Fluid mechanics
The mechanics of fluids is the branch of physics included within the mechanics of continuums that studies the movement of fluids, as well as the forces that cause it. The fundamental characteristic that defines to fluids is their inability to resist shear stresses (causing them to lack definite shape). It also studies the interactions between the fluid and the boundary that limits it.
Can be divided into fluid statics, the study of fluids at rest; and fluid dynamics, the study of the effect of forces on the motion of fluids. It is a branch of continuum mechanics, matter that models matter without using the information that it is made up of atoms; that is, it models matter from a macroscopic point of view and not from a microscopic point of view. Fluid mechanics, especially fluid dynamics, is a very active field of research, typically mathematically complex. Many problems are partially or totally unresolved and are best approached by numerical methods, usually using computers. A modern discipline, called computational fluid dynamics (CFD), is dedicated to this approach. Particle Imaging Velocimetry, an experimental method for visualizing and analyzing fluid flow, also takes advantage of the highly visual nature of fluid flow.
Note that gases can be compressed, while liquids lack this characteristic (the compressibility of liquids at high pressures is not exactly zero but close to zero) although they take the shape of the container that contains them. The compressibility of a fluid depends on the type of problem, in some aerodynamic applications, even when the fluid is air, it can be assumed that the volume change of the air is zero.
History
The history of fluid mechanics traces the history of knowledge in that field—a branch of physics that studies the movement of fluids and the forces that act on them—from ancient times. The ancient Greeks developed many of the basic concepts of the field, while most of the concepts and theories used in modern physics were discovered in the Europe of the seventeenth and eighteenth centuries.
Before being studied, fluid mechanics were widely used for everyday applications such as irrigation in agriculture or the construction of channels and sources, etc. The sedentarization of humans entailed the necessary invention of means to control water: small-scale irrigation was born around 6500 B.C., at the end of the Neolithic period, and large hydraulic works (channels, irrigation by gravity) begin to be found at 3000 B. C. By that time instruments had already been invented to measure the level of floods, pantanous areas were drained, and dams and dams were built to protect themselves from floods in the Nile, Amarillo and Euphrates rivers. It is possible that the oldest aqueducts will be built in Crete in the second millennium BC and in Palestine in the 11th century BC.
The study of water and its mechanical behavior did not move from concrete applications to theory until quite late. In the Alexandria in the third century BC, Archimedes studied with the disciples of Euclides and, back to Syracuse, formulated the principles that are at the origin of the static of the fluids in particular with its eponymous principle. In the centuryI Heron of Alexandria continued the work on fluid static by discovering the principle of pressure and above all the flow.
Throughout the Late Antiquity, large hydraulic works were continued and perfected with aqueducts, water distribution and sanitation systems, as well as fountains and bathrooms. These works were described by Frontino. As in most of the sciences in Europe during the Middle Ages, the hydrostatic and hydraulic knowledge of the former Greek-Roman Empire was lost in part, being preserved and developed in the Islamic world, whose Golden Age saw for the first time the translation of the works of Archimedes and Euclides,·
From the point of view of the hydraulic buildings, although the Middle Ages, because of the Mongol invasions, saw the irrigation system of Mesopotamia disappear, causing the collapse of the local population, also in the centuryVII He saw that under the Sui dynasty the first stage of the works of the Grand Canal will unite the north and south of China.
Fluid mechanics were once again studied in Europe only with the studies of Leonardo da Vinci in the centuryXVwho described the multiple types of flow and formulated the principle of mass conservation or principle of continuity, taking Heron's relief. It was he who laid the foundations of discipline and introduced many notions of hydrodynamics, such as the current lines. Intrinsically understanding the issue of flow resistance, he designed parachute, anemmeter and centrifugal pump.
Although Simon Stevin (1548-1620) discovered the great principles of the hydrostatic, thus completing the work of Archimedes, he did not nevertheless manage to present them sufficiently beautiful and orderly; it was the work of Blaise Pascal to give those discoveries an irreproachable form. It could be said that if Stevin discovered the hydrostatic paradox and the principle of equal pressure in a liquid, Pascal was the one who, in his "Récit de la grande experiment de l'quilibre des liqueurs" of 1648, gave for the first time a homogeneous and orderly presentation of those fundamental principles of the hydrostatic. The manifestations of the hydrostatic paradox are used in the teaching of the phenomenon. One of the best known experiments is the explosion of the barrel of Pascal.
Book II Mathematica Principia Newton, which deals with body movements in resistant environments, leaves no substantial scientific knowledge in this field. However, according to Clifford Truesdell, Newton's work provided both disciplines with a curriculum followed for fifty years. Even the work of Alexis Claude Clairaut (1713-1765) and Jean le Rond D'Alembert (1717-1783) the laws of fluid mechanics did not begin to be established.
Only with the arrival of mathematics to physics the mechanics of fluids gained depth. In 1738 Daniel Bernoulli established the laws applicable to non-viscous fluids using the principle of mechanical energy conservation. The birth of the differential calculus allowed Jean le Rond D'Alembert to expose in 1749, in 137 pages, the bases of the hydrodynamics when presenting the principle of the internal pressure of a fluid, the field of velocities and the partial derivatives applied to the fluids. Leonhard Euler then completed D'Alembert's analysis of internal pressure and incompressible fluid dynamic equations: in 1755, he published a treaty with partial differential equations describing the perfect incompressible fluids. A little earlier, in 1752, D'Alembert had noticed the paradox bearing his name that showed that the equations contradicted the practice: a body submerged in a fluid would move without resistance according to theory, which the observation contradicted directly. It was resolved by the introduction by Henri Navier in 1820 of the concept of friction in the form of a new term in the mathematical equations of fluid mechanics. George Gabriel Stokes arrived in 1845 at an equation that allowed to describe a viscous fluid flow. The Navier-Stokes equations will mark the rest of the history of fluid mechanics.
This suite took shape in the second half of the centuryXVIII and the first of the centuryXX.:
- developments in the incompressible or understandable domains with the creation of the concept of cape by Ludwig Prandtl that will be very fruitful, particularly for naval aerodynamics and hydrodynamics,
- study of the new domain that constitutes the supersonic,
- by Henry Darcy and of water-to-air interfaces by Moritz Weber,
- study of instability and turbulence, a chapter still far from being closed. This field saw the appearance of schools founded by a precursor: Prandtl in Göttingen or the Russian school of Kolmogorov mathematicians.
During this period Ludwig Boltzmann opened a new chapter with the statistical description of the gases at the microscopic level, which will be developed by Martin Knudsen for the inaccessible domain to a description under the hypothesis of the continuum; David Enskog and Sydney Chapman will show how to pass the gases from the molecular level to the continuum, thus allowing the calculation of the transport coefficients (difusion, viscosity, molecular conduction) from the potential. All these theoretical works were based on the previous fundamental works of mathematicians such as Leonhard Euler, Augustin Louis Cauchy or Bernhard Riemann.
In addition, the development of numerous test facilities and measuring devices enabled many results. Not all of them could be explained by the theory and there was a large number of dimensional numbers that explained and justified the tests carried out on models in a wind tunnel or in a mask tank. Two scientific worlds coexisted and very often ignored until the end of the centuryXIX. That gap will disappear under the momentum of people like Theodore von Kármán or Ludwig Prandtl in the early centuryXX.. All these developments were supported by developments in industry: industrial hydrodynamics, naval and aeronautical construction.
The numerical calculation, which was born in the second half of the centuryXX., will allow the emergence of a new branch of fluid mechanics, the mechanics of computational fluids, based both on the appearance of increasingly powerful calculators and new mathematical methods that allow the numerical calculation. The computing power allows the realization of "numerical experiments" that compete with the means of proof or allow the easier interpretation of these. This type of approach is commonly used in turbulence study.
The second important fact in this period is the considerable increase in the number of persons involved in research and development. The discoveries have become more of teamwork than of individuals. These teams are mainly Americans: Europe (mainly France, UK and Germany) has lost its leadership.
The industrial fields that justify these developments are weather, weather, geophysics, or even oceanography and astrophysics. These domains exist only through the numerical calculation, at least for the first two.
Basic assumptions
As in all branches of science, fluid mechanics starts from the hypothesis based on which all concepts are developed. In particular, in fluid mechanics it is assumed that fluids verify the following laws:
- conservation of the mass and the amount of movement.
- first and second law of thermodynamics.
Continuous medium hypothesis
The continuum hypothesis is the fundamental hypothesis of fluid mechanics and in general of all continuum mechanics. In this hypothesis, it is considered that the fluid is continuous throughout the space it occupies, thus ignoring its molecular structure and the discontinuities associated with it. With this hypothesis it can be considered that the properties of the fluid (density, temperature, etc.) are continuous functions.
The way to determine the validity of this hypothesis is to compare the mean free path of the molecules with the characteristic length of the physical system. The quotient between these lengths is called the Knudsen number. When this dimensionless number is much less than unity, the fluid in question can be considered a continuous medium. Otherwise, the effects due to the molecular nature of matter cannot be neglected and statistical mechanics must be used to predict the behavior of matter. Examples of situations where the continuum hypothesis is not valid can be found in the study of plasmas.
Fluid particle concept
This concept is closely linked to that of the continuum and is extremely important in fluid mechanics. The elementary mass of fluid that is at a point in space at a given instant is called a fluid particle. Said elementary mass must be large enough to contain a large number of molecules, and small enough to be able to consider that there are no variations in the macroscopic properties of the fluid inside it, so that in each fluid particle we can assign a value to these properties. It is important to note that the fluid particle moves with the macroscopic speed of the fluid, so it is always made up of the same molecules. Thus, a certain point in space at different moments of time will be occupied by different fluid particles.
Lagrangian and Eulerian descriptions of fluid motion
When describing the motion of a fluid, there are two points of view. A first way to do it is to follow each fluid particle in its movement, so that we will look for some functions that give us the position, as well as the properties of the fluid particle at each instant. This is the Lagrangian description.
A second way is to assign to each point in space and at each instant, a value for the fluid properties or magnitudes regardless of which fluid particle occupies, at that instant, that differential volume. This is the Eulerian description, which is not tied to fluid particles but to the points in space occupied by the fluid. In this description, the value of a property at a point and at a given instant is that of the fluid particle that occupies that point at that instant.
The Eulerian description is the most common, since in most cases and applications it is more useful. We will use this description to obtain the general equations of fluid mechanics.
General Equations of Fluid Mechanics
The equations that govern all fluid mechanics are obtained by applying the conservation principles of mechanics and thermodynamics to a fluid volume. To generalize them we will use the Reynolds transport theorem and the divergence theorem (or Gauss's theorem) to obtain the equations in a more useful form for the Eulerian formulation.
The three fundamental equations are the continuity equation, the momentum equation, and the conservation of energy equation. These equations can be given in their integral formulation or in their differential form, depending on the problem. This set of equations given in their differential form is also called the Navier-Stokes equations (Euler's equations are a particular case of the Navier-Stokes equations for fluids without viscosity).
There is no general solution to this set of equations due to its complexity, so for each specific problem in fluid mechanics these equations are studied looking for simplifications that facilitate the resolution of the problem. In some cases it is not possible to obtain an analytical solution, so we have to resort to computer generated numerical solutions. This branch of fluid mechanics is called computational fluid mechanics. The equations are the following:
Continuity Equation
For incompressible fluid with constant density, it is required that the fluid element have constant density when moving only along a current line, that is, that the substantial derivative with respect to time is zero.
- Integral form: ddt∫ ∫ Ω Ω ρ ρ dΩ Ω +∫ ∫ ▪ ▪ Ω Ω ρ ρ (v⋅ ⋅ n)d▪ ▪ Ω Ω =0{displaystyle {frac {d}{dt}}int _{Omega }rho ;dOmega +int _{partial Omega }rho (mathbf {vcdot n}) dpartial Omega =0}
- Differential form: ▪ ▪ ρ ρ ▪ ▪ t+► ► ⋅ ⋅ (ρ ρ v)=0{displaystyle {frac {partial rho }{partial t}}}+nabla cdot (rho mathbf {v})=0}
Momentum Equation
Suppose a volume element with a fluid moving in an arbitrary direction through the six faces of the volume element. It is the equation of a vector with components for each of the three coordinate directions x, y, or z. Momentum moves in and out of the control volume by two mechanisms: convection (ie, due to global fluid flow) and molecular transport (ie, velocity gradients). In most cases the only important forces will be those from the fluid pressure p and the gravitational force per unit mass g. The pressure of a moving fluid is defined by the equation of state p=f(p,T) and is a scalar quantity. One must also consider the rate of accumulation of momentum.
- Integral form: ddt∫ ∫ Ω Ω ρ ρ vdΩ Ω +∫ ∫ ▪ ▪ Ω Ω ρ ρ v(v⋅ ⋅ n)d▪ ▪ Ω Ω =∫ ∫ ▪ ▪ Ω Ω Δ Δ ⋅ ⋅ nd▪ ▪ Ω Ω +∫ ∫ Ω Ω ρ ρ fdΩ Ω ############### ##########################################################################################################################################################################################################################################
- Differential form: ▪ ▪ ▪ ▪ t(ρ ρ v)+► ► ⋅ ⋅ (ρ ρ v v)=− − ► ► p+ρ ρ f+► ► ⋅ ⋅ Δ Δ .{displaystyle {frac {partial }{partial t}}left(rho mathbf {v} right)+nabla cdot (rho mathbf {v} otimes mathbf {v})=-nabla authbf {p} +rho mathbf} !
Analyzing each term of the equation of motion, it is obtained that: the rate of increase of momentum per unit volume plus the rate of gain of momentum by convection per unit volume are equal to the force of pressure that acts on the element per unit volume minus the rate of gain of momentum by viscous transport per unit volume plus the gravitational force acting on the element volume. Another interpretation of the terms of the equation of motion arises from using the substantial derivative with which it is obtained that the mass per unit of volume multiplied by acceleration is equal to the sum of the force of pressure on the element per unit of volume, force viscous force on the element per unit volume and the gravitational force on the element per unit volume. With which it is concluded that a volume element that moves with the fluid is accelerated by the forces that act on it. Therefore the momentum balance is an equivalent form of Newton's Second Law. It is necessary to take into account that the equations are valid for continuous media and that if the partial derivative is used, it corresponds to the case in which the element moves at the speed of the fluid. When shear stresses are replaced by Newton's laws of viscosity and the equation of motion is combined with the equation of continuity, the variation of viscosity with density, the equation of state and the initial and limit conditions, the difference equations are obtained that allow to describe the pressure, density and velocity components point by point. For constant viscosity and density, the density is left out of the derivative and we are left with the Navier-Stokes Equation, this condition corresponds to an incompressible fluid with zero velocity vector divergence in the current line. When viscous effects are negligible, the viscous force on the fluid element per unit volume can be taken to be zero and Euler's Equation results.
Energy Conservation Equation
- Integral form: ddt∫ ∫ Ω Ω ρ ρ (e+12v2)dΩ Ω +∫ ∫ ▪ ▪ Ω Ω ρ ρ (e+12v2)v⋅ ⋅ nd▪ ▪ Ω Ω =∫ ∫ ▪ ▪ Ω Ω n⋅ ⋅ Δ Δ ⋅ ⋅ vd▪ ▪ Ω Ω +∫ ∫ Ω Ω ρ ρ f⋅ ⋅ vdΩ Ω − − ∫ ∫ ▪ ▪ Ω Ω q⋅ ⋅ nd▪ ▪ Ω Ω ##
- Differential form:ρ ρ DDt(e+12v2)=− − ► ► ⋅ ⋅ (pv)+► ► ⋅ ⋅ (Δ Δ ♫⋅ ⋅ v)+ρ ρ f⋅ ⋅ v+► ► ⋅ ⋅ (k► ► T){displaystyle rho {frac}{Dt}}left(e+{frac {1}{2}{2}{2}right)=-nabla cdot left(pmathbf {v} right cdot left(cdot mathf+v}
Non-viscous and viscous fluids
A not viscous fluid It has no viscosity, .. =0{displaystyle nu =0}. In practice, a non-viscous flow is an idealization, which facilitates mathematical treatment. In fact, purely invisible flows are only known in the case of superfluidez. Otherwise, fluids are generally viscous, a property that is usually more important within a boundary layer near a solid surface, where the flow must coincide with the non-slip condition on the solid. In some cases, the mathematics of a fluid-mechanical system can be treated by assuming that the fluid outside the limit layers is not viscous, and then coinciding with its solution in a thin layer of laminar.
For fluid flow over a porous boundary, the velocity of the fluid can be discontinuous between the free fluid and the fluid in the porous medium (this is related to the Beavers and Joseph condition). Furthermore, it is useful at low subsonic velocities to assume that the gas is incompressible, that is, the density of the gas does not change even if the velocity and static pressure change.
Fluid Dynamics
Fluid dynamics is a subdiscipline of fluid mechanics that deals with the flow of fluids - the science of liquids and gases in motion. Fluid dynamics offers a systematic framework - underlying these practical disciplines - encompassing empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. Solving a fluid dynamics problem often involves calculating various fluid properties, such as velocity, pressure, density, and temperature, as a function of space and time. It has several subdisciplines of its own, including aerodynamics (the study of air and other gases in motion) and hydrodynamics (the study of liquids). moving). Fluid dynamics has a wide range of applications, including calculating forces and motions in aircraft, determining the rate of mass flow of oil through pipelines, predicting changing weather patterns, understanding of nebulae in interstellar space and the modeling of explosions. Some fluid dynamics principles are used in traffic engineering and crowd dynamics.