Non-Newtonian flow in incompressible fluids

THE COLLEGE OF AERONAUTICS

C R A N F I E L D

NON-NEWTONIAN F L O W IN I N C O M P R E S S I B L E FLUIDS

P a r t III. Some p r o b l e m s in t r a n s i e n t flow

b y

CRMFIELD

Won-Newtonian Flow in Incompressible Fluids Part III Some problems in transient flow.

b y -A. Kaye, M.-A.

SlMjARY

Some transient flows are investigated, using a rheological equation of state for an incompressible fluid of the form:

p.. -PS.. =2/"'

(^

s.. - ^ s . . - A dt'

•«0

where fi is a function of Ji and J^, the invariants of the Cauchy-Green deformation tensor S.., which relates the defoliation at the present

time t with that at some past time t' . J} is also a function of t and t'. It is found that a material obeying the above equation of state will show elastic and stress relaxation effects and may be of some use in invest-igating these properties in concentrated polymer solutions and melts.

List of symbols 1

a) Introduction 2

b ) A special form of the equation of state when the deformations

are homogeneous 3

c) Stress relaxation 6

d) Simple shear flow 10

e) Elongational flow

Ik

t)

Shear stress l6

g) Stress with constant principal directions 25

h) Constrained flow 28

i) Discussion 35

References 55

Appendix I 56

Appendix II 57

Appendix III 57

Appendix IV 38

Figures

List of Symbols

This list defines the additional symbols used in this Note, and should be talcen in conjunction with the list in College of Aeronautics Note No. 13i|-.

fi the acceleration of a particle at x.

Fi the body force per unit volume on the body at x. V. the velocity of a particle at x.

1 1

1 the matrices defining a given homogeneous A. .(t) or A(t) or A

A..(t') or A(t') or A' ^ deformation. ij ' =^ ' =

I the unit matrix.

P the matrix of the stress tensor.

s

the invariants of the tensor S. ."•'•

. a matrix whose only non-zero terms are the

' leading diagonal terms a, b, c.

^ the extension ratios on deformation of the sides >- of a parallelopiped whose sides are parallel to J the principal directions.

the transient stress.

the transient normal force.

constants in the definition of a particular form of n.

Lt A

tr-5 o , "•

, constants used for particular forms of A T t-t'

a. or a (a.' or a') the position of a particle Just before (or • ' • ' ' ' • " just after) a change in stress.

0 the shear stress in constrained flow. F(t) a function defining the amount of shear in

constrained flow. J l ' J 2 "

diag (a

Xi. I \ 2 }

(PiJ^T

(z

M,

èo

V, a , \ N, m b , c J 3 -, b -, c X3 , n )

a) Introduction

In Part I of this series (CoA Note 134) a rheological equation of

state for an incompressible fluid was obtained by a formal generalisation

of the rheological equation of state for an incompressible elastic solid

capable of sustaining large strains. The complete physical significance

of this generalisation is not understood but, since the elastic equation

of state has been used to describe the elastic behaviour of nabber-like

polymers at large strains, it seems possible that this equation of state

for an incompressible fluid may be used to describe the properties of

materials such as polymer melts, concentrated polymer solutions and

even 'solid' rubber-like polymers which show time-dej)endent properties.

In Part II (CoA Note 15^) the equation of state was used to obtain

solutions to some problems in steady flow in which the velocity and stress

at any point in the fluid were independent of time.

In this Note (Part III) we investigate some problems in which the

stress and velocity are not independent of time. In particular we

investigate problems in which there is an instantaneous change in the

deformation, the rate of deformation, or the applied stress. Such

transient flow phenomena are of interest because they provide further

insight into the behaviour of the class of fluids which may be

character-ized by the general equation of state, and also because many experiments

involve flow of this type; for example, creep under constant load, stress

relaxation under constant deformation, and elastic recoveiy.

Many polymeric substances show marked elastic recovery in the sense

that the sudden removal of an applied stress, even after the steady flow

under this stress has been achieved, results in further deformation, even

when the inertia forces may be neglected. It is important to develop a

general formalism for the description and inter-relation of all these

phenomena.

The mathematical problem can be represented as that of finding a

solution to the equation of motion

op. .

^ -

P^i-^i (5.1)

o

subject to the equation of continuity

1

for the rheological equation of state

when the boundary conditions are specified and may be discontinuous with respect to time t.

In these equations, p.. is the stress tensor at the point x. referred to the rectangular cartesian co-ordinate system OX.. F. is the body force per unit volume at x. and f. is given by

ÖV ÖV.

where v. is the velocity of the particle at x.. The equation of state (3'5) is discussed in detail in Part I, where a definition of n, Jl, J2, S. ."^ and S. . will be found. The equation of continuity

1 J -J-J

is the mathematical expression of the assumption that the material is incompressible. The general problem is, of course, too difficult and it will be simplified by making the following assimiptions; (l) body forces do not exist (F. = O) and (2) inertial forces are negligible (f. = 0 ) . In addition, we shall consider only homogeneous defonnations (see Appendix l ) . For slow moving viscous liquids it is reasonable to ignore inertial forces and most physical experiments are designed to produce a defoimation as nearly homogeneous as possible.

The effect of equating F. and f. to zero in the equation of motion is to reduce it to the form

ÖP,

-ii

-ÖXj = 0 (3.5)

and it is shown in section (b) that, for homogeneous defoimation, the stress, p. .^ is not a function of position, but only of time; tlierefore, the equation of motion of the form (3-5) must be satisfied. Hence any homogeneous deformation, for which the equation of continuity is

satisfied, is a possible deformation, and the stress may be calculated directly from the equation of state. Similarly, any stress which is not a function of position gives rise to a homogeneous deformation which may be calculated directly from the equation of state, using as a

subsidiary condition the fact that the material is incompressible. b) A special form of the equation of state when the deformations

are homogeneous.

We now consider only those deformations, (homogeneous deformations) which convert any set of parallel planes into another set of parallel planes, and in which the origin of co-ordinates remains fixed. It can be shown, (Appendix l ) , that the most general defonnation of this

type can be expressed by,

A. . X. = A^ . x'. (3,6) where x. and x^ are, as in Part I, the rectangular cartesian

co-1 1 ' ' °

ordinates of a particle at times t and t' respectively and

A. . = A..(t) (3.7) A'. . = A. .(t') (3.8) ¥e denote the matrix whose element in the i row and j column

is A.. by A, and define A' in a similar manner. If x is the column matrix of x. and x' the column matrix of Xj_,then

A x = A' x' (3.9)

or

X = A'^ A' x' (3-10)

The matrix of

ÖX

7 is A ^ A' (3.11)

»^o

and therefore the matrix of

s.. = T-r r-f is

_{ij ox' dx' = = ^ =} A"^ A'(A''^ A') _{= ' = = = =}

=

A"^ A' A' A'^ _\y

(3-12)

_J

a a

Similarly, the matrix of

S7^ is A A'"i A'"^ A (3-15)

J.J — — — —

For homogeneous deformations the equation of state (5«5) can he expressed in matrix form where P is the matrix of p. ., I is the matrix of 6 . . by

P - pi = / 2 ( ^ A"i A' A' A~i - ||- A A'"i A'"^ A)dt' (3-14)

Noting that A is a function of t only, this has the alternative form,

P - pi = A"l I r 2 |§r- A' A' dfjl"! - A[ r2 1 ^ A'-l

A"^ dfAk

- 00 - 00

It must be remembered that, in general, f^ is a function of t and Jl, Jg are functions of t; therefore the integrands are not simply functions of t' unless °^ , o|^ are constants. We also note that the

oJi 0J2

condition J3 a 1 implies, for all t,

det A = constant (5«16) and this constant may be taken to be one.

Moreover, since A, Ji and J2 axe not functions of position, P is not a function of position. In fact J^, Ja and J3 are defined by"

Jl = S = trace S.. ^^ act; «p pa

J3 = det S.^ (5.17) and, if J'l Jg and J3 axe similarly defined invariants of S.T-'-, it can

be shown that (5.18) (5.19) (5.20)

Ji

Jl

J 3 s = =

Ja

J3 J3

1

J3

The incompressibility condition gives J3 and therefore Jl JI = s: = J3 = 1 Ja Jl

(5.21)

(5.22)

(5.25)

and hence, using (5.12) and (5.15)? for a homogeneous deformation we have

Jl = trace S. . = trace A " ^ A' A' A " ^ (5.2i^)

•^ ij = = = = = \-' /

c) stress relaxation

i) Consider a deformation defined by A where A = I t < o

A = diag [XÏ^, X p , Xs^) t > o 1 (3.26)

XJXPXS — 1

where Xi, Xa, X3 are constants independent of time. It can be seen, by examining the equation ( A 9 ) in this special case, that the physical meaning of this deformation is that the body is at rest until t = 0, when it is given an instantaneous extension of amount Xj_ along the Xj_ direction. If the stress is now measured as a function of time, this represents a stress relaxation experiment.

Therefore, A A = I t < o

1

_(5.27)

A A = diagfXi^, '^z', Xs^} t > o J

t > o f J l = xf + x i + X§ t > o f J l = 5 ,^ ^QN

"1 - o -p '1 k5.2oj

t ' < o i J2 = Xi^ + X2 + X3 t ' > o ' J2 = 5

Inserting these results in the equation of state (5.15) we obtain,

for t less than zero,

I = p ' l (3.29)

and for t greater than zero,

r ° -N ^ ° >

I - p I = diagtxf, xi, xf} J 2 § | ^ t ' - diagtxif X2? X^^} j 2 | § ^ t '

+ diag{x!, xi, xf} \ ^ l^iagCxI? Xsf x;^}dt'

"o

- diag(xlf X2? \T^ ƒ 2 | ^ i a g ( x f , x i , x D d t ' (3.50)

o

where we have used the fact that diagonal matrices commute; hence we obtain,for t > o,

and,

, o ^ o

P - p"I = diag{x!, xi, X§} / 2 fj^t'. diag{xlf xif X^^) / 2 ^ t '

(5.51)

p . . p H . j 2 | a ^ t ' - / S ^ t ' (5.52)

o "b

I f such a s t r e s s r e l a x a t i o n experiment i s c a r r i e d out, the q u a n t i t i e s

which can be measured p r a c t i c a l l y axe P n - P22> Paa - P33* sxii P n - P33.

5qr-dt' and / T T ^ t ' which;, of coiirse, depend on t .

— 00 " 0 0

Since, in this stress relaxation experiment, Ji and J2 can be varied o

independently by changing Xi, XQ, X3, a measurement of / §~dt' and

ƒ

il^*'

/ ^Yr~^'^' for various Ji and J2 will be sufficient to determine ü completely

- c o

as a function of Ji, J2 and T = ( t - f ) . Thus, if we consider the case where ü is given by (2.20) (Part II) and the initial deformation is a

simple elongation, that is,

_i Xi = X X2 = X3 = X 2 (5.55) we find, P22 - P33 = P21 = Pi3 = P32 = 0 '11 - ^22 o/^ 2 1\ 1^1 -"Kit , 1 Co -Kstj

I (5.54)

and we note that p n - paa falls to zero as t tends to infinity, as we would expect for a liquid. A measurement of P n - P22 a^s a function of t for various values of X will enable Ci, C2, Ki and K2 to be found.

The equation of state (5.5) may be useful in investigating the

properties of 'solid' substances. By a solid we mean that, in the stress relaxation experiment outlined above, the stress does not become isotropic as t tends to infinity, as it must if it is of the form given by (l.lO) and (2o22). The equation of state could be altered by suitably changing the form of Q in equation 2.22, as a function of t-t', in such a manner that the stress does not tend to zero as t tends to infinity. In this case, an experiment on homogeneous stress relaxation may throw some

light on the remarks by Cifferi and Flory (1961) on the connection between the C2 term (see equation (2.20)) and hysteresis.

il) Stress relaxation in simple shear Consider the deformation defined by

A A } "" = 1 \ 0 u < 0 : o

f)

t > 0

1

J

(5.55) X. = x'. If t < o t'< o or t > o t'> o

(3.56)

^i = xi - axz xé = X2 if t' < o t > o (5.57) X3 = X 3

By examining the physical meaning of equations (3.36) and (3-57) we see that the deformation is such that the body is at rest until t = 0, when it is given an instantaneous simple shear of amount a.

We find, for t > o, that /1+a^ a 0 X

s = ( a l o ) t ' < o )

V 0 0 1 / j

a = Ï t' > o ,

(3.38) 1 -Ci 0

S ^ = f -a 1+a^ 0 ) t' <

o \

0 0 1

S"i = I t' > o

J (3.59) (5.J+O) jj_ = J2 = 3 + a^ t' < o t > o 1 Jl = J2 = 3 t ' > o t > o - J

and the equation of s t a t e becomes, for t > o,

P o ^ ^

Q S + I

a O v p O ^ . l K x O

p - p"I = / 2 ^ ( a 1 0 )dt' - / 2 5 ^ ( -a 1+a^ 0 )dt'

,^00 ^ - ^ A 0 0 1 / - s i ^-^2 V 0 0 1

where Vy— and Yf" ^^^ evaluated with j-i_ = J2 = 3 + a^

Hence,

o

Pil - P33 = a^ ƒ 2 ^ dt' (5.42)

00 -O

P22 - P33 = -«^j 2 §§^ dt' (5.U5)

— 00

P^- = «

{Jj

§§r ^*'

* £

2 ^ at-] (5.V,)

Thus a simple shear stress relaxation experiment is sufficient to

find fl when Ji = J^- It is probably more convenient to carry out the experiment either by the torsion of a cylinder of the fluid between parallel plates or the torsion of the fluid between a cone and plate. Although these deformations are not homogeneous, they give essentially the same information as a simple shear experiment and it will be

convenient to quote the results here.

In the parallel plate experiment, the nonaal force Z"' per unit area on the plate is given by

— 00 T "

and the tangential force, T, per unit area on the plate is given by

- oo ^ "o •'

where Ji = J2 = 3 + — 2 ~ and oo is the angle the cylinder is twisted L

about its axis; L is the distance between the ends of the cylinder, which are held at a fixed distance apart, and r is the distance between a point on the end plate and the axis of rotation.

In the cone and plate experiment, the noimal force Z"" per unit area on the plate is given by

• • 0 0 — 00 " " 0 0

(5.i^7)

(D.

o

f P

o öa_ ^+, , r oM- ^4-/1

^ = r i j ^ife^*'-^j ^Sr^*'^ (5-«')

_./

where Ji = Jg = 3 + r?* CD is the angle the cone turns through about P o

its axis and p is the complement of the semivertical angle of the cone. Hence, in both these experiments, measurements of Z" and T as functions of radius will enable Q to be found as a function of J i ^ 2 a,nd t-t', when Jl = J2.

d) Simple shear flow

i) Instantaneous application of simple shear flow

Let us envisage a situation in which the fluid has been at rest until t = o, when a deformation corresponding to a simple shear flovr is suddenly applied. We proceed to investigate how the stress veiries with time.

It is easy to see that in this case we have, A = I t < o (3.49) = A = / 1 -tG

( ° ^

_{V 0 0} 0 ) t> 0 J 1 /•

where G is the shear rate. Obviously, for t < o, p.. is Just an isotropic pressure. For t > o we find

t' < o Jl = J2 = 3 + t^G^ 1 t' > o Jl = J2 :- 3 + (t-t' )^G^ J Inserting the value of A into (5-15) we find,

(5.50)

l+t^G^ tG 0 N po . .,0 / l + ( t - t ' ) ^ 2 ( t - t ' ) G 0 P - p I = ( tG 1 O W 2{f^) d t ' - / 2 ( 2 ^ ) ( ( t . t ' ) G 1 0 )dt'

0 0 1 yJ ^-^i t ' < o J ^-^i t'>o^ 0 0 ;

/ I -tG Ox rO r ° ^n / I - ( t - t ' ) G 0 - ( -tG l+t^G^ 0 ) / 2 ( 2 ^ ) d t ' + / 2 ( 2 ^ ) ( . ( t - t ' ) G l + ( t - t ' ) 2 G 2 0 V 0 0 1 yJ '"^^ t ' > o ^^ ""^^ t'>oV 0 0 1 idt' * ? . ^ ^ / l + ( t - t ' ) ^ ^ ( t - t ' ) G Ox pt 1 - ( t - t ' ) G 0 2(da_; ( ( t - t ' ) G 1 0 ) d t ' - / 2 ( S ^ ) ( - ( t - t ' ) G l + ( t - t ' ) ^ 2 0 , ^"^1 t'>oV 0 0 1 / ^ „ ^'^^ t'>6^ 0 0 1^' (5.51)

where (^^)^y^ is the value of ^ ^ when Ji = Jo = 5 + t ^ '

aJi t'<o oJi

•^

and (!§-)-(./>o is the value of

XT'

when Ji = J2 = 5 + (t-t'2)o

/2V2

^Öfl X /Öfi

^^ ^^h'<o'

^ÖJI^t'>o

^^^^

Similar meanings.

We recognise the last pair of Integrals as the stress that Is

obtained when a simple shear flow has been maintained for an infinite

time. The other integrals therefore represent the transient stress,

due to starting the flow at t=o, which decays to zero as t tends to

infinity.

This transient stress (P. .)m cannot be evaluated until the form

of n is known. Using the form (2.20) (Part II) we find,

(PII-P)T = - "KJi jlj" + 1 ^ e-"^!*

\

G^ (3.52)

(P33-P)rj, = 0

Once again, the most convenient way to examine these simple shear

transients is to carry out experiments in a cone and plate or parallel

plate viscometer. It can be shown that the transient force on the plate

of a cone and plate viscometer is given by,

^z ;^ - 4C2 p2 e I K3 +

K |

J

+

^f.f}{...e-3Yi,.|).^,e-^^(i,.|,)].

(5.55)

where cu is the angular velocity of the cone. Now the sign of (Z ')„ is

-Kat _ f 1 ^ t 1 -Kit ^ f 1 ^ t 1 /, c),\

C^\^•^^j-e

Ci-jjq +j^J- (5.5i^)

It can be shown that this function of t can change sign as t goes from 0 to CO. We observe that (Z")^ becomes zero as t tends to infinity; we would expect this, since the steady state force distribution is produced at large times. Tlie fact that (Z"")„ may change sign as t

increases meems that the force on the plate may go thnrough a maximum,

which is higher than the steady state value of the force when the transient has died away. This effect has been observed by Lodge and Adams during experiments on polymethylmethacrylate dissolved In dimethyl phthalate.

Considering now the shear stress in the case of simple shear, using the form of ü given by (2.20) (Part II), the shear stress is given by,

Pi2 = j ^ G J l - e ^. + ^ G J i . e J- (5-55)

This shows that the shear stress grows monotonically from 0 at t=o to the steady state value of

2Ci X 2C2

,Ki - K2 - (5.56)

at large times.

If we define the viscosity ri(t) = TT^f we see that the viscosity Increases monotonically with time, becoming constant after a very long time. The variation of viscosity with time is usually called thixotropy; an increase of viscosity with time we call negative thixotropy, and a fall of viscosity with time we call positive thixotropy. It is more usual to observe positive thixotropy. We can express the time behaviour of Pi2 in equation (5.54) 'by saying that if fi talies the form given by (2.20) (Part II), then the fluid shows negative thixotropy. Of course, whether the variation of Pi2 with time could be observed experimentally would depend on the relative values of Ci, C2, Ki and K2.

It can be shown (see Appendix II) that if fi takes the form given by (2.6) (Part II) then it is not possible for the liquid to show positive thixotropy if S£o and S61 are non-negative for all T .

However, this is not true for all forms of n, as we can see by an example.

Let

ü

= M e-"(^^-*' ){Ji - 3) - I e--(^-^')(Jl - 5)^ (5.57)

M, N , m, n > o

Evaluating the shear stress, using equation (5.51)» we find

MG f, -mt,

6 N G ^

r-, -nt, , 5NG^

r^.

-nt ^2 2 -nt,

P12 = ^ (1 - e } -

- ^

{1 - e } + ^ ^ ^ [2tn e + t^n^e )

(5.58)

The steady state viscosity Is given by,

„ / i i £ U M . % ; (3.59)

t—> 00

1 ^

]•

m^ n

We note that the steady state viscosity falls with G, as is usually

found in practice. Obviously we must have

M 6NGf ^ /^

^r^\

-5 - — T — > O (5.60)

m"^ n* \^ /

or the form (5.57) can only be valid for sufficiently small G. Now it

can be shown that, if m > n and condition (5.60) is obeyed, then Pi2>

given by (5.57)* does not tend monotonically to the steady state value

(5.59) ^ut first passes it and then slowly falls to the value given by

(5.59). Moreover, if m » n the initial rise may only take a very

short time and experimentally we would only observe the positive thixotropy.

Figs. 1 and 2 demonstrate this effect.

11) Instantaneous stopping of simple shear flow

In this case, A is defined by,

/I -tG 0 X

A = ( 0 1 0 ) - c » < t < o 1 (5.61)

"^ Vo 0 1 / j

A = I t > o

Obviously for t < o we get the stress for steady flow. For t > o

we get,

rO ^ / l + t ' ^ 2 - t ' G Ox r ° ;^n / ^ t ' G 0 X P - p ' l = / 2 # - ( - t ' G 1 0 ) d t ' - / 2 ^ ( t ' G l + G ^ t ' S 0 ) d t '

= ^ = J ÖJ1V. 0 0 1 / J ^J^V 0 0 1 /

, — 00 — 00

(5.62)

I

where ^ j ~ and

'rj-

are evaluated with

Jl = J2 = 5 + G^t'^ (5.65)

This is obviously a transient which starts at t=o as the steady state

stress, and decays to an isotropic stress as t tends to infinity.

Evaluating this for the simple form of

Q,

in equation (2.20) (Paxt II)

we find,

, _^ fl ^ 2G^ 1 -Kit ^„ fl 1 -K2t

Pii-P' = 2Ci-{j^ + j ^ j e -2C2Jj^J-e

/ on fi-l -Kit on fl ^ 2G^ -Kpt

P22P' = 2Ci..jje .2C2{j^ + j^Je

-^ -^/ _ pr fl-l-^-Kit _ M -^ - K 2 t

P33-P - 2Ci-^^^je -

2C2|K2J-^

(5.64)

n f ^ C i - K i t , 2Cp -K2t1 P 3 1 = P 3 2 = 0

\Je

first note that the transient obtained when the simple shear is

removed is different from that obtained when it is applied. Lodge (1956)

has shown that in the case of the removal of simple shear flow, the stress

decays in such a manner that the normal stress, P11-P22* falls to zero

less rapidly than does the shear stress, Pi2* when

Q

= Sio (t-f).

This result is not true for all forms of fi in the general equation

of state.

e) Elongational flow

In Part II we have examined elongational flow (Trouton flow) in the

steady state and we now consider the transients associated with starting

and stopping this type of flow.

i) Instantaneous application of elongational flow

In this case we have

A = I t < o „, -^ = = a t a t I ^ . , - a t 2 2 , ^ ( A = d i a g ( e , e , e } t > o ^ F o r t > o we have

(5.65)

, , ^ T 2 a t , _ - a t _ - 2 a t , ^ a t /_ /-/-% t ' < o J l = e + 2e J 2 = e + 2e ( 5 . 6 6 )

(3.67)

Inserting this value of A into (5.15) we find,

„ _ J. f 2at -at -at,f

P

„/öü

P - pl = diag{e , e , e

^ I ^i^:^^^—

yf f

2(g-) dt'

^^L

^"^^t'<o

-r2(|§-) diag(e-2-^',e-^', e-^')dt'|

- i

"^ "•

t'>o J

-dla^(e-2-Se-Se-*}|r2(^) dt'

\J^

^•^^t'<o

- ƒ °2(|^) ^ diag(e2^*', e-^^', e"^^' )dt' ]

t'>o

J

t

t'>o

. L . , 2at -at -at,

P r.rèn \

•,. / -2at' at' at'., ,^,

+ ^diag(e , e , e } /

2{—-)

diag(e , e , e )dt'

' . 1

^^"^

t'>o

- diag(e•2^^ e ^ ^ e-^) / % ( | ^ ) ^ ^ ^diag(e2-^', e ^ * ' , e^^')dt' j

(5.68)

where (^^T") i^ the value of ^ ^ when Ji and J2 are given by (5.66)

ÖJ1 ^/^-o ÖJ1

(-^^r—) is the value of r ~ when Ji and J2 axe given by (5.67)

OJI ^;>Q (3Jl

and

{^-)

and (|j-) are similarly defined.

t'<o ^ t'>o

The last two integrals in (5.68) give the stress due to the steady

state flow already obtained in equation (2.I9) (Part II). For the

transient flow, assuming the form of

ü

in equation (2.20) (Part II), we

obtain P22 = P33 P12 = P23 = P31 = 0

/ V __ -Kit f 2at/l

_ 1 _ \

-at,l 1 v 1

(Pil - P 2 2 ) T = 2 C I e je (- - ^ ^ - e (_ - j ^ ) j

-K2t f -2at,l _ 1 _ \ at,l 1 \ 1 /, -co\

- 2C2 e -{e (j^ -

^ ; ^ ^ )

- e (j^ -

^ J

j- (5.69)

and we note that at t = o, this reduces to minus the stress for steady

Trouton flow, as it should.

il) Instantaneous stopping of elongational flow

We now consider the situation when a steady elongational flow is

suddenly stopped. In this case we have

/ + / . ^- ^ -at ^2* 7 2 S

- o o < t < o A = d i a g ( e , e ' , e ' ) -^

(5.70)

t > o A = I J

Obviously for t < o we have the steady state stress; and for t > o we

have

4-/^ T -2at'^ ^ at' _ 2at'^ ^ -at'

,^

„^

t'<o Jl = e + 2e J2 = e + 2e (3.71)

t'>o Jl = J2 = 5 (5.72)

By inserting the value of A in equation (5.15), we find, for t>o,

o

T. X r o cin =. I - 2 a t ' a t ' a t ' v , , ,

P - p l = / 2 vT" <lisë ( s ) ^ > e ) d t '

= = J o J i — 00 P o ^fi =• I 2 a t ' - a t ' - a t ' v , , , ,_ __v

" / ^Wk^^^^ '^ '^ ^ (5.73)

— 00

This gives, in the simple case when

Q

is of the form (2.20) (Part II)

Pii - P22 - 2Ci e -|Ki-2a Ki+aJ ^^^ e iK2+2a " K2-a J

P22 - P33 = P23 = P31 = P12 = 0 (3.74)

Again we note the difference between the transients in the ' switch

on' and'switch off' cases.

(f) Shear stress

We have so far considered the application of transient deformations.

We now consider the application of impulsive stresses and investigate the

deformations that arise from them.

i) Instantaneous application of a shear stress

We have seen that the stress in the case of uniform steady simple

shear is given by equation (2.3) (Part II). We investigate the

deformation when the stress P is isotropic for all t<o, and equal to the

simple steady state shear stress for all t>o. That is,

P = pi when t < o (5*75) and P is given by equation (2.5) (Part II) when t > o.

Since for^an isotropic stress there is no deformation, for t < o we get

A = J

For t > o , A ( t ) i s t o be d e t e r m i n e d , and i t must s a t i s f y t h e e q u a t i o n ~ o t

P - pi =

A"^

I r 2 ^ I dt' + r 2 ^ A' X'dt' I T^

- 00 o O

4-- A | r 2 | | ^ I dt' + f 2^1'''4-- A'4--i dt' I A (5.76)

- 00 ' o

This equation for A is subject to the condition

detA = 1 (5.77) A need not change continuously with t. In fact,,Lt A = I need not be

equal to.Lt A. We Investigate the instantaneous^Sange In A when the stress is applied. Mathematically, we look for a solution of (5.60) for A (t) when t -» Oj.

Let A = ,Lt A (t) (5.78) =0 t-* o^ = ^ ' ^-^ ' and t h e r e f o r e o o P - p i = A - i r M 2 ^ d t ' - 2 A / 2 ^ d t ' ( 5 . 7 9 ) ^= =0 =0 J è J i =0 = 0 j c)J2 ^-^ '^^ — 00 — 00 s u b j e c t t o detA = 1 /G%2+N -M G(Ni+Mi) 0 X

and P = ( G(Ni+fii) No-G%2-Mo 0 ) ( 5 . 8 o )

V 0 0 NO-MQ-'

J

We now consider the special case when n is given by (2.6), in this case ,Lt ^ 2 ^ dt' = r ^ ^ . , - ^ <it' = N' t - o . J d J i o

]- (5.81)

— 00

t^f^h^""' = ^ó

t= 1 — 00 ( s e e e q u a t i o n s (2.^+) and ( 2 . 5 ) ) .

Hence we required to find a solution of

p' - pi = A"^ A "^ N' - A A M' (3.82)

= -^s =0 so o =0 =0 o

\y I

subject to condition (3.77)

/<}%é+N6-M6 G(Ni+Mi) 0 X

where P' = ( G(Ni+Mi) No-G^e-^^o 0 ) (3.85)

~ V 0 0 N^-M^

.-We consider two special cases; when n = Sio(Ji-5)(Case l ) , and when

fi = S6i(J2-5)(Case 11). The more general case is discussed in Appendix

III.

Case_I

A is not uniquely determined by equation (5-82). It is instructive

to find a solution in the form

/P 1 0 X

A ^ = ( 0 r 0 ) (3.84)

^-0 0 s /

It will be found that it is possible to satisfy (3.82) when

f2 = S£o(Ji -5) with this form of A . This solution can be interpreted

physically by examining the meaning of A"-"-. A particle which was at a

position a' Just before time zero moves to a Just after time zero where

A a = a' (5.85)

or

ai = pai + qa2

a2 = ra| |- (5-86)

a3 = sa3

j

or

aix a2 =

aV

^1

= 0

Vo

Vr

1 0

°\

0 \ y

f^

(°

^ • 0 0 r 0

0 )

s /

^a . ^ Va

(5.87)

This deformation has the physical meaning that a unit cube of

material, whose edges are parallel to the axes, is given two successive

defoliations at t=o; firstly, an Instantaneous deformation so that it

becomes a rectangular parallelopiped with edges of length p, r, s, parallel

to the co-ordinate axes, and secondly, a simple shear whose angle is

tan"^ J (See Fig. III).

We find, using (5-84), (5-85) and (5-82) with

ü

= Sio(Ji-5) and

condition (5.77) that

s = r (5.89)

p = Vr^ (5.90)

^ = ^ (5.91)

"o

where r is a root of

It is shown in Appendix IV that this cubic In r^ has only one

positive root and that this positive root is greater or less than one

according to whether N2NÓ - Ni^ is negative or positive. Lodge (1958)

has shown that when

Q,

takes the form

n = S[o(Ji-5) Sio(t-t') > o all t-t' > o (5.95)

then

NéN^ - Nl^ > o (5-9^)

In this case, r < 1 and therefore p > 1; thus the instant the stress

is applied there is an elongation in the Xi direction with equal

instant-aneous contractions in the X2 and X3 directions, followed by an Instantinstant-aneous

simple shear.

Case_2

When

Ü

takes the form Sèi(j2-5)^ let us suppose

/a d Ox

AQ = (o b o ) (5.95)

Vo o c /

where a, b, c, and d are to be determined. We find from (5.82) and (3.77)

that

a = c b = l/a^ ad = - ï ^ (5.96)

where a is a root of

Again we see that this has only one positive root. Whether this root is greater or less than one is determined by the sign of M^ M Q - Mi^. If we suppose this is positive, then a < 1. It may be shown, in the same manner as Lodge (1958), that Me M^ - Mi > 0 when Q takes the form,

n = sói(J2-5) sèi(t-t')>o aii(t-t')>o (5.98)

In this particular case we have,

0 < a < l 0 < c < l b > l ( 5 . 9 9 )

By considering the meaning of A , we see that the Instant stress is applied there is an elongation in the Xi direction, an elongation in the X3 direction, a contraction in the X2 direction, and an instantaneous simple shear.

It seems plausible, therefore, that when fi is given by (2.6), in addition to the simple sheeir there will be an elongation in the Xi direction and a contraction in the X2 direction, with the behaviour in the X3 direction being dependent on the relative magnitudes of the M'l

and N [ .

li) Instantaneous removal of shear stress

Let us suppose that the stress (2.5) needed to maintain a simple shear flow has been applied for all time in the past and is suddenly removed at time zero.

I n t h i s c a s e ,

/ I -Gt Ox

A = ( 0 1 0 ) t < o ( 5 . 1 0 0 ) \ o 0 1 y

since the shear stress has been applied for an Infinite time and there-fore the steady state deformation must have been established.

On removal of the steady state shear stress we must be left with an isotropic pressure.

E = p' I t > o (5.101) A(t) for t > o is unknown and to be determined. For t > o, A is

determined by,

t *

I(p'-p) = A-ij

f 2 ^ A'

A' dt'jp- |[ r 2 1 ^ A'-i A'"^ dt'JA

and

det A = 1 (5-105)

Again we only consider the behaviour at the instant the stress Is

removed for the special case when

Q,

= Sio(Ji-5) + S6i(J2-5)

Let. A^ = t^^A

If t-o in equation (5.102) we get,

^N^+NéG^ NiG 0

(5.10U)

(pi - Po'i =• £"

- I o (

with det A = 1 .

= 0 ( NiG N6 V 0 0 / M^ -MiG -MiG M^+M^^ ^ 0 0 0 ^

N^y

A?

= 0 ^o

Let us consider the special cases when ft = Sio(Ji-5) (case l) and

n » Sèi(J2-5) (case 2 ) .

Case_l

Lodge (1958) has i n v e s t i g a t e d the p a r t i c u l a r case when,

Ü = S i o ( J i - 5)

Let us suppose t h a t ,

r

a d o X

o b o ) (5.105)

= 0

"^o o c

where a, b, c and d are to be determined. We find from equations (5.101»-)

and (5-105) that

r r 2 1 V 3

a = jl + (NPS - N i ^ ) ^ !

If we examine the meaning of A , we find that when the shearing forces are removed, the liquid suffers an instantaneous deformation which may be considered to be composed of two successive deformations such that a unit cube of material whose edges are parallel to the co-ordinate axes first changes to a rectangular parallelopiped whose edges axe parallel to the co-ordinate axes and whose lengths are

" 1 1 1 J

—f —, —, this parallelopiped being then sheared by an angle - tan"-^ —.

€1 D C Q,

Since N ^ ^ - N i ^ o, we have

a > l b = c < l (5.107) and we find that there is an instantaneous contraction in the direction

of flow and Instantaneous expansions in the two perpendicular directions followed by an instantaneous simple shear recovery of angle tan"-^ EL.

a Case_2

When a i s of the form S6i(J2 - 5)> l e t ,

/P q. o \

A 7 = 0 r o ) (5.108)

=0 0 0 s

We then find,

r = j l + 1 ^ (MéM^ - Mi^) j V ^

^ " V.

p = s = j l + ^ (MéM^ - Mi^) j ^ (3.109)

M'G r f"^ 1

q = _ 3 ^ J 1 + ^ ( M ^ ' - Mi2)

M' 1 M'^ ' ^ ° "• ' j

when,

r > l o < p = s < l (3.110)

The meaning of A shows that, as well as an instantaneous simple

shear, there are equal contractions in the Xi and X3 directions, and an expansion in the X2 direction immediately the stress is removed.

It has been pointed out by Lodge (1960) that when

the instantaneous solution is true for all time. That is, when the stress is removed, there is an instantaneous deformation, after which the fluid does not move. It can be shown that a similar state of affairs exists for a more general equation of state defined by

fl = Ci e-^i^(Ji - 5) + C2 e-^2t(j^ . 5) (5_13_2) if Ki = Ka.

The above analysis has a certain practical significance. Consider a liquid which has been pumped through a long pipe and which suddenly emerges from the end of the pipe. While it was in the pipe it was being sheared, the walls exerting shearing force on the liquid. When the liquid emerges from the end of the pipe the shearing force is

suddenly removed, and the situation considered by the above analysis is realized. Of course the comparison is only descriptive, since the

geometries axe different and the defonuation of the liquid flowing through the tube is certainly Inhomogeneous. However, it is found in practice that an expansion usually takes place as the liquid emerges from the pipe. This expansion is to be expected, in a qualitative manner, if the liquid was governed by an equation of state for which fl = Sio(Ji-5). g) Stress whose principal directions remain constant in time

We choose the co-ordinate axes to be along the principal directions; the stress P is given by diag. (Pi(t), P2(t), P3(t)).

i) Suddenly applied stress

We consider what happens when such a stress is suddenly applied to the fluid; that is, we have,

P = pi t < o and hence A = I all t < o (5.115) P = diag (Pl, Pa, P3) t > o

and A(t) is to be determined for t > o. For t > o, the equation which determined A is

d i a ^ P i ( t ) , P2(t), P3(t)^ - pi = A"i| ƒ 2 ^ I dt'

— 00

+ f^^A' A' d t ' j p _ | | r 2|§^l dt' + r 2 ^ A ' " l A'"l dt'

(5.1li^)

\A

J

-Consider first the instantaneous deformation. If t-o , A-A ,

+' = =o'

p.(t)-ï'.(0) and fl is of the form (2.6), we obtain

diag |Pi(0), P2(0), P3(0)j - pi =

£"•

A ; I N ^ -

l^ A^

M ; (5-115)

where det A^ = 1 (5-116)

and N', M' are given by equations (2.7) and (2.8). Obviously A is

diagonal, say

AQ =

^^^'(^> ^>

^ ) (5.117)

and we obtain

P.(o) = x f N^ - M ; X ° " ^ 1 = 1, 2, 5 (5.118)

0 0 0 , .

with Xi X2 X3 = 1 (5-119)

X. is the instantaneous extension on applying the stress; if we

consider the particular case of simple elongation, that is, P(0) =

dlag(P, 0, 0 ) , we find that

X? = X, X i = ^ X g = y ^ (5.120)

o 'o

where X is the positive root of

P = (xj _ i-)(N^ + i-Mi) (5.121)

For t > o we consider a special case.

P = diag (P, 0, 0) P is constant for all t")

(^

12?")

P = p'l t < o i

and fl is given by the special form (2.20).

Inserting these values in equation (5.1ll<-) and noting that

A = diag

(x~Ht),

X^(t),

xht))

(3.125)

P = fX2 . h 2Cj^-Kit ^ (l^ ,-,) 2C2 .Kgt

P = ^x

' ^) ^

- ^x^ - ^^ icT^

where O O (5.121^) X = x(t) X' = x(t') (5.125) This equation is to be solved for X(t). We note that as t-o ,

(5.12!»-) tends to the equation (5.I21) where N' and M' axe evaluated

using (2.20) and X-ï». . Now equation (5.12li-) has an approximate solution for large values of t. We have already solved the problem of steady state elongation flow. Part II, and obviously the solution of (5.124) must tend to this solution, for after a long time, the fluid will have

' forgotten' the effect of starting the flow at time zero. Mathematically we expect that there will be an asymptotic solution to equation (5.12lj-)

of the form, at X = Be (5.126) where a is given by ^ = (Ki+a)(K2-2a) "^ (K2+2SfK2-a) (5-127) and B is a constant.

That this is an asymptotic solution can be verified analytically. Inserting (5.126) in equation (5.121»-) we find,

X -(Ki-2a)t ^-(Ki+a)tx x^-(K2+2a)t ^-(K2-a)tx

^"^^ (^Ki-2a Ki+a

J ' ^^^ \

K2+2a

K^ J

32 32at _ i ^ - a t ^ ^-Kit _ A ^-2at _ ^ ^at^g2 e-K2t

(5.128) at This cannot be satisfied by any constant value of B, so X = Be is not an exact solution. However, we can choose B so that (5.126) is a good fit for large values of t. For large values of t,

and if (K2-a) is greater than(Ki-2a) say,

^-(K2-a)t^ ^-(Ki-2a)t ^^^^^^^ and hence we get the best fit for large t by talcing,

i.e., equation (5.121»-) has an asymptotic solution of the form

li) Instantaneous removal of stress

We have considered the case of the sudden application of stress whose principal directions remain constant in time. We now consider what happens when such a stress is suddenly removed. We have

P = diag(Pi(t), P2(t), P3(t)) t < o \ (3.135)

P = p'l t > o J

The value of A for t < o must be found by solving

dlag(Pi, P2, P3) - Pl = A " /

[ 2 ^ A'

A'dt'Vi - / r 2 1 ^ A'"i A'"^dt'

^ - 0 0 '^ - 00 (5.15i^) For t > o, we have f t

(p-p' )I = A ' / ƒ 2 1 ^ A' A' dt'^ A"i _ / r

2^1''^

A'-i dt'^ A

- CO -• 00

(3.135)

Now A must be diagonal, say

A = diag (Ai(t), A2(t), A3(t))

and for the Instantaneous deformation we have, letting t-o ,

and

l(p-P') = d i a / • ^ ^ D S ' ; ^ V l a g ( n i , Uzi ^3) - dia^J^^ ^02^ A§2jdiag(mi, mg, m^)

(5.157) subject to

det A =» 1 where

diag(ni, ns, n3) = Lt r°diag(Ai^, Aaf A32) 2

_{ti^ J -x«ev^l- > ^ 2 , ^3 ; ^ § j ^}

^

dt' (5.158)

— CO

and

diag(mi, m 2 , mj) = ^ ƒ diag(Ar^, Aa"^, A3~^)2 ^ dt' (5-159)

- CO

It must not be forgotten that n. and m^^ are not only functions of the flow history up to time zero, but also functions of Ai, A2, A3.

If the Instantaneous change is elongation Xj^ along the x. axis, then

A° = irr (5.li^0)

To find X. we must solve

p-p' = x f n^ - x?"^ m^ 1 = 1, 2, 5 (not summed) (5.llH) subject to

X? xi X§ = 1 (5.lij-2) If we consider the particular case when P = diag(P, 0, O ) , where P

is constant for all time, we have simple elongational flow with the stress suddenly removed. Using the form of fl given by equation (2.20) we

obtain

C P - P O I

= diae(xï- x r x§-)

^^^^,.

l i s , f f e )

- dla«(xï- x i - xS-)diag (if|£|;, | 5 s , g ? j ) (3 -143)

o o o / 1 1 \

diag(Xi, X2, X3) = diag f X Q , y^" y^^" ) (5-1^) and we find an equation for X ,

The solution of this quartic gives X , the Instantaneous extension along the axis Xi.

Again it can be shown that, if Ki = K2 in the particular form of fl in equation (2.20), then A is a solution for all time.

h) Constrained flow

We have considered so far those problems in which either the deformation is given for all time and the stress is to be determined, or those in which the stress is defined and the deformation is to be determined. However, it is possible that a situation might arise in which only some of the components of the stress tensor are defined, and the deformation is restricted by constraints. Such a situation is commonly investigated in some recovery experiments. For example, if a liquid is sheared in a concentric cylinder viscometer by rotating the

inner cylinder, and the torque is suddenly removed from the inner cylinder, we axe effectively defining the shear stress for all time. Since the distance between the cylinders is fixed, the liquid is constrained in such a manner that only a simple shear recovery is possible. Of course, the problem is not restricted to recovery; the transients associated with the sudden application of a constant sheax stress can be similarly

Investigated.

We now investigate this situation mathematically. 1) Application of a shear stress to a constrained system

Let us suppose we apply a shear stress Pi2^ which is given by, P12 = 0 t < o

(5.146)

P12 = Cf t > o

to a system which is constrained in such a manner that only a simple shear deformation is possible. Physically we may think of the liquid as being constrained between two infinite parallel plates whose distance apart is fixed. We must have

A = I t < o (5.147) since there is no force on the liquid for all t < o.

and

where F(t

/

A = (

;) is to

^ 1

0

~ 0

be

-F(t)

1

0

V

determined,

t > 0 (5.1^)

since the only allowed deformation is

a simple shear. Moreover, we wish to find p n ,

^ssf

P33 as functions

of t. This problem has been considered by Lodge (1958.B) for the case

where fl = Sio(Ji-5). We shall follow his method. Evaluating

S^i

and

S^T^ from (5.12) and (5.15) and using (5-1^), we find for t > 0.

^1+F^ F 0 X r^ Nn / 1 + ( F - F ' ) 2 F - F ' 0

P-pI = / 2{m-.) ( F 1 0 )dt' + / 2m-) ( F-F' 1 0 )dt'

^ - ^00 "^"^^ t'<oV 0 0 1 / 4, ^ ^ t'>oV 0

0

1

/

r° >n A

~F 0 \ r^ >n / I -(F-F') 0 X

— 00 ^

where

F = F(t)

F' - F(t')

(5.IU9)

(3.150)

t > o t' < o Jl = J2 = 5 + F^ (5-151)

t > o t' > o Jl = J2 = 5 + (P-F')^ (5.152)

(KT~) is evaluated using equation (5-151)

^^^

t'<o

'

' -

"

^ is evaluated using equation (5.152)

(s~^^

and ('rrr-) are evaluated in a similar manner.

0J2 t'<o 0J2 t'>o

/Pll P12 0 X

P = (Pi2 P22 0 ) (5.155)

\ 0 0 P33 /

where Pi2 is known and equal to

a

and p n - P33 and P22 - P33 are to be

determined. We consider the special case when fl is given by (2.6), i.e.,

From (5.149) we find

a

= ƒ 2F(t)|sio(t-t') + S6i(t-t') jdt' + r2|'F(t) - F(t')|{'sio(t-t')+S6i(t.t')|dt'

- 00 o

(5.155) This is an integral equation for F(t). We note that t-o. F(t) -•F(o), where

a = F(o) I 2{Sio(o-t') + Sèi(o-t'))dt' (5-156)

— 00

or from (2.7) and (2.8),

a = F(o){N^ + M y (3-157)

^(°) = N ^ W ^ ° (5.158)

o o

Physically there is an Instantaneous shear of amount F ( O ) the instant the shear stress is applied. Following Lodge (1958 B ) we find that (3.155) has an asymptotic solution for large t of the form

^(^) - ^(°)

INTTMI

• ^ ^ 2(Ni ^ Mi)^ =- } (5.159)

dPf t')

and we note t h a t the shear r a t e -r-^—'-, for l a r g e t i s given .by

M i l _ w . ) ! ^ ° " ^^! (5.160)

dt - ^^°^ (Ni + Mi)

or, using (5.158),

f = N r ^ = ^ - ^ (5-161)

This of course is the steady value we would expect from (2.5). From (5.149) we find

o t

P11-P33 = ƒ 2 Sio(t-t') F2(t)dt' + ƒ 2 Sio(t-t'){F(t) - F(t')}2dt' (5.I62)

- 00 o

and hence

G2Ni(Ni + Mi)2 ^ • /, .^, \

P11-P33 =

^^^^-^ ^"^

as t -* o (3.164)

P11-P33 = G^N4 as t-»oo (5.165)

a s a s a s a s

t

t t t

-*°+

-* 00

^ ° +

- * 00 G%6{Ni + Mi}2 ^ , , ,^^v P33-P22 = 2i_J^ _ i i _ a s t - > 0, ( 5 - 1 6 6 ) P33-P22 = G ^ è a s t -* 00 ( 5 - I 6 7 )

These results axe plotted in Fig. IV. It will be noted that 7 1 ,

the extrapolated value of the asymptotic form of F(t) at t = o, i.e.

^^

2(Ni + Mi) U'-LDO;

is probably more easily measured experimentally than F ( o ) .

2 ) Removal of shear stress from a constrained system

Consider a system in which a constant shear stress

a

has been applied

for a very long time and is suddenly removed at time t = o. The system

is constrained so that only a simple shear recovery is possible. That

is, we must have

P12 = a t < o (5^^g^) P12 = o t > o

and therefore, since the constant stress has been maintained for a very long time, we have

A -Gt 0 x^^ A = ( 0 1 0 ) t < o (5.170)

Vo 0 1 /

A -r-(t) Ox^

A = (0 1 0 ) t > o (5.171)

VO 0 1 /

,

7^n

/1+(F;:-Gt)^ F"'-Gt' O

P - pi = / 2(^7-) ( F"'-Gt' 1 O )dt'

"^•^l t'<o V O O ;

• 2{§-) f

(F---F--') 1 0)dt'

-I

^'^i t'>o V O O 1 /

_o

. / 2(2-^) ( -(F---Gt' ) 1+(F---Gt' )2 O )dt'

-'

^'^^

t'<o V o O 1 /

-(F^-F';!'' ) O

~ f2{^)

(^-(F''"-F""') 1+(F""-F"'''')2 o V * ' (5-172)

^o

^"^^

t'>o V 0 0 1 /

where F" = F"'(t) F'"' = F'''(t')

(5-175)

t > o t' < o Jl = J2 = (F'"' - Gt')2 + 5

t > o t' > o Jl = J2 = {F'-'' - F'"''}^ + 5

and

(TTT-)

, etc., have their usual meanings.

^ 1 t'<o

For t > o we have

/Pil 0 O X

P = O P22 O ) (5.17^^)

VO O P33 /

where P n , P22^ P33 are to be determined. Again, considering the special

case when fl takes the form (2.6), we find that F"{t) is determined by

0 = ƒ 2(F"-Gt')|sio(t-tO + Sèi(t-t')jdt' + ƒ 2(F"-F''^')|sio(t-t')+Sèi(t-t')|dt'

- 00 o

(5.175)

and it can be shown that

^ ^ ° ^ " (N6 + Mè) - N^ + M^ ^5.17b;

and, the total recovery, 71", is

^ M - A ^ - ^ ^

(3.177)

From (3.172) we see that P n - P33 and P22 - P33 are determined by

o t Pll - P33 = ƒ 2 Sio(t-t')|F"'''(t) - Gt'j^dt' + ƒ 2 Sio(t-t'>j'F''(t) - F''-(t')Wt' -* 00 o

(3.178)

= r 2 Sii(t-t'>|F"(t) - Gt'l^dt' + r 2 S6i(t-t'>[F-''(t)-F'''"(t')Vdt'

P33 - P22 - 00

(3-179)

The instantaneous change in P n - P33, when the stress is removed, is found by taking Lt t-» o

tLt_^(pli - P33) = G^N4 - ^ ^ ^ ^ — I ^ J N è N i + 2M6Ni - MiNsj (5.I80)

tLt_^ (P33 - P22) = G ^ é - ^ ^ f V ï l ^ l ^ i ^ ó + 2Ny4i - M^Ni j (5.181) We note that the Instantaneous change in the normal force is not

necessarily the same for the sudden removal of shear stress as for the sudden application of shear stress. Lodge, using fl = Sio(t-t')(Ji-5), found these instantaneous changes to be the same.

i) Discussion

It was pointed out in CoA Note 154 that equations of state may be divided into two classesj those which have as their basis microrheology

(that is a description of the macroscopic rheological properties of a material in terms of nicroscopiic elements which may even be molecular) and those which are. purely phenomenological. It is possible to subdivide both these types into two categories; the first we shall call the strain history type and the second the point derivative type. In the point derivative type of equation of state the stress at the current time t at a given particle is a polynomial function of the matrices representing the rate of strain tensor, 2nd rate of strain tensor, etc., where the rate of strain and its derivatives axe evaluated at the current time and at the given particle. A typical example of this type of equation of state is the Stokesian fluid defined by

p. . = P6. . + U A ( ^ ) + V A ( ^ ) A, (.1) (5.182)

where

A(^^= ^ ^ Ï : 1 (5.185)

and V. is the velocity of the partiele at x. and time t.

The equations of state discussed in this paper and that of Lodge (1956) are of the strain history type. That is, the stress is a function of all the strains between any time t'ln the past and the current time t. It should be possible to relate these two types of equations of state.

This Note discusses transient behaviour In liquids whose flow properties are governed by an equation of state (5.5) which is of the

strain history type. In order to simplify the mathematics we have - *' considered only homogeneous deformations, and it has been necessary to

assume that we may ignore inertial effects. The most useful deformation to study is, of course, shear flow, since in general this is the deformation most often found in practice.

Two major differences between Newtonian and non-Newtonian fluids are that, even in the absence of Inertial forces, non-Newtonian fluids usually show elastic effects and stress relaxation effects. In this Note it has been shown that, by considering the variation of stress on changing the

deformation, the equation of state (5-5) predicts stress relaxation effects and, by considering the change in deformation on rapidly altering the

stress, the equation of state also predicts elastic effects.

Moreover, by considering special systems, it is shown that the elastic effect may be divided into two parts; an instantaneous deformation followed by a time-dependent deformation. These deformations may be compared with experiment. Pollett (1958) has in fact measured the total elastic recovery in polyvinyl chloride melts when a shear flow is suddenly stopped. The elastic properties of the fluids obeying the point derivative type of equation of state are discussed by Ericksen (i960). While he is able to give a mathematical Justification for calling the fluid described by such an equation of state elastic, it is difficult to interpret his results in terms of the deformation one would obtain if the stress produced by a given deformation were suddenly removed. The strain history equations of state are, in this sense, more useful in describing elastic effects.

Since, in a physical system, inertia is always present, in practice it will never be possible to produce an Instantaneous elastic recovery, although the recovery may be so rapid initially that it may be possible experimentally to Identify the first recovery with the instantaneous

recovery of the theory. Although in designing a physical experiment some effort may be made to keep the deformation homogeneous, in general the deformation will be inhomogeneous. The problem of predicting the defonn-ation on the sudden change of the boundary conditions becomes very difficult and in fact a deformation which is a continuous function of position may not exist - see Lodge (1958) for example.

However, if experiments could be interpreted in terms of the theory, and this is possible with well-designed experiments, stress relaxation and elastic recovery experiments would be invaluable in defining the form of fl in equation (5-5)- An examination of equations (2.5), (2.4) and (2.5) of Part II shows that even if all the stresses needed to maintain a state of simple shear flow (or an equivalent shear flow) are measured for all shear rates, then these measurements are not sufficient

to define fl completely. However, it has been pointed out in this Note that stress relaxation experiments do give sufficient information.

Most experiments in this field have investigated the steady state behaviour of non-Newtonian fluids. Experiments on stress relaxation and elastic recovery are needed. A further Note will deal with the interpretation of the existing experimental results in terms of this equation of state. References

1. Clferri, A. and Flory, P.J.

J. Appl. Phys., Vol. 30, 1959^ P- 1498. 2. Ericksen, J.L. The behaviour of certain vlsco-elastic

materials in laminar shearing motions. In Bergen, E., Ed. Viscoelastlcity -phenomenological aspects.

Academic Press, I96O. 5. Lodge, A.S.

4. Lodge, A.S.

5. Lodge, A.S. 6. Lodge, A.S.

Trans. Faraday Soc. Vol. 52, 1956, p. 120. A theory of elastic recovery in concentrated solutions of elastomers.

In Mason, P. and Wookey, N., Eds. Rheology of elastomers.

Pergamon Press, 1958.

Rheologlca Acta. Band 1, I958 B, p. I58. Colloque International de Rheologie.

Paris, i960.

257-A^ENDIX_I

To prove that a homogeneous deformation can be represented by A x = A' x^

L_i£ J iL_i

Consider a transformation which transforms the point x. into X., the

1 i'

origin being fixed. The most general transformation which converts planes into planes is

X, = aiiXi 4- aigXp + ai3X3 _ a2iXi + a2sXg + a23X3 ^^o ^ biXi + b2X2 + bsXs + e ^ ~ biXi + b2X2 + b3X3 + e

(Al) Consider a set of planes defined by

PiXi + P2X2 + P3X3 = Q (A2) where Pj^ is fixed and Q varies. Substituting, we get

Xi (aiiPi + a2iP2 + a3iP3 - Qb^) + Xa (ai2Pi + a22P2 + a32P3 - Qb2)

+ X3 (ai3Pi + a23P2 + a33P3 - Qb3) = Qe ( A 5 ) If the transformed set of planes is to remain a set of parallel

planes as Q varies, we must have the coefficients of X. constant. That is,

bi = b2 = b3 = 0

The transformation has become

-1 = ^^ij ^J ^^ij =

-f-

(A^)

Now we consider a transformation in which a particle at x? at times t is continuously deformed into the point x. at t. Hence, "^

x° = ^^ij(t^.t)x. ( A 5 ) and if x( is the position at t' ,

x° = .ij(t,,t')xj , (A6)

whence,

n..(t ,t')x'. = Li..(t ,t)x. (A7) ^^ij^ o' '' ,1 ^^i.r o' -* .1 ^"'''

Now, in the fluids we consider, there is no preferred configuration

so |j...(t,t ) is not a function of any special t , so that

^^ij(*)^J = ^^ij(*')-J (^)

and in the original notation,

A.^ X. = A^j xj (A9)

APPENDIX II

The__thixotroplc properties of'"tlie-liquid with an equation of state in

i^IiïIOI§i2Si:52Iï3aï!2i:ÏI

The shear stress as a function of time, when a simple shear flow is

started at t = o in a liquid which has been at rest for all t < o, is

given by equation (5-51).

Using fl = Sio(Ji-5) + Sèi(J2-5)* where Sio(t-t') and S^i(t-t') are

positive for all (t-t') > o, we find

(Pi2)^ = ƒ t'G Sio(t-t')dt' + ƒ t'G Séi(t-t')dt' (AIO)

— 00 — 00

and obviously Pi2 is monotonie in t if (pi2)rp is monotonie in t.

But

(Pi2)^ = - G ƒ (T-t){Sio(T) + Sèi(T))dT when T = t-t' (All)

t

00

i(|iilT=

G J

{Sio(T)4-Sé>l(T))dT (A12)

and, since Sio ^^^ ^oi ^^® positive for all T , the right hand side is

positive for all t. Hence (pi2)rn^ and therefore Pi2 are monotonie and

any liquid obeying the equation of state where fl is given by (2.6) can

show only negative thixotropy.

APPENDIX^III

On the solution of

P' - pl =

A " ^ X ' ^ N '

-

A A M '

s -^s =0 =0 o =0 =o o

flow is suddenly applied, may be solved (at least numerically) in the following manner.

Let

X = A A (A15) = =0 =o ^"^^/ then

l^

= è?

èo''

(^^^

and hence P' = pi + N' X"^ - M ' X (A15)

but if X is diagonal so is P'. Hence if the principal values of P

are T I , ~ T 2 > T3 and those of~X are Xi, X2, X3, we have ~

T. = P + N^ X. - 3^ 1 = 1, 2, 5 (AI6) subject to det X = 1 / i7">

or Xi X2 X3 = 1

where

L P' L = diag(Ti, T 2 , T3) C A I 8 )

L X L = diag(xi, X2, X3) (AI9) L being the orthogonal transformation which diagonalized P' . Since

p'is known L'and T . can be found. From equation (AI6) and ( A 1 7 ) ^ X.

can be determined. Equation (AI9) enables X to be found. APPENDIX_IV On the roots of pS + J,-

,4 /NèG^ 1

.

jNfG^ 1 _

Let r^ = X (A 20) ^ = a (A21) o

^ff

= P > o (A22)

o

Then If

f(x) = x^ + ox^ - px - 1 (A25)

We now see that

f(o) = - 1

f(oo) = + 00

and hence there are either one or three real positive roots.

Now

f'(x) =

3^^ + 2cix - ^

(A24)

The turning points are given by

f'(x) = o (A25)

or

_5x

_{= - a ±}

_Va^

₊

_5P^

_(A26)

and hence the turning values are given by one negative and one positive

value of X. The positive value corresponds to a minimum of f(x) and

there is therefore only one real positive root of

f(x) = o (A27)

Moreover,

f(l) = a -p

= ^ | N é N 6 - N i ^ G ^ (A28)

If N2NÓ - N i ^ o then the positive root of f(x) = o lies between 0 and 1

Si«oa* State

FIG. m DIAGRAM O F T H E INSTANTANEOUS CHANGE IN

D E F O R M A T I O N ON SUDDENLY REMOVING (OR A P P L Y I N G ) THE STRESS N E E D E D T O MAINTAIN S I M P L E SHEAR F L O W . Sheer rei« F I G . II G R A P H O F STEADY S T A T E VISCOSITY AS A F U N C T I O N O F SHEAR R A T E F O R A LIQUID FOR WHICH aa o t Sn. 2 ^ - Me-""^- ( J l - 3)Ne-°^ ^

CNi+MP' f> -- f» G'N;CN!-I-M'.3^ DEFORMATION CN>lvCD SHEAR STRESS ff' - GCN,tM,a G'Ni ï,« GCN^M^3 2 C N > M ! 3 y , . jf, T I M E

-F I G . IV. CONSTRAINED SHEAR -F L O W . BEHAVIOUR O -F T H E D E F O R M A T I O N AND S T R E S S FOR THE SUDDEN A P P L I C A T I O N AND SUDDEN R E M O V A L O F A CONSTANT SHEAR S T R E S S