F1 Flow of incompressible fluid and gases assumed to be incompressible

The incompressibility of fluids and the compressibility of gases are the reason for the separation of problems of the equlibrium of fluids and gases. However, flows of fluids and gases can be treated together, since also a gas can be considered to be incompressible as long as its flow velocity remains below a certain level (small compared with the velocity of sound in the gas under consideration). Its change in volume is less than 0.5% and practically irrelevant as long as the velocity is less than 0.5% of the velocity of sound. Then its motion differs in no essential aspect from that of an incompressible fluid (Ludwig Prandtl 1875-1933). In what follows, it is assumed that the velocities of gases are below this limit.

Pressure in a flow

So far, we have only dealt with the pressure in a fluid at rest. As soon as the pressure of a mass at rest causes acceleration of the mass, a fluid flows and the pressure at a definite location depends on the local velocity: It increases as the velocity decreases and decreases as the velocity increases.

How to follow a flow? It is difficult to follow an individual fluid particle, whence we will only investigate the changes in time of the velocity and the pressure in a fluid at a given point in space. At each point in space, you find a velocity in magnitude and direction and, in general, these quantities change from point to point. They are represented by stream lines: Curves the tangents of which at each point have the direction of the velocity. An area of the shape of a tube, which is formed by stream lines, is called a stream tube, its content of fluid a stream thread.

A stream thread flows in a stream tube like in a channel with solid walls - the fluid moves only along the stream lines, not at right angle to them. If there enters or leaves at each instant an equal amount of fluid, that is, the velocity does not anywhere depend on the time, the flow is called stationary. The strength of the flow - the amount of fluid passing through a cross-section in a second - is then the same at each cross-section along the thread, whence the velocities at different locations are inversely proportional to the cross-section: The smaller the area of the section, the larger in the same ratio is the velocity.

If you imagine the space filled by the fluid subdivided into thin stream tubes, you have an instantaneous image of the flow; naturally, it can change all the time. Only during stationary flow does the flow pattern not depend on the time, that is, the flow tubes do not change their shapes. In general, in an elementary treatment, attention is concentrated on a single stream thread: You either deal with one exhibiting mean conditions or you consider the entire flow as a collection of threads and then average the results. Changes in density can be neglected, since the compressibility of the fluid is much too small to demand attention. An elementary treatment also neglects friction. Water has so little friction that the results can be applied to real conditions.

Equation of Daniel Bernoulli 1738

What can we discover about the pressure in a stationary fluid flow? Consider in the flow tube of Fig. 225, which we imagine to be like a downward channel, the fluid between the cross-sections A and B and follow it over the time interval dt; during this time, the cross-section A moves to A', the cross-section B to B'. Let fA be the area of A and the velocity in it vA, and for B correspondingly fB and vB. Since the flow is stationary, one has fAvA = fBvB. We now ask: How changes the kinetic energy during the time interval dt? This change is equal to the work performed during dt by the force which is causing the motion.

We shall first find the expression for the change of the kinetic energy of the fluid mass and assume - certainly approximately correctly - that all particles, which at a definite instant of time lie between cross-sections (A and B) of the stream tube, which are very close to each other, will also during any arbitrary later time interval lie between two such cross-sections (A' and B'). According to this assumption, the mass which was lying between A and B reaches the space between A' and B'; in the space between A' and B nothing changes during the time dt, as every fluid particle is only replaced by another one which takes its velocity; however, the fluid between A' and B does not act on the outside. (By the above assumption, you make it possible, to take into consideration only the events at those cross-sections.)

The change of kinetic energy during dt is reflected in the change of the velocity (from vA to vB) of the fluid masses which pass during dt through the cross-sections A and B. If the cross-section A has the area fA and the velocity in it is vA, there passes during dt the volume fAvAdt with the mass (s/g)fAvAdt (s being the weight of unit volume) and the kinetic energy (s/g)fAvAdtvA2. The corresponding expression at the cross-section B is (s/g)fBvBdtvB2. The kinetic energy of the fluid mass, lying initially between A and B, changes during dt by

(s/g)fAvAdtvA2 - (s/g)fBvBdtvB2.

Next, consider the work done. It comprises the work which gravity and the pressure have performed on the fluid. For this purpose, we need again only consider the events at the cross-sections A and B. Gravity performs on a particle of weight q, which sinks from the height z (above the horizontal plane of reference) to the height z', the work q(z - z'). The work done on the entire fluid mass is therefore Sq(z - z'), where the sum is taken over all fluid particles. However, the particles between A' and B do not partake in the action, whence we extend the sum only over the fluid particles between A and B.

Let zA and zB denote the heights of the centres of gravity of the fluid masses, which pass through the cross-sections A and B during dt, and qA and qB the amounts of fluid, then Sqz - Sqz' = qA z A - qB zB. Thus, the work done by gravity during dt is: s fAvAdtzA - s fvBdtzB. Letting pA and pB denote the pressures per unit area at A and B and taking into consideration that the pressure at B is directed in the opposite direction to that at A, the work of the pressures during dt is: pAfA vAdt - pBfB vBdt. Hence follows the equation

(s/g)fBvBdtvB2-(s/g)fAvAdtvA2=s fAvAdtzA-s fBvBdtzB+pAfAvAdt-pBfBvBdt.

Each term contains dt and fBvB or fAvA, which are equal to each other. Dividing by and shifting terms with negative signs to the other side, we find:

v2B/2g + pB/s + zB = v2A/2g + pA/s + zA or v2/2g + p/s + z = const;

letting s = rg, (r is the mass per unit volume), we obtain:


the form in which we will use this equation below. It is Bernoulli's equation, the basic equation of Hydrodynamics.

The term v2/2g is called the velocity height, because it is exactly the height by which a body falls to reach the velocity v, the term p/s the pressure height, the height of a fluid column which by its weight generates the pressure p, z the location height, the height of the point of consideration of the fluid above a horizontal reference plane. Hence Bernoulli's equation says: The sum of velocity, pressure and location heights is the same at each point along a stream line. In practice, you do not only compute with this formula and relationship for single stream lines, but also for stream tubes with final cross-sections by averaging the true heights.

You can apply to the fluid flow between A and B (Fig. 225) the Momentum Theorem of Mechanics and investigate the change of momentum, which the fluid, initially contained in the bounded space, experiences during unit time when it is moved steadily. Here too you only need to consider the momentum at the cross-sections A and B, in order to find the Momentum Theorem of Hydrodynamics: The resultant of the forces, acting on a bounded space filled with fluid, equals in magnitude and direction the excess of the momentum of the exiting fluid over that of the entering fluid during unit time.

You can now draw important conclusions regarding the force exerted by the fluid on solid bodies without detailed knowledge of the velocity distribution. For example, you can compute the force, exerted by the fluid on a submerged and moving body as well as treat other tasks which are too difficult for elementary treatment. We will not discuss the momentum theorem in detail and turn now to the problems of pressure and velocity in moving fluids, a treatment of which is possible on the basis of Bernoulli's equation.

Consequences and application of Bernoulli's equation (Pressure and velocity in fluid flows)

Through every cross-section in a steady flow passes at each instant the same amount of fluid, whence the quantity which passes through a cross-section per second is inversely proportional to the area of the cross-section. What about the pressure at different cross-sections?

Let f and f0 be the areas of two cross-sections, to which correspond the pressures p and p0, the velocities v and v0, and the heights z and z0, related to some reference plane. Bernoulli's Equation then yields for steady fluid flow

p - p0+ rg(z - z0) + (v2 - v20) = 0.

Since fv = f0v0, you can introduce the cross-section into this equation. The computation yields the result (Fig. 225a): In a {smaller/larger} cross-section - that is, where the velocity is {larger/smaller} - the pressure is (smaller/larger}.

But you can also perceive this without computation (Fig. 226): Through a, horizontal tube, narrowing towards the right hand side, flows steadily towards the right hand side an incompressible, frictionless fluid. There must pass every second through each cross-section the same amount of fluid. Therefore the velocity of the fluid's particles is smallest at L and grows steadily towards R. The fluid particles can receive this acceleration only through the pressure forces acting on them. In order that at an instant the cylindrical fluid quantity F will be accelerated towards the right hand side, a larger pressure must act at its back A than at its front B, that is, the pressure at B is smaller than at A. By repetition of this argument, you see that the pressure in the tube decreases steadily from L towards R.

We find the same pressure distribution (decrease of pressure from L to R) by an analogous argument regarding the reverse flow of the fluid. The fluid particle F then moves slower from R to L. Fluid particles can therefore only experience acceleration during motion from points of higher pressure to points of lower pressure or, more briefly stated, when the pressure drops in the direction in which they move.

Flutter of flags

If a wall, along which a flow passes, has a wavy form (Fig. 226b), the cross-section of the flow is more slow at B and faster at T, whence the flow generates excess pressure at B, deficit pressure at T. However, everywhere along the wall the pressure is directed in such a way that each curve would be enlarged, if the wall were flexible. This is characterized by a weakly waved, thin, really deflectable wall, both sides of which have a flow along them (Fig. 226c); excess and deficit pressures at both sides of the wall support one another. This process explains the flutter of flags in wind, the generation of waves on the surface of water, when the wind reaches a certain velocity, etc.

You see the distribution of pressure, which follows from Bernoulli's Equation at a tube (Fig. 225a) with gentle narrowing and widening of its cross-section (for the reason of gentle) after attaching manometer tubes: The shorter and longer fluid columns indicate the smaller and larger pressures at these locations.

A widening tube with flow through it is employed in water-, steam- and gas-jet-appliances. For example, at the narrow end (Fig. 230) , the flow provides suction for air; the rising pressure at the widening end takes the sucked-in air away. Among these, you have Bunsen's water-jet-air-pump of Fig. 227, suction installations like that of Fig. 228 and the Bunsen-burner of Fig. 229, in which the gas jet exiting at A sucks air through the opening L with which it mixes before being lighted. Others are the inhalation instrument and injectors, for example, the steam-jet-injector (Fig. 230).

Mercury-vapour-jet-air pump

Langmuir 1916 built such an airpump, the suction velocity of which exceeded that of all pumps built earlier (Fig.231). The vapour rises from the heated mercury A through the pipe B and sucks air through the ring-shaped slot S around the exit L from the receiver shut at F. The sucked air flows away in the the vapour stream. You first remove air with an auxiliary pump (pre-vacuum pump) from the receiver until the pressure drops to 0.05 - 0.1 mm.You then activate the main pump. The vapour jet, entering the pre-vacuum space C, spreads lin tufts.

Unless you provide precautions against the vapour hitting the wall of the tube, into which it entered, and forming by deposition mercury drops, these would again vaporize and reduce the suction. Langmuir avoids this by energetic condensation of the particles approaching the wall (flow of cool water through K1K2), a special design of the pre-vacuum space C for very fast removal of the condensation heat. According to him, this is most important, whence also the name condensation pump.The pump sucks 1500 - 3000 cm/sec.

Measurement of pressure and flow velocity

Bernoulli's Equation relating the pressure and velocity of a flow permits computation of 1) the velocity, if you can measure the pressure, and 2) the pressure, if you can measure the velocity. If there exists in a uniform fluid flow (with velocity v0) an obstacle, there arises in front of it retardation of fluid particles within which the flow spreads in all directions (Fig. 232) At the centre of the impounding, the stagnation point P, the velocity v vanishes.









At the stagnation point, Bernoulli's Equation yields p=p+vr/2, where p0 is the undisturbed pressure in the same horizontal plane. You call the increase in pressure p - p0 stagnation or velocity or dynamical pressure. The total pressure p is measured with the Pitot Tube (Fig. 233), named after its inventor Henri Pitot in 1730 (Histoire de l'Academie des Sciences, 1732, p 376), one leg of which is placed facing the flow, the other vertically upwards. In the first leg, the flow stops, whence the pressure rises. The fluid arriving at velocity v raises the water level above the free surface by D, which is in agreement with the velocity height v/2g in Bernoulli's Equation.

The static pressure p is measured with a pressure probe (Fig. 234); in the sidewards cuts, the openings of which are parallel to the direction of the flow, and inside the probe the fluid is practically at rest. The pressure in the probe, since is it constant in the section at rest and must change in the interface steadily into that in the moving fluid, is equal to the pressure in the passing fluid. The stagnation pressure p - p is measured with Prandtl's stagnation instrument which combines the Pitot Tube and Pressure Sonde (Fig. 235). In order to simplify the reading of the, as a rule, small differences in height D, which would be very inconvenient just above the free surface of the water, the fluid is sucked in both tubes simultaneously to a convenient height.

The flow velocity (air, water) in a pipe line is measured with a stagnation disk (Fig. 236). A venturi nozzle is also employed for its measurement (Fig. 237); you measure (manometrically) the difference between the (lower) pressure at the narrowest section f0, which arises from the increase of velocity, and the pressure at the not reduced opening f in front.

Outflow velocity (Torricelli's Law). Discharge quantity

Bernoulli's Equation also yields a theoretical expression for the velocity, at which a fluid escapes through the lower section of an open vessel (Fig. 238), given the constant height (z - z) and the pressure difference at the cross-sections. If the velocity at A is so small that it can be neglected compared with that at B, it is assumed that the pressure at the lower section is equal to that at the upper section and, if the vertical distance between the sections is h, then

rgh - rv2/2 = 0, whence v = (2gh)1/2,

a formula which Torricelli discovered in 1641. The fluid escapes down below as if it had fallen freely from the height h above. The velocity depends only on the height h, and not on the direction in which the jet escapes. - If p and p0 are not equal, that p - p0 is an excess pressure, acting on the free surface, and v0 if the mean velocity of the free surface, that is

v = {v0 + 2[gh + (p - p0)/2}1/2.

A jet which escapes from a circular opening (the form of it influences that of the jet) is behind the smallest cross-section like a solid, transparent cylinder, further on it becomes turbid and displays now and then nodes and bulges. These nodes and bulges (Felix Savart 1791-1841) can be shown to be an optical illusion: The jet does not form there a contiguous structure, it decomposes as a result of the ever increasing velocity into drops, which change during the fall their shape and oscillate about a spherical one (Fig. 239). By eye, you cannot distinguish the drops, which follow rapidly one another, and you see nodes and bulges.

The independence of the escape velocity of the direction suggests experiments, which establish the formula v = (2gh)1/2. You let the fluid escape horizontally from a vessel (Fig. 240). With water columns of 2 - 6 m height, the observed escape velocities have been found to be very close to their calculated values, but always somewhat smaller due to fluid friction at the wall of the vessel and the air, and due to internal friction of the fluid.

According to the formula v = (2gh)1/2, the velocity v depends only on h, all fluids should escape from an opening equally fast, as long as h is constant (like all bodies fall equally fast and reach in the course of time the same velocity). However, this does not apply to fluids. In fact, we have silently assumed that the work, performed by gravity on the escaping fluid, converts completely into kinetic energy, that is, we have not taken into account that the fluid particles undergo friction, one with another, and along the wall of the vessel, whence their motion is retarded. The more viscous a fluid, the larger is the impediment and the larger must be h, so that v reaches the same value (the smaller is v as h stays constant). Hence, the actual escape velocity differs the more from the theoretical value, the more viscous is the fluid, for example, more for castor oil than for alcohol.

If you measure in grams the amount of fluid , which escapes from a vessel during 1 second, and compute how much should escape, corresponding to the formula v = (2gh)1/2, the size of the opening and the specific weight, you will find in the mean that only two thirds actually escape, because during the computation you have assumed that the opening a b and the jet (Fig. 241) have everywhere the same cross-section. Actually, the jet contracts near the opening; Its smallest cross-section at c d is only 0.6 of that at the opening (vena contracta). In fact, the fluid flows from all sides to the opening, as is readily confirmed experimentally. The individual stream tubes only become parallel at a certain distance from the opening.

Surface tension has here a remarkable role. At the opening, the outer layer of fluid particles acts like an elastic ring around the jet. For example, the influence of the surface tension manifests itself, if you vaporize alcohol or ether near the jet . The escape velocity will then increase noticeably, because the surface tension is reduced.

A modification of the outlet (Fig. 242) changes considerably the shape of a jet and the quantity discharged; a short cylinder stops the jet from contraction and makes the discharge quantity close to that computed. A long tube has a greater effect. Torricelli's formula than fails completely and you must rely on experiments. Beyond all, frictional resistance becomes important: It increases with the surface of the tube and the velocity of escape. One part of the pressure, exerted on the fluid, is then spent on overcoming this resistance. Hence the escape velocity becomes smaller than what it would have been, if the entire pressure had generated velocity. For example, if water escapes from a vessel through a long tube, it escapes with a much smaller velocity than would correspond to the pressure height; you notice that from the parabola which the fluid describes on leaving the tube.

The pressure acting on moving water at a given location in a tube is the smaller the further it is away from the vessel. You see this when you install along the tube a b thin, vertical tubes (Fig. 243). As long as the tube is closed at b, the water in these tubes has the same height - namely that in the vessel itself; as soon as you open b, you obtain the shown distribution.

Escape of a gas under pressure

As long as in a space, filled with gas, the pressure everywhere is the same, there is no flow. According to the Kinetic Theory, there exists a state of equilibrium, although not one of rest: Through a plane, which you imagine having been placed through the space, pass each second as many particles from A to B (Fig. 244) as from B to A, so that the particles do not form anywhere clusters at the expense of another location: The centre of gravity of all particles remains at rest. This prevails as long as the pressure throughout the space is equal. However, if for some reason or other the pressure in A becomes larger than in B, a partition (in place of the imagined plane) experiences an excess pressure. If you puncture it, the gas flows from A to B until the pressure on both sides is the same. Hence the content of gas in B is increased at the expense of A, that is, more gas particles pass from A to B than from B to A: The centre of gravity of the gas is also shifted from A towards B.

The velocity v, at which a gas escapes from a space, is measured by the volume, which escapes in a given time interval from an opening of given size; v depends on the density r of the gas and the excess pressure p - p0. This equation only applies while p - p0 is very small and the opening, through which the gas escapes, is very small compared with the size of the space, containing the gas. Nevertheless, the formula is valuable: The relationship between the escape velocity v and the gas density r allows you to measure the density without having to know the escape velocity.

Measurement of the density of gases through their escape velocity (Bunsen)

If v, v1 and s, s1 are, respectively, the velocities and densities of two gases which escape from vessels under the excess pressure p - p0 = P, then

v = 2P/s and v1= 2P/s1, whence v2/v12 = s1/s,

that is, the gas densities are inversely proportional to the squares of the escape velocities. For example, if v1 = 2v, then s1 = s/4, whence the gas which escapes twice as fast as another gas (under the same conditions) has one quarter of the specific weight of the first gas.

Thomas Graham 1805-1869 has derived this result experimentally and Bunsen has based on it a method for the measurement of the density of gases; however, he does not measure (by means of a special apparatus) the escape velocity, but the time, which is required by equal quantities of both gases for their escape from a vessel. Since the gas with double the density requires only half the time as the other gas, you have the result: The densities of the gases are related one to another as the squares of the time intervals t and t1, during which equal amounts of gas escape:

s1/s = t21/t2.

For example, if a certain amount of air (density s) requires t = 36.9 sec and the same amount of carbolic acid (density s1) requires t1= 45.3 sec, then

s1/s = (45.3/36.9)2 = 1.507,

that is, the density of carbolic acid s1 is 1.507 that of air. The method is reliable and requires only a few cm of gas, an advantage which is valuable for gas measurements. It is especially convenient for industrial purposes (measurement of domestic gas).

Diffusion of gas through porous bodies

The velocity of escape of a gas from a vessel depends also greatly on the size of the opening. The conditions, under which the formula above applies, also include that the opening is in very thin wall, that is, the gas escapes only though an opening in a very thin wall, so that gas really escapes through an opening, not through a channel. (In Bunsen's apparatus, it escapes through a hole in a thin platinum foil.)

If a gas flows through a capillary, the formula does not apply; you have a new law (O.E.Meyer 1834-1909), if the length of the capillary is approximately 4000 times its diameter. If part of the wall is a porous plate (graphite, unglazed clay), so that the gas can escape through a large number of capillary channels, it diffuses through the plate. Through a porous subdivision between two gases (Fig. 244 above), the gases diffuse into each other without there having to be a pressure difference.

Graham's Law of diffusion rate

The rate at which a gas passes through a porous plate is, other things being equal, inversely proportional to the density (Thomas Graham); for example, for oxygen, which is 16 times as dense as hydrogen, it is only one quarter of the velocity of hydrogen.

You can demonstrate the difference between the diffusion rates of air and hydrogen by means of an unglazed clay cell (Fig. 245): You conduct the hydrogen through an air tight fitted pipe and the clay cell until all the air has been ejected; you then close the tap. The clay wall borders inside against hydrogen, outside against air; the hydrogen diffuses much faster than the air, whence the pressure sinks inside, as is shown by the rising of the fluid.

The difference between diffusion rates can also be employed to separate from each other two mixed gases with different densities. Thus, you can reduce the content of

nitrogen in atmospheric air - a mixture of 21% oxygen and 79% nitrogen, the densities of which are interrelated like 16:14, - by conducting the air through an unglazed clay tube through a vacuum (Fig. 246). This method, changed and improved according to the the purpose it is to serve, enabled separation of the isotopes of neon (Gustav Hertz 1932).

Diffusion pump (Gaede)

The mercury-vapour-pump of Wolfgang Gaede 1878-1945 1913 employs diffusion and is the protoype of the pumps of modern high vacuum techniques. The air to be removed and the mercury vapour diffuse into each other and the mercury vapour entering into the air is condensed rapidly in a vacuum chamber.

The principle of the pump is shown in Fig. 247: C is a porous clay wall; on its left hand side is the mercury vapour entering through the tube A B, on its right hand side the air which communicates through the tube E with the recipient. To start with, the pressure in the space is reduced with the aid of an auxiliary pump to 0.1 mm mercury column (pre-vacuum). The air diffuses from the right hand side to the left hand side through C and flows away in the vapour jet A B, the mercury vapour diffuses from the left hand side to the right hand side and condenses immediately, because the gas trap D is submerged in liquid air. This is the basic idea.

In the pump (Fig. 248), Gaede employs instead of a clay wall with its pores a steel pipe C, through which the mercury valour passes and which has a slit S as wide as the pores - that is, approximately the mean path length of the molecules. Through the slit, air diffuses from the outside inwards (into the vapour stream) - it flows away with the vapour - and mercury vapour from the inside outwards (into the vacuum space); the refrigeration section K1K2 condenses it immediately. The pressure in the recipient becomes eventually 10-6 to 10-8 mm mercury (Gaede's rotating pump 7.10-5); theoretically, there exists no lower bound for the possible vacuum. The highest suction velocity of the pump is 80 cm/sec (rotating pump 130) and it remains constant also at the lowest pressures. It is substantially higher in the mercury-vapour-jet- condensation-pump.

continue step back