. With reference to a fluid the pressure at any point any direction is an expression used to denote the average pres sure per unit of area on a plane surface imagined as contan the point, and perpendicular to the direction in question, whe the area of that surface is indefinitely diminished.

744. At any point in a fluid at rest the pressure is | same in all directions: and, if no external forces act, t pressure is the same at every point. For the proof of thand most of the following propositions, we imagine, accoriz. to § 564, a definite portion of the fluid to become solid, with changing its mass, form, or dimensions.

Suppose the fluid to be contained in a closed vessel pressure within depending on the pressure exerted on it vessel, and not on any external force such as gravity.

it by

745. The resultant of the fluid pressures on the eli

sure proved

directions.

of any portion of a spherical surface must, like each of its com- Fluid pres ponents, pass through the centre of the sphere. Hence, if we equal in all suppose (§ 564) a portion of the fluid in the form of a planoconvex lens to be solidified, the resultant pressure on the plane side must pass through the centre of the sphere; and, therefore, being perpendicular to the plane, must pass through the centre of the circular area. From this it is obvious that the pressure is the same at all points of any plane in the fluid. Hence, by § 561, the resultant pressure on any plane surface passes through its centre of inertia.

Next, imagine a triangular prism of the fluid, with ends perpendicular to its faces, to be solidified. The resultant pressures on its ends act in the line joining the centres of inertia of their areas, and are equal (§ 551) since the resultant pressures on the sides are in directions perpendicular to this line. Hence the pressure is the same in all parallel planes.

But the centres of inertia of the three faces, and the resultant pressures applied there, lie in a triangular section parallel to the ends. The pressures act at the middle points of the sides of this triangle, and perpendicularly to them, so that their directions meet in a point. And, as they are in equilibrium, they must be, by § 557, proportional to the respective sides of the triangle; that is, to the breadths, or areas, of the faces of the prism. Thus the resultant pressures on the faces must be proportional to the areas of the faces, and therefore the pressure is equal in any two planes which meet.

to statics of

Collecting our results, we see that the pressure is the same. at all points, and in all directions, throughout the fluid mass. 746. One immediate application of this result gives us a Application simple though indirect proof of the second theorem in § 557, solids. for we have only to suppose the polyhedron to be a solidified portion of a mass of fluid in equilibrium under pressures only. The resultant pressure on each side will then be proportional to its area, and, by § 561, will act at its centre of inertia; which, in this case, is the Centre of Pressure.

Centre of

pressure.

of the prin

747. Another proof of the equality of pressure throughout Application a mass of fluid, uninfluenced by other external force than the ciple of pressure of the containing vessel, is easily furnished by the energy criterion of equilibrium, § 289; but, to avoid complica

energy

Proof by energy of the

tion, we will consider the fluid to be incompressible. Sup equality of a number of pistons fitted into cylinders inserted in the se

fluid pres

sure in all directions.

Fluid pressure depending on ex

ternal forces.

of the closed vessel containing the fluid. Then, if A be tiarea of one of these pistons, p the average pressure on it, z tidistance through which it is pressed, in or out; the en criterion is that no work shall be done on the whole, ie, the A1р1x1+Å‚μ‚x2+...=Σ(Apx)=0,

as much work being restored by the pistons which are fort out, as is done by those forced in. Also, since the fluid is compressible, it must have gained as much space by for out some of the pistons as it lost by the intrusion of the others This gives

The last is the only condition to which x, x, etc., in the tr equation, are subject; and therefore the first can only le satisfied if

P1=P=Ps=etc.,

that is, if the pressure be the same on each piston. Upon the property depends the action of Brahmah's Hydrostatic Press

If the fluid be compressible, the work expended in compressing it from volume V to V-SV, at mean pressure p, is pô V.

If in this case we assume the pressure to be the same throughout, we obtain a result consistent with the energy criterion. The work done on the fluid is Σ(Apx), that is, in consequere of the assumption, p≥(Ax).

748. When forces, such as gravity, act from external matt upon the substance of the fluid, either in proportion to t density of its own substance in its different parts, or in prportion to the density of electricity, or of magnetic polarity, of any other conceivable accidental property of it, the pressur will still be the same in all directions at any one point, !. will now vary continuously from point to point. For the pceding demonstration (§ 745) may still be applied by siz taking the dimensions of the prism small enough; since 1.pressures are as the squares of its linear dimensions, and effects of the applied forces such as gravity, as the cubes

equal pressure are per

pendicular

of force.

749. When forces act on the whole fluid, surfaces of equal Surfaces of pressure, if they exist, must be at every point perpendicular to the direction of the resultant force. For, any prism of the to the lines fluid so situated that the whole pressures on its ends are equal must (§ 551) experience from the applied forces no component in the direction of its length; and, therefore, if the prism be so small that from point to point of it the direction of the resultant of the applied forces does not vary sensibly, this direction must be perpendicular to the length of the prism. From this it follows that whatever be the physical origin, and the law, of the system of forces acting on the fluid, and whether it be conservative or non-conservative, the fluid cannot be in equilibrium unless the lines of force possess the geometrical property of being at right angles to a series of surfaces.

750. Again, considering two surfaces of equal pressure infinitely near one another, let the fluid between them be divided into columns of equal transverse section, and having their lengths perpendicular to the surfaces. The difference of pressures on the two ends being the same for each column, the resultant applied forces on the fluid masses composing them must be equal. Comparing this with § 488, we see that if the applied forces constitute a conservative system, the density of matter, or electricity, or whatever property of the substance potential they depend on, must be equal throughout the layer under system of consideration. This is the celebrated hydrostatic proposition servative. that in a fluid at rest, surfaces of equal pressure are also surfaces of equal density and of equal potential.

And are sur density and

faces of equal

of equal

when the

force is con

only external

751. Hence when gravity is the only external force con- Gravity the sidered, surfaces of equal pressure and equal density are (when force. of moderate extent) horizontal planes. On this depends the action of levels, syphons, barometers, etc.; also the separation of liquids of different densities (which do not mix or combine chemically) into horizontal strata, etc. etc. The free surface of a liquid is exposed to the pressure of the atmosphere simply; and therefore, when in equilibrium, must be a surface of equal pressure, and consequently level. In extensive sheets of water, such as the American lakes, differences of atmospheric pressure, even in moderately calm weather, often produce considerable deviations from a truly level surface.

Rate of increase of pressure.

752. The rate of increase of pressure per unit of length i the direction of the resultant force, is equal to the intensity the force reckoned per unit of volume of the fluid. Let F the resultant force per unit of volume in one of the columns § 750; p and p' the pressures at the ends of the column, i length, S its section. We have, for the equilibrium of the column, (p'—p) S=SIF.

Hence the rate of increase of pressure per unit of length is F If the applied forces belong to a conservative system, fr which V and V' are the values of the potential at the ends of the column, we have (§ 486)

where p is the density of the fluid. This gives

Hence in the case of gravity as the only impressed foro the rate of increase of pressure per unit of depth in the fa.

is

P,

in gravitation measure (usually employed in hydrostatic

In kinetic or absolute measure (§ 224) it is gp.

=

If the fluid be a gas, such as air, and be kept at a constant temperature, we have p cp, where c denotes a constant, the reciprocal of H, the "height of the homogeneous atmosphere," defined (§ 753) below. Hence, in a calm atmosphere of uniform

where po is the pressure at any particular level (the sea-level, fr instance) where we choose to reckon the potential as zero.

When the differences of level considered are infinitely small a comparison with the earth's radius, as we may practically regard them, in measuring the heights of mountains, or of a ballo, the barometer, the force of gravity is constant, and ther differences of potential (force being reckoned in units of we g are simply equal to differences of level. Hence if de height of the level of pressure p above that of p., we have, in the preceding formulæ, V=x, and therefore

p=p1~~; that is,