7.2 Vector Fields, Divergence and Curl or Rotation 87
Fig. 7.4 Planar
squeeze-stretch field
In this case, the gradient field is
vμ=eμeνrν−uμuνrν=xeμ−yuμ. (7.26)So far, all vector fields presented can be derived from a scalar potential function.
However, there exist also vector fields for which this is not the case. A simple example
is discussed next.
(vi) Solid-like Rotation or Vorticity Field
A circular flow with a constant angular velocitywis described by
vμ=εμνλwνrλ. (7.27)This flow field is calledsolid-likesince it corresponds to the motion of the points on
a solid disc, rotating about an axis normal to the disc and running through the point
r=0. The axial vectorwis parallel to the rotation axis. The fieldvhas a non-zero
vorticity∇×v. For this reason it is also referred to asvorticity flow fieldor just
vorticity field. This kind of vector field cannot be represented as the gradient of a
scalar potential function! (Fig7.5).
(vii) Simple Shear Flow
Leteanduagain be two orthogonal unit vectors,e·u=0. The simple vector field
vμ=eμuνrν=yeμ, (7.28)withx =eνrνandy =uνrν, is called asimple shearfield. When the vectorv
is associated with the displacement of a part of a solid, the field describes a shear
deformation. In fluids, such a flow field can be realized in aplane Couettegeometry,
where a fluid is confined between parallel flat plates, normal tou, and one plate
moves parallel toe. Such a flow is also calledsimple shear flow. The vector field
(7.28) is essentially a linear combination of the planar squeeze-stretch field (7.23)
and the circular field (7.27). The simple vector field (7.28) cannot be obtained as the
gradient of a scalar field (Fig.7.6).