In

fluid dynamicsIn physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...

,

**circulation** is the

line integralIn mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...

around a closed curve of the

fluidIn physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....

velocityIn physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

. Circulation is normally denoted

.

If

is the fluid velocity on a small element of a defined curve, and

is a vector representing the differential length of that small element, the contribution of that differential length to circulation is

:

where

is the angle between the vectors

and

.

The circulation around a closed curve

is the line integral:

The dimensions of circulation are length squared, divided by time.

Circulation was first used independently by

Frederick LanchesterFrederick William Lanchester, Hon FRAeS was an English polymath and engineer who made important contributions to automotive engineering, aerodynamics and co-invented the field of operations research....

,

Wilhelm KuttaMartin Wilhelm Kutta was a German mathematician.Kutta was born in Pitschen, Upper Silesia . He attended the University of Breslau from 1885 to 1890, and continued his studies in Munich until 1894, where he became the assistant of Walther Franz Anton von Dyck. From 1898, he spent a year at the...

, and Nikolai Zhukovsky.

## Kutta–Joukowski theorem

The

liftA fluid flowing past the surface of a body exerts a surface force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction. It contrasts with the drag force, which is the component of the surface force parallel to the flow direction...

force acting per unit span on a body in a two-dimensional inviscid flow field can be expressed as the product of the circulation (

) about the body, the fluid density (

), and the speed of the body relative to the free-stream (

). Thus,

This is known as the Kutta–Joukowski theorem.

This equation applies around airfoils, where the circulation is generated by airfoil action, and around spinning objects, experiencing the

Magnus effectThe Magnus effect is the phenomenon whereby a spinning object flying in a fluid creates a whirlpool of fluid around itself, and experiences a force perpendicular to the line of motion...

, where the circulation is induced mechanically.

Circulation is often used in

computational fluid dynamicsComputational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with...

as an intermediate variable to calculate forces on an

airfoilAn airfoil or aerofoil is the shape of a wing or blade or sail as seen in cross-section....

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 layerIn physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity of the fluid are considered in detail. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal...

. 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 by

Stokes' theoremIn differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...

:

but only if the integration path is a boundary, not just a closed curve. Thus vorticity is the circulation per unit area, taken around an infinitesimal loop.

## See also

- Vorticity
- Biot-Savart law
- Kutta condition
The Kutta condition is a principle in steady flow fluid dynamics, especially aerodynamics, that is applicable to solid bodies which have sharp corners such as the trailing edges of airfoils...

- Kutta–Joukowski theorem
The Kutta–Joukowski theorem is a fundamental theorem of aerodynamics. It is named after the German Martin Wilhelm Kutta and the Russian Nikolai Zhukovsky who first developed its key ideas in the early 20th century. The theorem relates the lift generated by a right cylinder to the speed of the...

- Kelvin circulation theorem