Derived Fields - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 143

by the following decomposition of the gradient tensor:

T = D + R + S (14.2)

where D = γ d Id is a multiple of the identity matrix, R is an anti-symmetric matrix,

and S is a traceless , symmetric matrix. There are three fundamental fluid motions
besides translation, and they are isotropic scaling , rotation , and anisotropic stretching
or pure shearing . Interestingly, these motions correspond to the three components
described in Eq. ( 14.2 ). D describes the isotropic stretching. When

γ d > 0, the

particle's volume will increase, while when 0, the particle will lose volume

when it travels. R represents rotations, i.e., spinning around the center of the particle.
This is related to the vortices in the flow. S corresponds to the anisotropic stretching. In
this case the particle is under pure shearing , which refers to simultaneous expansion
along some axis or axes and contraction along perpendicular directions without
changing the volume. Pure shearing is linked to the rate of angular deformation, rate
of mixing of multiple interacting fluid materials, and energy dissipation.
While these fields can be studied independently, in this context it is often important
to study their interaction. For example, for two-dimensional cases, i.e., T is a 2

γ d <

2
matrix, Zhang et al. [ 28 ] introduce the notion of eigenvalue manifold and eigenvector
manifold . We will examine these concepts in detail.
In 2D, the components in Eq. ( 14.2 ) can be written as follows:

×

= γ d 10 = γ r 0 = γ s cos

1
10

− θ sin θ

D , R , S (14.3)

01 sin θ − cos θ

where γ d , γ r , and γ s ≥ 0 are the strengths of isotropic scaling, rotation, and pure

shearing, respectively. decodes the orientation of the shearing. Note that the eigen-

values of T are purely decided by

θ

γ d , γ r , and γ s . Zhang et al. [ 28 ] treat the triple

(γ d ,γ r ,γ s ) as a vector and consider the configurations corresponding to unit vec-

tors. Such vectors form a hemisphere which they refer to as the eigenvalue mani-
fold (Fig. 14.9 : left). There are five canonical points on this manifold (Fig. 14.9 :
colored dots), corresponding to

(γ d = 1 ,γ r = 0 ,γ s = 0 ) (pure expansion),

(γ d =− 1 ,γ r = 0 ,γ s = 0 ) (pure contraction), (γ d = 0 ,γ r = 1 ,γ s = 0 ) (pure

counterclockwise rotation), (γ d = 0 ,γ r =− 1 ,γ s = 0 ) (pure clockwise rotation),

and (pure shearing). A configuration is said to be dom-

inated by one of these five canonical motions,

(γ d = 0 ,γ r = 0 ,γ s = 1 )

, if the point corresponding to the
this configuration has the smallest geodesic distance to the canonical motion

μ

μ

.The
partition of the eigenvalue manifold in turn leads to a partition of the domain of
tensor field T , although the map is not bijective.
Figure 14.10 illustrates this with an example vector field that is generated by com-
bining two counter-rotating Sullivan vortices. Notice that the flow is predominantly
expanding in the middle (yellow), contracting on the outside (blue), rotating (red
and green), and pure shearing (white) elsewhere. Note that a region of predominant
expansion motion cannot be directly adjacent to a region of predominant contraction.
Similarly, a counterclockwise rotation region cannot be adjacent to a region domi-