Lissajous curve: Difference between revisions

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{{Continuum mechanics| cTopic=Fluid mechanics}}
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In [[fluid dynamics]], '''circulation''' is the [[line integral]] around a closed curve of the [[velocity field]]. Circulation is normally denoted Γ ([[Greek alphabet|Greek]] [[uppercase]] [[gamma]]). Circulation was first used independently by [[Frederick Lanchester]], [[Martin Wilhelm Kutta|Wilhelm Kutta]], and [[Nikolai Yegorovich Zhukovsky|Nikolai Zhukovsky]].
 
==Definition==
 
If '''V''' is the fluid [[velocity]] on a small element of a defined curve, and ''d'''''l''' is a vector representing the [[Differential (infinitesimal)|differential]] length of that small element, the contribution of that differential length to circulation is ''d''Γ:
 
:<math>d\Gamma=\mathbf{V}\cdot \mathbf{dl}=|\mathbf{V}||d\mathbf{l}|\cos \theta</math>  
 
where ''θ'' is the angle between the vectors '''V''' and ''d'''''l'''.
 
The circulation around a [[closed curve]] ''C'' is the [[line integral]]:<ref>{{cite book
  | title = Introduction to Fluid Mechanics
  | author1 =  Robert W. Fox
  | author2 =  Alan T. McDonald
  | author3 =  Philip J. Pritchard
  | edition = 6
  | publisher = [[John Wiley & Sons|Wiley]]
  | year = 2003
  | isbn = 0-471-20231-2
  }}</ref>
 
:<math>\Gamma=\oint_{C}\mathbf{V}\cdot d\mathbf{l}</math>
 
The [[dimensional analysis|dimension]]s of circulation are [[length]] [[square number|square]]d, divided by [[time]]; L<sup>2</sup>⋅T<sup>−1</sup>, which is equivalent to velocity times length.
 
== Kutta–Joukowski theorem ==
{{main|Kutta–Joukowski theorem}}
 
The [[lift (force)|lift]] force acting per unit span on a body in a two-dimensional inviscid flow field can be expressed as the product of the circulation Γ about the body, the fluid density ''ρ'', and the speed of the body relative to the free-stream '''V'''. Thus,
 
:<math>l = \rho V \Gamma\!</math>
 
This is known as the Kutta–Joukowski theorem.<ref>{{cite book
  | title = Foundations of Aerodynamics
  | author1 = A. M. Kuethe
  | author2 = J. D. Schetzer
  | edition = 2
  | publisher = [[John Wiley & Sons]]
  | year = 1959
  | isbn = 0-471-50952-3
  }}</ref>
 
This equation applies around airfoils, where the circulation is generated by airfoil action, and around spinning objects, experiencing the [[Magnus effect]], where the circulation is induced mechanically.
 
Circulation is often used in [[computational fluid dynamics]] as an intermediate variable to calculate forces on an [[airfoil]] or other body. When an airfoil is generating lift the circulation around the airfoil is finite, and is related to the vorticity of the [[boundary layer]]. Outside the boundary layer the vorticity is zero everywhere and therefore the circulation is the same around every circuit, regardless of the length of the circumference of the circuit.
 
== Relation to vorticity ==
 
Circulation can be related to [[vorticity]]:
 
:<math>\mathbf{\omega} = \nabla\times\mathbf{V}</math>
 
by [[Stokes' theorem]]:
 
:<math>\Gamma=\oint_{\partial S}\mathbf{V}\cdot d\mathbf{l}=\int\!\!\!\int_S \mathbf{\omega} \cdot d\mathbf{S}</math>
 
only if the integration path is a [[boundary (topology)|boundary]] (indicated by "∂") of a [[closed surface]] ''S'', not just a closed curve. Thus vorticity is the circulation per unit area, taken around an infinitesimal loop. Correspondingly, the [[flux]] of vorticity is the circulation.
 
==See also==
 
* [[Biot-Savart law#Aerodynamics applications|Biot-Savart law]]
* [[Kutta condition]]
* [[Kutta–Joukowski theorem]]
* [[Kelvin circulation theorem]]
 
==References==
{{reflist}}
 
[[Category:Fluid dynamics]]

Revision as of 13:18, 17 February 2014

' Hire a Certified Water Damage Restoration Company It is paramount that you hire a company that specializes in treating water damaged properties. This includes over-processing with dyes or any kind of chemicals such as perms or relaxers as well. Our bodies are engineered much like finely precisioned automobiles. Normally, key decision makers are part of the team as well as engineers, maintenance personnel, security and health officers, and people from finance, purchasing, and the legal department. You would think that a technologically developed island like Manhattan might have some water damage prevention methods to help you avoid flooding in your home, apartment or business, and it does.

The fact about wilderness land in the West is that people who violate it are breaking land use principles they claim to cherish. It could be the worst disaster which you can ever experience. The most common complaints associated with a sex hormone imbalance at mid-life is weight gain, reduced libido (sex drive), energy reduction, thinning hair, and muscle weakness or pain. Following the principle of diversity in design, the use of different controls at each entry point helps to attain this objective. Whenever you are bored of the same old design, you can dismantle your stuff and move them at your convenience.

As I stated earlier, I believe healing is possible if you embrace the right and perfect tools to accomplish it. This ring trues when dealing with the purchase especially on the internet of Granite Work Surfaces. That is why it is vital for everyone to know the dos and don'ts for water damage cleanup. It is very essential to make a video of the damage caused by water. Some situations of flooding are traced back to the use of failed pumps.

You can achieve the shabby chic look for the wrought iron piece yourself by painting it white and then sanding off some portions of it to give it a slightly worn out look. With the increasing demand for wood fuels and building materials, this process helps to meet these demands without cutting down the natural forests. There are some restoration professionals that come with special services also like removal bugs from the bed and many more. Nitrous oxide has two hundred and ninety six times more impact on global warming than the same unit mass amount of carbon dioxide (CO2). They have a reputation of more than 17 years of building restoration.

Whenever on-site restoration is not really feasible, the actual broken carpeting is actually relocated for a Restoration1 warehouse*, handled using fungicide as well as deodorant, in that case saved upon shelves until such time as prepared with regard to re-installation across brand new mat. Since I made the adjusters job easy the invoice was paid in full. Able Restoration understands just how water damage can occur in your attic. 4) If you find at least two companies make phone calls to the company and compare prices. Should a flood occur to cause water damage in Georgia, it is important to get in touch with the insurance agent or the company directly.

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