The inelastic fluids (which form drops) and the elastic fluids (Gases)

E1 Equilibrium of drop forming substances

Solids and fluids

Substances have the aggregate forms: Solid, fluid, gas, according to their obvious differences in the cohesion of their particles. However, there does not exist a sharp border between solids and fluids: There are many grades of coherence. The fact that also fluids have cohesion is established by their formation of drops. This property separates them from solid as well as gaseous substances, whence they are referred to as drop forming fluids (gases are elastic fluids). Even firm (rigid) bodies are not altogether firm, because they change their shape when forces act on them. However, as long as the change does not exceed the limit of elasticity, it soon ceases, even though the force continues to act. In this deformed state, the solid body can be conceived to be completely firm, as long as only this force acts on it. However, a body which on application of a force changes shape in time, is not called solid, but viscous. In this respect, fluids differ from solids: If only a very small force is required for a progressive change of shape and it acts long enough, the substance is called a viscous fluid, although it may appear to be rigid. But if the required force must have a certain magnitude (that is, an arbitrarily small force is insufficient), a substance is still called solid, even if it is very soft.

A tallow candle is much softer than a stick of sealing wax. However, if you place during the hot season both of them horizontaly supported by their ends, the sealing wax bar will eventually bend due to its weight, while the tallow candle will remain straight, whence the tallow candle is made out of a firmer substance, even though it is soft, and the sealing wax is a fluid, although a very tough one.

In order to change the form of a soft, firm body permanently, a large force is required, which has an immediate reaction. If you are dealing with a viscous fluid, you only need time; already the smallest force will cause a detectable reaction, as long as it acts long enough; for the same reaction, if the force only acts for a brief time, it must be very large. A block of pitch can be so hard that it does not react if you beat it with your fingers' knuckles, and yet it will in the course of time become flatter by its weight and spread out like water. The inclusion of pitch and similar materials among fluids is justified by the following experiment of Otto Obermeyer 1843-1873 1877:

Place a piece of pitch into a groove on top of a piece of cork and a stone on top of the pitch; after a few days, the pitch will have flown into the groove and taken on its form, but the stone will lie in the groove and the cork on top of the pitch.

Ideal and real fluids

If you simply speak of fluids, what should you understand by this? A fluid as viscous as pitch or one which is as little viscous as water or alcohol? The word viscous already describes a property of a fluid. However, you can also think of a fluid without viscosity. We said above: Already the smallest force provokes a noticeable reaction of a viscous fluid, provided it is given enough time. A fluid without viscosity is an ideal fluid, one which reacts immediately to an arbitrarily small force with a change of shape, that is, displacement of its particles, does not require work. We will discuss this aspect below.

In reality, the mobility of fluid particles is always affected by their friction - internal friction, viscosity. This explains, for example, the behaviour of a fluid which is rotated in a vessel and then allowed to continue on its own: The fluid will gradually rotate less fast and eventually come to a stop, first at the wall of the vessel ( as a result of the friction along the wall) and subsequently inside.

We will now assume that our fluids are ideal, that is, do not have internal friction. Friction only acts when a fluid moves; we will deal first with fluids at rest.

An ideal fluid changes its shape as soon as a very small force acts for a brief instant. Its other characteristic property is: It is incompressible. Water - and this is what we think of when we speak of fluids, whence we use the terms Hydrostatics and Hydrodynamics in Mechanics of Fluids - is elastic and under high pressure compressible, but even at the highest pressure so little that we can disregard this property and conceive the ideal fluid to be incompressible - volume preserving.

Equilibrium of a resting fluid

How can fluids be at rest, since gravity acts at all times everywhere? Every water surface in a pond at rest or a drop of water tells you that this is possible.

Let A be a fluid mass at rest (Fig. 172), although a force acts on it (gravity) and P the force which acts on a particle at its surface. If the force is parallel to the surface, it will displace the particle along it, whence in order for the particle to be at rest, the force should not have a component parallel to the surface, that is, it must be perpendicular to the surface, because only then it would not have a component (Q) parallel to it. The surface of a fluid at rest must be everywhere perpendicular to the direction of the force acting there. Indeed, the free surface (not one which coincides with a wall) of a fluid at rest (in a vessel, in a pond), which is only acted upon by gravity, is horizontal, that is, perpendicular to the force of gravity; fluid masses, which you deprive of the action of gravity, form perfect spheres up to a diameter of 10 cm (Plateau's experiment ).

Propagation of pressure inside a fluid

As a result of the ready mobility of their particles, fluids transfer pressure acting on them in all directions at equal strength to individual fluid particles, so that they move unless an equally large counter-pressure balances it. This mode of propagation of pressure distinguishes fluids from solids as clearly as does the mobility of their particles: For example, if you press with a weight on the upper end of a rigid cylindrical body, the pressure propagates from layer to layer to its base and becomes an increase of weight; there occurs no sideways action on the cylindrical wall. This situation already changes when the body consists of loose grains (small shot or sand) which the cylindrical wall holds together. The grains close to the weight attempt to escape the pressure from above, push in between their neighbours, transfer in the process the pressure to them; these neighbours do the same to their neighbours, etc.; in this manner, the pressure is transmitted in all directions from particle to particle and to the wall of the vessel and back again. If the wall is elastic, it will bulge and thus indicate the presence of pressure; if holes are drilled into it, the presence of pressure becomes more obvious in that the grains are flung out through them.

What is true for individual grains is even more applicable to completely freely movable fluid particles. The pressure which acts from inwards at some location on the vessel's wall (Fig. 173) or somewhere inside the fluid is, of course, the larger, the larger is the location, because it has then to withstand the pressure of a larger number of particles. Obviously, the internal pressure on equally large areas is equally large, that is, at areas of different size, it is proportional to the area; it is proportional to the magnitude of the external pressure (Principle of Blaise Pascal ).

You measure the pressure in terms of the force which acts perpendicularly on 1 cm (Fig. 174). Load the mobile piston P with a cross-section of 1 cm with 1 kg*; you can only keep the piston P' with a cross-section of 100 cm at rest, if you load it with 100* kg. If the weight is smaller, P' rises as a result of the higher pressure exerted by P.

Hydraulic Press

The principle of the propagation of pressure is employed in Industry in the hydraulic press (Fig. 175), invented by Pascal and improved by better packing of the piston by Bramah 1749-1814 1795: You press with the small piston a by means of the one-armed lever O on the water in the barrel A, the pressure propagates through the water in the pipe, drawn in dashes, to the large piston C and pushes it upwards until it is stopped. A safety valve S stops the pressure in the pipe from exceeding the admissible value. C is pressed upwards by a pressure which is related to that at a in the ratio of the cross-sections of C and a.

For example, if you apply at O the force of 0.2 kg* and the long lever arm is 10 times as long as the short one, a is pressed downwards with 20 kg*. If the cross-section of C is ten times as large as that of a, C is acted upon by 200 kg*. The hydraulic press allows to generate very high pressures, to lift extremely large loads, to forge iron, to extract oil and sugar in oil and beet sugar extraction factories, etc.

Compressibility. Principle of Piezometer of Oberstedt

The volume of water decreases minimally if high pressures are applied (at 8C and 705 atms; its unit volume is reduced by 1/47106 for every atmosphere (atm), whence water can be assumed to be incompressible. All drop-forming fluids are only compressible by very small fractions of their original volume. The piezometer (Greek: piezein = press), employed for measurements of pressure, consists of a thick-walled, thermometer-like, glass container for the fluid to be tested, a pressure pump connected to it and a thermometer (Oberstedt 177-1851 1822, later improved). Fig. 176 shows the essentials of the method of measurement of the compressibility of water. The vessel A is initially filled completely with water (without any air in it) and dipped with its capillary into mercury. All lie under the water. A large pressure is applied to the water with a compression pump . It propagates to the mercury, drives the mercury into the capillary; its volume is negligible compared with that of the water and shows how small is the compressibility of water.

Pressure inside a fluid

As long as an ideal fluid is at rest, each particle rests with respect to all the other particles. In this respect, the fluid behaves like a rigid body, whence, in order to be in equilibrium, external forces must meet the conditions of equilibrium for forces acting on rigid bodies. However, they must satisfy one more condition which is characteristic for liquid bodies.

Consider Fig. 177. Let a part of a fluid have been separated from the entire mass by a closed surface with an infinitely thin rigid wall. Only the area elements df and df' can displace normally to their planes, so that they move like in channels with cross-sections df and df' perpendicularly to the wall. The element df' lies horizontal, as this will simplify later considerations without changing the process; let the element df be arbitrarily oriented in space. Displace now the element df by d n inwards, when df' will move simultaneously outwards, because the fluid is incompressible, whence dfd n = df 'd n'. These are motions which are compatible with the conditions of the system, that is, they are virtual ones which can be executed. By the earlier considerations, an ideal fluid does not require work for a change of shape by external forces, whence the sum of the performed work by the external forces vanishes for all virtual displacements. The change of the configuration of the mass points in the fluid mass merely involves that the borders of the fluid mass displace at df and df'; inside the fluid, every particle is replaced by another equally valued particle. Hence we need only deal with the events at df and df'.

According to Fig.172, the surface of a fluid at rest must be at each point at right angle to the direction of the force acting there. Let p be the force per unit area of df and p' the force which we must let act on df' in order to maintain equilibrium; then the virtual works of each of the two forces are pdfd n and - p'df 'd n' (it is negative because d n' and p' have opposite directions to each other). If the fluid were not acted upon by gravity - you can make a fluid weightless - you would continue your argument: Since the sum of the virtual work must vanish, you must have pdfd n = p'df 'd n', and since dfd n = df 'd n', you find p' = p.

However, apart from p' and p, there acts also the weight of the fluid. If there is a fluid column of height h above df ' and the weight of a unit volume is g , then gravity presses on df ' with hg vertically downwards, its virtual work on df ' is hgdf'd n' and you have the equation

pdfd n + hgdf'd n' - p'df 'd n' = 0,

and since dfd n = df 'd n', you find p' = p + hg; the direction of the fluid elements do not enter into the relationship! Hence:

1. The pressure on an element of area at a given point in the fluid does not depend on the orientation of the area element in space.
2. The pressure is the same at all points on a horizontal plane within the fluid, that is, at equal depth, and it grows with the depth proportionally to the difference in depth.


The ability of the action of a pressure to propagate from one location uniformly is inseparable from the properties of fluids. Moreover, if a fluid is solely under the influence of gravity and its upper layers press with their weight on the lower ones, the pressure, which a layer receives from above, propagates in all directions and with the same magnitude. Nevertheless, the equilibrium is not disturbed anywhere in the fluid. The pressure is therefore obviously everywhere balanced by a pressure which is equally large and has the opposite direction; it is called buoyancy.

You can envisage buoyancy with the aid of Fig. 178. B is a glass tube which is open at both ends. You close its lower end by the plate CD, which you must press against the end of the tube by pulling the thread. But if you lower the tube with the plate in place deep enough into the water, it remains there without you pulling the thread. You can fill the vessel, formed by the tube and the plate, with water up to the level EF before the plate will drop. It falls when the weight of the water column above it and its own weight become larger than the buoyancy acting on it.

Pressure also propagates to the wall of a vessel and is counteracted by it - assuming, of course, that the wall is strong enough. (Otherwise its strength is surpassed by the pressure of the fluid and it bursts like the drum which Pascal (1647) exploded by water pressure with the aid of a thin, 10 m high tube.)

You can always display the pressure on the wall of a vessel by eliminating the wall's counter pressure by opening it, for example, at D in Fig. 179, when the fluid will be ejected through the opening. If there were no pressure, it would run down along the outside like when you overfill a vessel. The simultaneous action on the vessel's wall in the opposite direction at d manifests itself by setting the whole vessel in motion, provided that it can move easily, for example, rests on a float or is suspended like a pendulum. This motion is the action of the force of reaction. This process finds application in gardens in rotating water sprayers.

Bottom Pressure

If the pressure1 is propagated only in the direction of the force generating it, that is, of gravity, every area would only experience the pressure of the weight of the fluid above it; for example, at the horizontal section HH (Fig. 180), a would experience the pressure of the column aa, A the weight of the column AB, etc. However, the upper end of the column aa experiences also a pressure from above, although there is no fluid above it, because there acts on every unit area of the plane of the fluid, of which a is a part, the pressure of the weight of the fluid column g b on g which also propagates to the location a. The base a therefore experiences, besides the weight of the column aa, an additional pressure as if a column of height g b were on top of it, that is, as if above it were a fluid column, which reaches from it to the top of the fluid. Hence:

1. The pressure at a point in a fluid only depends on how far it lies below its pressure free level (disregarding the air pressure); the pressure increases with the depth and has its maximum at the bottom.

2. Points at equal depth below the pressure free surface experience the same pressure; a horizontal section of the fluid experiences the force on 1 cm, multipled by the number of cm of its area.

If the fluid is water (Fig. 180), the cross-section HH lies 50 cm below the free surface and has an area of 800 cm, then the force per 1 cm equals the weight of 50 cm, that is, 50 g*, and the force acting on the entire cross-section is 50g*800 = 40 kg*. If the fluid is mercury, the force per 1 cm is 5013.59 g* and the entire cross-section is pressed upon by 50*13.59800 = 543.6 kg*.

Hence the force is as large as if the entire cross-section had to carry a vertical column of fluid, which extends to the free surface and has everywhere the same horizontal cross-section; the magnitude of the force acting on a cross-section is only determined by the size of the cross-section, the specific weight, that is, the weight of 1 cm of the fluid and the depth of the cross-section below the free surface - but not on the amount of fluid which lies actually above the cross-section. That is also true for the pressure on the horizontal bottom of a vessel, bottom pressure. In all three cases of Fig. 181 act on the entire bottom - assuming the specific weight, the area of the bottom and the depth of the fluid to be the same - a force of the weight of the fluid column ABCD.

Hydrostatic Paradox (Stevin 1548-1620 1587)

In Fig. 181, the force at the bottom is seen to be in the second case larger, in the third case smaller than the weight of the fluid and is only equal to it in the case of vertical walls. It seems to be paradoxical that a fluid at rest, subject only to gravity, can exert on the bottom of a vessel another force than its weight would make one expect and that nevertheless an ordinary scale yields correctly the weight of the vessel with the fluid. However, you must take into consideration that not only the force on the bottom is transmitted to the weighing scale, but also those components which the side walls, rigidly connected to the bottom, experience in the direction of gravity. The pressure on the sides yields in the downwards converging vessel c an additional downwards component; in the upwards converging vessel b, it yields a (subtractive) upwards component. The force which acts on the scale, resulting from the force on the bottom and the force on the side, thus becomes always equal to the weight of the fluid.

If you do not change the area of the bottom, the depth of the fluid and the specific weight, then indeed the force on the bottom is always the same, namely equal to the weight of the fluid which lies perpendicularly above the bottom. This can be proved.

The vessel in Fig. 182 has the horizontal bottom a, which is not rigidly connected to the sides, but only pressed against the lower rim of the wall, namely by the upwards pressure of the weight on the scale. You thus have a vessel, the strength of which you can change arbitrarily according to the magnitude of the weight on the scale.

As long as the weight remains unchanged, you must always fill the vessel to the same height o, whether it is M or P or Q, before the bottom is pressed away, if it only has the same content of fluid. In the case of a vessel with vertical sides, which has everywhere horizontal cross-sections congruent to the bottom section, the force on the bottom equals the weight of the fluid. Naturally, you must find the weight by a scale. The set-up in Fig.182 can only be employed as a balance when you connect the bottom rigidly to the vessel and suspend the vessel as weighing scale from the beam (that is, when it moves with the beam).

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