사용자:Astronauthang/유체역학

Fluid mechanics is the study of fluids and the forces on them. (Fluids include liquids, gases, and plasmas.) Fluid mechanics can be divided into fluid kinematics, the study of fluid motion, and fluid dynamics, the study of the effect of forces on fluid motion, which can further be divided into fluid statics, the study of fluids at rest, and fluid kinetics, the study of fluids in motion. It is a branch of continuum mechanics, a subject which models matter without using the information that it is made out of atoms, that is, it models matter from a macroscopic viewpoint rather than from a microscopic viewpoint. Fluid mechanics, especially fluid dynamics, is an active field of research with many unsolved or partly solved problems. Fluid mechanics can be mathematically complex. Sometimes it can best be solved by numerical methods, typically using computers. A modern discipline, called computational fluid dynamics (CFD), is devoted to this approach to solving fluid mechanics problems. Also taking advantage of the highly visual nature of fluid flow is particle image velocimetry, an experimental method for visualizing and analyzing fluid flow.

유체역학(流體力學, 영어: fluid mechanics)이란 유체와 그에 작용하는 힘에 대해 연구하는 학문이다. (유체란 액체, 기체, 플라스마를 포함한다.) 정지상태에 있는 유체에 대해 연구하는 유체 정역학, 운동중에 있는 유체에 대해 연구하는 유체 동역학으로 나뉜다. 또는 유체에 따라서 액체일 때는 수력학, 기체일 때는 기체역학으로 분류하기도 한다. 최근에는 컴퓨터의 발달과 함께 전산유체역학이 새롭게 떠오르고 있다.

역사

유체역학에 관한 연구는 적어도 아르키메데스가 유체정역학과 부력을 연구하고 아르키메데스의 원리로 알려져 있는 그의 유명한 법칙을 발견한 고대 그리스부터 시작되었다. 유체역학의 본격적인 발전은 레오나르도 다 빈치 (발견과 실험), 에반젤리스타 토리첼리 (기압계), 아이작 뉴턴 (점성), 블레즈 파스칼 (수력학)로 부터 시작되었고 다니엘 베르누이가 1738년 <유동체역학>이라는 책을 통해 유체역학을 수학적이로 풀어나가는 길을 여므로써 그 발전을 이어나갔다. 비점성 유동은 더욱 다양한 수학자들에게 분석되었고 (레온하르트 오일러, 달랑베르, 라그랑주, 라플라스, 푸아송) 점성 유동은 푸아죄유와 하겐과 같은 많은 공학자들에게 탐구되었다. 뿐만 아니라 수학적으로는 클로드 루이 나비에와 조지 가브리엘 스토크스의 나비에-스토크스 방정식에 의해 나타낼 수 있다. 또한 경계층은 루드비히 프란틀에 의해 연구되었고 다양한 학자들이 유체의 점도와 난류에 대한 지식을 발전시켰다.

연속체 역학과의 관계

유체역학은 연속체 역학의 한 분야이다. 역학적인 관점에서 유체는 전단응력을 받지 않는 물질이다. 이 때문에 멈춰있는 유체는 담겨져 있는 용기의 형태를 갖는다.

Assumptions

현실세계에 대한 다른 수학적 모델들과 마찬가지로 유체역학 또한 연구하는 물질에 관한 몇 가지 기본적인 가정들이 있다. 가정들로 부터 이 가정들이 성립하는 경우에 만족되는 식을 유도할 수 있다. 예를 들어 3차원에서 비압축성 유체를 생각해보자. 질량이 보존된다는 가정은 고정된 폐곡면에서 (예를 들어 구가 있다) 외부에서 내부로 지나는 질량의 변화율이 다른 경로로 가는 질량의 변화율과 같아야 한다는 말과 같다. (다시 말해, 내부와 외부의 질량은 일정하다). 이는 곡면에 대한 적분 방정식으로 나타낼 수 있다.

유체역학에서 모든 유체는 다음을 따른다고 가정한다:

• 질량 보존의 법칙
• 에너지 보존 법칙
• 운동량 보존 법칙
• 연속체 가설

또한 유체를 비압축성 유체로 가정하는 것도 때로 유용하다. (다시 말해, 유체의 밀도가 변하지 않는다). 기체는 그렇지 않지만, 액체는 때로 비압축성 유체로 다루기도 한다.

비슷하게 유체의 점성을 때로는 0으로 가정한다. 기체를 때로 무점성 유체로 가정하기도 한다.

The continuum hypothesis

 이 부분의 본문은 Continuum mechanics입니다.

Fluids are composed of molecules that collide with one another and solid objects. The continuum assumption, however, considers fluids to be continuous. That is, properties such as density, pressure, temperature, and velocity are taken to be well-defined at "infinitely" small points, defining a REV (Reference Element of Volume), at the geometric order of the distance between two adjacent molecules of fluid. Properties are assumed to vary continuously from one point to another, and are averaged values in the REV. The fact that the fluid is made up of discrete molecules is ignored.

The continuum hypothesis is basically an approximation, in the same way planets are approximated by point particles when dealing with celestial mechanics, and therefore results in approximate solutions. Consequently, assumption of the continuum hypothesis can lead to results which are not of desired accuracy. That said, under the right circumstances, the continuum hypothesis produces extremely accurate results.

Those problems for which the continuum hypothesis does not allow solutions of desired accuracy are solved using statistical mechanics. To determine whether or not to use conventional fluid dynamics or statistical mechanics, the Knudsen number is evaluated for the problem. The Knudsen number is defined as the ratio of the molecular mean free path length to a certain representative physical length scale. This length scale could be, for example, the radius of a body in a fluid. (More simply, the Knudsen number is how many times its own diameter a particle will travel on average before hitting another particle). Problems with Knudsen numbers at or above unity are best evaluated using statistical mechanics for reliable solutions.

Navier–Stokes equations

 이 부분의 본문은 Navier–Stokes equations입니다.

The Navier–Stokes equations (named after Claude-Louis Navier and George Gabriel Stokes) are the set of equations that describe the motion of fluid substances such as liquids and gases. These equations state that changes in momentum (force) of fluid particles depend only on the external pressure and internal viscous forces (similar to friction) acting on the fluid. Thus, the Navier–Stokes equations describe the balance of forces acting at any given region of the fluid.

The Navier–Stokes equations are differential equations which describe the motion of a fluid. Such equations establish relations among the rates of change of the variables of interest. For example, the Navier–Stokes equations for an ideal fluid with zero viscosity states that acceleration (the rate of change of velocity) is proportional to the derivative of internal pressure.

This means that solutions of the Navier–Stokes equations for a given physical problem must be sought with the help of calculus. In practical terms only the simplest cases can be solved exactly in this way. These cases generally involve non-turbulent, steady flow (flow does not change with time) in which the Reynolds number is small.

For more complex situations, such as global weather systems like El Niño or lift in a wing, solutions of the Navier–Stokes equations can currently only be found with the help of computers. This is a field of sciences by its own called computational fluid dynamics.

General form of the equation

The general form of the Navier–Stokes equations for the conservation of momentum is:

$\rho {\frac {D\mathbf {v} }{Dt}}=\nabla \cdot \mathbb {P} +\rho \mathbf {f}$

where

• $\rho \$ is the fluid density,
• ${\frac {D}{Dt}}$ is the substantive derivative (also called the material derivative),
• $\mathbf {v}$ is the velocity vector,
• $\mathbf {f}$ is the body force vector, and
• $\mathbb {P}$ is a tensor that represents the surface forces applied on a fluid particle (the stress tensor).

Unless the fluid is made up of spinning degrees of freedom like vortices, $\mathbb {P}$ is a symmetric tensor. In general, (in three dimensions) $\mathbb {P}$ has the form:

$\mathbb {P} ={\begin{pmatrix}\sigma _{xx}&\tau _{xy}&\tau _{xz}\\\tau _{yx}&\sigma _{yy}&\tau _{yz}\\\tau _{zx}&\tau _{zy}&\sigma _{zz}\end{pmatrix}}$

where

• $\sigma \$ are normal stresses,
• $\tau \$ are tangential stresses (shear stresses).

The above is actually a set of three equations, one per dimension. By themselves, these aren't sufficient to produce a solution. However, adding conservation of mass and appropriate boundary conditions to the system of equations produces a solvable set of equations.

Newtonian versus non-Newtonian fluids

A Newtonian fluid (named after Isaac Newton) is defined to be a fluid whose shear stress is linearly proportional to the velocity gradient in the direction perpendicular to the plane of shear. This definition means regardless of the forces acting on a fluid, it continues to flow. For example, water is a Newtonian fluid, because it continues to display fluid properties no matter how much it is stirred or mixed. A slightly less rigorous definition is that the drag of a small object being moved slowly through the fluid is proportional to the force applied to the object. (Compare friction). Important fluids, like water as well as most gases, behave — to good approximation — as a Newtonian fluid under normal conditions on Earth.

By contrast, stirring a non-Newtonian fluid can leave a "hole" behind. This will gradually fill up over time – this behaviour is seen in materials such as pudding, oobleck, or sand (although sand isn't strictly a fluid). Alternatively, stirring a non-Newtonian fluid can cause the viscosity to decrease, so the fluid appears "thinner" (this is seen in non-drip paints). There are many types of non-Newtonian fluids, as they are defined to be something that fails to obey a particular property — for example, most fluids with long molecular chains can react in a non-Newtonian manner.

Equations for a Newtonian fluid

 이 부분의 본문은 Newtonian fluid입니다.

The constant of proportionality between the shear stress and the velocity gradient is known as the viscosity. A simple equation to describe Newtonian fluid behaviour is

$\tau =-\mu {\frac {dv}{dy}}$

where

$\tau$ is the shear stress exerted by the fluid ("drag")

$\mu$ is the fluid viscosity – a constant of proportionality

${\frac {dv}{dy}}$ is the velocity gradient perpendicular to the direction of shear.

For a Newtonian fluid, the viscosity, by definition, depends only on temperature and pressure, not on the forces acting upon it. If the fluid is incompressible and viscosity is constant across the fluid, the equation governing the shear stress (in Cartesian coordinates) is

$\tau _{ij}=\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}\right)$

where

$\tau _{ij}$ is the shear stress on the $i^{th}$ face of a fluid element in the $j^{th}$ direction

$v_{i}$ is the velocity in the $i^{th}$ direction

$x_{j}$ is the $j^{th}$ direction coordinate.

If a fluid does not obey this relation, it is termed a non-Newtonian fluid, of which there are several types.

Among fluids, two rough broad divisions can be made: ideal and non-ideal fluids. An ideal fluid really does not exist, but in some calculations, the assumption is justifiable. An Ideal fluid is non viscous- offers no resistance whatsoever to a shearing force.

One can group real fluids into Newtonian and non-Newtonian. Newtonian fluids agree with Newton's law of viscosity. Non-Newtonian fluids can be either plastic, bingham plastic, pseudoplastic, dilatant, thixotropic, rheopectic, viscoelatic.

See also

• Aerodynamics
• Applied mechanics
• Secondary flow
• Bernoulli's principle
• Communicating vessels

References

• Batchelor, George K. (1967), 《An Introduction to Fluid Dynamics》, Cambridge University Press, ISBN 0521663962 
• Kundu, Pijush K.; Cohen, Ira M. (2008), 《Fluid Mechanics》 4 revis판, Academic Press, ISBN 978-0-123-73735-9 
• Currie, I. G. (1974), 《Fundamental Mechanics of Fluids》, McGraw-Hill, Inc., ISBN 0070150001 
• Massey, B.; Ward-Smith, J. (2005), 《Mechanics of Fluids》 8판, Taylor & Francis, ISBN 978-0-415-36206-1 
• White, Frank M. (2003), 《Fluid Mechanics》, McGraw–Hill, ISBN 0072402172 

External links

• Free Fluid Mechanics books
• Annual Review of Fluid Mechanics
• CFDWiki – the Computational Fluid Dynamics reference wiki.
• Educational Particle Image Velocimetry – resources and demonstrations
• prof. DSc Ivan Antonov, Technical University of Sofia, Bulgaria.

틀:Physics-footer 틀:Chemical engg ko:유체역학