1 strain rate, rotation rate and isotropy in the previous lecture the strain rate tensor ij and the...

1

STRAIN RATE, ROTATION RATE AND ISOTROPY

In the previous lecture the strain rate tensor ij and the rotation rate tensor rij were defined as

i

j

j

iij

i

j

j

iij x

u

x

ur,

x

u

x

u

2

1

The shear stress tensor ij was represented as

vijijij p

where ijv denotes the viscous stress tensor, and ij

v was related only to the strain rate tensor ij, in the most general linear form

klijklvij A

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.

2


A deformable body may be deformed in two ways: extension shearing. Extensional deformation is illustrated below:

Shear deformation is illustrated below:

3

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 u1t, and the right boundary moves a distance [u1 + (u1/x1)x1]t

11

111 x

x

uuandu

1x

1x

1u 11

11 x

x

uu

tu1

txx

uu 1

1

11

1x

4

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 =

tutxx

uux 11

1

111

tu1

txx

uux 1

1

111

[(new length) – (old length)]/(old length)/(t) =

1

1

1

1111

111

x

u

tx

xtutxxu

ux

5

Note that u1/x1 > 0 for an elongating body and u1

/x1 < 0 for a shortening body. The corresponding extensional strain rates in the x2 and x3 directions are, respectively,


3

3

2

2

x

uand

x

u

These extensional strain rates relate to the diagonal components of the strain rate tensor 11, 22 and 33 as follows:

3

3

3

3

3

333

2

2

2

2

2

222

1

1

1

1

1

111

x

u

x

u

x

u

2

1

x

u

x

u

x

u

2

1

x

u

x

u

x

u

2

1

6

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,

A B

C

1x

2x

22

111 x

x

uuandu

11

222 x

x

uuandu

2x

1x

2u 11

22 x

x

uu

22

11 x

x

uu

1u

7

1x

2x


The body undergoes shear deformation over time t. That is:• Point A moves a distance u1t in the x1 direction and u2t 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.

tu1

txx

uu 2

2

11

tu2 txx

uu 1

1

22

2x

1x

8

1x

2x


Recall that for small angle , sin . The angles and created in time t are defined below. These can be approximated as:

tu1

txx

uu 2

2

11

tu2 txx

uu 1

1

22

2x

1x

tx

u

x

tutxxu

u

2

1

2

122

11

tx

u

x

tutxxu

u

1

2

x

211

22

9

1x

2x


The strain rate due to shearing can be defined as the angle increase rate ( + )/t, or thus

tu1

txx

uu 2

2

11

tu2 txx

uu 1

1

22

2x

1x

1

2

2

11

2

2

1

x

u

x

u

t

txu

xu

t

)(

10

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


t

)(

2

1

x

u

x

u

2

1

1

2

2

112

The corresponding components of the strain rate tensor due to shearing in the x2-x3 and x1-x3 planes are correspondingly

1

3

3

113

2

3

3

223

x

u

x

u

2

1

x

u

x

u

2

1

11


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

i

j

j

iij x

u

x

ur

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.

sd

sd

C

The circulation is defined as:

C

sdu

12


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:

HU

1

2

3

4

L

00)UL(0UL

sdusdusdusdu

sdu

4321

C

sd

sd

sd

sd

Thus the is no circulation around a circuit in rectilinear flow.

x

y

13


Now we consider the case of plane Couette flow with

Note that u = 0 where y = 0 and u = U where y = H. Now

HU

1

2

3

4

L

UL0)UL(00

sdusdusdusdu

sdu

4321

C

sd

sd

sd

sd

Thus there is circulation, and it is negative (i.e. directed in the clockwise direction).

0,0,

H

yU)w,v,u(

x

y

14


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

dA

d

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 2dr, the area of the circle is (dr)2, and the peripheral velocity is dr. Thus

dr

2)dr(

dr2dr

dA

d

)dr(dA

dr2drd

2

2

15


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

1

2

3

4

x

sd

sd

sd

sd

(u,v)

x

y

y2

1

y

uu

y2

1

y

uu

x2

1

x

vv

x2

1

x

vv

Thus

y)x2

1

x

vv(x)y

2

1

y

uu(

y)x2

1

x

vv(x)y

2

1

y

uu(d

16


Reducing,

y

1

2

3

4

x

sd

sd

sd

sd

(u,v)

x

y

And thus since = d/dA,

yxdA

yxy

u

x

vd

y

u

x

v2

17


Now we return to constant rectilinear flow and plane Couette flow.

For rectilinear flow (u, v, w) = (U, 0, 0) and

x

y

U

0y

u

x

v2

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

H

U

y

u

x

v2

The illustrate red paddle will thus rotate in the clockwise direction as it moves with the flow, with angular speed = U/(2H).

x

y

U

18


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

j

kijkii x

u2orux2

kx

u

x

uj

x

u

x

ui

x

u

x

u

uuuxxx

kji

ux

2

1

1

2

1

3

3

1

3

2

2

3

321

321

Or expanding out

Thus for example

y

u

x

v

x

u

x

u

2

1

1

23

19


The rate of rotation tensor rij is directly related to the vorticity vector i and thus to the angular velocity i. That is,

kijk

12

13

23

3

2

2

3

3

1

1

3

2

3

3

2

2

1

1

2

1

3

3

1

1

2

2

1

i

j

j

iij

0

0

0

0x

u

x

u

x

u

x

u

x

u

x

u0

x

u

x

u

x

u

x

u

x

u

x

u0

x

u

x

ur

(Note: do not confuse the Levi-Civita third-order tensor ijk with the strain rate tensor ij.)

20


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

klijklvij A

The relation can be simplified by assuming that it is a) isotropic, so that the form of the relation is invariant to coordinate

rotation, and has the same physics in any direction, and b) symmetric, so that ij

v = 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 ij

p denotes the part of the stress tensor associated with pressure,

ijpij p

21


The most general second-order isotropic tensor Aij takes the form

ijij CA

where again C1, C2 and C3 are arbitrary scalars. Thus the relation

reduces to

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)

jkil3jlik2klij1ijkl CCCA

klijklvij A

ji3ij2klklij1vij CCC

But for an incompressible fluid

0x

u

k

kkkklkl

22


Thus

But

i

j

j

iij x

u

x

u

2

1

ji3ij2vij CC

or

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

ijijvijijijij

vij 2pp,2

i

j

j

iijij x

u

x

up

The above relation defines the constitutive relation for a viscous Newtonian fluid. Note that the form guarantees symmetry in ij.

23


Reference

Aris, R. (1962) Vectors, Tensors and the Basic Equations of Fluid Mechanics. Prentice-Hall.

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