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A note on kinematical tensors
293
Journal of Non-Newtonian Fluid Mechanics, 3 (197711978) 293-296 @ Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
Short Communication A NOTE ON KINEMATICAL TENSORS
K. WALTERS Department (Received
and W.M. WATERHOUSE
of Applied September
Mathematics,
University
College of Wales, Aberystwyth
(U.K.)
1,1977)
1. Introduction In the development of non-Newtonian fluid mechanics, there has been a certain degree of choice concerning the kinematical tensors which have been used in the construction of suitable rheological equations of state. In this note, we develop a convenient closed-form relationship between the two most popular sets of such tensors. We use rectangular Cartesian coordinates throughout and introduce the two deformation measures Gij and F” given by
and
(2) where xii is the position at time t’ of the fluid element that is instantaneously at xi at time t. It is usual to express these tensors in the form G
. =
c
fj
[--ft
0
-
k!
t’)]kA(k)ti
7
t--l)k+l(t - Ok
ji’“=z 0
(31
Bt;),
k!
where Ack) and B(k) satisfy the recurrence relations (5)
294
ui being the velocity vector, and A!?) (7) u = &,, B&, = -6,. Although the existence and importance of the Atk) and B(k) kinematical tensors is implied in the pioneering work of Oldroyd [ 11, by popular usage the Atk) are known as Rivlin-Ericksen tensors and the B(k) as White-Metzner tensors. In the next section, we shall prove a theorem, (suggested by a well known result in the theory of symmetric functions [2]) which relates the Ack) and B(k) tensors. That such a relationship must exist is self evident. Any merit in the present work is associated with the convenient closed-form relationship which is obtained. We shall find it convenient to use matrix notation and use will be made of the result (in the obvious notation) FG=I,
(8)
where Z is the unit matrix. 2. Thebrem
fie AtWand ~~~~ kinematical tensors are related by the following “matrix determinant”
B(k) =
(-l)k+lk!
A(1)
Z
0 . . . . . . . . . . . . .
0
$4’2’
A(l)
Z . . . . . . . . . . . . .
0
$4’3
;,A’2’
A(l)
Z .. . . .
0 . . . . .
. . . . . . . .
&A(k)
Z
0
A(1)
Z
. . . . . . . . . . . . . . . . . . . . . . . . . .
A(11
This result is also valid if A and B are interchanged, provided the (-l)k+l factor is omitted and I is replaced by -Z in the determinant.
Proof If L is:the square matrix
(9)
295
where D, E, F, G, H, J, K, M, N are themselves square matrices, we denote the “matrix determinant” of L by
L=
D
E
F
G K
H M
J,
(11)
N
which is to be evaluated by the standard methods of determinant expansion. We further require that multiplication from left to right is preserved, i.e. the expansion is carried out down columns of the matrix only. Consider now the two series T = Z + sdl) + . . . ~“a(“) + . . . ,
(12)
T’ = Z + sb(,) + ... s”b(,) + ... ,
(13)
where a@) and btkj are square matrices, s is a small parameter and T, T are mutual inverses so that 7T =z.
(14)
Equating coefficients of powers of s gives (for k > 1) b(k) + &7#_-1) + PV7#_2) + . .. a(k-lVQl) + Jk) = 0,
(15)
from which it is easily deduced that 41)
= -l&‘l
(16)
and
42) =
a(l)
z
a(2)
a(l)
I
I
(17)
*
We now proceed by induction and suppose a(l) 0 .................... Z a(2)
a(l)
Z
0 .......... ..
. b (?I)= t--l)n
: #rl) p)
0 0 . . . . .
(p-2)
(+l'
Z
@-1)
g(2)
a(1)
................
From (15) (with k = n + 1) and (US), we have
296 a(l)
b
(n+l)
=
(-q”+la(l)
Z
. . (p-1)
Z
a(l)
(-l)“a’2’
0
. a(n)
+
0 . . . . .
&)
. . . . . . . . .
0 . . . . .
0
I
. .
-. 1 Ll(n-l)
+ . . . 9 (-_1)2a(n)l&
atn-2)
+ (-1)
. . . . . . . . . a(l) I
@+l),
(19)
i.e.
bc,+l) = (-,)“+I
a(1)
Z
0 ............. 0
aw
aw
Z
0 .
. .
a(n+l)
.
a(n)
........ ..
a’l’
Z
a(2)
aw
(20)
We have thus shown that if (18) is valid for n = m, it is also valid for n = m + 1. But we have shown that it is valid for n = 1 and II = 2 from (16) and (17). Hence by induction our supposition (18) is correct. If we now put T = G and T’ = F we see from (3), (4), (12) and (13) (with s = t - t’) that a(n) - (--lY lq”“’ n! ’
(21)
(22) Substituting (21) and (22) into (18) we obtain (9) and the proof of the theorem is complete. Acknowledgments We have benefitted from helpful discussions with Professor J. Heading, Professor J.G. Oldroyd, Mr. J. Astin and Mr. R.W. Williams. References 1 J.G. Oldroyd, Proc. Roy. Sot., A200 (1950) 523. 2 D.E. Littlewood, A University Algebra, Heinemann, London, 1950, p. 81.
Journal of Non-Newtonian Fluid Mechanics, 3 (197711978) 293-296 @ Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands
Short Communication A NOTE ON KINEMATICAL TENSORS
K. WALTERS Department (Received
and W.M. WATERHOUSE
of Applied September
Mathematics,
University
College of Wales, Aberystwyth
(U.K.)
1,1977)
1. Introduction In the development of non-Newtonian fluid mechanics, there has been a certain degree of choice concerning the kinematical tensors which have been used in the construction of suitable rheological equations of state. In this note, we develop a convenient closed-form relationship between the two most popular sets of such tensors. We use rectangular Cartesian coordinates throughout and introduce the two deformation measures Gij and F” given by
and
(2) where xii is the position at time t’ of the fluid element that is instantaneously at xi at time t. It is usual to express these tensors in the form G
. =
c
fj
[--ft
0
-
k!
t’)]kA(k)ti
7
t--l)k+l(t - Ok
ji’“=z 0
(31
Bt;),
k!
where Ack) and B(k) satisfy the recurrence relations (5)
294
ui being the velocity vector, and A!?) (7) u = &,, B&, = -6,. Although the existence and importance of the Atk) and B(k) kinematical tensors is implied in the pioneering work of Oldroyd [ 11, by popular usage the Atk) are known as Rivlin-Ericksen tensors and the B(k) as White-Metzner tensors. In the next section, we shall prove a theorem, (suggested by a well known result in the theory of symmetric functions [2]) which relates the Ack) and B(k) tensors. That such a relationship must exist is self evident. Any merit in the present work is associated with the convenient closed-form relationship which is obtained. We shall find it convenient to use matrix notation and use will be made of the result (in the obvious notation) FG=I,
(8)
where Z is the unit matrix. 2. Thebrem
fie AtWand ~~~~ kinematical tensors are related by the following “matrix determinant”
B(k) =
(-l)k+lk!
A(1)
Z
0 . . . . . . . . . . . . .
0
$4’2’
A(l)
Z . . . . . . . . . . . . .
0
$4’3
;,A’2’
A(l)
Z .. . . .
0 . . . . .
. . . . . . . .
&A(k)
Z
0
A(1)
Z
. . . . . . . . . . . . . . . . . . . . . . . . . .
A(11
This result is also valid if A and B are interchanged, provided the (-l)k+l factor is omitted and I is replaced by -Z in the determinant.
Proof If L is:the square matrix
(9)
295
where D, E, F, G, H, J, K, M, N are themselves square matrices, we denote the “matrix determinant” of L by
L=
D
E
F
G K
H M
J,
(11)
N
which is to be evaluated by the standard methods of determinant expansion. We further require that multiplication from left to right is preserved, i.e. the expansion is carried out down columns of the matrix only. Consider now the two series T = Z + sdl) + . . . ~“a(“) + . . . ,
(12)
T’ = Z + sb(,) + ... s”b(,) + ... ,
(13)
where a@) and btkj are square matrices, s is a small parameter and T, T are mutual inverses so that 7T =z.
(14)
Equating coefficients of powers of s gives (for k > 1) b(k) + &7#_-1) + PV7#_2) + . .. a(k-lVQl) + Jk) = 0,
(15)
from which it is easily deduced that 41)
= -l&‘l
(16)
and
42) =
a(l)
z
a(2)
a(l)
I
I
(17)
*
We now proceed by induction and suppose a(l) 0 .................... Z a(2)
a(l)
Z
0 .......... ..
. b (?I)= t--l)n
: #rl) p)
0 0 . . . . .
(p-2)
(+l'
Z
@-1)
g(2)
a(1)
................
From (15) (with k = n + 1) and (US), we have
296 a(l)
b
(n+l)
=
(-q”+la(l)
Z
. . (p-1)
Z
a(l)
(-l)“a’2’
0
. a(n)
+
0 . . . . .
&)
. . . . . . . . .
0 . . . . .
0
I
. .
-. 1 Ll(n-l)
+ . . . 9 (-_1)2a(n)l&
atn-2)
+ (-1)
. . . . . . . . . a(l) I
@+l),
(19)
i.e.
bc,+l) = (-,)“+I
a(1)
Z
0 ............. 0
aw
aw
Z
0 .
. .
a(n+l)
.
a(n)
........ ..
a’l’
Z
a(2)
aw
(20)
We have thus shown that if (18) is valid for n = m, it is also valid for n = m + 1. But we have shown that it is valid for n = 1 and II = 2 from (16) and (17). Hence by induction our supposition (18) is correct. If we now put T = G and T’ = F we see from (3), (4), (12) and (13) (with s = t - t’) that a(n) - (--lY lq”“’ n! ’
(21)
(22) Substituting (21) and (22) into (18) we obtain (9) and the proof of the theorem is complete. Acknowledgments We have benefitted from helpful discussions with Professor J. Heading, Professor J.G. Oldroyd, Mr. J. Astin and Mr. R.W. Williams. References 1 J.G. Oldroyd, Proc. Roy. Sot., A200 (1950) 523. 2 D.E. Littlewood, A University Algebra, Heinemann, London, 1950, p. 81.