7.2 Vector Fields, Divergence and Curl or Rotation 89
The scalar∇λvλ=∇·v:=divvis called thedivergenceof the vectorv. The cross
product(∇×v)of the nabla operator and a vectorvis also denoted by curlvor rotv,
pronounced ascurlorrotation. In component notation it is given by
(∇×v)λ=ελσ τ∇σvτ. (7.32)Whenvstands for the flow velocity of a streaming fluid, the quantity( 1 / 2 )(∇×v)
is referred to as thevorticityof the flow field. A sphere suspended in a plane Couette
flow picks up an angular velocity equal to the vorticity, provided it is allowed to
rotate freely.
The symmetric traceless part, also called deviatoric part of the gradient of the
vector field, is given by
∇νvμ =1
2
(∇νvμ+∇νvμ)−1
3
(∇λvλ)δμν. (7.33)Vector fields with∇·v =0, examples (ii) and (iii), have field lines with sources
and sinks. Fields with∇·v=0, all other cases above, are calledsource freevector
fields. Vector fields with∇×v=0, examples (i)–(v), are calledvortex-free. Vectors
of this kind can be derived from a scalar potential field. Frequently, vector fields are
classified according to whether their divergence and their rotation are zero or not
equal to zero, more details later.
The divergence, the rotation and ∇v can be calculated for the special vector
fields treated above in Sect.7.2.1. The cases (i)–(iii) and (vii) are considered next,
the calculations for the vector fields (iv)–(vi) are transferred to the following exercise.
Divergence, Rotation and Symmetric Traceless Part of the Gradient Tensorfor
the vector fields (i)–(iii) and (vii) of Sect.7.2.1:
(i) Homogeneous Field
vμ=eμ=const.Obviously, one finds∇νvμ=0, consequently∇·v=0,∇×v=0, and∇v=0.
(ii) Linearly Increasing Field
vμ=eμeκrκ=xeμ.Due to∇νrκ=δνκone obtains here,∇νvμ=eμeν, and consequently
∇·v= 1 , ∇×v= 0 , ∇νvμ =eνeμ.(iii) Radial and Cylindrical Fields
vμ=rμ.