Curl

rotation, of a vector field $ \mathbf a ( M) $

The vector field given by the "rotational component" of this field. If $ \mathbf a ( M) $ is the velocity field of the particles of a moving continuous medium, the curl is equal to one-half the angular velocity of the particles. The curl is denoted by $ \mathop{\rm curl} \mathbf a $ (sometimes by $ \mathop{\rm rot} \mathbf a $). In orthogonal Cartesian coordinates $ x, y, z $ the curl is defined by the expression

$$ \left \{ { \frac{1}{2} } \left ( \frac{\partial w }{\partial y } - \frac{\partial v }{\partial z } \right ) ; \ { \frac{1}{2} } \left ( \frac{\partial u }{\partial z } - \frac{\partial w }{\partial x } \right ) ; \ { \frac{1}{2} } \left ( \frac{\partial v }{\partial x } - \frac{\partial u }{\partial y } \right ) \right \} $$

where $ u( M), v( M), w( M) $ are the components of $ \mathbf a ( M) $.

An integral curve of the curl of a vector field is known as a vortical line. Any surface generated by a one-parameter family of vortical lines is known as a vortical surface. A very important special case of vortical surfaces are vortical tubes, formed by the vortical lines issuing from all points of some closed curve. If this curve is infinitely small, the generated vortical surface is said to be a vortical thread. Vortical surfaces are also known as vortical layers, these layers being regarded as consisting of a geometrical surface with a facing of vortical lines. When intersecting the vortical layer the velocities of the liquid particles undergo a tangential discontinuity which is proportional to the curl at the respective point.

According to the fundamental Helmholtz theorem in hydrodynamics, if the volume forces have a potential, then, during the flow of a homogeneous ideal non-compressible liquid or a barotropic gas, those particles of the medium located at a certain moment of time on a vortical line will also be on the vortical line at all successive moments of time. Thus, vortical surfaces, and in particular vortical tubes and vortical lines, are persistent in time. Any vortical tube may be characterized by some number — the strength of the tube — which is equal to the flux of the curl vector field across an arbitrary cross-section of the tube. This number is independent of the form of the cross-section, since $ \textrm{ div curl } \mathbf a = 0 $. It means that the vortical tube may be either closed (a vortical ring) or may have a beginning and an end on the boundaries of the liquid. The strength of a vortical tube in an ideal liquid remains unchanged with time.

These properties of vortical tubes, discovered by H. Helmholtz, can be very simply demonstrated with the aid of W. Thomson's (Lord Kelvin's) concept of the circulation $ \Gamma $ of the velocity $ \mathbf v $ along a closed contour $ ( L) $:

$$ \Gamma = \oint _ { (L) } | \mathbf v | \cos \ ( \widehat{ {\mathbf v d \mathbf s }} ) ds, $$

where $ d s $ is the arc element of the contour $ L $ and $ ( \widehat{ {\mathbf v d \mathbf s }} ) $ denotes the angle between $ \mathbf v $ and $ d \mathbf s $. The study of the properties of velocity circulation leads to the theorem of Lagrange on the constancy of non-turbulent motion with time.

The principal task of the theory of curls is to determine the velocity field of the liquid motion along a given curl vector field. If the region occupied by the liquid is infinite in all directions and if the region $ ( D) $ occupied by the curl is bounded by a closed vortical surface, the velocity field can be found with the aid of the vector potential

$$ \Pi = \frac{1}{4 \pi } {\int\limits \int\limits \int\limits } _ { (D) } \frac{ \mathop{\rm curl} \mathbf v }{r} d \tau , $$

from the formula

$$ \mathbf v = \mathop{\rm curl} \Pi . $$

If the task consists of determining velocities along vortical lines in a bounded space, its solution is very difficult, involving integral equations with singular kernels. For a complete solution of this problem see , [7].

In the important special case of plane-parallel motion

$$ u = u ( x, y),\ \ v = v ( x, y),\ \ w = w ( x, y), $$

two components $ \alpha , \beta $ of the curl are zero, while the third component $ \gamma $ represents all of the curl which, in that particular case, is perpendicular to the plane $ XOY $. A tiny area, known as a curl point, is formed at the intersection of a curl thread with the plane $ XOY $. If several curl points are present in a liquid, there results a motion of the points themselves, due to the velocities they generate in the liquid. The equation of motion of curl points has the form of canonical equations of mechanics.

References

[1]	P. Appell, "Traité de mécanique rationelle" , 3 , Gauthier-Villars (1909)
[2]	H. Villat, "Théorie des rotations" , Leningrad-Moscow (1936) (In Russian; translated from French)
[3]	L. Lichtenstein, "Grundlagen der Hydromechanik" , Springer (1929) MR0228225 Zbl 55.1124.01
[4]	L.M. Milne-Thomson, "Theoretical hydrodynamics" , Macmillan (1950) Zbl 0164.55802 Zbl 0089.42601 Zbl 0065.40604 Zbl 0019.37501 Zbl 64.0848.12
[5]	H. Lamb, "Hydrodynamics" , Cambridge Univ. Press (1906) MR1317348 Zbl 36.0817.07
[6a]	N.M. Gyunter, Izv. Akad. Nauk SSSR Ser. 6 , 20 : 13–14 (1926) pp. 1323–1348
[6b]	N.M. Gyunter, Izv. Akad. Nauk SSSR Ser. 6 , 20 : 15–17 (1926) pp. 1503–1532
[7]	N.M. Gyunter, Zh. Leningrad. Fiz.-Mat. Obshch. , 1 : 1 (1926) pp. 12–36

Comments

See also Rotation of a vector field. A mathematical treatment of the vector differentiation operator curl can be found in, e.g., [a3]–[a5] (see also Vector analysis).

References

[a1]	J. Serrin, "Mathematical principles of fluid mechanics" , Handbook of Physics , 8 : 1. Fluid dynamics , Springer (1959) MR108116
[a2]	J.N. Newman, "Marine hydrodynamics" , M.I.T.
[a3]	M. Spivak, "Calculus on manifolds" , Benjamin (1965) MR0209411 Zbl 0141.05403
[a4]	R.L. Bishop, S.I. Goldberg, "Tensor analysis on manifolds" , Dover, reprint (1980) MR0615912 Zbl 0218.53021
[a5]	B.D. Craven, "Functions of several variables" , Chapman & Hall (1981) MR0636505 Zbl 0485.26004