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Concept
Circulation (physics)
Summary
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. It is usually denoted Γ (Greek uppercase gamma).
If V is a vector field and dl is a vector representing the differential length of a small element of a defined curve, the contribution of that differential length to circulation is dΓ:
Here, θ is the angle between the vectors V and dl.
The circulation Γ of a vector field V around a closed curve C is the line integral:
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.
Circulation can be related to curl of a vector field V and, more specifically, to vorticity if the field is a fluid velocity field,
By Stokes' theorem, the flux of curl or vorticity vectors through a surface S is equal to the circulation around its perimeter,
Here, the closed integration path ∂S is the boundary or perimeter of an open surface S, whose infinitesimal element normal dS = ndS 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.
Kutta–Joukowski theorem
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 , and the speed of the body relative to the free-stream :
This is known as the Kutta–Joukowski theorem.
Official source
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