Dynamics of the fluids

The dynamic of the fluids is the motion study of the Fluide S, which they are Liquide or Gaz. The resolution of a problem of dynamics of the fluids normally requires to calculate various properties of the fluids such as for example the Speed, the Pression, the Densité and the Température as functions of space and time.

Current prospects

One knows perfectly the equations which control the fluids: they are the Navier-Stokes equations, or the derivatives. But at present, the problem is not there any more. In spite of their (relative) simplicity, these equations can generate extremely complex behaviors, like the Turbulence. One can approach these chaotic phenomena only one statistical point of view, by using an arsenal of theoretical methods (bases of filters, fractales…) - but there remains difficult to envisage starting from the equations, the fine behaviors of turbulence. However the very great majority of the flows which surround us (water, air) are turbulent - is to say the practical importance of this problem.

It is also interesting to study the transition between a simple behavior from the fluids (laminar flow) and a chaotic behavior (turbulent flow).

Study of the phenomena

The study of these phenomena is very often numerical today: one simulates solutions of the equations, which resemble real flows indeed - except that it is as if one had a perfect system of measurement, which could all measure without anything to disturb.

Another way of research very much used is the study out of blower . By putting a model to be studied in a strong flow of air, and by studying the flow by various means (velocity measurement of flow by Anemometer or Tube of Pitot, measurement of the efforts by dynamometers, visualization of the threads of current), one can make many calculations and improve the aerodynamic parameters of the object).

In parallel, the studies of hydrodynamics on the harbor ships, oil installations at sea or works often use basins in which one can represent realistic Vague S. As out of blower, the tests are generally carried out on a small-scale model.

That thus poses, in both cases, of the problems of similarity which require initially a critical analysis of the phenomena to highlight the relevant parameters and those that one can neglect, then their taking into account using numbers without dimension.

There also exists of other experimental methods to study flows: Strioscopy, Laser-Doppler,…

Applications of the dynamics of the fluids

The dynamics of the fluids and its under-disciplines like the Aerodynamic , the Hydrodynamic , and the Hydraulique have very diverse applications. For example, they are used in calculation of the force S and the moments in the Aéronautique or for the weather forecasts .

The concept of fluid is surprisingly general. For example, some of the basic mathematical concepts concerning the Gestion of the traffic are derived by regarding the traffic as a continuous fluid.

The assumption of continuity

The gases are composed of Molécule S which run up between them like full objects. The assumption of continuity, however, regards the fluids as being Continus. I.e. one admits that properties such as the density, the pressure, the temperature, and speed are taken for being well defined in points infinitely smalls, and do not change from one point to another. The discrete and molecular nature of a fluid is thus ignored.

These problems for which the assumption of continuity does not give answers with desired exactitude are solved thanks to the Mécanique statistics. In order to determine if it is necessary to employ conventional liquid dynamics (a under-discipline of the Mécanique of the continuous mediums) or statistical mechanics, the Nombre of Knudsen is evaluated to solve the problem. The problems for which the number of Knudsen is equal or higher than 1 must be treated by statistical mechanics to give reliable answers.

Equations of the dynamics of the fluids

The fundamental axioms of the dynamics of the fluids are the laws of conservation like the Conservation of the masses, the Conservation of the momentum (more known under the name of second law of Newton), and the Conservation of energy. That constitutes the base of the Newtonian Mécanique and are also important in mechanical relativist.

The most important equations are the Navier-Stokes equations, which are differential equations Non-linéaire describing the movement of the fluids. These equations, when they are not simplified do not have analytical solutions and are thus useful only for digital simulations. These equations can be simplified in various ways what returns the equations easier to solve. Certain simplifications make it possible to find solutions analytical with problems of dynamics of the fluids.

Choice of a description of the fluid

Mathematically to describe the properties of a fluid moving, two systems cohabit, one and the other presenting of the advantages in typical locations. It is of the Lagrangian Description and the Description eulérienne. While the first consists in observing the modifications of the properties of a particle Fluide which one follows in his movement, the second is placed in a fixed point of the medium being studied and observes the modifications of the properties of the fluid which ravels in this point. Two descriptions are mathematically dependant by the relation:

\ frac {D} {Dt} = \ frac {\ partial} {\ partial T} + \ left (\ overrightarrow {v}. \ overrightarrow {grad} \ right)

Where the term \ frac {D} {Dt} , called " derived totale" or " derived particulaire" , the derivative in description Lagrangienne (" represents; Ressentie" by a particle moving), and the partial term \ frac {\} {\ partial T} represents the derivative in description Eulérienne (" Vue" by an observer in a fixed point).

Compressible and incompressible flow

A fluid is called compressible if the density switchings of the fluid have significant effects on the unit of the solution. In the contrary case, it is about an incompressible fluid and the density switchings are ignored.

In order to know if the fluid is compressible or incompressible, one calculates the Mach number. Roughly, the effects of compression can be been unaware of for the Mach numbers in lower part of 0,3. Almost all the problems implying of the Liquide S are in this category and are defined like incompressible.

The incompressible Navier-Stokes equations are simplifications of the Navier-Stokes equations in which the density is regarded as constant. They can be used to solve the problems implying of the incompressible fluids.

In acoustics, the Speed of sound in the air being finished, the fluid “air” must be treated like compressible. Indeed, let us suppose that the air is an incompressible fluid, then it would move in block and would propagate any modification of local pressure at an infinite speed. The speed of sound c in a compressible fluid is written besides like function of its compressibility \ chi:

c^2= (\ rho_0 \ chi) ^ {- 1}

Viscosity

The problems due to the Viscosité are those in which frictions of the fluid have significant effects on the solution. If frictions can be neglected, the fluid is called nonviscous .

The Reynolds number can be employed to estimate which type of equation is adapted to solve a given problem. A Reynolds number high indicates that the forces of Inertie are more important than the forces of friction. However, even when the Reynolds number is high, certain problems require to take into account the effects of viscosity. In particular, in the problems where one calculates the forces exerted on a body (like the wing S of a plane), it is necessary to take into account viscosity. As illustrated by the Paradox of Alembert, a body immersed in a nonviscous fluid is subjected to no force.

The equations normally used for the flow of a nonviscous fluid are the equations of Euler. In the numerical dynamics of the fluids, one employs the equations of Euler when one is far from the body and equations taking account of the Boundary layer when one is near the body.

The equations of Euler can be integrated along a line of flow to lead to the equation of Bernoulli. When the flow is irrotational and nonviscous everywhere, the equation of Bernoulli can be employed to solve the problem.

Constant and nonconstant flow

Another simplification of the equations of the dynamics of the fluids is to regard all the properties of the fluid as being constant in time. This is called then a stationary Flux and applicable to many problems, such as is pushed or the trail of a wing or a flow crossing a pipe. In the particular case of a stationary flow, the Navier-Stokes equations and Euler are thus simplified.

If a fluid is at the same time incompressible, nonviscous and stationary, it can be solved with the potential flow rising from the equation of Laplace. The problems of this class have solutions which are combinations of linear flows elementary.

When a body is accelerated in a fluid, the Notion of added mass is introduced.

Laminar flow and turbulence

The Turbulence is a flow dominated by movements, and an apparent random aspect. When there are no turbulences it is said that the flow is laminar.

The turbulence of the fluids obeys the Navier-Stokes equation. However, the problems of flow are so complex that it is not currently possible to numerically solve them on the basis of the basic principles. Turbulence is rather modelled using one of many the model of turbulence and is coupled with a resolvor of flow which supposes that flow is laminar apart from the area of turbulence. The study of the Reynolds number makes it possible to determine the turbulent or laminar character of a flow.

Other approximations

There is a great number of other possible approximations faces with the problems of the dynamics of the fluids. A flow of Stokes is the flow of a fluid whose Reynolds number is very low, so that the inertias can be neglected vis-a-vis the forces of friction. The approximation of Boussinesq neglects compressive forces except calculating the forces of buoyancy.

Dependant articles

Fields of study

  • Acoustic (uses the derivatives of the dynamics of the fluids)
  • Aérodynamique
  • Élasticité
  • Aéronautique
  • Étude of the dynamics of the fluids by computer
  • Géosciences
  • Mesure of flows
  • Hémodynamique
  • Hydraulique
  • naval Hydrodynamique
  • Rhéologie

Mathematical equations

Type of flow of the fluids

Properties of the fluids

Numbers without dimension describing a flow

Simple: Fluid dynamics

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