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Outstanding Problems. Future Trends
3 14 C h a p t e r 11
Outstanding Problems. Future Trends 11.1
GENERAL
We have seen i n p r e v i o u s c h a p t e r s t h a t t h e b a s i c p r o b l e m s a s s o c i a t e d w i t h t h e numerical
s i m u l a t i o n o f n o n - N e w t o n i a n f l o w a r e now w e l l
both i m p l i c i t d i f f e r e n t i a l
and e x p l i c i t
understood
In principle,
0
i n t e g r a l m o d e l s c a n be h a n d l e d a n d t h e r e
i s a g r o w i n g l i t e r a t u r e on t h e a d a p t a t i o n o f f i n i t e d i f f e r e n c e a n d f i n i t e t e c h n i q u e s t o t a k e a c c o u n t o f t h e new c h a l l e n g e s o f n o n - N e w t o n i a n ( C h a p t e r 9 ) and memory e f f e c t s
(Chapters 6 , 7 , 1 0 ) .
r e s o l v e d and e x i s t i n g a l g o r i t h m s
f o r visco-ineiastic
element
viscosity
The f o r m e r c h a l l e n g e i s now m o d e l s , c h a r a c t e r i z e d by
one s h e a r - d e p e n d e n t v i s c o s i t y f u n c t i o n , c a n be used w i t h c o n f i d e n c e i n t h o s e s i t u a t i o n s w h e r e memory e f f e c t s c a n be i g n o r e d . the status o f algorithms
The same c a n n o t be s a i d a b o u t
p r e s e n t l y a v a i l a b l e f o r memory f l u i d s
e m p h a s i z e d i n C h a p t e r s 2 , 3 , 6 and 1 0
f o r reasons
o
Much o f t h e p r e v i o u s d i s c u s s i o n on t h e s i m u l a t i o n o f v i s c o e l a s t i c c o n c e n t r a t e d on t h e s i m p l e u p p e r - c o n v e c t e d M a x w e l l model equivalent
(2.80) with a = 0, this
f l o w has
( 2 . 7 2 ) and i t s
preoccupation w i t h s i m p l i c i t y being
integral justified
by t h e k n o w l e d g e t h a t m o s t o f t h e new p r o b l e m s posed by f l u i d - m e m o r y e f f e c t s a r e p r e s e n t i n t h e Maxwell m o d e l
I n d e e d , t h e M a x w e l l model may have more d i f f i c u l -
0
t i e s t h a n more c o m p l i c a t e d d i f f e r e n t i a l
m o d e l s ( c f . C h a p t e r s 6 and 1 0 )
o
Further-
m o r e , i t has a l r e a d y been m e n t i o n e d i n C h a p t e r 10 t h a t t h e b a s i c p r o b l e m o f extending e x i s t i n g numerical ly straightforward derivatives
a l g o r i t h m s t o more c o m p l i c a t e d m o d e l s i s a r e l a t i v e -
procedure.
For example, models i n v o l v i n g h i g h e r
time
( l i k e t h e O l d r o y d Β m o d e l ) c a n be accommodated u s i n g t h e d e c o m p o s i t i o n
techniques discussed i n §10.8.
F u r t h e r m o r e , one o f t h e main problems
associated
w i t h a n y c o m p l i c a t e d integral model i s t h e d e t e r m i n a t i o n o f t h e r e l e v a n t m a t i o n t e n s o r s from a knowledge o f t h e d i s p l a c e m e n t f u n c t i o n s .
defor-
I n t h e sense
t h a t s u c h p r o b l e m s have t o be f a c e d even f o r t h e s i m p l e M a x w e l l model ( i n integral
f o r m ) , we may c o n c l u d e t h a t e x t e n s i o n s t o more c o m p l i c a t e d
models w i l l
simply involve the extra detail
integral
i n t h e model a n d , a c c o r d i n g l y , we
do n o t a n t i c i p a t e a n y new i s s u e s o f p r i n c i p l e . This o p t i m i s t i c numerical
s t a n c e i m p l i e s t h a t one o f t h e u n r e s o l v e d p r o b l e m s i n t h e
s i m u l a t i o n o f n o n - N e w t o n i a n f l o w may soon be t r a c t a b l e .
the occasional
l a c k o f a g r e e m e n t between n u m e r i c a l
r e s u l t s under c o n d i t i o n s where converged n u m e r i c a l
We r e f e r t o
p r e d i c t i o n s and e x p e r i m e n t a l s o l u t i o n s c a n be o b t a i n e d
( s e e , f o r e x a m p l e , C r o c h e t and Bezy 1 9 8 0 , W a l t e r s a n d W e b s t e r 1 9 8 2 ) .
Before
s e e k i n g more s o p h i s t i c a t e d r e a s o n s f o r t h e d i s c r e p a n c y between t h e o r y and e x p e r i m e n t , i t i s c l e a r l y a d v i s a b l e t o q u e s t i o n w h e t h e r t h e m o d e l s used up t o now i n t h e s i m u l a t i o n s a r e t o o n a i v e and s i m p l e
e
A l r e a d y , t h e r e i s some e v i d e n c e
to support the view t h a t r e l a t i v e l y minor m o d i f i c a t i o n s
to the rheological
models
3 1 5 can a l t e r d r a m a t i c a l l y t h e r e s u l t i n g n u m e r i c a l 1 9 8 3 , see a l s o
predictions
( K e u n i n g s and C r o c h e t
§10.11).
The second u n r e s o l v e d p r o b l e m i n t h e n u m e r i c a l flow concerns the degradation o f the numerical
simulation of
non-Newtonian
s o l u t i o n and/or the lack
of
c o n v e r g e n c e o f a l g o r i t h m s f o r modest v a l u e s o f t h e e l a s t i c i t y numbers W ( o r SR) in all
e x i s t i n g a l g o r i t h m s f o r f l u i d s w i t h l o n g - r a n g e memory.
Here, there
l e s s r e a s o n f o r o p t i m i s m and t h e w h o l e q u e s t i o n d e s e r v e s s p e c i a l 11.2
is
consideration.
NUMERICAL SIMULATION BREAKDOWN
T h e r e i s no d o u b t t h a t p r e s e n t l y t h e o u t s t a n d i n g p r o b l e m i n t h e
numerical
s i m u l a t i o n o f n o n - N e w t o n i a n f l o w c o n c e r n s t h e u p p e r l i m i t above w h i c h t h e numerical a l g o r i t h m f a i l s limit
( s e e , f o r e x a m p l e , C r o c h e t and W a l t e r s 1 9 8 3 a , b )
i s r e l a t i v e l y l o w and o f t e n i n a r e g i o n w h e r e t h e s o l u t i o n s b e f o r e
down a r e no more t h a n p e r t u r b a t i o n s a b o u t t h e N e w t o n i a n c a s e . information is of relevance in t h i s (i)
published works.
following
I t applies to
d i f f e r e n c e and f i n i t e e l e m e n t t e c h n i q u e s , t o d i f f e r e n t i a l (ii)
finite
and i n t e g r a l
I t even a p p l i e s t o t h e s i m p l e s e c o n d - o r d e r model
rheo-
(2.56).
M i n o r changes i n t h e c o n s t i t u t i v e e q u a t i o n a n d / o r t h e a l g o r i t h m
employed
can l e a d t o h i g h e r l i m i t i n g v a l u e s o f W ( s u i t a b l y m o d i f i e d i n d e f i n i t i o n accommodate t h e new c o n s t i t u t i v e e q u a t i o n s ) o r SR. O l d r o y d Β model
( 2 . 7 7 ) i n p l a c e o f t h e u p p e r - c o n v e c t e d M a x w e l l model
(2.72)
As W a p p r o a c h e s i t s c r i t i c a l
value W
c n
-
t
it
process.
is o f t e n (but not
always) stress
components a r e t h e n more s e v e r e l y a f f e c t e d t h a n t h e v e l o c i t y c o m p o n e n t s , l a r g e and e r r o n e o u s s t r e s s g r a d i e n t s .
The s p u r i o u s o s c i l l a t i o n s
Mesh r e f i n e m e n t and c o r n e r s t r a t e g i e s a f f e c t t h e c r i t i c a l is d i f f i c u l t
t o d i s c e r n an o v e r a l l
yielding
have no p h y s i c a l
b a c k g r o u n d , and t h e i r w a v e l e n g t h depends upon t h e mesh used f o r t h e (iv)
can
s o l u t i o n s can be
observed t h a t spurious o s c i l l a t i o n s appear i n the f i e l d v a r i a b l e s ; the
breakdown, but i t
the
H o w e v e r , i t must be e m p h a s i z e d t h a t s u c h i m p r o v e m e n t s ,
a l t h o u g h w e l c o m e , do no more t h a n d e l a y t h e b r e a k d o w n (iii)
to
F o r e x a m p l e , t h e use o f
sometimes e x t e n d t h e r a n g e o f W f o r w h i c h c o n v e r g e d n u m e r i c a l obtained (Chapter 1 0 ) .
break-
connection:
A l i m i t on W i s f o u n d i n a l l
l o g i c a l models.
The
The
Q
discretization.
conditions
consistent trend in
for
published
works. 11.3 11.3.1
POSSIBLE REASONS FOR BREAKDOWN : AN EVALUATION Choice o f r h e o l o g i c a l
model
From w h a t has a l r e a d y been w r i t t e n , i t
i s c l e a r t h a t changing the
model can do no more t h a n m o d i f y t h e b r e a k d o w n p r o c e s s . remove i t a n d , i n any c a s e , i t w o u l d s t i l l the numerical
rheological
It is unlikely
s o l u t i o n f o r one o f t h e s i m p l e s t p o s s i b l e m o d e l s f a i l s t o
a t a v e r y modest W v a l u e !
to
l e a v e u n r e s o l v e d t h e p r o b l e m o f why converge
3 1 β Π.3.2
Stress
evaluation
On t h e b a s i s o f t h e c o n s t i t u t i v e e q u a t i o n s e a s y t o show t h a t t h e t e n s o r Τ ' = Τ + n Ι Λ - j Q
throughout the flow domain.
( 3 . 5 ) o f a Maxwell f l u i d , i t must remain p o s i t i v e
is
definite
The s i g n i f i c a n c e o f t h e p o s i t i v e d e f i n i t e n e s s
of
t h e t e n s o r Τ ' has been e m p h a s i z e d i n a s e r i e s o f p a p e r s by R e g i r e r and R u t k e v i c h ( 1 9 6 8 ) and R u t k e v i c h ( 1 9 6 9 , 1 9 7 0 ) .
I t has been shown by D u p r e t and M a r c h a l
(1983) t h a t , f o r a Maxwell f l u i d , t h e l o s s o f t h e p o s i t i v e d e f i n i t e c h a r a c t e r t h e above t e n s o r i n a s u b r e g i o n o f t h e f l o w c o i n c i d e s w i t h t h e breakdown o f numerical a l g o r i t h m .
The l o s s o f p o s i t i v e d e f i n i t e n e s s o f t h e t e n s o r T , due t o 1
an e r r o n e o u s e v a l u a t i o n o f t h e s t r e s s f i e l d , p r o d u c e s a l o c a l change o f of the set of p a r t i a l
of
the
differential
e q u a t i o n s w h i c h may e x p l a i n t h e
character
numerical
p r o b l ems. I n many v i s c o e l a s t i c particularly difficult boundaries.
flow problems, the evaluation of the stress f i e l d
is
in view o f regions o f high stress gradients along
solid
A l o n g smooth b o u n d a r i e s w i t h n o r m a l and t a n g e n t v e c t o r s d e n o t e d by
η and s , t h e s h e a r s t r e s s T
n s
i n a M a x w e l l f l u i d i s g i v e n by n y w h i l e t h e Q
component i s g i v e n by Ζ λ - ^ γ .
The q u a d r a t i c d e p e n d e n c e o f T
2
$ s
T
$ s
upon γ g i v e s
r i s e t o h i g h s t r e s s g r a d i e n t s w h i c h r e q u i r e a h i g h d e g r e e o f mesh r e f i n e m e n t for a correct stress 11.3.3
evaluation.
Mesh r e f i n e m e n t
Up t o t h e p r e s e n t t i m e , t h e r e l a t i v e l y s m a l l s i z e o f c o m p u t e r s has a thorough study o f numerical
prevented
e r r o r s and t h e r e l a t e d p r o b l e m o f mesh r e f i n e m e n t .
C e r t a i n l y , i t may be n a i v e t o s i m p l y t a k e o v e r e s t a b l i s h e d p r a c t i c e w i t h N a v i e r - S t o k e s e q u a t i o n s t o t h e more c o m p l i c a t e d f i e l d o f n o n - N e w t o n i a n mechanics.
Too much i s o f t e n demanded o f o u r n u m e r i c a l
the fluid
s i m u l a t i o n s and t h e r e
i s g r o w i n g e v i d e n c e t o s u g g e s t t h a t some w o r k e r s may o c c a s i o n a l l y r e a d t o o much i n t o t h e f i n e r d e t a i l s o f t h e i r s i m u l a t i o n s .
We have now r e a c h e d t h e
stage
i n t h e d e v e l o p m e n t o f t h e s u b j e c t when t h e w h o l e q u e s t i o n o f mesh r e f i n e m e n t and n u m e r i c a l e r r o r s must be g i v e n p r i o r i t y ( c f D a v i e s e t a l A l t h o u g h mesh r e f i n e m e n t i s e s s e n t i a l
if
1983).
i n a c c u r a t e converged s o l u t i o n s
t o be a v o i d e d , we must c o n c l u d e t h a t a t t h e p r e s e n t t i m e t h e r e i s
little
e v i d e n c e t o s u g g e s t t h a t mesh r e f i n e m e n t per se w o u l d r e s o l v e t h e
breakdown
are
p r o b l em. 11.3.4
Corner
strategies
We d e v o t e d § 6 . 2 . 4 t o a f u l l
discussion of re-entrant corner
w i t h i n the f i n i t e difference context.
singularities
Much w o r k , b o t h n u m e r i c a l and a n a l y t i c a l ,
3 1 7
is s t i l l
required in this
evidence a v a i l a b l e
i m p o r t a n t area and t h e r e i s s u f f i c i e n t
experimental
( s e e , f o r example, W a l t e r s and Webster 1982) t o emphasize
how i m p o r t a n t i t i s t o s p e c i f y c o r n e r c o n d i t i o n s a c c u r a t e l y .
C h a p t e r 10 has
argued t h a t t h e problem also deserves s e r i o u s c o n s i d e r a t i o n w i t h i n t h e f i n i t e element c o n t e x t and t h e s e a r c h f o r a s u i t a b l e s i n g u l a r element i s a p r e s s i n g subject o f current
research
0
Crochet (1982a) demonstrates t h a t numerical without b o u n d a r y s i n g u l a r i t i e s .
breakdown o c c u r s i n f l o w
problems
Although a correct corner strategy i s essential
f o r a c c u r a c y , t h e r e f o r e , i t i s u n l i k e l y i n i t s e l f t o remove t h e breakdown p r o b l e m . 11.3 5 0
Bifurcation
T h e r e i s a w e a l t h o f r e c e n t l i t e r a t u r e on t h e g e n e r a l
problem o f
bifurcation
( s e e , f o r e x a m p l e , I o o s s and J o s e p h 1 9 8 0 ) , a n d t h e r e i s no d o u b t t h a t evidence i s a v a i l a b l e i n Newtonian f l u i d mechanics
sufficient
( s e e , f o r e x a m p l e , C I i f f e and
G r e e n f i e l d 1982) t o warrant g i v i n g serious c o n s i d e r a t i o n to t h i s p o s s i b i l i t y t h e more c o m p l e x f i e l d o f n o n - N e w t o n i a n
in
f l u i d m e c h a n i c s , w h e r e we a r e c o n c e r n e d
w i t h a t l e a s t t w o p a r a m e t e r s R and W ( s e e , f o r e x a m p l e , K i m - E . e t a l 1 9 8 3 ) . What we s a y b e l o w c o n c e r n i n g b i f u r c a t i o n may a l s o be e x t e n d e d t o i n c l u d e limit
p o i n t s i n t h e parameter s p a c e , o r p o i n t s a t w h i c h a s o l u t i o n ceases t o
exist. It is essential ated with the f u l l discretized
i n t h i s c o n t e x t t o d i s t i n g u i s h between a b i f u r c a t i o n
associ-
s e t o f differential e q u a t i o n s and a b i f u r c a t i o n i n t h e
{algebraic) f o r m o f t h e s e e q u a t i o n s .
b i f u r c a t i o n i n the former w i l l
We may a n t i c i p a t e t h a t a
be a c c o m p a n i e d by a s i m i l a r phenomenon i n t h e
d i s c r e t i z e d a l g e b r a i c e q u a t i o n s ; t h e c o n v e r s e may n o t be t r u e . M e n d e l s o n e t a l . ( 1 9 8 2 ) e x a m i n e t h e c r e e p i n g f l o w (R = 0 ) o f a s e c o n d - o r d e r f l u i d i n a 4 / 1 p l a n a r c o n t r a c t i o n u s i n g f i n i t e e l e m e n t s w i t h Newton and c o n t i n u a t i o n w i t h r e s p e c t t o t h e p a r a m e t e r W.
iteration
They f i n d t h a t as W a p p r o a c h e s
W ^ c r i t ' t h e J a c o b i a n m a t r i x does n o t become s i n g u l a r , t h u s g i v i n g no e v i d e n c e o f solution bifurcation
in the discrete regime.
T h i s r e s u l t , h o w e v e r , does n o t
r u l e out the p o s s i b i l i t y o f 3-dimensional
f l o w s a r i s i n g as b i f u r c a t i o n s
the family o f 2-dimensional
calculated.
Another t h e o r e t i c a l
flows already
possibility
i s t h e onset o f time periodic f l o w s ,
i n p r i n c i p l e s h o u l d be d e t e c t a b l e by t i m e i n t e g r a t i o n , , r e s u l t s f o r Oldroyd f l u i d s
which
preliminary
(see Chapter 7 ) , however, c o n f i r m t h e e x i s t e n c e o f
steady flow s o l u t i o n s , w i t h s i m i l a r indication of
Townsend's
from
breakdown a t W - j . c r
t
T h e r e i s as y e t no
time-periodicity.
F i n a l l y i n t h i s s e c t i o n we m u s t m e n t i o n t h e c l a s s i c a l o f a n y model f l u i d .
Several authors
s t a b i l i t y of the flow
( s e e , f o r e x a m p l e , Chan Man Fong a n d
W a l t e r s 1 9 6 5 ) have i n v e s t i g a t e d t h e t e m p o r a l and s p a t i a l
r e s p o n s e o f model
3 I 8
rheological fields.
fluids to sinusoidal
perturbations i n v e l o c i t y f o r simple
flow
S i n c e , i n computer c a l c u l a t i o n s , t h e v e l o c i t y increments w i l l
a range o f F o u r i e r modes, t h e o n s e t o f c l a s s i c a l
contain
i n s t a b i l i t y being t h e cause o f
breakdown i s a p o s s i b i l i t y w o r t h y o f e x p l o r a t i o n „ 11.3.6
I n t e r a c t i o n between d i s c r e t i z a t i o n e r r o r and n o n l i n e a r
iteration
A p o s s i b l e c a u s e o f t h e breakdown c a n be a t t r i b u t e d t o d i s c r e t i z a t i o n and t h e i r e f f e c t on t h e n o n l i n e a r c o u p l i n g o f t h e g o v e r n i n g e q u a t i o n s .
errors For
c o n v e n i e n c e o f p r e s e n t a t i o n , we i l l u s t r a t e t h i s p o s s i b i l i t y by c o n f i n i n g tion to the f i n i t e difference
atten-
( ψ , ω , § ) f o r m u l i s m d i s c u s s e d i n C h a p t e r 6»
We have t o s o l v e t h e f o l l o w i n g n o n - d i m e n s i o n a l
equations
( c f . 6 . 1 , 6 . 4 and
6.5): T
l k
= S
l k
+ 2d
i k
ν ψ = -ω 2
ν ω 2
+R
,
( U J )
,
(Π.2)
9ψ 3ω 3ψ 9ω _9χ 8y " 3y 3 x J
-2—
9x3y
S
x
*
x
-S
y
y
+
together with suitable constitutive equations. t h e M a x w e l l model
I n t h e case o f p l a n a r f l o w o f
( 2 . 7 2 ) , these give r i s e t o three hyperbolic equations.
The
s o l u t i o n o f t h e s e e q u a t i o n s , t h e m s e l v e s , may be a f a c t o r o f i m p o r t a n c e i n t h e breakdown p r o c e s s ( c f . § 1 1 3 . 2 ) , b u t i t i s u n l i k e l y t h a t t h i s f a c t o r i s f u n d a 0
m e n t a l , s i n c e f o r a g i v e n fixed v e l o c i t y f i e l d i t i s o f t e n p o s s i b l e t o o b t a i n r e s p e c t a b l e c o n v e r g e d s o l u t i o n s f o r t h e s t r e s s components f o r a h i g h v a l u e o f W„ T h i s means t h a t t h e p r e c i s e f o r m o f t h e c o n s t i t u t i v e e q u a t i o n s per se does n o t a p p e a r t o be o f m a j o r s i g n i f i c a n c e , o n l y t h e p o s s i b l e n o n l i n e a r i n t e r a c t i o n o f t h e s e e q u a t i o n s w i t h ( 1 1 . 2 ) and ( 1 1 3 ) i n t h e i t e r a t i o n p r o c e s s . 0
breakdown p r o b l e m i s p r e s e n t f o r a s p e c i a l c a s e o f t h e e x p l i c i t
Indeed t h e second-order
model g i v e n by S
l k
= - 2Wd
l k
,
and f o r i l l u s t r a t i o n
(11,4) p u r p o s e s we now c o n c e n t r a t e o n t h i s m o d e l .
Further, the
breakdown i s p r e s e n t f o r R = 0 ( i e . c r e e p i n g f l o w ) a n d t h e r e i s no p a r t i c u l a r 0
m e r i t i n r e t a i n i n g t h e i n e r t i a terms i n ( 1 1 . 3 ) i n t h e present c o n t e x t o f a t t e m p t i n g t o l o c a t e a mechanism f o r b r e a k d o w n . We now c o n s i d e r t h e f l o w p r o b l e m p r o v i d e d by ( 1 1 . 2 ) , ( 1 1 . 3 ) w i t h R = 0 a n d (11.4)
0
For plane f l o w s , t h e Tanner-Giesekus theorem ( s e e , f o r example, Tanner
1 9 6 6 ) e n s u r e s t h a t t h e N e w t o n i a n v e l o c i t y f i e l d f o r (W = 0 ) must s a t i s f y
(11.2)
319 - (11.4) with R = 0, W f 0.
I t i s easy t o show t h a t s u b s t i t u t i o n o f
(11.4)
i n t o ( 1 1 . 3 ) r e n d e r s t h e r i g h t hand s i d e o f t h i s e q u a t i o n z e r o , c o n s i s t e n t the Tanner-Giesekus Interestingly,
it
with
theorem. i s found t h a t e x i s t i n g f i n i t e d i f f e r e n c e a l g o r i t h m s
adap-
t e d t o t h e p l a n a r s e c o n d - o r d e r p r o b l e m a l s o b r e a k down a t some m o d e s t W v a l u e and a p o s s i b l e r e a s o n f o r t h e breakdown i s h i g h l i g h t e d .
Analytically,
N e w t o n i a n v e l o c i t y f i e l d must s a t i s f y t h e s e c oi nk d - o r d e r p r o b l e m . numerical noise i s exaggerated i n computing S
the
H o w e v e r , any
f r o m ( 1 1 4 ) and even more so i n i k t a k i n g t h e second d e r i v a t i v e s o f t h e components o f S to provide the ' f o r c i n g function'
in the v o r t i c i t y equation ( 1 1 . 3 ) .
z e r o , as we have n o t e d , b u t i t
0
T h i s f o r c i n g f u n c t i o n s h o u l d be
i s q u i t e e v i d e n t l y not zero ( n u m e r i c a l l y )
the s o l u t i o n f a i l s to converge.
The r e a s o n f o r t h e breakdown i n t h i s
r e l a t e d p r o b l e m s may t h e r e f o r e be due t o n o t h i n g more t h a n n u m e r i c a l
noise
the forcing function of (11.3) i n t e r a c t i n g with the nonlinear i t e r a t i o n I f t h i s reason f o r the numerical
breakdown can be u n e q u i v o c a l l y
when
and in
scheme
0
associated
w i t h t h e n u m e r i c a l n o i s e g e n e r a t e d i n t h e r i g h t hand s i d e o f ( 1 1 , 3 ) , t h e n e x t s t e p m u s t be t h e i m p l e m e n t a t i o n o f f i l t e r i n g t e c h n i q u e s Davies 1983) t o c o u n t e r a c t t h e e f f e c t o f
(see, for
example,
noise.
W i t h f i n i t e e l e m e n t s , a r e l a t e d r e s u l t has been f o u n d by M e n d e l s o n e t a l . (1982) i n t h e i r study o f the f l o w o f a second-order f l u i d i n a planar tion.
Using a numerical
contrac-
t e c h n i q u e s i m i l a r t o MIX4 ( § 1 0 „ 9 ) , M e n d e l s o n e t a l .
f o u n d t h a t t h e n u m e r i c a l n o i s e p r o d u c e s a l a r g e v o r t e x w h i c h s h o u l d n o t be there.
F u r t h e r c a l c u l a t i o n s w i t h r e f i n e d meshes have shown t h a t , on t h e
s p e c i f i c problem o f the planar c o n t r a c t i o n , a b e t t e r d i s c r e t i z a t i o n the i n t e n s i t y of the spurious 11.3.7
decreases
vortex.
Algorithms
Ample e v i d e n c e has been g i v e n i n C h a p t e r 10 t h a t t h e l i m i t i n g v a l u e o f W depends upon t h e m e t h o d used f o r d i s c r e t i z i n g t h e p a r t i a l and upon t h e n o n l i n e a r i t e r a t i o n a l g o r i t h m .
differential
equations
Much w o r k r e m a i n s t o be done i n
u n d e r s t a n d i n g how n u m e r i c a l c o n v e r g e n c e depends on t h e s e l e c t i o n o f
algorithm,
o f f i n i t e e l e m e n t s , and o f m e s h . 11.4
CONCLUDING REMARKS
Numerical s i m u l a t i o n o f non-Newtonian f l o w i s c l e a r l y a p r o v o c a t i v e area o f c u r r e n t r e s e a r c h , w h i c h i s endowed w i t h more t h a n i t s u s u a l s h a r e o f often unresolved,
interesting,
problems.
To end t h i s book we can do no b e t t e r t h a n a d v i s e t h e r e a d e r t h a t p r o g r e s s i n t h i s s u b j e c t m u s t depend on c a r e f u l
meaningful
numerical experimentation
on t h e most r e l e v a n t g u i d e l i n e s t h a t c u r r e n t m a t h e m a t i c a l
t h e o r y can o f f e r .
based We
320 must s t r i v e t o u n d e r s t a n d n o t o n l y t h e n a t u r e o f o u r n u m e r i c a l a l g o r i t h m s , also the a n a l y t i c p r o p e r t i e s o f our governing e q u a t i o n s .
All t h i s , of
must be c a r e f u l l y b a l a n c e d a g a i n s t o u r i n c r e a s i n g k n o w l e d g e o f f l u i d s gleaned from l a b o r a t o r y
experiments.
but
course,
non-Newtonian
Outstanding Problems. Future Trends 11.1
GENERAL
We have seen i n p r e v i o u s c h a p t e r s t h a t t h e b a s i c p r o b l e m s a s s o c i a t e d w i t h t h e numerical
s i m u l a t i o n o f n o n - N e w t o n i a n f l o w a r e now w e l l
both i m p l i c i t d i f f e r e n t i a l
and e x p l i c i t
understood
In principle,
0
i n t e g r a l m o d e l s c a n be h a n d l e d a n d t h e r e
i s a g r o w i n g l i t e r a t u r e on t h e a d a p t a t i o n o f f i n i t e d i f f e r e n c e a n d f i n i t e t e c h n i q u e s t o t a k e a c c o u n t o f t h e new c h a l l e n g e s o f n o n - N e w t o n i a n ( C h a p t e r 9 ) and memory e f f e c t s
(Chapters 6 , 7 , 1 0 ) .
r e s o l v e d and e x i s t i n g a l g o r i t h m s
f o r visco-ineiastic
element
viscosity
The f o r m e r c h a l l e n g e i s now m o d e l s , c h a r a c t e r i z e d by
one s h e a r - d e p e n d e n t v i s c o s i t y f u n c t i o n , c a n be used w i t h c o n f i d e n c e i n t h o s e s i t u a t i o n s w h e r e memory e f f e c t s c a n be i g n o r e d . the status o f algorithms
The same c a n n o t be s a i d a b o u t
p r e s e n t l y a v a i l a b l e f o r memory f l u i d s
e m p h a s i z e d i n C h a p t e r s 2 , 3 , 6 and 1 0
f o r reasons
o
Much o f t h e p r e v i o u s d i s c u s s i o n on t h e s i m u l a t i o n o f v i s c o e l a s t i c c o n c e n t r a t e d on t h e s i m p l e u p p e r - c o n v e c t e d M a x w e l l model equivalent
(2.80) with a = 0, this
f l o w has
( 2 . 7 2 ) and i t s
preoccupation w i t h s i m p l i c i t y being
integral justified
by t h e k n o w l e d g e t h a t m o s t o f t h e new p r o b l e m s posed by f l u i d - m e m o r y e f f e c t s a r e p r e s e n t i n t h e Maxwell m o d e l
I n d e e d , t h e M a x w e l l model may have more d i f f i c u l -
0
t i e s t h a n more c o m p l i c a t e d d i f f e r e n t i a l
m o d e l s ( c f . C h a p t e r s 6 and 1 0 )
o
Further-
m o r e , i t has a l r e a d y been m e n t i o n e d i n C h a p t e r 10 t h a t t h e b a s i c p r o b l e m o f extending e x i s t i n g numerical ly straightforward derivatives
a l g o r i t h m s t o more c o m p l i c a t e d m o d e l s i s a r e l a t i v e -
procedure.
For example, models i n v o l v i n g h i g h e r
time
( l i k e t h e O l d r o y d Β m o d e l ) c a n be accommodated u s i n g t h e d e c o m p o s i t i o n
techniques discussed i n §10.8.
F u r t h e r m o r e , one o f t h e main problems
associated
w i t h a n y c o m p l i c a t e d integral model i s t h e d e t e r m i n a t i o n o f t h e r e l e v a n t m a t i o n t e n s o r s from a knowledge o f t h e d i s p l a c e m e n t f u n c t i o n s .
defor-
I n t h e sense
t h a t s u c h p r o b l e m s have t o be f a c e d even f o r t h e s i m p l e M a x w e l l model ( i n integral
f o r m ) , we may c o n c l u d e t h a t e x t e n s i o n s t o more c o m p l i c a t e d
models w i l l
simply involve the extra detail
integral
i n t h e model a n d , a c c o r d i n g l y , we
do n o t a n t i c i p a t e a n y new i s s u e s o f p r i n c i p l e . This o p t i m i s t i c numerical
s t a n c e i m p l i e s t h a t one o f t h e u n r e s o l v e d p r o b l e m s i n t h e
s i m u l a t i o n o f n o n - N e w t o n i a n f l o w may soon be t r a c t a b l e .
the occasional
l a c k o f a g r e e m e n t between n u m e r i c a l
r e s u l t s under c o n d i t i o n s where converged n u m e r i c a l
We r e f e r t o
p r e d i c t i o n s and e x p e r i m e n t a l s o l u t i o n s c a n be o b t a i n e d
( s e e , f o r e x a m p l e , C r o c h e t and Bezy 1 9 8 0 , W a l t e r s a n d W e b s t e r 1 9 8 2 ) .
Before
s e e k i n g more s o p h i s t i c a t e d r e a s o n s f o r t h e d i s c r e p a n c y between t h e o r y and e x p e r i m e n t , i t i s c l e a r l y a d v i s a b l e t o q u e s t i o n w h e t h e r t h e m o d e l s used up t o now i n t h e s i m u l a t i o n s a r e t o o n a i v e and s i m p l e
e
A l r e a d y , t h e r e i s some e v i d e n c e
to support the view t h a t r e l a t i v e l y minor m o d i f i c a t i o n s
to the rheological
models
3 1 5 can a l t e r d r a m a t i c a l l y t h e r e s u l t i n g n u m e r i c a l 1 9 8 3 , see a l s o
predictions
( K e u n i n g s and C r o c h e t
§10.11).
The second u n r e s o l v e d p r o b l e m i n t h e n u m e r i c a l flow concerns the degradation o f the numerical
simulation of
non-Newtonian
s o l u t i o n and/or the lack
of
c o n v e r g e n c e o f a l g o r i t h m s f o r modest v a l u e s o f t h e e l a s t i c i t y numbers W ( o r SR) in all
e x i s t i n g a l g o r i t h m s f o r f l u i d s w i t h l o n g - r a n g e memory.
Here, there
l e s s r e a s o n f o r o p t i m i s m and t h e w h o l e q u e s t i o n d e s e r v e s s p e c i a l 11.2
is
consideration.
NUMERICAL SIMULATION BREAKDOWN
T h e r e i s no d o u b t t h a t p r e s e n t l y t h e o u t s t a n d i n g p r o b l e m i n t h e
numerical
s i m u l a t i o n o f n o n - N e w t o n i a n f l o w c o n c e r n s t h e u p p e r l i m i t above w h i c h t h e numerical a l g o r i t h m f a i l s limit
( s e e , f o r e x a m p l e , C r o c h e t and W a l t e r s 1 9 8 3 a , b )
i s r e l a t i v e l y l o w and o f t e n i n a r e g i o n w h e r e t h e s o l u t i o n s b e f o r e
down a r e no more t h a n p e r t u r b a t i o n s a b o u t t h e N e w t o n i a n c a s e . information is of relevance in t h i s (i)
published works.
following
I t applies to
d i f f e r e n c e and f i n i t e e l e m e n t t e c h n i q u e s , t o d i f f e r e n t i a l (ii)
finite
and i n t e g r a l
I t even a p p l i e s t o t h e s i m p l e s e c o n d - o r d e r model
rheo-
(2.56).
M i n o r changes i n t h e c o n s t i t u t i v e e q u a t i o n a n d / o r t h e a l g o r i t h m
employed
can l e a d t o h i g h e r l i m i t i n g v a l u e s o f W ( s u i t a b l y m o d i f i e d i n d e f i n i t i o n accommodate t h e new c o n s t i t u t i v e e q u a t i o n s ) o r SR. O l d r o y d Β model
( 2 . 7 7 ) i n p l a c e o f t h e u p p e r - c o n v e c t e d M a x w e l l model
(2.72)
As W a p p r o a c h e s i t s c r i t i c a l
value W
c n
-
t
it
process.
is o f t e n (but not
always) stress
components a r e t h e n more s e v e r e l y a f f e c t e d t h a n t h e v e l o c i t y c o m p o n e n t s , l a r g e and e r r o n e o u s s t r e s s g r a d i e n t s .
The s p u r i o u s o s c i l l a t i o n s
Mesh r e f i n e m e n t and c o r n e r s t r a t e g i e s a f f e c t t h e c r i t i c a l is d i f f i c u l t
t o d i s c e r n an o v e r a l l
yielding
have no p h y s i c a l
b a c k g r o u n d , and t h e i r w a v e l e n g t h depends upon t h e mesh used f o r t h e (iv)
can
s o l u t i o n s can be
observed t h a t spurious o s c i l l a t i o n s appear i n the f i e l d v a r i a b l e s ; the
breakdown, but i t
the
H o w e v e r , i t must be e m p h a s i z e d t h a t s u c h i m p r o v e m e n t s ,
a l t h o u g h w e l c o m e , do no more t h a n d e l a y t h e b r e a k d o w n (iii)
to
F o r e x a m p l e , t h e use o f
sometimes e x t e n d t h e r a n g e o f W f o r w h i c h c o n v e r g e d n u m e r i c a l obtained (Chapter 1 0 ) .
break-
connection:
A l i m i t on W i s f o u n d i n a l l
l o g i c a l models.
The
The
Q
discretization.
conditions
consistent trend in
for
published
works. 11.3 11.3.1
POSSIBLE REASONS FOR BREAKDOWN : AN EVALUATION Choice o f r h e o l o g i c a l
model
From w h a t has a l r e a d y been w r i t t e n , i t
i s c l e a r t h a t changing the
model can do no more t h a n m o d i f y t h e b r e a k d o w n p r o c e s s . remove i t a n d , i n any c a s e , i t w o u l d s t i l l the numerical
rheological
It is unlikely
s o l u t i o n f o r one o f t h e s i m p l e s t p o s s i b l e m o d e l s f a i l s t o
a t a v e r y modest W v a l u e !
to
l e a v e u n r e s o l v e d t h e p r o b l e m o f why converge
3 1 β Π.3.2
Stress
evaluation
On t h e b a s i s o f t h e c o n s t i t u t i v e e q u a t i o n s e a s y t o show t h a t t h e t e n s o r Τ ' = Τ + n Ι Λ - j Q
throughout the flow domain.
( 3 . 5 ) o f a Maxwell f l u i d , i t must remain p o s i t i v e
is
definite
The s i g n i f i c a n c e o f t h e p o s i t i v e d e f i n i t e n e s s
of
t h e t e n s o r Τ ' has been e m p h a s i z e d i n a s e r i e s o f p a p e r s by R e g i r e r and R u t k e v i c h ( 1 9 6 8 ) and R u t k e v i c h ( 1 9 6 9 , 1 9 7 0 ) .
I t has been shown by D u p r e t and M a r c h a l
(1983) t h a t , f o r a Maxwell f l u i d , t h e l o s s o f t h e p o s i t i v e d e f i n i t e c h a r a c t e r t h e above t e n s o r i n a s u b r e g i o n o f t h e f l o w c o i n c i d e s w i t h t h e breakdown o f numerical a l g o r i t h m .
The l o s s o f p o s i t i v e d e f i n i t e n e s s o f t h e t e n s o r T , due t o 1
an e r r o n e o u s e v a l u a t i o n o f t h e s t r e s s f i e l d , p r o d u c e s a l o c a l change o f of the set of p a r t i a l
of
the
differential
e q u a t i o n s w h i c h may e x p l a i n t h e
character
numerical
p r o b l ems. I n many v i s c o e l a s t i c particularly difficult boundaries.
flow problems, the evaluation of the stress f i e l d
is
in view o f regions o f high stress gradients along
solid
A l o n g smooth b o u n d a r i e s w i t h n o r m a l and t a n g e n t v e c t o r s d e n o t e d by
η and s , t h e s h e a r s t r e s s T
n s
i n a M a x w e l l f l u i d i s g i v e n by n y w h i l e t h e Q
component i s g i v e n by Ζ λ - ^ γ .
The q u a d r a t i c d e p e n d e n c e o f T
2
$ s
T
$ s
upon γ g i v e s
r i s e t o h i g h s t r e s s g r a d i e n t s w h i c h r e q u i r e a h i g h d e g r e e o f mesh r e f i n e m e n t for a correct stress 11.3.3
evaluation.
Mesh r e f i n e m e n t
Up t o t h e p r e s e n t t i m e , t h e r e l a t i v e l y s m a l l s i z e o f c o m p u t e r s has a thorough study o f numerical
prevented
e r r o r s and t h e r e l a t e d p r o b l e m o f mesh r e f i n e m e n t .
C e r t a i n l y , i t may be n a i v e t o s i m p l y t a k e o v e r e s t a b l i s h e d p r a c t i c e w i t h N a v i e r - S t o k e s e q u a t i o n s t o t h e more c o m p l i c a t e d f i e l d o f n o n - N e w t o n i a n mechanics.
Too much i s o f t e n demanded o f o u r n u m e r i c a l
the fluid
s i m u l a t i o n s and t h e r e
i s g r o w i n g e v i d e n c e t o s u g g e s t t h a t some w o r k e r s may o c c a s i o n a l l y r e a d t o o much i n t o t h e f i n e r d e t a i l s o f t h e i r s i m u l a t i o n s .
We have now r e a c h e d t h e
stage
i n t h e d e v e l o p m e n t o f t h e s u b j e c t when t h e w h o l e q u e s t i o n o f mesh r e f i n e m e n t and n u m e r i c a l e r r o r s must be g i v e n p r i o r i t y ( c f D a v i e s e t a l A l t h o u g h mesh r e f i n e m e n t i s e s s e n t i a l
if
1983).
i n a c c u r a t e converged s o l u t i o n s
t o be a v o i d e d , we must c o n c l u d e t h a t a t t h e p r e s e n t t i m e t h e r e i s
little
e v i d e n c e t o s u g g e s t t h a t mesh r e f i n e m e n t per se w o u l d r e s o l v e t h e
breakdown
are
p r o b l em. 11.3.4
Corner
strategies
We d e v o t e d § 6 . 2 . 4 t o a f u l l
discussion of re-entrant corner
w i t h i n the f i n i t e difference context.
singularities
Much w o r k , b o t h n u m e r i c a l and a n a l y t i c a l ,
3 1 7
is s t i l l
required in this
evidence a v a i l a b l e
i m p o r t a n t area and t h e r e i s s u f f i c i e n t
experimental
( s e e , f o r example, W a l t e r s and Webster 1982) t o emphasize
how i m p o r t a n t i t i s t o s p e c i f y c o r n e r c o n d i t i o n s a c c u r a t e l y .
C h a p t e r 10 has
argued t h a t t h e problem also deserves s e r i o u s c o n s i d e r a t i o n w i t h i n t h e f i n i t e element c o n t e x t and t h e s e a r c h f o r a s u i t a b l e s i n g u l a r element i s a p r e s s i n g subject o f current
research
0
Crochet (1982a) demonstrates t h a t numerical without b o u n d a r y s i n g u l a r i t i e s .
breakdown o c c u r s i n f l o w
problems
Although a correct corner strategy i s essential
f o r a c c u r a c y , t h e r e f o r e , i t i s u n l i k e l y i n i t s e l f t o remove t h e breakdown p r o b l e m . 11.3 5 0
Bifurcation
T h e r e i s a w e a l t h o f r e c e n t l i t e r a t u r e on t h e g e n e r a l
problem o f
bifurcation
( s e e , f o r e x a m p l e , I o o s s and J o s e p h 1 9 8 0 ) , a n d t h e r e i s no d o u b t t h a t evidence i s a v a i l a b l e i n Newtonian f l u i d mechanics
sufficient
( s e e , f o r e x a m p l e , C I i f f e and
G r e e n f i e l d 1982) t o warrant g i v i n g serious c o n s i d e r a t i o n to t h i s p o s s i b i l i t y t h e more c o m p l e x f i e l d o f n o n - N e w t o n i a n
in
f l u i d m e c h a n i c s , w h e r e we a r e c o n c e r n e d
w i t h a t l e a s t t w o p a r a m e t e r s R and W ( s e e , f o r e x a m p l e , K i m - E . e t a l 1 9 8 3 ) . What we s a y b e l o w c o n c e r n i n g b i f u r c a t i o n may a l s o be e x t e n d e d t o i n c l u d e limit
p o i n t s i n t h e parameter s p a c e , o r p o i n t s a t w h i c h a s o l u t i o n ceases t o
exist. It is essential ated with the f u l l discretized
i n t h i s c o n t e x t t o d i s t i n g u i s h between a b i f u r c a t i o n
associ-
s e t o f differential e q u a t i o n s and a b i f u r c a t i o n i n t h e
{algebraic) f o r m o f t h e s e e q u a t i o n s .
b i f u r c a t i o n i n the former w i l l
We may a n t i c i p a t e t h a t a
be a c c o m p a n i e d by a s i m i l a r phenomenon i n t h e
d i s c r e t i z e d a l g e b r a i c e q u a t i o n s ; t h e c o n v e r s e may n o t be t r u e . M e n d e l s o n e t a l . ( 1 9 8 2 ) e x a m i n e t h e c r e e p i n g f l o w (R = 0 ) o f a s e c o n d - o r d e r f l u i d i n a 4 / 1 p l a n a r c o n t r a c t i o n u s i n g f i n i t e e l e m e n t s w i t h Newton and c o n t i n u a t i o n w i t h r e s p e c t t o t h e p a r a m e t e r W.
iteration
They f i n d t h a t as W a p p r o a c h e s
W ^ c r i t ' t h e J a c o b i a n m a t r i x does n o t become s i n g u l a r , t h u s g i v i n g no e v i d e n c e o f solution bifurcation
in the discrete regime.
T h i s r e s u l t , h o w e v e r , does n o t
r u l e out the p o s s i b i l i t y o f 3-dimensional
f l o w s a r i s i n g as b i f u r c a t i o n s
the family o f 2-dimensional
calculated.
Another t h e o r e t i c a l
flows already
possibility
i s t h e onset o f time periodic f l o w s ,
i n p r i n c i p l e s h o u l d be d e t e c t a b l e by t i m e i n t e g r a t i o n , , r e s u l t s f o r Oldroyd f l u i d s
which
preliminary
(see Chapter 7 ) , however, c o n f i r m t h e e x i s t e n c e o f
steady flow s o l u t i o n s , w i t h s i m i l a r indication of
Townsend's
from
breakdown a t W - j . c r
t
T h e r e i s as y e t no
time-periodicity.
F i n a l l y i n t h i s s e c t i o n we m u s t m e n t i o n t h e c l a s s i c a l o f a n y model f l u i d .
Several authors
s t a b i l i t y of the flow
( s e e , f o r e x a m p l e , Chan Man Fong a n d
W a l t e r s 1 9 6 5 ) have i n v e s t i g a t e d t h e t e m p o r a l and s p a t i a l
r e s p o n s e o f model
3 I 8
rheological fields.
fluids to sinusoidal
perturbations i n v e l o c i t y f o r simple
flow
S i n c e , i n computer c a l c u l a t i o n s , t h e v e l o c i t y increments w i l l
a range o f F o u r i e r modes, t h e o n s e t o f c l a s s i c a l
contain
i n s t a b i l i t y being t h e cause o f
breakdown i s a p o s s i b i l i t y w o r t h y o f e x p l o r a t i o n „ 11.3.6
I n t e r a c t i o n between d i s c r e t i z a t i o n e r r o r and n o n l i n e a r
iteration
A p o s s i b l e c a u s e o f t h e breakdown c a n be a t t r i b u t e d t o d i s c r e t i z a t i o n and t h e i r e f f e c t on t h e n o n l i n e a r c o u p l i n g o f t h e g o v e r n i n g e q u a t i o n s .
errors For
c o n v e n i e n c e o f p r e s e n t a t i o n , we i l l u s t r a t e t h i s p o s s i b i l i t y by c o n f i n i n g tion to the f i n i t e difference
atten-
( ψ , ω , § ) f o r m u l i s m d i s c u s s e d i n C h a p t e r 6»
We have t o s o l v e t h e f o l l o w i n g n o n - d i m e n s i o n a l
equations
( c f . 6 . 1 , 6 . 4 and
6.5): T
l k
= S
l k
+ 2d
i k
ν ψ = -ω 2
ν ω 2
+R
,
( U J )
,
(Π.2)
9ψ 3ω 3ψ 9ω _9χ 8y " 3y 3 x J
-2—
9x3y
S
x
*
x
-S
y
y
+
together with suitable constitutive equations. t h e M a x w e l l model
I n t h e case o f p l a n a r f l o w o f
( 2 . 7 2 ) , these give r i s e t o three hyperbolic equations.
The
s o l u t i o n o f t h e s e e q u a t i o n s , t h e m s e l v e s , may be a f a c t o r o f i m p o r t a n c e i n t h e breakdown p r o c e s s ( c f . § 1 1 3 . 2 ) , b u t i t i s u n l i k e l y t h a t t h i s f a c t o r i s f u n d a 0
m e n t a l , s i n c e f o r a g i v e n fixed v e l o c i t y f i e l d i t i s o f t e n p o s s i b l e t o o b t a i n r e s p e c t a b l e c o n v e r g e d s o l u t i o n s f o r t h e s t r e s s components f o r a h i g h v a l u e o f W„ T h i s means t h a t t h e p r e c i s e f o r m o f t h e c o n s t i t u t i v e e q u a t i o n s per se does n o t a p p e a r t o be o f m a j o r s i g n i f i c a n c e , o n l y t h e p o s s i b l e n o n l i n e a r i n t e r a c t i o n o f t h e s e e q u a t i o n s w i t h ( 1 1 . 2 ) and ( 1 1 3 ) i n t h e i t e r a t i o n p r o c e s s . 0
breakdown p r o b l e m i s p r e s e n t f o r a s p e c i a l c a s e o f t h e e x p l i c i t
Indeed t h e second-order
model g i v e n by S
l k
= - 2Wd
l k
,
and f o r i l l u s t r a t i o n
(11,4) p u r p o s e s we now c o n c e n t r a t e o n t h i s m o d e l .
Further, the
breakdown i s p r e s e n t f o r R = 0 ( i e . c r e e p i n g f l o w ) a n d t h e r e i s no p a r t i c u l a r 0
m e r i t i n r e t a i n i n g t h e i n e r t i a terms i n ( 1 1 . 3 ) i n t h e present c o n t e x t o f a t t e m p t i n g t o l o c a t e a mechanism f o r b r e a k d o w n . We now c o n s i d e r t h e f l o w p r o b l e m p r o v i d e d by ( 1 1 . 2 ) , ( 1 1 . 3 ) w i t h R = 0 a n d (11.4)
0
For plane f l o w s , t h e Tanner-Giesekus theorem ( s e e , f o r example, Tanner
1 9 6 6 ) e n s u r e s t h a t t h e N e w t o n i a n v e l o c i t y f i e l d f o r (W = 0 ) must s a t i s f y
(11.2)
319 - (11.4) with R = 0, W f 0.
I t i s easy t o show t h a t s u b s t i t u t i o n o f
(11.4)
i n t o ( 1 1 . 3 ) r e n d e r s t h e r i g h t hand s i d e o f t h i s e q u a t i o n z e r o , c o n s i s t e n t the Tanner-Giesekus Interestingly,
it
with
theorem. i s found t h a t e x i s t i n g f i n i t e d i f f e r e n c e a l g o r i t h m s
adap-
t e d t o t h e p l a n a r s e c o n d - o r d e r p r o b l e m a l s o b r e a k down a t some m o d e s t W v a l u e and a p o s s i b l e r e a s o n f o r t h e breakdown i s h i g h l i g h t e d .
Analytically,
N e w t o n i a n v e l o c i t y f i e l d must s a t i s f y t h e s e c oi nk d - o r d e r p r o b l e m . numerical noise i s exaggerated i n computing S
the
H o w e v e r , any
f r o m ( 1 1 4 ) and even more so i n i k t a k i n g t h e second d e r i v a t i v e s o f t h e components o f S to provide the ' f o r c i n g function'
in the v o r t i c i t y equation ( 1 1 . 3 ) .
z e r o , as we have n o t e d , b u t i t
0
T h i s f o r c i n g f u n c t i o n s h o u l d be
i s q u i t e e v i d e n t l y not zero ( n u m e r i c a l l y )
the s o l u t i o n f a i l s to converge.
The r e a s o n f o r t h e breakdown i n t h i s
r e l a t e d p r o b l e m s may t h e r e f o r e be due t o n o t h i n g more t h a n n u m e r i c a l
noise
the forcing function of (11.3) i n t e r a c t i n g with the nonlinear i t e r a t i o n I f t h i s reason f o r the numerical
breakdown can be u n e q u i v o c a l l y
when
and in
scheme
0
associated
w i t h t h e n u m e r i c a l n o i s e g e n e r a t e d i n t h e r i g h t hand s i d e o f ( 1 1 , 3 ) , t h e n e x t s t e p m u s t be t h e i m p l e m e n t a t i o n o f f i l t e r i n g t e c h n i q u e s Davies 1983) t o c o u n t e r a c t t h e e f f e c t o f
(see, for
example,
noise.
W i t h f i n i t e e l e m e n t s , a r e l a t e d r e s u l t has been f o u n d by M e n d e l s o n e t a l . (1982) i n t h e i r study o f the f l o w o f a second-order f l u i d i n a planar tion.
Using a numerical
contrac-
t e c h n i q u e s i m i l a r t o MIX4 ( § 1 0 „ 9 ) , M e n d e l s o n e t a l .
f o u n d t h a t t h e n u m e r i c a l n o i s e p r o d u c e s a l a r g e v o r t e x w h i c h s h o u l d n o t be there.
F u r t h e r c a l c u l a t i o n s w i t h r e f i n e d meshes have shown t h a t , on t h e
s p e c i f i c problem o f the planar c o n t r a c t i o n , a b e t t e r d i s c r e t i z a t i o n the i n t e n s i t y of the spurious 11.3.7
decreases
vortex.
Algorithms
Ample e v i d e n c e has been g i v e n i n C h a p t e r 10 t h a t t h e l i m i t i n g v a l u e o f W depends upon t h e m e t h o d used f o r d i s c r e t i z i n g t h e p a r t i a l and upon t h e n o n l i n e a r i t e r a t i o n a l g o r i t h m .
differential
equations
Much w o r k r e m a i n s t o be done i n
u n d e r s t a n d i n g how n u m e r i c a l c o n v e r g e n c e depends on t h e s e l e c t i o n o f
algorithm,
o f f i n i t e e l e m e n t s , and o f m e s h . 11.4
CONCLUDING REMARKS
Numerical s i m u l a t i o n o f non-Newtonian f l o w i s c l e a r l y a p r o v o c a t i v e area o f c u r r e n t r e s e a r c h , w h i c h i s endowed w i t h more t h a n i t s u s u a l s h a r e o f often unresolved,
interesting,
problems.
To end t h i s book we can do no b e t t e r t h a n a d v i s e t h e r e a d e r t h a t p r o g r e s s i n t h i s s u b j e c t m u s t depend on c a r e f u l
meaningful
numerical experimentation
on t h e most r e l e v a n t g u i d e l i n e s t h a t c u r r e n t m a t h e m a t i c a l
t h e o r y can o f f e r .
based We
320 must s t r i v e t o u n d e r s t a n d n o t o n l y t h e n a t u r e o f o u r n u m e r i c a l a l g o r i t h m s , also the a n a l y t i c p r o p e r t i e s o f our governing e q u a t i o n s .
All t h i s , of
must be c a r e f u l l y b a l a n c e d a g a i n s t o u r i n c r e a s i n g k n o w l e d g e o f f l u i d s gleaned from l a b o r a t o r y
experiments.
but
course,
non-Newtonian