s law (Equation 3), we see that from this law
, and from Equation (7), which shows that the behavior of the pressure
head is the opposite of the velocity head as the relation between the flow areas at the top and at the bottom changes, which is the essence of Bernoulliâs law.
In Section V of Hydraulics, the hydraulic forceâthe force responsible for the local accelerationâwas equated by J. Bernoulli as follows. The displacement of the layer Fm when the elementary displacement at the
exit is is given by, where is the thickness of the layer Fm; that is, the vertical projection of the
segment that links the faces of the element along the conduit center line. As before, the effects of the ver-
tical motive force of this layer, translated to the first area h gives the equivalent, which upon
integration through the total length of the axis of the conduit, for any layer and for any acquired velocity v, noting
that herein not only h and w but also must be considered constant, gives. This is the hy-
draulic force, and accounts the required motive force for the actual acceleration of the liquid flowing out. This force is responsible for what is known today as the local acceleration.
When the gravitation force, is the only motive force available to compose the hydrostatic force and the hydraulic force, in Section VI, J. Bernoulli finally obtains:
for a liquid with unity density, claiming that this is the most general equation for the determination of the veloc
ity with which the liquid flows out at any moment. It is understood that denotes the sum of all the contained between the two extremes of the conduit. Of course, this is a quantity that depends only on the
geometry of the conduit under consideration.
Based on this general equation, J. Bernoulli shows with a numerical example that the transit time from rest to the steady state efflux, from rather wide vessels through narrow orifices is imperceptible, and can safely be considered constant. This is an interesting conclusion regarding the importance of the hydraulic force in such problems.
The remaining of Part II, is basically dedicated to other applications of the general equation in problems such as: the derivation of an expression for the pressure at the walls of the conduit (Sections XI & XII), the variation of the exit velocity during the emptying of vessels (Sections LI & LII)âto be seen in more details later on in the present work, the making of clepsydras (Sections LIV?LIX), the examination of Newtonâs cataract (Section LX) etc.
In Section VII, Part 2 of Hydraulics, J. Bernoulliâs writes6, where z is the height from which a given body, by falling under natural gravity g, would acquire the velocity v. This allowed J. Bernoulliâs re-writing his general non-steady equation as:
6Here we find a rare instance of a precise quotation of Torricelliâs law as is known today:.
Equation (9), for the case where; that is, when the cross-sectional areas are normal to the axis of the vessel reduces to:
In Section X, Part 2, we also find for the first time the interpretation of internal pressureâcalled immaterial force by J. Bernoulliâ:
ââŠ in portions of fluid acting mutually on each other, the immaterial force lying between must be considered just as elastic air, which extends itself not only to opposite directions, but also into all surrounding regions; from which now it is easily understood that from this immaterial force itself the pressure, which is the subject here, develops. This certainly is exerted on the walls of a conduit, by which in turn it must be confined while it acts freely forward and backward on the portions of the liquid wherein it existsâŠâ
J. Bernoulli approaches can be considered remarkable for the time for the following reasons.
1) For reaching the same law proposed by his son D. Bernoulli for steady sate flow with a physical model more attached to reality, that is, from the pressure developed for accelerating the fluid in the contraction of a conduit.
2) For writing expressions with dimensionally homogeneous quantities (such as Torricelliâs law), which were not the standard of the day.
3) For giving for the first time the correct interpretation for the internal pressure in a fluid.
For these reasons, his contributions to the problem of discharge can be considered far superior than that of D. Bernoulli, and he would naturally have deserved a place in the hall of the greatest contributors to fluid mechanics, and particularly to the problem of discharge, without the need to compete with his son for pioneering the work in the area. Therefore, it is not appropriate to say that J. Bernoulli predated the work of D. Bernoulli; what in fact might had happen, is that after the publication of Hydrodynamics he saw that he had missed a great opportunity to be the pioneer in an area where he felt he had better approaches than those of his sonâs, perhaps since an earlier time than that when Hydrodynamics appeared. As mentioned earlier, the superiority of J. Bernoulli approach to the problem of discharge over that of D. Bernoulli was even recognized by Euler. Unfortunately for Johann, Bernoulliâs law was named after Daniel.
2.5. DâAlembertâs Approach
In the Preface of the Traité de lâ équilibre et du mouvement des fluides (DâAlembert, 1744) , DâAlembert makes remarks to some results and approaches used by D. Bernoulli in Hydrodynamics. For example, he judged that the application of the principle of conservation of living forces, considering the fluid particles as elastic corpuscles was an induction without force, and that it was in need of a more clear and exact demonstration. He claims that despite the fact that this was already done by him in his Traité de Dynamique (DâAlembert, 1743) , he would prove it again in a more extended and detailed form.
In a letter to Euler of July 7, 1745, D. Bernoulli of dâAlembertâs work says (see Truesdell, 1955: p. XXXVII, footnote 2 ):
ââŠ I have seen with astonishment that apart from a few little things there is nothing to be seen in his hydrodynamics but an impertinent conceit. His criticisms are puerile indeed, and show that he is no remarkable man, but also that he never will beâŠâ
Therefore, if DâAlembert was to produce something new and meaningful, of course he could not follow D. Bernoulliâs footsteps, at least as far as mechanical principles are concerned. In the Traité de Dynamique (DâAlembert, 1743: pp. 50-51) , a âgeneral principleâ comes in the form of a âgeneral problemâ, in which DâAlembert proposes to find the motion that each body should take in a system of bodies arranged mutually in any manner whatever; in which a particular motion is impressed on each of the bodies, that it cannot follow because of the action of the others. For finding the motion of the several bodies the following principle is given: âDecompose the motions a, b, c, etc. impressed on each body into two others, etc. which are such that if the motions etc. were impressed alone on the bodies they would retain these motions without interfering with each other; and that if the motions etc. were impressed alone, the system would remain at rest; it is clear that etc. will be the motions that the bodies will take by virtue of their actionâ. To show the power of his principle, DâAlembert solves several dynamic problems, one which is Huygensâs center of oscillation for the compound-pendulum, without recoursing to the principle of the living forces.
If one denotes ma an applied force on the mass point m of a connected system, and the actual acceleration of this mass point, then, in which corresponds to the destroyed force, which is a fictitious force applied to the point mass that maintains the system in equilibrium. Therefore, for the equi-
librium of a system of N point masses. For the cases in which there are no impressed forces on the system,; and hence,. As we shall see in the following, for a continuous
system of particles such as a fluid, the integral form of these summation expressions are used by DâAlembert in the Traité de lâ équilibre et du mouvement des fluides.
A theorem presented by DâAlembert in §22 of the Traité de lâ équilibre et du mouvement des fluides (DâAlembert, 1744) , which is a geometric interpretation of his dynamic principle, forms the basis of the various developments presented in this Traité. By dividing the vessel to the right of Figure 7 into horizontal layers (tranches), each one submitted to a particular accelerative force represented in the left side of this figure (not necessarily the gravitational force), he says that âthe fluid in such state cannot be in equilibrium, unless the area is not zero; that is, the sum of the positive areas equals the sum of the negatives.â
In §25 of the Traité de lâ équilibre et du mouvement des fluides (DâAlembert, 1744) , by calling the undetermined accelerative force of each layer, dâAlembert deduces that the state of equilibrium would then require
that (x is the co-ordinate along the vessel), which would correspond to, where and
are the infinitesimal thickness and velocity of the layer, and is an infinitesimal time. This is recognized as the continuous form of dâAlembertâs dynamic principle for the case when there are no impressed forces on the system. By considering that the layer thickness can be associated with the mass of this fluid element
and the acceleration a of the element, then, which is recognized as Newtonâs
second law for the equilibrium.
DâAlembert had rejected the concept of force and thought of mechanics as âthe science of effects, rather than the science of causesâ. ââŠ All we see distinctly in the movement of a body is that it crosses a certain space and that it employs a certain time to cross it. It is from this idea alone that one should draw all the principles of mechanicsâŠâ (DâAlembert, 1743: p. XVI) . DâAlembert was born ten years before Newton died in 1727 and Newton, in turn, was born in the year which Galileo died. According to Hankins (Hankins, 1970: pp. 152-153) , when dâAlembertâs Traité de Dynamique was published in 1743, Newtonâs general laws of motion were not yet regarded as the major synthesis of all those that had gone before, up to the point that Newton is mentioned only
Figure 7. Vessel (to the right) and representation of the accelerative force (to the left) for the presentation of the fluid state of equilibrium (a reproduction of Figure 8 from the Traité de lâ équilibre et du mouvement des fluides).
three times in the Traité de Dynamique, and not associated with any important principles. The search for mechanical principles continued, particularly with dâAlembert, who rejected âobscure and metaphysicalâ entitiesâ particularly âforcesâ and âmotive causesâ. This is way the expression, involving only kinematic quantities is basic and used throughout his Traité de lâ équilibre et du mouvement des fluides.
In the remaining of Book I of the Traité de lâ équilibre et du mouvement des fluides, DâAlembert discuss various cases o fluid equilibrium such as: fluid equilibrium with solids in it, pressure distribution in the fluid layers with gravity in a constant direction, equilibrium of fluids of different densities, law between gravity and density in different layers of a fluid with constant gravity in a given direction, equilibrium of a fluid where the layers vary in any way in gravity and density, âadherenceâ of fluids, equilibrium of a fluid with a curved upper surface (figure of the earth), the equilibrium of elastic fluids (various cases).
I will now focus on some paragraphs at the beginning of Book II of the Traité de lâ équilibre et du mouvement des fluides, entitled âOn the Movement of Fluids Encircle in Vesselsâ. After have beginning Chapter I demonstrating two theorems, in §86 DâAlembert remarks that âit is evident that the fluid will remain in equilibrium, if each layer is not animated by an infinitesimally small velocityâ. These two terms can be interpreted as what is now referred to local and convective infinitesimal increments in the velocity of the layer.
Before presenting the case âMovement of a portion of a fluid without weight inside an indefinite vesselâ, dâAlembert presents as preparation, a vessel of any shape (to the left of Figure 8), and defines a corresponding curve (to the right of Figure 8), constructed in such way that, as for example, the intermediate YX, is given by, in which GH is a constant line in the upper part of the vessel. He then calls N the area,
which is given by the integral, where is the height and y is the width of any indeterminate
layer, for which the product is constant (he is invoking continuity here). The infinitesimal area is then given by:
. (11a, b, c)
In §90, DâAlembert poses Problem I as follows: ââŠ Suppose that a given quantity of Fluid CDLP (Figure 32) homogeneous and without weight, is put into movement by any cause such as by the impulsion of a piston, which moves according to AB inside an undefined vessel; it is required to find the velocity of this Fluid in each
Figure 8. Vessel (to the left) and representation of a curve given by (to the right) as preparation for the later developments (a reproduction of Figure 32 from the Traité de lâéquilibre et du mouvement des fluides).
instantâŠâ Since the velocity of the layer CD is given by then, from the velocity u of the layer GH,
itâs possible to find all others.
DâAlembert equilibrium condition requires that. From this expression, and by assuming that a moving layer has a velocity V in an instant, following that in which the velocity was v, it follows that
. Since, then, or
, and, where u is the velocity in the layer GH, which is fixed.
Then, the velocity in any other layer such as CD, is the same as GH to CD. Since, and from Equation (11c) he finally gets:
In §100, DâAlembert gives a solution to the problem for the fluid subjected to a gravity p. In this case, his basic expression, transforms into This is recognized as the continuous form of dâAlembertâs dynamic principle for the case when the impressed force on the system is the gravitational force. When written in the form, is equivalent to Newtonâs second law of motion, where p is the external gravitational force acting on the fluid. For this case DâAlembert gets:
which, according to dâAlembert, when combined with, will give the value of u in each instant.
In §105, DâAlembert gives a solution for the velocity of the layer PL. This solution begins by rewriting Equation (13) as. Then, by substituting for the expression given by Equation (11c), and by calling, s is the subimity7; he finally gets:
In § 113, and based on Equation (14), DâAlembert gives the solution for a cylindrical vessel with an aperture made in its bottom (Figure 9), in which the water is subjected to the gravity p, as:
When PL is a very small aperture, and then; that is, Torricelliâs law is recovered not only for cylindrical vessels, but for all vessel shapes, when PL is a very small aperture.
DâAlembert claims that Equation (14) is the same found by D. Bernoulli, via the principle of living forces. Indeed, this is the same equation developed by D. Bernoulli in §8, Third Chapter of Hydrodynamics, in which it
is written as. This equation can be rewritten as
. In fact, apart from the differences in nomenclature and symbology, these
equations are all equal to J. Bernoulliâs general non-steady flow equation [Equation (10)] as well.
7As noted before with J. Bernoulli, here appears again a precise quotation of Torricelliâs law written as, where (the pesanteur) is the gravity g. This shows that by the middle of the 17th century the standard form of writing Torricelli´s law was already in use.
8There is an error in the second-hand side of this equation, in which was used, instead of.
By calling, which corresponds to the velocity of the water flowing out, then the above equation
Figure 9. Vessel with an aperture made in the bottom, for which Equation (15) was developed (a reproduction of Figure 40 from the Traité de lâ équilibre et du mouvement des fluides).
reduces to. As we saw before, when the lower surface is an orifice of area
which is very small, then; that is, Torricelliâs law is recovered in this case. Also, as D. Bernoulli points out quite correctly: ââŠ. when the orifice is not very small, by no means the shape of the vessel can be neglectedâŠâ Of course, this conclusion is valid for the non-steady flow cases, since the exit velocity for steady-flows, depends only on the ratio of the lower to the upper cross-sectional areas.
In most of the developments in Book I & II of the Traité de lâ équilibre et du mouvement des fluides, DâAlembert essentially replicates problems already dealt with by Daniel and Johann Bernoulli, but using his own principles and approaches instead. He criticizes D. Bernoulli for using the principle of the conservation of living forces when the change in velocities is not infinitesimal (DâAlembert, 1744: p. 109) , where he might be wrong, particularly when D. Bernoulli breaks the pipe in the development of his law because, as we saw earlier, the conservation of living forces was derived by Huygens for elastic impacts, and the breaking of the pipe can be considered a sudden impact in the water initially at rest in the pipe. DâAlembert also saw the approaches of J. Bernoulli with some reserves. For example, he remarks that various points in the Theory of J. Bernoulli needed demonstrations, and criticizes him by using a âvery complicated methodâ for determining the pressure at the bottom of the vase (DâAlembert, 1744: pp. 159-160) â.
In the Traité de lâ équilibre et du mouvement des fluides there are rare moments that dâAlembert talks about pressure, and worse, he did not seem to have realized and grasped the very important basic relation between pressure and velocity in fluid flows, as was done very explicitly by D. Bernoulli in Hydrodynamics. This is perhaps a consequence of his aversion to forces, and particularly to pressure forces. We will only find for the first time a specific development of the relation between pressure and velocity much later in his Essai dâune nouvelle théorie de la résistance des fluides (DâAlembert, 1752: p. 24) , in a result that apart from differences in nomenclature and symbology, resembles J. Bernoulliâs (Equation (5)); that is, Bernoulliâs law.
2.6. A Non-Steady Flow Application Example
The shape of the vessel is only important in non-steady flows and has a direct impact in the value of M (see Eq
uation 10), which for vessels with cross-sectional areas normal to the vertical axis is given by, where
x is the vertical axis and y is the cross-sectional area of the vessel at the x co-ordinate. However, in Equation (10), we see now time dependence appearing explicitly in this equation. The following application by J. Bernoulli will show how the time dependence is recovered from this equation.
The influence of the vessel shape, which appears equally in the Bernoullis and DâAlembert formulations, can be seen in the problem given by J. Bernoulli in Section XXVII, which concerns the calculation of the time that it takes for the exit velocity to reach the steady-state value from rest.
Under these conditions and for, Equation (10) yields
The time differential will be equal to
For cylindrical or prismatic vessels of cross-sectional area h and height a, for which, the latter expression upon integration yields
which implies an infinite time for the steady-state velocity to be achieved. The apparent anomaly suggested by the infinite time required to establish the steady-state flow arises from the assumption of fluid incompressibility.
However, the steady-state velocity does attain 99 per cent of in a finite time given by
As numerical examples, letâs consider:, , and three ratios of of 0.1, 0.05, and
0.01, for which the latter expression yields equal to 0.033, 0.0165, and 0.0033 seconds, respectively.
These results confirm a Corollary given by J. Bernoulli in Section XXVIII of Hydraulics ââŠ The efflux from rather wide vessels through narrow orifices can safely be considered as constant the instant after beginning of motion.â
Before closing this section, it should be pointed out that this problem forms the basis of the so-called Rigid Column Theory in hydraulic transients, in which the same expression given above for the time to reach the steady-state is simply obtained by applying Newtonâs 2nd law to the water that is accelerated in a pipeline, due to the sudden opening of a valve at the end9.
3. Eulerâs Contributions
Eulerâs main contributions to the problem of discharge appeared almost simultaneously in two publications of 1754, namely, Sur Le mouvement de lâeau par dês tuyaux de conduit (Euler, 1754) , and Tentamen theoriae de frictione fluidorum (Euler, 1761) 10. In the former publication, Euler presents the theory of a pump delivering water through a conduit to an elevated reservoir; in the latter publication, he presents the theory for water pipe friction. I shall now present the highlights of the first publication, to later on get into more details of the second publication.
In both memoirs, Euler strategy consists in finding the accelerations necessary to perform the necessary work, and according to Newtonâs second law, equates them to the forces available to perform it. In the pump memoir, since the flow is non-steady, the accelerations that come into play, in modern terms, are the local and the convective accelerations; and the forces responsible for these accelerations are the gravitational and pressure forces. In the friction memoir, he only considers the steady-state case, in which only the convective acceleration comes into play, and of course, with the additional friction force. As far as mechanical principles are concerned, Euler does not make any elaborations, and simply applies Newtonâs second law quite naturally in both memoirs. The pressure forces are equate quite naturally as well, stripped from any physical or metaphysical considerations.
3.1. Eulerâs Theory of Pump Delivering Water to an Elevated Reservoir
In the introduction of Sur Le mouvement de lâeau par dês tuyaux de conduite (Euler, 1754) , Euler mentions the works of the Bernoullis and DâAlembert on hydraulics, but considers their approaches âless than generalâ because they did not resorted to the âAnalysis of infinitesimalsâ. Moreover, since these authors have considered the motion of water in vessels of general shapes, one would try to find in vain the application of a specific case, and the Practitioners would find even less resources to conduct their projects. He then proposes to develop this subject, âonly as far as to its application in practice is concerned, which would allow finding the clarifications that are necessaryâ.
According to Eckert (2002) , the origins of this paper may be traced to Euler´s involvement by the request of the King of Prussia, Frederick the Great, to calculate the hydraulics for a fountain in his Park at Sanssouci in Potsdam. Eulerâs involvement began in 1749, when a new effort was made in the park to improve the water- raising machinery and the tubes for the pipeline to the elevated reservoir.
Euler then presents the complete theory of a one-cylinder pump to deliver water through a piping system to an elevated reservoir (see Figure 10). Here, since the flow is non-steady, Euler first find expressions for the local and convective accelerations in a general conduit cross-section, which according to Newtonâs second law, equate to accelerative forces, namely gravity and pressure. Consequently, by balancing these forces on two infinitely near cross-sections of a fluid element in a circular conduit, the following differential equation is obtained:
where p is pressure (in fact the pressure head), z and a are the diameters of the conduit and the cylinder bore, respectively, y is the height at ZY, r is the piston excursion to acquire the velocity v (in terms of sublimity), s is the co-ordinate along the conduit, and C is a constant.
According to Truesdell (1955: p. XLV) , Eulerâs method is simpler and clearer, and for the first time he had interpreted the pressure force in its modern sense, as the product of pressure by the cross-sectional area. Eulerâs method of balancing all forces against the accelerations is the method still in use today in any introductory course in fluid mechanics.
The memoir on pump-pipe flow gives a good idea of Eulerâs methods and approaches to physical problems,
Figure 10. Eulerâs sketch of a one-cylinder pump to deliver water through a piping system to an elevated reservoir [a reproduction from Sur Le mouvement de lâeau par dês tuyaux de conduite (Euler, 1754: p.148) ].
and particular to fluid flow problems. However, this can be considered a specific and more complex problem than the ones that I have considered so far. Therefore, for a fair comparison with the approaches of the other authors considered here, I shall analyze another one of Eulerâs contribution, in which the flow from an elevated reservoir is driven only by gravity, but taking into account the loss of head along the conduits. This contribution appears in Tentamen theoriae de frictione fluidorum (Euler, 1761) , in which he developed a theory to estimate the loss of head along water conduits.
3.2. Eulerâs Theory on Fluid Friction
Tentamen theoriae de frictione fluidorum (Euler, 1761) was built under the wrong assumption that, as for the case of solid friction, the fluid frictional force that appears at the wall of the conduit is proportional to the pressure. As turned out to be much later realized, Euler indeed did not recognized that the friction force is essentially a viscous effect, and hence independent of the applied pressure.
Eulerâs developments in this paper are only for steady flow cases (see Figure 11). Here, by adopting for the fluid friction the same law as for the solid friction, by which the friction force is supposed to be proportional to the applied normal force, Euler wrote the fluid frictional force as, where is a non-dimensional friction coefficient (to be experimentally obtained), p is the pressure (in fact the absolute pressure head), and is the cross-section area of the conduit. By considering an infinitesimal fluid element between any two cross-sec- tions of any conduit, he wrote the two other forces that act in the element in the s direction along the conduit as, for the pressure force, and, for the gravitational force, where x is the vertical co-ordinate oriented downwards. By using Newtonâs second law, Euler then equates these three forces to the convective acceleration of the fluid element and got:
where contrary to the accepted symbology in fluid mechanics, v is not the velocity, but instead the sublimity, which is the height from which a heavy body would have to fall, to acquire the same velocity as the water would have at the orifice at the end of the conduit, with an area of.
Equation (17), without the friction term in the second hand side, is recognized as what is now referred to the steady-state incompressible Eulerâs equation for the ideal, non-viscous fluid, written in the intrinsic co-ordinate s along a streamline. As it is, Equation (17) is in non-dimensional form, with all its terms written in units of length; and, as for the kinematics with Galileo, Euler was successful in geometrizing a fluid dynamic problem as well.
It should be noted that the integration of Equation (17) without the friction term, leads to Bernoulliâs equation in the form, where a is the vessel height, and and are the pressures at the
cross-sections z and h, respectively. This is an expected result because the integration of the incompressible Eulerâs equation for the ideal, non-viscous fluid along a streamline results in Bernoulliâs equation, which is considered the most appropriate formal form for the derivation of this equation.
The solution for p requires repeated integration by parts of Equation (17), in which the exponential functions that appear in the process are written as power series. Following this procedure, in §34 of Tentamen theoriae de frictione fluidorum, Euler gives the solution for p and v for a vessel of any shape, which is modeled by a composition of piecewise straight tubes as shown in Figure 12. For this vessel the following symbology applies.
, amplitude and vertical angle = 0;
, amplitude and vertical angle =;
, amplitude and vertical angle =;
, amplitude and vertical angle =;
For the state of the compression the following symbology is given by Euler.
In, head of atmospheric pressure in terms of a water column of 30 feet11;
The pressure at these junctions are then obtained as follows.
Figure 12. Vessel made by a composition of piecewise straight tubes, which was used by Euler as a model of a vessel of any shape [a reproduction of Figure 4 from Tentamen theoriae de frictione fluidorum (Euler, 1761) ].
The exit velocity at the orifice in the end of the vessel is the given by:
In the case of no friction, , and then:
and for results in:
This last result clear shows that for the case of no friction, the exit velocity for a given height q is independent of the shape of the vessel, and depends only on the first and the last flow areas.
Based on Equation (18), Euler then gives solutions for v for the following cases; Case I-vertical straight tube of uniform section; Case II-vertical straight tube composed of two parts of unequal sections; Case III-horizontal straight tube of uniform section; Case IV-inclined tube of uniform section, including a table of the gradient required before a stream of a given depth will begin to flow; Case V-inclined tube, analyzed together with a vertical tube followed by an horizontal tube with an orifice that projects water vertically.
On passing, it is interesting to note that a situation arose from Case I, discussed in §50, 51 of Tentamen theoriae de frictione fluidorum, that would occur for the case where the length of the vertical tube exceeded twice the atmospheric pressure head (of 30 ft.), when the pressure at the tube-wall would become negative, and the flow would leave behind the vacuum (i.e., cavitation). In this case, Euler points out that the calculations are no longer valid. It should be noticed that earlier, in 1738, Daniel Bernoulli in his book Hydrodynamics, and in 1744, dâAlembert in his book Traité de LâÉquilibre et du Mouvement des Fluides, had both anticipated the possibility of a theoretical negative wall pressure.
An analysis of Eulerâs Case III without friction, allows us to recover D. Bernoulliâs results (see Figure 13).
For this case, Equation (18) for gives. The pressure p in any point of the horizontal
pipe would then be given by, where here, , and are the amplitudes of the
reservoir, the pipe, and the orifice, respectively. Substituting the expression for v, we will have that in general,
the pressure in the pipe would be given by. For a reservoir of large amplitude compared with the amplitude of the orifice, we will have that the pressure in the pipe will be given by, which is recognized as Bernoulliâs law.
Before closing this section, I shall discuss the result, when k is smaller, but of the same
size of f, in which case. This situation would be perceived as a paradox, and it was also discussed by Euler in §41 of Tentamen theoriae de frictione fluidorum. The explanation given there is that there is an implicit assumption in the developments that the vessel is assumed to be constantly full, and therefore the water poured into the vessel would initiate its movement through the vessel not from rest; and hence, there should be no surprise if it escapes the orifice with a greater velocity than that given by the height a of the vessel.
A physical model of such situation, was given by J. Bernoulli in Section XXXIII, Part II of Hydraulics, in which a large pan of very small height is attached to the top of the vessel (see Figure 14). In this case, the height of the pan does not add to the height a of the vessel, with the water flowing into the conduit with the required velocity to maintain it constantly full.
When,; in this case, the water is continually accelerated as in the falling of heavy bodies, and the state of uniformity is never reached within the vessel. When, even less state of uniformity would be reached, with the water separating from the walls of the vessel.
4. Lagrangeâs Theory of Fluid Motion
To see how was Lagrangeâs approach to the problem of discharge, I need to bring first to this discussion another one of Eulerâs papers, that in which the famous Eulerâs equations were derived. These appeared in the Principes generaux du mouvement des fluides (Euler, 1757) . It is not the case to revisit this paper here, not only because
this has already been done elsewhere12, but also because this publication does not deal directly with the problem of the discharge.
Lagrange, in a memoir of 1781 (Lagrange, 1869) , presented a theory of fluid motion, and develops a general method for the solution of discharge problems. In Part I of the memoir, Lagrangeâs developments follow quite closely Eulerâs steps in the above mentioned paper. However a substantial difference marks both approaches, since the Lagrangian specification of the flow field is that of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time, whereas the Eulerian specification of the flow field focuses on specific locations in the space through which the fluid flows as time passes. The Lagrangian and Eulerian specifications of the flow field are sometimes loosely denoted as the Lagrangian and Eulerian frame of reference.
However, in general both the Lagrangian and Eulerian specification of the flow field can be applied in any observerâs frame of reference, up to the point that one finds no differences in the resulting differential equations of motion. This is so true because Lagrangeâs equation of motion, developed in Part I of the memoir, is equivalent to that of multiplying Eulerâs equation by an element of trajectory whose components are, obtaining:
where t is the time, are the components of the velocity vector, are the components of the curl of the velocity vector, (x is the vertical co-ordinate pointing upwards, and g is the gravity), is the pressure, and is the fluid density.
Here, each fluid particle has its own velocity components, which are in general functions of. The position of each particle at a given time t is given by the integrals of, and by three arbitrary constants, which depend on the initial position of the particle. This is what distinguishes the Lagrangian from the Eulerian frame of reference, since in this latter frame of reference are fixed points in space through which the fluid flow as the time t passes.
Lagrange then introduces the velocity potential, which transforms Equation (21) for an incompressible fluid into:
which allows the determination of the pressure. This equation, together with the incompressibility equation:
allows the determination of, from which
Before presenting Lagrangeâs approach to the problem of discharge, it should be noticed that he missed the opportunity to obtain Bernoulliâs equation from Equation (21) (which unlike Euler in the Principes generaux du mouvement des fluides Lagrange did not obtained it in his memoir), by recognizing that along a streamline (for rotational or irrotational motions), and that for irrotational motions. In both cases, the right-hand side of Equation (21) equals zero, which upon integration yields:
which is recognized as the steady-state Bernoulliâs equation, where. C is in general a different constant for each streamline for rotational motions, and is a unique constant for the whole flow field in irrotational motions.
In Part II of the memoir, Lagrange gives a general solution for the fluid motion in vertical vessels of any shape, by expanding the velocity potential in power series of z, on the assumption that this co-ordinate is very small in comparison with x and y. He then develops a particular method for the solution of plane motions of an incompressible fluid of unity density, subject to the gravity g, in narrow vessels whose walls are given by, in which and are both functions of x and âvery smallâ, such that their higher orders expansions could be neglected.
By just retaining the first term in the power series expansion for, Lagrange got for the vertical velocity component p and for the horizontal velocity component r the following expressions
Again, by retaining just the first term of the power series expansion he got an expression for the pressure as
In these expressions, (the vessel width), , and and are arbitrary functions of t,
which would be determined according to particular cases.
The equation that represents the condition in which the same fluid particles would remain at the exterior surfaces of the fluid is
From these equations, which form the basis for the solution of discharge problems, Lagrange then develops specific methods for the solution of four cases of discharge; Case I-an infinite vessel in which a given quantity of fluid flows; Case II-a vessel of a given length in which the fluid flows through the bottom; Case III-an infinite vessel which is kept full at the same height by a new fluid poured in continuously; Case IV-a vessel of a given length which is kept full at the same height.
It is not the case here to go deeper into the Lagrangeâs procedures for the solution of these cases, not only because their lengthy and rather complicated, but also because they all end up in integral equations, which are to be solved according to particular vessel shapes in each case; in other words, the formulations are not for the practical use. But the idea is that once these integral equations are solved, it is possible to get the x position of the upper and lower surfaces and, respectively, , and, from which the velocity components p and r, and the pressure can be found. In Case I, for instance, the final result would give the positions of the lower and upper surfaces, as well as the velocity components of the fluid particles at these surfaces, as a given quantity of fluid descends in a vessel with undefined limits. These are the results that one would expected from the Lagrangian frame of reference, in which the observer follows individual fluid particlesâin this case, the particles that belong to the free surfacesâas they move through the infinite vessel space.
Lagrange begins his memoir by praising the work of dâAlembert, but omits the work of Euler, which, as we saw above, he follows closely in the first part of the memoir. Through the end of the memoir, after having dealt with the problem of discharge, Lagrange found that his solutions conform to those of the first Authors, namely D. Bernoulli in Hydrodynamics, J. Bernoulli in Hydraulics, and DâAlembert in the Traité des Fluides, which have based their works on the supposition that the fluid layers maintain their parallelism as they descend through the vessel.
5. Summary and Conclusion
In a time when the laws of mechanics were not yet certain, Bernoullis and DâAlembert faced the discharge problem using their own principles and approaches to reach essentially the same end. D. Bernoulli adopted the principle of conservation of the living forces, and the equality between the potential ascent and actual descent. J. Bernoulli, although recognizing that the principle of living forces was true, but still not accepted by all philosophers, adopted Newtonâs concept of a motive force impressed on a body, which, in modern terms, equated to the variation of its kinetic energy. He then introduced a translation technique to develop his formulation of the discharge problem, which consisted in translating the effects of motive forces from intermediate layers to the uppermost surface of the water contained in a vessel. DâAlembert, on its turn, because of his rejection to the concept of force, worked not on the problem of discharge in terms of the impressed force, but rather on its effect, in terms of the travelled space and velocity.
It was shown that the formulations thus obtained by these authors, apart from differences in nomenclature and symbology, essentially led to the same results, and that they were applied to solve the same steady and non- steady flow problems. It was possible to conclude that after D. Bernoulliâs great success with Hydrodynamics in 1738, the two other authors felt that they were at least as capable if not superior to D. Bernoulli in accomplishing the proposed tasks, but that was too late for J. Bernoulli and DâAlembert, because Hydrodynamics was soon considered a landmark. These developments occurred at about the same time, in a kind of competition for priority in which Euler seemed to have tacitly accepted the role of presiding over the disputes, in a time when these ferocious discussions on priority were rather common.
However, it was Euler who brought the fluid mechanics problem of discharge to a new and definitive level with two publications at about the same time: one about a one-cylinder pump to deliver water through a piping system to an elevated reservoir, and another considering the forces that arose on the walls of the conduit due to friction. For the first time, the pressure force and the friction force appeared explicitly in the formulations of fluid flow through conduits. In these publications, the pressure in its modern sense has made its appearance. And by using Newtonâs second law, he equated the flow convective acceleration in a conduit of any shape to the applied forces, namely, gravity, pressure and friction. However, the friction force was built under the wrong assumption that, as for the case of solid friction, the fluid friction force was proportional to the pressure. As turned out to be much later realized, Euler indeed did not recognized that the friction force was essentially a viscous effect, and hence independent of the applied pressure.
Finally, Lagrangeâs memoir on the theory of fluid motion of 1781 is presented as a sequel to these first theoretical constructions. Despite the fact that Lagrange did not approach the subject with the eyes of practical applications, he had contributed to the subject with new mathematical resources, like the introduction of the velocity potential, and the application of power series expansion in fluid mechanics.
1This nomenclature comes from the fact that one of the goals in problems of this type is the calculation of the flow velocity in conduits, which allows the determination of the volume of flow per unit of timeâthe flow-rateâ, or the discharge.
2There are controversies about dates raised by J. Bernoulli, who claims having developed his theory as earlier as 1732. In fact, Hydraulics by J. Bernoulli was actually published in 1743, but it was pre-dated to 1732. For an account on the disputes between Daniel Bernoulli and Johann Bernoulli, see the preface by Hunter Rouse on Hydrodynamics (Bernoulli, 1968a) & Hydraulics (Bernoulli, 1968b) .
3According to the records in âThe Euler Archiveâ http://eulerarchive.maa.org/, a treatise with the title Tentamen theoriae de frictione solidorum (!) was read to the Berlin Academy on December 2, 1751, and it was presented to the St. Petersburg Academy on June 17, 1754.
4Sublimity (Lat. sublimitas, Ital. sublimità), is a parameter introduced by Galileo in the Dialogues, being the height from where a heavy body would have to fall (free falling or on an inclined plane) to reach at the end of its descent the said velocity. With this parameter, Galileo was able to complete geometrize the kinematics of free falling bodies.
5Torricelli states that his mentor Benedetto Castelli (1578-1643), knew that the quatities of water issuing from equal apertures are proportional to the square root of their respective heights. However, according to Poleni (Raccolta DâAutori Italiani Che Trattano Del Moto DellâAcque (Edizione Quarta), Tomo VI, 1823. POLENI, Giovanni; Del Moto Misto DellâAqua), Torricelli had said that he was taught by Castelli that the quantities of water are in direct proportion to their heights, and therefore found appropriate to correct this in his book on the motion of waters.
10For a Portuguese translation by the author access http://eulerarchive.maa.org/.
11Note that Euler is working in terms of absolute pressures in these developments.
12See Calero (2008: pp. 426-435) and Truesdell (1955: pp. IX-CXXV) . Eulerâs work is also discussed in the perspective of eighteenth century fluid dynamics research by Darrigol and Frisch (2008). https://www-n.oca.eu/etc7/EE250/texts/darrigol-frisch.pdf