Английская Википедия:Circulation (physics)
In physics, circulation is the line integral of a vector field around a closed curve. In fluid dynamics, the field is the fluid velocity field. In electrodynamics, it can be the electric or the magnetic field.
Circulation was first used independently by Frederick Lanchester, Martin Kutta and Nikolay Zhukovsky.Шаблон:Citation needed It is usually denoted Шаблон:Math (Greek uppercase gamma).
Definition and properties
If Шаблон:Math is a vector field and Шаблон:Math is a vector representing the differential length of a small element of a defined curve, the contribution of that differential length to circulation is Шаблон:Math: <math display="block">\mathrm{d}\Gamma = \mathbf{V} \cdot \mathrm{d}\mathbf{l} = \left|\mathbf{V}\right| \left|\mathrm{d}\mathbf{l}\right| \cos \theta.</math>
Here, Шаблон:Math is the angle between the vectors Шаблон:Math and Шаблон:Math.
The circulation Шаблон:Math of a vector field Шаблон:Math around a closed curve Шаблон:Math is the line integral:[1][2] <math display="block">\Gamma = \oint_{C}\mathbf{V} \cdot \mathrm d \mathbf{l}.</math>
In a conservative vector field this integral evaluates to zero for every closed curve. That means that a line integral between any two points in the field is independent of the path taken. It also implies that the vector field can be expressed as the gradient of a scalar function, which is called a potential.[2]
Relation to vorticity and curl
Circulation can be related to curl of a vector field Шаблон:Math and, more specifically, to vorticity if the field is a fluid velocity field, <math display="block">\boldsymbol{\omega} = \nabla\times\mathbf{V}.</math>
By Stokes' theorem, the flux of curl or vorticity vectors through a surface S is equal to the circulation around its perimeter,[2] <math display="block">\Gamma = \oint_{\partial S} \mathbf{V}\cdot \mathrm{d}\mathbf{l} = \iint_S \nabla \times \mathbf{V} \cdot \mathrm{d}\mathbf{S}=\iint_S \boldsymbol{\omega} \cdot \mathrm{d}\mathbf{S}</math>
Here, the closed integration path Шаблон:Math is the boundary or perimeter of an open surface Шаблон:Math, whose infinitesimal element normal Шаблон:Math is oriented according to the right-hand rule. Thus curl and vorticity are the circulation per unit area, taken around a local infinitesimal loop.
In potential flow of a fluid with a region of vorticity, all closed curves that enclose the vorticity have the same value for circulation.[3]
Uses
Kutta–Joukowski theorem in fluid dynamics
In fluid dynamics, the lift per unit span (L') acting on a body in a two-dimensional flow field is directly proportional to the circulation, i.e. it can be expressed as the product of the circulation Γ about the body, the fluid density <math>\rho</math>, and the speed of the body relative to the free-stream <math>v_{\infty}</math>: <math display="block">L' = \rho v_{\infty} \Gamma</math>
This is known as the Kutta–Joukowski theorem.[4]
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. In airfoil action, the magnitude of the circulation is determined by the Kutta condition.[4]
The circulation on every closed curve around the airfoil has the same value, and is related to the lift generated by each unit length of span. Provided the closed curve encloses the airfoil, the choice of curve is arbitrary.[3]
Circulation is often used in computational fluid dynamics as an intermediate variable to calculate forces on an airfoil or other body.
Fundamental equations of electromagnetism
In electrodynamics, the Maxwell-Faraday law of induction can be stated in two equivalent forms:[5] that the curl of the electric field is equal to the negative rate of change of the magnetic field, <math display="block">\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
or that the circulation of the electric field around a loop is equal to the negative rate of change of the magnetic field flux through any surface spanned by the loop, by Stokes' theorem <math display="block">\oint_{\partial S} \mathbf{E} \cdot \mathrm{d}\mathbf{l} = \iint_S \nabla\times\mathbf{E} \cdot \mathrm{d}\mathbf{S} =
- \frac{\mathrm{d}}{\mathrm{d}t} \int_{S} \mathbf{B} \cdot \mathrm{d}\mathbf{S}.</math>
Circulation of a static magnetic field is, by Ampère's law, proportional to the total current enclosed by the loop <math display="block">\oint_{\partial S} \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 \iint_S \mathbf{J} \cdot \mathrm{d}\mathbf{S} = \mu_0 I_\text{enc}.</math>
For systems with electric fields that change over time, the law must be modified to include a term known as Maxwell's correction.
See also
References
- ↑ Шаблон:Cite book
- ↑ 2,0 2,1 2,2 Шаблон:Cite web
- ↑ 3,0 3,1 Anderson, John D. (1984), Fundamentals of Aerodynamics, section 3.16. McGraw-Hill. Шаблон:ISBN
- ↑ 4,0 4,1 Шаблон:Cite book
- ↑ Шаблон:Cite web
