- Email: [email protected]
Dynamics of Polymeric Liquids: Vol. I. Fluid Mechanics
269
Journal of Non-Newtonian Fluid Mechanics, 4 (1978) 269-275 @ Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
Short Communication ON CO-ROTATIONAL EQUATIONS
AND OTHER RATE TYPE CONSTITUTIVE
R.R. HUILGOL School of Mathematical
Sciences,
Flinders
University,
Bedford
Park, S.A. 5042 (Australia)
(Received March 12,1978)
Introduction Recently, Bird and his coworkers [1,2] have commented at length on the usefulness of the co-rotational rheological model, developed by Goddard [ 3,4]. In reading Goddard’s paper [4], two points seemed worth exploring and this short note is a result of this examination. The first point concerns the frame indifference or objectivity or the invariance property of the stress functional introduced by Goddard. It is more restrictive than that which arises in the simple fluid theory [ 51 and I have removed this limitation here. Secondly, by the rate-type nature is meant that just as Walters [6] showed that the Oldroyd theories of 1950 [7] were related to a functional of dC,(r)/dr, where Ct(r) is the relative strain history (< r < t), it is possible to show the relationship between Goddard’s theory and the derivative of Ct(r). We also show the way in which the generalized Maxwell model used by Ho and Denn [ 81 is related to the derivative of C,(r)-l. All calculations are assumed to be done in a fixed Cartesian coordinate system in what follows. Derivatives of the strain history From No11 [9], we know that dF(r)/dr = L(T)F(T),
dFij(r)/dr = Lik(r)Fw(r),
(1)
where F(r) is the deformation gradient computed with respect ot a fixed reference configuration, d/dr is the material derivative and L(r) is the gradient of the velocity z, at time T: &k(T)
=
aui(x, 7) ax
k
270
The relative deformation gradient Ft(r) is given by [lo]
Ft(7)= F(T)F(t)-1,
(3)
and the relative strain Ct(r) by G(r) = K(r)=&(r), (4) where the superscript T stands for the transpose. From eqn. (1) and eqn. (3), - ., it follows that fl,(r)/dr
= I(r)FA
and from
F F1 = 2,we get
[Mt(r)/dr]
F&)-l
(5) + Ft(T)
d(&(r)-‘)/dr
= 0.
(6)
Hence d(Ft(r)-‘)/dr
= -Ft(r)-‘[dFt(r)/dr]
F,(T)-1
= -F,(T)-l L(T).
(7)
Using eqn. (4) and eqn. (5), one derives the following [ll,
p. 1971:
dCt(r)/dr = Ft(r)TA&)Ft(r),
(3)
and from eqns. (4) and (7): d(C,(T)+)/dT
=
-F,(T)-~A~(T)[F~(T)~]-~,
where A, is the first Rivlin-Ericksen
(9)
[ 121 tensor:
A,=L+L=.
(19)
It is trivial to verify that dFJr)/dt
=
-Ft(7)L(t).
(11)
Hence, we get (12)
=L(t) [&(7)-l;l+[&-t(T)-l)l L(t)=-
-$[-&(T)-l)] The tensor
ZYZ~(T)
The symmetric part of
W=L+LT,
(13)
L isD, i.e.,
(14) where the superscript T denotes transposition. The skew-symmetric part of
271
L is S2, i.e., 2ti = L - LT.
(15)
Given a(r), one can construct the mean-rotation que solution to the differential equation [ 31:
dQt(WdT= f%~)l)t(~) Qt(t) = 1.
tensor [4] Qt(r) as the uni-
I
(16)
Now, suppose we define Q(T) as an orthogonal tensor satisfying the equation:
= a(~) Q(T),
dQ(WT Q(0)
I
= 1.
(17)
Then
Q(T) = 1+ f
n(o) Q(o) do,
(18)
0
or, equivalently,
MT) = Q(t) + j Q(u) Q(o) do .
(19)
t
It is quite easy to see that the Qt(7) in eqn. (16) can be constructed as follows:
PAT) = Q(T)QWT-
(20)
This construction is very helpful in discussing the time derivatives of Qt(r) and H,(r), which is defined by eqn. (2.18) of Goddard [4] as: Z-&(r) = Qt(7)T D(7) Qt(T),
--oo < T d t .
(21)
From eqns. (20) and (21), we can derive a number of results, such as
dQt(T)ldt = -QtWT n(t), W (T)ldt = a(t) H,(T) -f&(T)
(22) a(t),
(23)
and
_cg” f&(T) (07*
= Qt(T)T[$$D(T)]
Qt(T),
where the co-rotational operator ANT
=
d(-)/dr +
(.)52(T)
-
Q/CDT
n=12 , 9 -a*,
(24)
is given by:
n(T)(-),
for any second order tensor, and d/dr denotes the material derivative.
(25)
272
The objectivity of k&(r) Consider an objective motion defined by [9]: x*(t) = A(t) x(t) + c(t).
(26)
Here A = A(t) is an orthogonal tensor and c = c(t) is a vector. Then, the tensors D* and 52* in the new frame are given by D* = ADAT,
(27)
S2*=AGtAT +kiAT.
(23)
Now, the tensor QT (7) corresponding to Qt(r) obeys:
dQT (7W
(29)
= a*(~) QI (7) I
Q?(t) = 1. I assert that
Q?(T) = 47) Qt(7) AWT.
(30)
For dQP(r)/dr = A(r) ADA
Qt(7) A(t)T +
+ A(T) a(7) AWT A(T)
Qt(7) AWT.
= a*(~) Q?(T)-
(31)
Therefore, on using eqns. (21), (27) and (30),
HP (7) = QT (T)~D*(T)QT (7)
= A(t) Q~(T)~A(T)~A(T) D(T) AA(TlT A(T) Qt(7) AWT
= A(t) f&(T) AWT, or the tensor
Ht(7)
(32)
is ObjeCkW.
The co-rotational stress functional It is a postulate in continuum mechanics that the stress tensor is objective and that the constitutive operator is frame indifferent [ 51. Hence, a constitutive functional of the form: TE(t)
=
s
(f&(T))
must satisfy:
A(t)
k (%T)) A(tlT =-:
-
(A(t) fh(~) AWT),
(34)
273
for all orthogonal tensors A(t). The restriction in eqn. (34) is broader than that given by Goddard [4] in his eqn. (2.21), and follows from eqn. (32). While our derivation is perhaps more direct, it can also be shown by the use of the various equations in [4] that eqn. (34) is correct. Connection with rate type theories To illustrate the connection between functionals of dC,(T)/dT, d(C,(T))-l/d7 and Ht(7) and rate type constitutive relations, we shall use specific single integral models only. Consider the consitutive relation for the extra stress tensor of an incompressible fluid: t
TE = -
s (Q/M
ewG_(t
-
Md
[dCt(Wd71d7.
(35)
Using eqn. (12), we obtain dTx/dt = -(r/&,)A1
- (l/X,)T,
- LT TE - TEL,
(36)
and thus eqn. (35) is equivalent to TE + XO(dTE/dt + TJ, + LTTE) = -+4,
,
(37)
i.e., (38)
TE + A,, 6T,/Gt = -q,,Al,
when 6 /ht is the co-deformational derivative [2] of Oldroyd [7]. This connection between eqn. (35) and eqn. (38) was first made in Walters [6] as far as I am aware. If we now consider TE = -
i (%l&) exp]lt
- 7)&l
[d(Ct(7)-1)/dTldT,
(39j *
we get from eqn. (13): TE + Xe[dTx/dt -L T, - TE LT] = -0Al,
(46)
which is the equation used by Ho and Denn [8].
* The negative sign means that on integration by parts, one recovers a Lodge model [ 131. I wish to thank Professor R.I. Tanner for this suggestion. Moreover, with a negative sign, eqn. (35) leads to a finitely linear viscoelastic fluid [lo].
214
Finally, if we use t
Tx=-2 .
s (n&A
expk--V - r)/hol K(r)dr,
(41)
.
we get the co-rotational model [ 1,2]: TE + X0CoTE/‘ot= -rjo AI,
(42)
where we have used eqns. (23) and (25). We close by pointing out that in order to calculate HJT) as defined in eqn. (21), one needs Qt(r) and D(r). In a general motion, we can find Qt(r) by the integration of eqn. (16). This leads to an iterative procedure, described in [3] or by Hochstadt [14, pp. 76-791. This procedure can be quite cumbersome and no simple method exists for finding Qt(r). The present paper shows that in one sense Goddard’s rate theory is a “second-order” theory because it involves the product of dCt(r)/dr and u,(T)-? m,(T)
= Qua
k(T)
u,(T)-‘k%(T)/dTl
&(T)-l%(T)TQt(T)
>
(43)
where I have employed eqn. (8), eqn. (21) and the polar decomposition of F*(r): Ft(r) = k(r)
I&(r).
(44)
In eqn. (44), Rt(7) is the relative rotation tensor and U*(T) is positive definite and symmetric. Remark: It is worth remembering that Qt(7) + Rt(7). Indeed if we write F(T) = R(T) U(T) and I;(t) = R(t) U(t), then &(r) f R(r) R(t)T
(45)
in general. Interestingly enough, on the other hand, dR,(r)/drl,,
(46)
t = n(t)
and in an objective motion, R?(T) = A(T) Rt(7) A(t)T.
(47)
From what has just been stated, one could use products of QJT), Rt(7), Ut(7), and the nth derivatives with respect to T of Ct(7) and its inverse, as desired, to form consitutive equations. For instance, models could be developed by using R,(T)~ A,(T) R,(T), II = 1,2, .... or sums of such products. References 1 R.B. Bid, 0. Hassager and S.I. Abdel-Khalik, AIChE J., 20 (1974) 2 R.B. Bird, Ann. Rev. Fluid Mech., 8 (1976) 13. 3 J.D. Goddard and C. Miller, Rheol. Acta, 5 (1966) 177.
1041.
275 4 5 6 7 8 9 10 11
J.D. Goddard, Trans. Sot. Rheol., 11 (1967) 381. W. Noll, Arch. Ratl. Mech. Anal., 2 (1958) 197. K. Walters, ZAMP., 21 (1970) 592. J.G. Oldroyd, Proc. Roy. Sot. London, A200 (1950) 523. T.C. Ho and M.M. Denn, J. Non-Newtonian Fluid Mech., 3 (1977/78) 179. W. Noll, J. Ratl. Mech. Anal., 4 (1955) 3. C. Truesdell and W. NOR Encycl. Phys. III/S, Springer-Verlag, New York, 1965. R.R. Huilgol, Continuum Mechanics of Viscoelastic Liquids, Hindustan Pub, Corp., Delhi and I-IaIsted Press, New York, 1975. 12 R.S. Rivlin and J.L. Ericksen, J. Ratl. Mech. Anal., 4 (1955) 323. 13 A.S. Lodge, Elastic Liquids, Academic Press, New York, 1964. 14 H. Hochstadt, Differential Equations, Dover, New York, 1975.
Note added in proof After this note was submitted, Professors R.C. Armstrong, R.B. Bird and 0. Hassager informed me that there is a considerable overlap between the results derived here and in their book: Dynamics of Polymeric Liquids, Vol. 1, Wiley & Sons, New York, 1977. The methods of derivation are different, however.
Journal of Non-Newtonian Fluid Mechanics, 4 (1978) 269-275 @ Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
Short Communication ON CO-ROTATIONAL EQUATIONS
AND OTHER RATE TYPE CONSTITUTIVE
R.R. HUILGOL School of Mathematical
Sciences,
Flinders
University,
Bedford
Park, S.A. 5042 (Australia)
(Received March 12,1978)
Introduction Recently, Bird and his coworkers [1,2] have commented at length on the usefulness of the co-rotational rheological model, developed by Goddard [ 3,4]. In reading Goddard’s paper [4], two points seemed worth exploring and this short note is a result of this examination. The first point concerns the frame indifference or objectivity or the invariance property of the stress functional introduced by Goddard. It is more restrictive than that which arises in the simple fluid theory [ 51 and I have removed this limitation here. Secondly, by the rate-type nature is meant that just as Walters [6] showed that the Oldroyd theories of 1950 [7] were related to a functional of dC,(r)/dr, where Ct(r) is the relative strain history (< r < t), it is possible to show the relationship between Goddard’s theory and the derivative of Ct(r). We also show the way in which the generalized Maxwell model used by Ho and Denn [ 81 is related to the derivative of C,(r)-l. All calculations are assumed to be done in a fixed Cartesian coordinate system in what follows. Derivatives of the strain history From No11 [9], we know that dF(r)/dr = L(T)F(T),
dFij(r)/dr = Lik(r)Fw(r),
(1)
where F(r) is the deformation gradient computed with respect ot a fixed reference configuration, d/dr is the material derivative and L(r) is the gradient of the velocity z, at time T: &k(T)
=
aui(x, 7) ax
k
270
The relative deformation gradient Ft(r) is given by [lo]
Ft(7)= F(T)F(t)-1,
(3)
and the relative strain Ct(r) by G(r) = K(r)=&(r), (4) where the superscript T stands for the transpose. From eqn. (1) and eqn. (3), - ., it follows that fl,(r)/dr
= I(r)FA
and from
F F1 = 2,we get
[Mt(r)/dr]
F&)-l
(5) + Ft(T)
d(&(r)-‘)/dr
= 0.
(6)
Hence d(Ft(r)-‘)/dr
= -Ft(r)-‘[dFt(r)/dr]
F,(T)-1
= -F,(T)-l L(T).
(7)
Using eqn. (4) and eqn. (5), one derives the following [ll,
p. 1971:
dCt(r)/dr = Ft(r)TA&)Ft(r),
(3)
and from eqns. (4) and (7): d(C,(T)+)/dT
=
-F,(T)-~A~(T)[F~(T)~]-~,
where A, is the first Rivlin-Ericksen
(9)
[ 121 tensor:
A,=L+L=.
(19)
It is trivial to verify that dFJr)/dt
=
-Ft(7)L(t).
(11)
Hence, we get (12)
=L(t) [&(7)-l;l+[&-t(T)-l)l L(t)=-
-$[-&(T)-l)] The tensor
ZYZ~(T)
The symmetric part of
W=L+LT,
(13)
L isD, i.e.,
(14) where the superscript T denotes transposition. The skew-symmetric part of
271
L is S2, i.e., 2ti = L - LT.
(15)
Given a(r), one can construct the mean-rotation que solution to the differential equation [ 31:
dQt(WdT= f%~)l)t(~) Qt(t) = 1.
tensor [4] Qt(r) as the uni-
I
(16)
Now, suppose we define Q(T) as an orthogonal tensor satisfying the equation:
= a(~) Q(T),
dQ(WT Q(0)
I
= 1.
(17)
Then
Q(T) = 1+ f
n(o) Q(o) do,
(18)
0
or, equivalently,
MT) = Q(t) + j Q(u) Q(o) do .
(19)
t
It is quite easy to see that the Qt(7) in eqn. (16) can be constructed as follows:
PAT) = Q(T)QWT-
(20)
This construction is very helpful in discussing the time derivatives of Qt(r) and H,(r), which is defined by eqn. (2.18) of Goddard [4] as: Z-&(r) = Qt(7)T D(7) Qt(T),
--oo < T d t .
(21)
From eqns. (20) and (21), we can derive a number of results, such as
dQt(T)ldt = -QtWT n(t), W (T)ldt = a(t) H,(T) -f&(T)
(22) a(t),
(23)
and
_cg” f&(T) (07*
= Qt(T)T[$$D(T)]
Qt(T),
where the co-rotational operator ANT
=
d(-)/dr +
(.)52(T)
-
Q/CDT
n=12 , 9 -a*,
(24)
is given by:
n(T)(-),
for any second order tensor, and d/dr denotes the material derivative.
(25)
272
The objectivity of k&(r) Consider an objective motion defined by [9]: x*(t) = A(t) x(t) + c(t).
(26)
Here A = A(t) is an orthogonal tensor and c = c(t) is a vector. Then, the tensors D* and 52* in the new frame are given by D* = ADAT,
(27)
S2*=AGtAT +kiAT.
(23)
Now, the tensor QT (7) corresponding to Qt(r) obeys:
dQT (7W
(29)
= a*(~) QI (7) I
Q?(t) = 1. I assert that
Q?(T) = 47) Qt(7) AWT.
(30)
For dQP(r)/dr = A(r) ADA
Qt(7) A(t)T +
+ A(T) a(7) AWT A(T)
Qt(7) AWT.
= a*(~) Q?(T)-
(31)
Therefore, on using eqns. (21), (27) and (30),
HP (7) = QT (T)~D*(T)QT (7)
= A(t) Q~(T)~A(T)~A(T) D(T) AA(TlT A(T) Qt(7) AWT
= A(t) f&(T) AWT, or the tensor
Ht(7)
(32)
is ObjeCkW.
The co-rotational stress functional It is a postulate in continuum mechanics that the stress tensor is objective and that the constitutive operator is frame indifferent [ 51. Hence, a constitutive functional of the form: TE(t)
=
s
(f&(T))
must satisfy:
A(t)
k (%T)) A(tlT =-:
-
(A(t) fh(~) AWT),
(34)
273
for all orthogonal tensors A(t). The restriction in eqn. (34) is broader than that given by Goddard [4] in his eqn. (2.21), and follows from eqn. (32). While our derivation is perhaps more direct, it can also be shown by the use of the various equations in [4] that eqn. (34) is correct. Connection with rate type theories To illustrate the connection between functionals of dC,(T)/dT, d(C,(T))-l/d7 and Ht(7) and rate type constitutive relations, we shall use specific single integral models only. Consider the consitutive relation for the extra stress tensor of an incompressible fluid: t
TE = -
s (Q/M
ewG_(t
-
Md
[dCt(Wd71d7.
(35)
Using eqn. (12), we obtain dTx/dt = -(r/&,)A1
- (l/X,)T,
- LT TE - TEL,
(36)
and thus eqn. (35) is equivalent to TE + XO(dTE/dt + TJ, + LTTE) = -+4,
,
(37)
i.e., (38)
TE + A,, 6T,/Gt = -q,,Al,
when 6 /ht is the co-deformational derivative [2] of Oldroyd [7]. This connection between eqn. (35) and eqn. (38) was first made in Walters [6] as far as I am aware. If we now consider TE = -
i (%l&) exp]lt
- 7)&l
[d(Ct(7)-1)/dTldT,
(39j *
we get from eqn. (13): TE + Xe[dTx/dt -L T, - TE LT] = -0Al,
(46)
which is the equation used by Ho and Denn [8].
* The negative sign means that on integration by parts, one recovers a Lodge model [ 131. I wish to thank Professor R.I. Tanner for this suggestion. Moreover, with a negative sign, eqn. (35) leads to a finitely linear viscoelastic fluid [lo].
214
Finally, if we use t
Tx=-2 .
s (n&A
expk--V - r)/hol K(r)dr,
(41)
.
we get the co-rotational model [ 1,2]: TE + X0CoTE/‘ot= -rjo AI,
(42)
where we have used eqns. (23) and (25). We close by pointing out that in order to calculate HJT) as defined in eqn. (21), one needs Qt(r) and D(r). In a general motion, we can find Qt(r) by the integration of eqn. (16). This leads to an iterative procedure, described in [3] or by Hochstadt [14, pp. 76-791. This procedure can be quite cumbersome and no simple method exists for finding Qt(r). The present paper shows that in one sense Goddard’s rate theory is a “second-order” theory because it involves the product of dCt(r)/dr and u,(T)-? m,(T)
= Qua
k(T)
u,(T)-‘k%(T)/dTl
&(T)-l%(T)TQt(T)
>
(43)
where I have employed eqn. (8), eqn. (21) and the polar decomposition of F*(r): Ft(r) = k(r)
I&(r).
(44)
In eqn. (44), Rt(7) is the relative rotation tensor and U*(T) is positive definite and symmetric. Remark: It is worth remembering that Qt(7) + Rt(7). Indeed if we write F(T) = R(T) U(T) and I;(t) = R(t) U(t), then &(r) f R(r) R(t)T
(45)
in general. Interestingly enough, on the other hand, dR,(r)/drl,,
(46)
t = n(t)
and in an objective motion, R?(T) = A(T) Rt(7) A(t)T.
(47)
From what has just been stated, one could use products of QJT), Rt(7), Ut(7), and the nth derivatives with respect to T of Ct(7) and its inverse, as desired, to form consitutive equations. For instance, models could be developed by using R,(T)~ A,(T) R,(T), II = 1,2, .... or sums of such products. References 1 R.B. Bid, 0. Hassager and S.I. Abdel-Khalik, AIChE J., 20 (1974) 2 R.B. Bird, Ann. Rev. Fluid Mech., 8 (1976) 13. 3 J.D. Goddard and C. Miller, Rheol. Acta, 5 (1966) 177.
1041.
275 4 5 6 7 8 9 10 11
J.D. Goddard, Trans. Sot. Rheol., 11 (1967) 381. W. Noll, Arch. Ratl. Mech. Anal., 2 (1958) 197. K. Walters, ZAMP., 21 (1970) 592. J.G. Oldroyd, Proc. Roy. Sot. London, A200 (1950) 523. T.C. Ho and M.M. Denn, J. Non-Newtonian Fluid Mech., 3 (1977/78) 179. W. Noll, J. Ratl. Mech. Anal., 4 (1955) 3. C. Truesdell and W. NOR Encycl. Phys. III/S, Springer-Verlag, New York, 1965. R.R. Huilgol, Continuum Mechanics of Viscoelastic Liquids, Hindustan Pub, Corp., Delhi and I-IaIsted Press, New York, 1975. 12 R.S. Rivlin and J.L. Ericksen, J. Ratl. Mech. Anal., 4 (1955) 323. 13 A.S. Lodge, Elastic Liquids, Academic Press, New York, 1964. 14 H. Hochstadt, Differential Equations, Dover, New York, 1975.
Note added in proof After this note was submitted, Professors R.C. Armstrong, R.B. Bird and 0. Hassager informed me that there is a considerable overlap between the results derived here and in their book: Dynamics of Polymeric Liquids, Vol. 1, Wiley & Sons, New York, 1977. The methods of derivation are different, however.