【文章內(nèi)容簡介】 ? ? ? ??? ? ? ? ? ? ?:1:2:3rz?11 22 22 33? ? ? ???: F o r f l ui d s t he f i rs t a nd s e c o nd no rma l s t res s d i f f e ren c e s , a nd , a reb o t h i n s he a r f l o w , a nd e q . 2. 3 5 s ho w s t ha t t he no rma l p res s ure e xe rt e d o n t he l( N ) N eub ricz aw t o nt e d ialin e ro di n c w i t h t he rad i usre ase s11 2222 33??????: F or f l u i d s w e h a ve a l rea d y i n d i c a t e d t h a t i s p rat i c a l l y a l w a ys n e g a t i vew i t h a ( P )n u meri c a l val u e muc h l a rg e r t h a n t h a t of . We s e e t h a t t h e n orma l s t res s e s ma y c a u sp ol yme re t hic e t ot a l n orma l p res s u re t o i n t h e rad i a l d i r e c t i on .de c re aseThe resultant formula derived in this example is: II. Use of Equations of Change to Analyze the Distribution of the Normal Pressure (N) (P) ? r? zP Examples amp。 : ConeandPlate Instrument FIG. . Coneandplate geometry 0303()2 WTR????? 1 222() FR? ????00W???:: TF F orc e req u i red t o ke e p t i p of c on e i n c on t a c t w i t h cT orq u e on p l a t ei rc u l a r p l a t e :?S h e a r ra te 0 ( , ) 0 r a d ( 6 )rv r v v????????( 1) S t e a d y , l a m i n a r, i s o t h e rm a l f l o w( 2) o n l y 。( 3)( 4) N e g l i g i b l e b o d y f o rc e s( 5) S p h e ri c a l l i q u i d b o u n d a ryA ssumpt i o ns :(From of ref 3) (homogeneous) 1 ( ) :??T h e f ir st n o r m a l st r e ss d if f e r e n ce co e f f icie n t ( ) :??S h e a r r a t e d e p e n d e n tv isco sit y 00: : : WR?A ng ul a r vel oc it y of c on eC on e a ng leRa d ius of c irc ul a r p la t e? ?2 2 200 R r d rT d? ?? ?????? ??Example: Uniaxial Elongational Flow m a x m a x m a x 0l n ( )t L L?? ??:H e nc ky s tra in0m a x: : LLIn i t i a l s a m p l e l e n g t hM a x i m u m s m a p l e l e n g t h( ) ( )zz rr F t A t??? ? ?( ) : ( ) : FtAtT o t a l f o r ce p e r u n it a r e a e xe r t e d b y t h e lo a d ce llI n st a n t a n e o u s co r ss se ct io n a l a r e a o f t h e sa m p leFIG. . Device used to generate uniaxial elongational flows by separating Clamped ends of the sample :T h e N o rm a l S t re s s D i f f e re n c eExptl. data see 167。 0000()() tzz rrFtAe ???????? ?? ? ?:? ?T he T r a ns i e nt Elo ng a t i on a l V i s c os i t y 00: : A? E lo n g at io n r at eI n it ial cr o ss sect io n al ar ea o f t h e sam p lezr? Supplementary Examples ? Capillary: o Example : Obtaining the NonNewtonian Viscosity from the Capillary ? Concentric Cylinders o Problem : Viscous Heating in a Concentric Cylinder Viseter ? Parallel Plates: o Example : Measurement of the Visetric Functions in the ParallelDisk Instrument o Problem : ParallelDisk Viseter o Problem : Viscous Heating in Oscillatory Flow . Basic Vector/Tensor Manipulations ? Vector Operations (Gibbs Notation) 1 ( A . 2 1 , 2 )0 ijijijij??? ? ????? ????, if, ifij?T he Kron ec ker d el t a Dot product: 1 2 31 2 3iii u u uu u??? ?? ? ? ??123xyz?????? ??u1?2?3?Vector: 1 2 3 ( ) ( ) ( )i i j j i j i ji j ijjii i ii i iiiu v u u u vu v u v? ? ? ???????? ? ? ? ??????? ????? ? ???L et1 1 2 3 2 3 1 3 1 21 3 2 1 1 3 2 2 1 3 ( A .2 3 , 4 , 5 )0 ijkijkijkijkijk???? ? ? ??? ? ? ??? ??, if , , or, if , , or, if any t wo ind ic es are al ikeijk?T he p erm ut a t i on s ym b ol Cross product: ( ) ( ) ( ) ( )i i j j i j i ji j iji j ij k kijku v u u u vuv? ? ? ???????? ? ? ? ??????? ???????31 ( A .2 1 5 )i j ijk kk? ? ? ??? ? ??U sing1 2 31 2 31 2 3u v u u uv v v? ? ???? Tensor Operations 1 1 11 1 2 12 1 3 132 1 21 2 2 22 2 3 233 1 31 3 2 32 3 3 33 i j ijij? ? ? ?? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ? ???? ? ?? ? ?? ? ???12 1321 2331 32112233pppp? ???????? ? ??? ? ?????? ? ??????Hydrostatic pressure forces Stress tensor or Momentum flux tensor Tensor: 1 2 3 3 1 3 2 3 3( , , )? ? ? ??1 1 1 1 2 1 3( , , )? ? ? ??2 21 22 23( , , )? ? ? ??FIG. The stress tensor Stresses acting on plane 1 The total momentum flux tensor for an inpressible fluid is: Normal stresses ( ) i j ij k ki j kij k i j kijkkjij j iijn nnn? ? ? ? ?? ? ? ??????? ??? ? ??? ?????????? ? ???L etnT h e t ot a l mo me n t u m f l u x t h rou g h ori e n t a ta s u rf a c e of i on i s :Example: Some Definitions amp。 Frequently Used Operations: ??222:::1 0 0: 0 1 00 0 1i j ij i j jiij ijii ii ii j ijijxx? ? ? ? ? ? ??? ? ? ???????????????? ? ? ? ? ???????????????????=T ra n s p o s e : G ra d i e n t : L a p l a c i a n : Un i t t e n s o r : ( ) ( )( ) ( )jijij iikkik iij k i jk l lijk lij jl i lijkjijk kijk iij jiijvvxxvvvvvvxv?????? ? ? ? ?? ? ? ???? ? ????????????? ? ???????? ? ??????????Cartesian coordinate Cartesian coordinate . Material Functions in Simple Shear Flows ? Remarks: ? A variety of experiments performed on a polymeric liquid will yield a host of material functions that depend on shear rate, frequency, time, and so on ?