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Mechanics and Fluid

1.1. Introduction

The mechanics of fluids is a type of mechanics: it looks at the movement of matter when under the influence of forces. Matter here is in the "fluid state".

This chapter is approached from the perspective of the foundations of the mechanics of point power. It will also later define what fluid is and which of this matter's main characteristics are useful to know. These characteristics shall then be brought to "life" in later chapters.

1.1.1. Mechanics: what to remember

1.1.1.1. Who is afraid of mechanics?

For some curious reason, this branch of physics appears frightening to many students, a curse that thermodynamics also shares. Somewhat recoiled from, the mechanical engineer occupies a special place in the academic world. Some people even wonder whether mechanical engineers are actually physicists who have a strong handle on mathematics, or are in fact mathematicians lost among physicists. These classifications have not been made any simpler by the addition of digital calculations.

It cannot be stressed enough that the appearance of mechanics gave birth to mathematical physics.

By pairing movement with mathematics, the Neoplanitician, Galileo, created kinematics. And then, with a stroke of genius, although perhaps slightly mythically, Isaac Newton created dynamics by incorporating the fall of an apple and the Moon's trajectory into one vision.

Descartes must not be left out of this Pantheon of emerging physics, for he created momentum, was engaged in heated debates with Newton and Leibnitz on this subject as well as others, and discovered kinetic energy through "life force". Leibnitz and Newton were also the precursors to the differential approach in mechanics.

1.1.1.2. Principles to remember

Like a game of chess, the starting rules of mechanics are the simplest. And, like a game of chess, not all paths lead to an easy victory.

a) Remember that a position vector is defined as a vector that links the starting point to another point in space. The coordinates of are evidently the point's three coordinates:

By definition, the point's speed is the derivative of the position vector in relation to the time:

which, when passing, accelerates the position vector's second derivative:

Remember that a vector is derived with regard to a scalar by deriving its components:

b) In 1687, Isaac Newton's Philosophiae Naturalis Principia Mathematica outlined three laws, which indeed can be reduced into two:

1. 1) The principle of inertia;
2. 2) Fundamental dynamics law;
3. 3) The principle of action and reaction.

Let us take these three principles further:

Law no. 2. Let us begin with the fundamental dynamics principle, when applied to a constant mass (m) material point:

The acceleration that a body undergoes in an inertial frame of reference is proportional to the resulting forces that it undergoes, and is inversely proportional to its mass.

In modern notation (the notion of the vector was acquired in the 20th Century), this is written as:

NOTE: Vectorial notation reminds us that a given speed contains three pieces of information: a direction (instantaneous movement support), a route and an hourly speed. A speed cannot be reduced to the datum of m.s-1. A speed vector not only tells me that my car is traveling at V = 130 km.hr-1 (hourly speed), but it also tells me that I am on a highway between Paris and Rome (direction) and that I am going from Paris to Rome (route). However, I would still need the position vector to tell me where the next exit is.

Therefore, an acceleration is also a vector, and there is no reason why it is not collinear to the speed. Central acceleration in a circular movement is (or should be) known to all secondary school students.

Law no. 1. The principle of inertia was actually discovered by Galileo: In the absence of an external force, all material points continue in a uniform, straight-lined movement.

NOTE: This is what Captain Haddock realizes in the "Explorers on the Moon", the illustrated Tintin adventure story by the famous Belgian author, Hergé.

This principle of inertia is in fact a consequence of the fundamental dynamics principle. If the result of forces applied to a material point is zero, then:

and:

It means a uniform straight-lined movement.

Law no. 3. If the first principle can be reduced to the second, the third principle of action and reaction is independent: Every body A exerting a force on a body B undergoes a force of equal intensity, but in the opposite direction, exerted by body B:

When solving a problem, to write that every force has an equal and opposite reaction is to write something new with regard to the fundamental dynamics principle.

These principles have been rewritten in various different forms, which lead to equations that are often much more directly applicable. A few of these equations are given in the following sections.

1.1.2. Momentum theorem

We can rewrite the fundamental dynamics principle by noting that mass is invariable:

A momentum vector has also been introduced:

And the fundamental dynamics principle is also found to be rewritten in terms of momentum:

In the course of mechanics, it is demonstrated that this equation applies in material points to the center of a group's mass, whether it is continuous or discontinuous and alterable or otherwise. m is therefore replaced by the total mass of the system's points and then represents the resultant of the forces applied to these points. This is what constitutes the center of mass theorem.

NOTE: It goes without saying that we do not intend to write a "digest" here on the course of fluid mechanics.

It would be impossible to attempt to reproduce a complete mechanics course. However, we must insist upon the consequences of these principles which will be directly applied when establishing fluid mechanics theorems. We will build upon the mechanics of point power, and if the reader deems it necessary, they can refer to a dedicated textbook to study system mechanics, which constitutes a more complex domain. Furthermore, in the appendix, we can find a reminder of fluid mechanics equations for a continuous fluid system. This script will be used when demonstrating Euler's first theorem.

We observe that while mass becomes variable with speed, it is this expression that remains valid in particular mechanics. This is also the case for relativist dynamics.

1.1.3. Kinetic energy theorem

Forced movement implies work. Here we will give mechanics an energetic dimension. The work of a force when applied to a material point during a time dt provides calculated work from the force and this point's small movement :

is a small vector, which indicates not only the small distance traveled, but also the carrying line of this movement or direction, and the movement's route. It is linked to speed by:

Remember the dynamic relation:

The work is written as:

It can be observed that

Finally, it becomes:

The work performed has helped to increase the quantity carried by the material point. This is how kinetic energy appears:

1.1.4. Forces deriving from a potential

In a frame of reference Oxyz, where Oz is vertical, the force of gravity applied to a mass of m = 1 kg will have the following components:

Furthermore, the operating gradient is defined by associating the vector with a function f(x, y z) by:

Therefore, can be written in the form of a gradient:

which, by definition, implies the following about the gradient:

By identifying:

Therefore, it can be said that is derived from the potential ?G. It is worth at least being aware of this.

In general terms, it is said that a force is derived from a potential ?(x, y, z) when

This property is not universal: in particular, friction forces or electromagnetic forces are not derived from a potential.

1.1.5. Conserving the energy of a material point

The work performed by a force derived from a potential during a time period of dt is written...