STRAIN RATE, ROTATION RATE AND ISOTROPY

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STRAIN RATE, ROTATION RATE AND ISOTROPY
In the previous lecture the strain rate tensor
?ij and the rotation rate tensor rij were defined
as
The shear stress tensor ?ij was represented as
where ?ijv denotes the viscous stress tensor, and
?ijv was related only to the strain rate tensor
?ij, in the most general linear form
This was done because we want to relate ?ijv to
how a body deforms, not how it rotates. But to
do this we must establish that ?ij does indeed
characterize deformation, and rij does indeed
characterize rotation.
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STRAIN RATE, ROTATION RATE AND ISOTROPY
A deformable body may be deformed in two ways
extension shearing. Extensional deformation is
illustrated below
Shear deformation is illustrated below
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STRAIN RATE, ROTATION RATE AND ISOTROPY
We first consider extensional deformation. A
moving deformable body has length ?x1 in the x1
direction. The velocity u1 is assumed to be
changing in the x1 direction, so that the values
on the left-hand side and right-hand side of the
body are, respectively is
In time ?t the left boundary moves a distance
u1?t, and the right boundary moves a distance u1
(?u1/?x1)?x1?t
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STRAIN RATE, ROTATION RATE AND ISOTROPY
The initial length of the body is ?x1. The
length of the body after time ?t is given as
The extensional strain rate is the rate of length
increase of the body per unit initial length per
unit time
(new length) (old length)/(old length)/(?t)
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STRAIN RATE, ROTATION RATE AND ISOTROPY
Note that ?u1/?x1 gt 0 for an elongating body and
?u1/?x1 lt 0 for a shortening body. The
corresponding extensional strain rates in the x2
and x3 directions are, respectively,
These extensional strain rates relate to the
diagonal components of the strain rate tensor
?11, ?22 and ?33 as follows
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STRAIN RATE, ROTATION RATE AND ISOTROPY
Now we consider shear deformation. Consider the
points A, B and C below. The velocities in the
x1 direction at points A and C are, respectively,
and the velocities in the x2 direction at points
A and B are, respectively,
C
A
B
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STRAIN RATE, ROTATION RATE AND ISOTROPY

• The body undergoes shear deformation over time
  ?t. That is
• Point A moves a distance u1?t in the x1
  direction and u2?t in the x2 direction
• Point B moves a distance u2 (?u2/?x1)?x1?t
  in the x2 direction and
• Point C moves a distance u1 (?u1/?x2)?x2?t
  in the x1 direction.

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STRAIN RATE, ROTATION RATE AND ISOTROPY
Recall that for small angle ?, sin? ? ?. The
angles ?? and ?? created in time ?t are defined
below. These can be approximated as
??
??
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STRAIN RATE, ROTATION RATE AND ISOTROPY
The strain rate due to shearing can be defined as
the angle increase rate ?(? ?)/?t, or thus
??
??
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STRAIN RATE, ROTATION RATE AND ISOTROPY
The strain rate due to shearing due to shearing
d(? ?)/dt in the x1-x2 plane is related to the
component ?12 of the strain rate tensor as
The corresponding components of the strain rate
tensor due to shearing in the x2-x3 and x1-x3
planes are correspondingly
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STRAIN RATE, ROTATION RATE AND ISOTROPY
It is thus seen that the strain rate tensor ?ij
does indeed characterize the rate at which a body
is deformed by elongation or shearing. We now
must establish that the tensor
does indeed characterize rotation.
We approach this indirectly, by first defining
circulation. Circulation ? is an integral
measure of the tendency of a fluid to rotate.
Let C denote some fixed closed circuit within a
fluid (across which fluid can flow freely), and
let denote an elemental arc length
tangential to the circuit that is positive in the
counterclockwise direction.
The circulation ? is defined as
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STRAIN RATE, ROTATION RATE AND ISOTROPY
To illustrate the idea of circulation, we
consider two simple examples. The first of these
is constant, rectilinear flow in the x direction
with velocity U, so that (u, v, w) (U, 0, 0).
The circuit has length L in the x direction and
length H in the y direction. The circulation
around the circuit is
H
U
Thus the is no circulation around a circuit in
rectilinear flow.
L
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STRAIN RATE, ROTATION RATE AND ISOTROPY
Now we consider the case of plane Couette flow
with Note that u 0 where y 0 and u U
where y H. Now
H
U
Thus there is circulation, and it is negative
(i.e. directed in the clockwise direction).
L
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STRAIN RATE, ROTATION RATE AND ISOTROPY
The concept of circulation is closely related to
the concept of vorticity. Consider a loop around
a region with area dA, such that the circulation
around the circuit is d?. The vorticity ? is
defined as
The vorticity of a fluid at a point is equal to
twice the angular velocity of the fluid particles
at that point. This can be seen by considering a
fluid particle rotating with angular speed ?, at
the center of a circle with radius dr. The arc
length of the periphery of the circle is 2?dr,
the area of the circle is ?(dr)2, and the
peripheral velocity is ?dr. Thus
?
dr
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STRAIN RATE, ROTATION RATE AND ISOTROPY
The vorticity can be related to the velocity
field as follows. Consider the elemental
rectangular circuit below. The components of
velocity at the center of the rectangle are (u,
v). The component of the velocity normal to the
circuit on segments 1, 2, 3 and 4
are 1. 2. 3. 4.
?y
(u,v)
Thus
?x
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STRAIN RATE, ROTATION RATE AND ISOTROPY
Reducing,
And thus since ? d?/dA,
?y
(u,v)
?x
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STRAIN RATE, ROTATION RATE AND ISOTROPY
Now we return to constant rectilinear flow and
plane Couette flow. For rectilinear flow (u, v,
w) (U, 0, 0) and
Therefore the illustrated red paddle will not
rotate as it moves with the flow.
For plane Couette flow (u, v, w) (Uy/H, 0, 0)
and
The illustrate red paddle will thus rotate in the
clockwise direction as it moves with the flow,
with angular speed ? U/(2H).
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STRAIN RATE, ROTATION RATE AND ISOTROPY
So far we have considered only 2D flows in the
(x, y) plane, in which case angular velocity and
vorticity are directed along the z axis. The
appropriate 3D extension is
Or expanding out
Thus for example
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STRAIN RATE, ROTATION RATE AND ISOTROPY
The rate of rotation tensor rij is directly
related to the vorticity vector ?i and thus to
the angular velocity ?i. That is,
(Note do not confuse the Levi-Civita third-order
tensor ?ijk with the strain rate tensor ?ij.)
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STRAIN RATE, ROTATION RATE AND ISOTROPY
Our goal is to relate the viscous stress tensor
?ijv to a measure of the rate of deformation not
the rate of rotation, of a fluid. Thus we relate
?ijv to ?ij rather than ?ui/?xj. The most
general linear relation between ?ijv and ?ij is

• The relation can be simplified by assuming that
  it is
• isotropic, so that the form of the relation is
  invariant to coordinate rotation, and has the
  same physics in any direction, and
• symmetric, so that ?ijv ?jiv.

We do not go through the complete details of
isotropy here. We showed in Lecture 2, however,
that pressure p is isotropic. More specifically,
where ?ijp denotes the part of the stress tensor
associated with pressure,
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STRAIN RATE, ROTATION RATE AND ISOTROPY
The most general second-order isotropic tensor
Aij takes the form
where C is an arbitrary scalar. (In the case of
?ijp, C -p.) It turns out that the most
general fourth-order isotropic tensor is (Aris,
1962)
where again C1, C2 and C3 are arbitrary scalars.
Thus the relation
reduces to
But for an incompressible fluid
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STRAIN RATE, ROTATION RATE AND ISOTROPY
Thus
But
But ?ij is a symmetric tensor, i.e. ?ij ?ji.
Further defining C2 C3 2?, where ? the
dynamic viscosity of the fluid, the relation
reduces to
or
The above relation defines the constitutive
relation for a viscous Newtonian fluid. Note
that the form guarantees symmetry in ?ij.
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STRAIN RATE, ROTATION RATE AND ISOTROPY
Reference Aris, R. (1962) Vectors, Tensors and
the Basic Equations of Fluid Mechanics.
Prentice-Hall.