Jean le Rond D’Alembert: A New Theory of the Resistance of Fluids

Springer Verlag (2018)
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Abstract

In the commentaries to this book we try to understand d’Alembert thoughts and how he contrives to translate his ideas on mechanics to the fluid realm with a new and radical point of view; how he arrives at the first two fundamental differential equations among the velocity components; and how he tries to reduce the resistance of a moving body, which is a change of its momentum, to the hydrostatical pressure, which is related to the gravity. All this knowing that his mechanics has no forces and no pressures as well, and that the fluids are aggregates of individual particles. The essay A New Theory of the Resistance of Fluids was a turning point in Fluid Mechanics because clearly, for the first time, the resistance is shown as the results of a fluid subjected to differential equations in a continuous mode instead of a set of impacts of individual particles. This contribution has been recognized by the scholars. However, only partial attention has been p aid to this work, which can be justified due to the difficulty in its reading and also because it was eclipsed by the publication, a few years later, of Euler’s three Memoirs that established modern hydrodynamics.

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Chapters

D’Alembert’s Dynamic Conceptions

D’Alembert published his Traité de Dynamique in 1743, when he was 26 years old. It was here where he expounded his ideas about dynamics, including what is known as his “general principle”. This is enunciated in the second part of the Traité, but the general ideas are in the first one.

Other Motions

Under this title three type of motions are included, the flow in a vessel, the river currents and the jet against a plate; they all are somewhat related to the general theory. Furthermore we add the motion by plane sections, although we have found it difficult to classify.

Other Resistances and Fluids

The resistance analysed until now comes from the transfer of momentum between fluid and body and it can be understood as being derived from the inertia of matter. However, the authors knew that there were other sources of resistance related to the properties of the fluids, as yet not well known, and... see more

Application of the Principles Outlined in This Essay in the Research of the Motion of a Fluid in a Vessel

148. As the principles I have given in this book to determine the laws of fluid resistance seem to me to have a lot of scope, I thought it would not be useless to show in some way how to apply them to the research of the movement of fluids in any vessels or channels. But as these researches are not ... see more

On the Pressure That a Fluid Exerts on a Body at Rest and Immersed in It

35. In order to determine the resistance that a fluid, either in motion or at rest, produces on a body that moves therein, we must first to determine the action that a moving fluid exerts against a body at rest. Because we will show in the next chapter that the whole theory of the resistance of flui... see more

On the Resistance of Fluids to the Bodies Moving Therein

80. All fluid wherein a body moves is elastic or non-elastic. I called elastic fluid one whose parts can be contracted so they occupy a lesser space than before their compression, and reciprocally they dilate so that they occupy a larger space than before their expansion. I called non-elastic fluid ... see more

Application of the Same Principles to Some Research on Streams in Rivers

156. Let be Bm the bottom of the river, CM its upper surface, and let it be assumed that the river flows from C to M, and that its riverbed is of equal width everywhere. Drawing at will the horizontal Qo and the vertical AQ, it is true that we can express the horizontal and vertical velocities of an... see more

The Preliminaries

As we have already commented, d’Alembert presents the general principle of dynamics in a somewhat different way from the Traité de Dynamique [§.1].

Principles of Dynamics and Hydrodynamics Necessary for the Understanding of the Subsequent Propositions

1. Let any system be composed of as many bodies as desired, which I designate as A, B, C, D, etc. and let us suppose that these bodies are impelled by any forces φ, ψ, π, υ, etc., being A by the force φ, B by the force ψ, etc., and that at any instant these bodies move with velocities V, U, v, u, et... see more

Reflections on Fluid Equilibrium

D’Alembert finished with an Appendix in the Essay containing “some reflections on the laws of the Equilibrium of Fluid that I have not thought necessary to include in the body of the work in order not to interrupt the sequence of matters, but they seem to me worthy of being submitted to the judgment... see more

General Considerations

These Commentaries incorporate the suggestions and comments done by Larrie Ferreiro and Manuel Sellés.

Oscillations of a Body Floating in a Fluid

118. Let there be a body DAd consisting of two equal and similar parts placed on either side of the axis AC, and which we consider for more convenience as a plane figure. Let us imagine that this body is placed on the surface of a fluid at rest, so that the axis AC is vertical and the immersed part ... see more

On the Action of a Fluid Stream That Exits from a Vessel and Strikes a Plane

137. About this question, that has some relation to the theory of fluid resistance, I thought it would be good to deal with it here, not only because its solution is easily deduced from my principles, but also because it will give me the opportunity to make some new observations about this matter, a... see more

Forces and Fluids in the Essay

Resistance of a Body Moving in a Fluid

This is the core of the Essay and Chaps 4 and 5 deal with it. We prefer to take this matter as a whole, leaving for a posterior analysis the parts of these chapters not clearly connected to the subject. This manner of dealing with it tries focus on the more essential items, which in our opinion woul... see more

General Principles of the Pressure Fluids, in Motion or at Rest

21. Let MNGH be a homogeneous fluid without weight, and that either it is of an indefinite extension or enclosed in a reservoir of any size and shape. Place a solid body BCDE anywhere one wishes inside this fluid, taking around that body a fluid portion limited by the surface FOKL and assuming that ... see more

General Principles of the Equilibrium of Fluids

13. Let ABCD be a fluid or any portion of fluid in which the particles are impelled by any forces, so that they are equilibrium; I say, if, from any point P placed inside that mass of fluid, are drawn the straight lines PA, PB to any two points A, B of the surface ABCD, the point P will be equally p... see more

The Oscillation of Floating Bodies

This chapter is dedicated to the oscillations of a body floating in a fluid, obviously a liquid. The procedure is to calculate the forces generated when the body is subjected to small perturbations, such as displacements and rotations, from the equilibrium position. The resulting forces and moments,... see more

Experiments and Theories

D’Alembert quotes experimenters in several places in the Essay using the results of their experiments in his own reflections as we have already seen. There are also mentions in the Introduction, but apart from this, he adds a complete section [§.75–79] concerning experimenters under the title “Refle... see more

Appendix

This Appendix will include some reflections on the laws of the Equilibrium of Fluids that I have not thought necessary to include in the body of the work in order not to interrupt the sequence of matters, but they seem to me worthy of being submitted to the judgment of the wise; and besides they hav... see more

Brief Analysis of the Contents of the Essay

The Essay’s Introduction

We have divided the Introduction in seven clearly differentiated parts.

Brief Analysis of the Contents of the Essay

Our main intention is to highlight the genuine contribution of d’Alembert to Fluid Mechanics. Therefore, instead of running through the Essay article by article, we have preferred to extract the core contribution and to bring it clearly to the light. Nevertheless this does not mean that we have reno... see more

Forces and Fluids in the Essay

The term force is one of the most widely employed in the Essay, more than two hundred and fifty times, only surpassed by the word pressure. Considering d’Alembert’s warning that he would only use the word force in order to avoid circumlocution, this repetition seems somewhat surprising.

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