Page 31 - Microtectonics

18 2 · Flow and Deformation

Box 2.4 Vorticity and spin
Vorticity is the ‘amount of rotation’ that a flow type possesses (Means
et al. 1980). The concept of vorticity can be illustrated with the ex-
ample of a river (Fig. B.2.3a, ×Video B.2.3). In the centre, flow is
faster than near the edges. If paddle wheels are inserted in the river
along the sides, they will rotate either sinistrally or dextrally; the
flow in these domains has a positive or negative vorticity. In the
centre, a paddle wheel does not rotate; here the vorticity is zero.
Rotation of material lines must be defined with respect to some ref-
erence frame (the edges of the river in Fig. B.2.3a, ×Video B.2.3)
and the same therefore applies to vorticity. In this book, we define
vorticity as the summed angular velocity of any two orthogonal
material lines in the flow with respect to the ISA (i.e. the ISA act as
our reference frame; Fig. B.2.3b).
If an external reference frame is used and ISA rotate in this
external reference frame, the angular velocity of the ISA is re-
ferred to as spin (Fig. B.2.3b; Lister and Williams 1983; Means
1994). Rotation of material lines in a randomly chosen external
reference frame can therefore have components of spin and vor-
ticity. Vorticity and spin can be shown as vectors parallel to the
rotation axis of the orthogonal material line sets (the axes of the
paddle wheels in Fig. B.2.3a, ×Video B.2.3).

Fig. B.2.3. Illustration of the concept of vorticity and spin. a If the
velocity of a river is fastest in the middle, paddle wheels inserted in
the river will rotate in opposite direction at the sides, but will not
rotate in the middle; they reflect the vorticity of flow in the river at
three different sites. b Vorticity is defined as the sum of the angular
velocity with respect to ISA of any pair of orthogonal material lines
(such as p and q); additional rotation of ISA (and all the other lines
and vectors) in an external reference frame is known as spin

2.6 Box 2.5 Vorticity and kinematic vorticity number 2.6
Deformation and Strain
It may seem unnecessarily complicated to define a kinematic
vorticity number W k when we can also simply use vorticity.
However, there is an obvious reason. W is normalised for Analogous to homogeneous flow, homogeneous deforma-
k
strain rate and is therefore a dimensionless number. This tion can be envisaged by the distribution patterns of
makes W more suitable for comparison of flow types than stretch and rotation of a set of lines connecting marker
k
vorticity. For example, imagine a river and a rock both particles (Fig. 2.7a–e). These values plot in two curves as
flowing with identical flow patterns. Vorticity in the river for flow, but these are now asymmetric (Figs. 2.5e, 2.7d).
–1
–1
is 0.2 s at a strain rate of 0.3 s . In the rock these values
s . In both cases,
are respectively 4 × 10 –14 –1 –14 –1 It is also necessary to define if the stretch and rotation of
s and 6 × 10
vorticity is vastly different. However, W k is in both cases 0.66. a line are given for the position of the line at the onset of,
The same principle applies for the kinematic dilatancy or after the deformation. Here, we use the former defini-
number A . tion. The maximum and minimum stretch values are
k
known as the principal stretches or principal strain values S 1
and S . They occur along lines that are orthogonal before
2
known as the kinematic vorticity number, and A as and after the deformation, the two principal strain axes
k
the kinematic dilatancy number. W is a measure of the (Fig. 2.7d,e). Since homogeneous deformation is a ten-
k
rotational quality of a flow type, while A is a measure sor, it can also be fully described by just four numbers.
k
of the rate at which a surface shrinks or expands with These are: (a) S and S which describe the strain or change
2
1
time. For example, simple shear flow without area change in shape that is part of the homogeneous deformation
has W =1 and A = 0. Pure shear flow has W =0 and (Sect. 9.2); (b) a number β describing the orientation of
k
k
k
k
A = 0 (Fig. 2.6). All possible flow pattern geometries the principal strain axes in a reference frame at the onset
k
can be defined by just W or A , while Ö defines how of deformation, and (c), ρ , the rotation of the principal
k
k
k
k
fast deformation is accumulated for a particular flow strain axes in the reference frame between the initial and
type and α describes its orientation in an external ref- the final state (Fig. 2.7e). Note that deformation is nor-
k
erence frame. mally composed of strain (which only describes a change