NOTATION T, s t r e s s f o r c e tensor; M, m o m e n t s t r e s s tensor; u, d i s p l a c e m e n t vector; w, rotation vector; X, ext e r n a l m a s s f o r c e vector; Y, e x t e r n a l m a s s m o m e n t vector; p, density; I, t e n s o r c h a r a c t e r i z i n g the i n e r t i a l p r o p e r t i e s of the m e d i u m during rotation; H, internal e n e r g y density; F, f r e e energy density; S, internal entropy density; K, kinetic energy; A, p o w e r of the e x t e r n a l m e c h a n i c a l forces; Q, power of the e x t e r n a l t h e r m a l sources; q, t h e r m a l flux vector; w, heat liberation density; | absolute t e m p e r a t u r e ; | initial t e m p e r a t u r e ; E, unit t e n s o r ; a B, v e c t o r a c c o m p a n y i n g the t e n s o r B; B +, s y m m e t r i c p a r t of the t e n s o r B; B-, a n t i s y m m e t r i c p a r t o f t h e ( e n s o r B;T, Laplace t r a n s f o r m in f.

LITERATURE 1.

2. 3. 4. 5. 6. 7. 8.

V. A. Pal'mov, "Fundamental equations of nonsymmetric elasticity theory," Prikl. Mat. Mekh., 28__,No. 3, 401-408 (1964). E. V. Kuvshinskii and E. P. Aero, "Continual theory of asymmetric elasticity. Taking account of internal rotation," Fiz. Tverd. Tela, 5, No. 9, 2591-2598 (1963). W. Nowaeki, "Couple-stresses in the theory of thermoelasticity," Bull. Acad. Polon. Sci., Ser. So]. Technol., 14, No. 8, 505-513 (1966). Vl. N. Smirnov, "Equations of generalized thermoelasticity of a Cosserat medium," Inzh.-Fiz. Zh., 3_~9, No. 4, 716-723 (1980). V. A. Pal'mov, Vibrations of Elastic-Plastic Solids [in Russian], Nauka, Moscow (1976). A. I. Lur'e, Nonlinear Elasticity Theory [in Russianl, Nauka, Moscow (]980). W. Nowacki, Elasticity Theory [Russian translation], Mir, Moscow (1975). H. Parkus, Unsteady Temperature Stresses [in Russian], Gos. Izd-vo Fiz.-Mat. Lit., Moscow (1963).

EQUIVALENCE

OF

RHEOLOGICAL

EQUATIONS

POLYMER PART

CITED

CERTAIN

TYPES OF

OF

STATE

FOR

MEDIA

I. GENERAL B.

M.

ANALYSIS UDC 532.135

Khusid

The conditions a r e e s t a b l i s h e d under which rheological r e l a x a t i o n equations and rheologieal int e g r a l equations will be equivalent.

According to the c l a s s i f i c a t i o n p r o p o s e d by C. T r u e s d e l l and W. Nell [ 1, 2], the t h e o l o g i c a l equations of s t a t e for a f t e r e f f e c t media fall into t h r e e groups: differential equations, relaxation (or s t r a i n r a t e ) equations, and integral equations. Equations of the differential type a r e applicable only to flow with a s m a l l D e b o r a h n u m b e r , i.e., to fluids with a r e l a x a t i o n t i m e much s h o r t e r than the t i m e s c a l e of flow. In o t h e r c a s e s one m u s t u s e either r e l a x a t i o n equations or i n t e g r a l equations of state. Many r e l a x a t i o n equations and i n t e g r a l equations of s t a t e have a l r e a d y been p r o p o s e d [4-7]. As a rule, they a r e p a r t l y b a s e d on m i c r o s c o p i c models of p o l y m e r fluids a n d on c e r t a i n a s s u m p t i o n s r e g a r d i n g the motion of the medium. They also include s e v e r a l p a r a m e t e r s which m u s t be evaluated e m p i r i c a l l y for any specific m a t e r i a l . The theological equations m o r e o r l e s s a g r e e with e x p e r i m e n t s . According to the bibliography on this subject, however, none of them adequately d e s c r i b e s the rheological c h a r a c t e r i s t i c s of v a r i o u s fluids in complex t r a n s i e n t s t r a i n states. This m a k e s it n e c e s s a r y to t r y various models for a given m a t e r i a l and then, a f t e r c o m p a r i s o n with the e x p e r i m e n t , s e l e c t the m o s t a p plicable ones. Such a d i v e r s i t y of rheological equations for fluids with m e m o r y i m p e d e s the p r o g r a m m i n g of n u m e r i c a l solution of hydrodynamic and t h e r m a l p r o b l e m s for t h e o l o g i c a l l y complex fluids. In the c a s e of r e laxation equations of s t a t e one f o r m u l a t e s the p r o b l e m s of hydrodynamics and heat t r a n s f e r in the f o r m of p a r tial differential equations, which can be solved by conventional methods of finite d i f f e r e n c e s . F o r i n t e g r a l .... B e l o r u s s i a n Polytechnic Institute, Minsk. T r a n s l a t e d f r o m I n z h e n e r n o - F i z i e h e s k i i Zhurnal, Vol. 42, No. 4, pp. 670-677, April, 1982. Original a r t i c l e submitted October 10, 1981.

470

0022-0841/82/4204-0470507.50

9 1982 Plenum Publishing C o r p o r a t i o n

equations of s t a t e these p r o b l e m s a r e f o r m u l a t e d in t e r m s of s y s t e m s of i n t e g r o d i f f e r e n t i a l equations. H e r e it b e c o m e s n e c e s s a r y to i n t e g r a t e o v e r t r a j e c t o r i e s of moving fluid p a r t i c l e s , which m a k e s it difficult to apply conventional methods of finite d i f f e r e n c e s . In view of this, t h e r e a r i s e s the question as to whether integral equations of s t a t e can be reduced to equivalent s y s t e m s of differential equations. A reduction of v a r i o u s t h e o logical equations to a single type would also r e v e a l the m o r e distinct d i f f e r e n c e s between the physical hypothes e s on the b a s i s of which v a r i o u s rheological equations of s t a t e have been c o n s t r u c t e d for a f t e r e f f e c t media. In this study will be c o n s i d e r e d this p r o b l e m of reducing the i n t e g r a l r h e o l o g i c a l equations of s t a t e of the (1) kind for an i n c o m p r e s s i b l e fluid to an equivalent s y s t e m of differential equations not containing higher than f i r s t d e r i v a t i v e s of the s t r e s s t e n s o r with r e s p e c t to t i m e t "

T:~Tcr

T~=

j'~t~(t, z)[f~(t, "r

(1)

H e r e s u m m a t i o n is p e r f o r m e d o v e r different p a i r s ( ~ , ~) and for different f i ' s , m o r e o v e r , the sets of c o r r e s p o n d i n g functions ~(~ can also differ. In this equation the i s o t r o p i e additive t e n s o r has been omitted, ~ is the r e l a x a t i o n function, and E is the unit t e n s o r . Integration is p e r f o r m e d o v e r the t r a j e c t o r y of a moving fluid p a r t i c l e . Most of the now available s e m i e m p i r i c a l i n t e g r a l equations of s t a t e for p o l y m e r m e l t s and conc e n t r a t e s a r e put in the (1) f o r m [ 4 - 7 ] . In analogy with the theory of relaxing lattice a c c o r d i n g to Lodge, Y a m a m o t o , et al. [8], one can i n t e r p r e t e x p r e s s i o n (1) as one d e s c r i b i n g the " r e s u l t a n t " tension T a ft on l a t t i c e s of kind ~ with nodes of group c~. Accordingly, the quantities ~ a ( t , ~) a r e p r o p o r t i o n a l to the n u m b e r of nodes f o r m e d at instant of t i m e ~ and still existing at instant of t i m e t, while the s y m m e t r i c t e n s o r s ~fi (% 7) E r e p r e s e n t s t r a i n produced by flow of the fluid in a /3-lattice at instant of t i m e T r e l a t i v e to its s t a t e at instant of t i m e t, with ~ fi(t, t) = E. We wiI1 now t r a n s f o r m each t e r m in s u m (1) into a s y s t e m of differential equations. An evaluation of the substantive d e r i v a t i v e with r e s p e c t to t i m e yields t

t

-Dt

Dl

[f~(t, ~ ) - - E l d ~ +

~t~(t, ,~) D~B(t'Dt "~) d~.

(2)

Equations (1) and (2) r e d u c e to a s y s t e m of f i r s t - o r d e r differential equations in time, if DP.~(t, ~)

Dt

"~[A;~(t) f~v(t, ~ ) + f~,(t, T)A,~(t)I,

D ~ (t, ~) Dt =-

X •

(3)

~)'

(4)

?

E x p r e s s i o n (3) t a k e s into account the s y m m e t r y of the ~2 f i - t e n s o r , s u p e r s c r i p t T denoting a transposition. In a c c o r d a n c e with the i n v a r i a n c e of the rheological Eq. (1) (principle of objectivity) [1, 2], a change of the r e f e r ence s y s t e m (Q (t) r e p r e s e n t i n g an a r b i t r a r y orthogonal tensor) X ~ X * = Y(t) + Q (t) (X - Z) will t r a n s f o r m the t e n s o r ~2fi(t, T) into

~B (t, ~) -~ ~ if, ~) = Q (t) ~ (t, ~) O~ (0

(5)

Rewriting Eq. (3) in the new r e f e r e n c e s y s t e m , with the aid of r e l a t i o n (5), we obtain the t r a n s f o r m a t i o n law f o r the t e n s o r AfiT (5fi7 is the K r o n e e k e r delta, 5fit = 0 forfi ~ ~ and 5fi~/ = 1 f o r fi = y ) A ~ - - + A ~9 == QA~Q~

+ ~DQ O~.

(6)

With the aid of the t r a n s f o r m a t i o n law for the v o r t i c i t y t e n s o r W = ~/2 ( V ~ ' - V v-'T) [1, 2], n a m e l y W ~ W* = Q W Q T + DQ/Dt QT, it is p o s s i b l e to r e p r e s e n t the t e n s o r Afl~ in the f o r m A~ = 8~W + B~,

B ~ ~ B ~9 : Q B ~vQ,v

(7)

w h e r e the t e n s o r Bfi 7 s a t i s f i e s the p r i n c i p l e of objectivity (6). Since the coefficients in s y s t e m (4) a r e not functions of T, one can e x p r e s s the function ~ (t, T) as ~c~ (t, ~) = Zy F~v(t, ~)g0~,(~) with F ~~ l, denoting the 1 / fundamental solution to s y s t e m (4) which s a t i s f i e s the condition Ec~T (% r) = fisT- I n s e r t i n g e x p r e s s i o n s (3), (4), and (7) into Eq. (2), and introducing the function p ~ =

~ ~(t,

z ) d , , we a r r i v e at the s y s t e m of f i r s t -

o r d e r d i f f e r e n t i a l equations in t i m e which is equivalent to Eq. (1)

Dp~ + X uc,~p~,= - - ~ , Dt

(8)

47]

?

?

1'

H e r e the notation of the J a u m a n n d e r i v a t i c e has been used o DT~ T~ WT~ + T~W. DI" All quantities in Eqs. (8) a r e to be evaluated at a given point at instant of t i m e t. T h e s e equations c o n s t i tute, of c o u r s e , a special c a s e of the g e n e r a l f o r m of r e l a x a t i o n equations for fluids which s a t i s f y the p r i n c i p l e of objectivity [ 1]. F o r m o s t rheologieal models found in the bibliography on this subject [4-7], •

= •

A~ : A ~

(9)

in Eqs. (3) and (4). According to the i n t e r p r e t a t i o n b a s e d on the theory of lattice relaxation, the f i r s t of r e l a tions (9) m e a n s that the p r o b a b i l i t y of breakup of group a n o d e s is p r o p o r t i o n a l to the n u m b e r of such nodes. t

In this e a s e # a (t, r) = ~a(~)e•

, where ~

defines the probability of breakup of one node. With the

second of relations (9) taken into account, Eq. (3) b e c o m e s Dft B(t, ~)

Dt

-

(10) t

It follows f r o m Eq. (10) (tr denotes the t r a c e of a t e n s o r ) that det ~2fl(t, r ) = exp [ - - f t r (A~ + A~)d~]. Since the T

eigenvalues of the s y m m e t r i c t e n s o r ~2fl(t, T) at t = r a r e equal to unity and since, det ~fl(t, T) > 0, t h e s e eigenvalues a r e always positive and there exists a p o s i t i v e - d e f i n i t e t e n s o r ~=~fi(t, r) [9]. This p e r m i t s us to use the r e p r e s e n t a t i o n T ft~ = toi3{%,

where coO = t t f i ~ ,

( 11 )

and ttfi is s o m e orthogonal tensor. Inserting the r e p r e s e n t a t i o n (11) into Eq. (10) yields

D~oi (t, ~) -Dt

c%(t, ~)A~(t),

{o~[t=T-----E.

(12)

Equations (10) and (12), which have been derived f r o m e x p r e s s i o n (1), d e t e r m i n e the t e n s o r s ~ f and ~ffl only at t _> r . We will now establish s e v e r a l p r o p e r t i e s of the t e n s o r ~fi(t, r ) . We a s s u m e a given mode of d e f o r m a tion of an A~}(t) lattice. Let us c o n s i d e r an instant of t i m e r _> 4. Since ~ t ( r , 4) {aft(t, ~ ) I t = r = E, hence the l i n e a r i t y of relation (12) m a k e s --I

{o~ (% ~)r

~) ----{% (t, "0,

t~T~.

(13)

Differentiating this equations with r e s p e c t to r and using r e l a t i o n (12), we obtain De% (t, ~) D~ -- A8 (~) c% (t, ~),

co~l,=t = E.

(14)

We next define the t e n s o r wfl(t, r) at t -~ r by the equality c% (t, ~) = {o7~ (T, t),

9 ~ t.

(15)

Differentiating e x p r e s s i o n (15) with r e s p e c t to r , with relation (12) taken into account, we again a r r i v e at Eq. (14). T h e r e f o r e , Eq. (14) defines the t e n s o r ~o/j(t, r) at all instants t and r . Since ~ofi (t, r) w ; l ( t , to) It=t0 = E at any to, hence the l i n e a r i t y of e x p r e s s i o n (14) m a k e s o}~(t, "~)- o}~(t0, T){%(t, to).

(16)

This e x p r e s s i o n extends relation (13) to t h r e e a r b i t r a r y instants of time. T e n s o r tot(t, r) is the m a t r i c a n t [9] of Eq. (14) and relation (16) d e s c r i b e s its b a s i c p r o p e r t i e s . According to r e l a t i o n (16), the distortions of a lattice at instant of t i m e r r e l a t i v e to its s t a t e at instant of t i m e t do not depend on the i n t e r m e d i a t e s t a t e at instant of t i m e t 0. P r o p e r t y (16) fully defines the t e n s o r wf(t, r ) . Starting with it, one can easily d e r i v e r e l a tion (15) and then Eqs. (12), (14), (10). Thus the second of relations (9) is equivalent to an a s s u m p t i o n that the r e l a t i v e d e f o r m a t i o n of a lattice has the multiplicative p r o p e r t y (16). On the b a s i s of Eq. (14), it is e a s y to d e m o n s t r a t e that changing the r e f e r e n c e s y s t e m will t r a n s f o r m the t e n s o r wp( t, r) as follows: w/j(t, r) oj;(t, r) = Q(r)wfi(t, T)Q T (t). In another study (10) the t e n s o r ~ f ( t , r) satisfying Eq. (14) and the t e n s o r ~ f ( t , T) defined by relation (11) a r e called nonholonomic or g e n e r a l i z e d d e f o r m a t i o n s . Relations (9) g r e a t l y simplify the t h e o l o g i c a l equations (8) and r e d u c e them to the f o r m of a Maxwell fluid

472

0

Ta~ + •

(I7')

+ B~ Ta~ + Ta~Bt~ = p= (B~ + B~), Dp,~ + • Dt

(1T')

= -- ~.

Such a r e the differential equations to which all i n t e g r a l rheologieal equations of the f o r m t

t

(18) with the t e n s o r oft(t, ~) satisfying the p r o p e r t y (16) can be reduced. The quantities ~ a , ~ a depend on the def o r m a t i o n m o d e at a given point at instant of t i m e t. Some theological models found in the bibliography on this subject c o r r e s p o n d to the i n t e g r a l equation t

f

~_~

~

Dz

which is obtained through integration of Eq. (18) by p a r t s . Functions ~c~ and 9)a a r e r e l a t e d through the equation ~t~+•

I n s e r t i n g this r e l a t i o n into Eq. (17") yields Pa = - ~ a -

A f e a t u r e distinguishing model (18)

f r o m model (19) is that the quantity Pa a p p e a r i n g as the s h e a r modulus in Eq. (17') can be d e t e r m i n e d f r o m Eq. (17") and depends on d e f o r m a t i o n h i s t o r y . In model (19), on the other hand, the quantity Pa depends only on the flow c h a r a c t e r i s t i c s at instant of t i m e t. When • q0a, ~ a a r e constants, then this difference between the two models (18), (19) vanishes and they b e c o m e fully equivalent. Rheological equations of the one (18) kind a r e g e n e r a l i z a t i o n s of e a r l i e r Lodge, Ward, Jenkins models and rheologieal equations of the (19) kind a r e g e n e r a l i z a t i o n s of e a r l i e r Oldroyd, Friedrichson, Walters models [4-7 ]. In o r d e r for Eqs. (17') and (17'T) to m e r g e into the single relation (18), it is n e c e s s a r y that the initial conditions for them be stipulated in the quiescent state, i.e., at t ~ co (the p r o b l e m of initial conditions for relaxation equations has been t r e a t e d m o r e thoroughly in [31). The t e n s o r lift in Eqs. (17') and (17") has been defined ambiguously. A t r a n s f o r m a t i o n BB-~ B~ = g~ - - b~E-- R~ ( T ~ - - p~E), p ~ - ~ / ~ = p~ - - ~ , ~ - ~ • = ~ + 2b~, T ~ - ~ T ~ = T ~ - - ~ E where D~

Dt

~ {r

d RB is an a r b i t r a r y a n t i s y m m e t r i c tensor, does not have the f o r m of Eqs. (17') and

{17"). Since the s t r e s s t e n s o r has been defined p r e c i s e l y , except for the i s o t r o p i c tensor, this t r a n s f o r m a t i o n will not a l t e r the r h e o l o g i c a l equation. In o r d e r f o r the i n t e g r a l equation (1) to be reducible to the s y s t e m (17'), (17") with a s y m m e t r i c t e n s o r Bfl, the t e n s o r 1~/~m u s t satisfy the equation lift - li~ = ( T a tf - p a E ) l~ft+ ttfi(Ta/~ - p a E ) . In a s y s t e m of coordinates f o r m e d bythe e i g e n v e c t o r s of the t e n s o r Taft we have 1Rftij = (Bftij - Bflij)/(T(~ftj + Taftj - 2 p a ). This e x p r e s s i o n m u s t be definite for any d e f o r m a t i o n mode and, s p e c i f ically, when Tafti + Taflj = 2pa. We will d e t e r m i n e the condition under which the g e n e r a l r e l a x a t i o n - t y p e equation 0

T + a T + u = ~E,

(20)

with the s y m m e t r i c t e n s o r Y, dependingon the flow mode at a given point at instant of t i m e t , b e c o m e s t r a n s f o r m e d into the s y s t e m of Eqs. ( I 7 ' ) and (17") with a s y m m e t r i c t e n s o r B. Inserting T = T - pE into Eq. (20) yields o

T+ •

Dp

--+• Dt

(21)

S y s t e m (21) c o r r e s p o n d s to the f o r m (17 r), (17") if t h e r e is a t e n s o r I3 which s a t i s f i e s the equation Y = liT + Tli. In a s y s t e m of coordinates f o r m e d by the e i g e n v e e t o r s of the t e n s o r T we have Bij = Yij/(Ti + Tj). This relation m u s t be definite for any d e f o r m a t i o n mode and, specifically, when Ti + Tj = 0. T h e r e f o r e , not e v e r y i n t e g r a l equation (1) and relaxation equation (20) is reducible to the s y s t e m (17'), (17") with a s y m m e t r i c t e n s o r ~fi. Let us examine c l o s e r the meaning of s y m m e t r i c t e n s o r s B~. F r o m r e p r e sentation (I1) and Eq. (14) we have D~DI = ~ (W + BI3)i% + ~

W + B~3)~ = 2~B~o~.~

(22)

It is evident h e r e that tiff = E and, consequently, no s t r e s s e s a p p e a r when Bft = 0. When Bfl = D, where D = t/2(V~ + V~T ) is the s t r a i n r a t e t e n s o r of the m e d i u m , then Aft = 7 ~ and Eq. (14) yields w = Ft(~) as the r e l a t i v e s t r a i n g r a d i e n t in the m e d i u m [ 1-3]. T h e s e p r o p e r t i e s as well as the t r a n s f o r m a t i o n law for t e n s o r

4 73

w f l m a k e it possible, upon a change of the r e f e r e n c e s y s t e m , to i n t e r p r e t t e n s o r

1]/3 as the s t r a i n r a t e t e n s o r of a B - l a t t i c e . The quantities ~
V~. (~ - - t) ~ n= 1

V~ '

n!

Dn QO Dz n

into integral Eqs. (18) and (19), r e s p e c t i v e l y :

'

('~--t)'~

-

T:2

ff,[$ n = l

t-

t

e x p ( - - I z<~d~)d'c,

(1:--t) '~ exp (--,t' • n,

--~

As t ~ - oo, we find that ~0a ~ q~0,a , ~o~ --* ~0,c~, z a ~

d'r.

"r

z0,o~ with s u b s c r i p t 0 denoting the values of t h e s e quan-

tities in the undeformed state and qa,n ~ ( - l~n'~ /~ n+l q a~n ~ ( - 1 ) n§ "~ , ~, # "t'Oia/vcOt~' ~"0, (~/~n+l / O,Od" When functions ~0o~ ~o~ depend on the d e f o r m a t i o n mode, then the quantities q a n , qo~,n can be conveniently e x p r e s s e d in the f o r m

(-- 1)~
with the functions d e t e r m i n e d f r o m the equations D~a~ Dt

+ •

= ~J~.

Using the multiplicativity r e l a t i o n (16), one can r e w r i t e the e x p r e s s i o n for t e n s o r Vfin in the f o r m (t o is an a r b i t r a r y instant of time) V~ (t) = ~ (t, /0) D~ ~ ( t ~ t) o)~(t, t0). Dt~

Differentiating this e x p r e s s i o n with r e s p e c t to t, we obtain the r e c u r r e n c e r e l a t i o n DV~ + A~V~ q- V~A~,

V~0

E,

Dt

analogous to the r e c u r r e n c e r e l a t i o n s for R i v l i n - E r i c k s e n and W h i t e - Metzner t e n s o r s [ 3]. Specifically we have VB1 = 2Bfi~ When the t e n s o r B B depends on the s t r e s s t e n s o r , then the d e r i v a t i v e D V B n / D t m u s t be c a l culated by elimination of the d e r i v a t i v e s of the s t r e s s t e n s o r with the aid of Eq. (17'). The preceding m a t h e m a t i c a l analysis thus r e v e a l s that m o s t rheological equations for flowing p o l y m e r m e d i a can be derived f r o m the equations of a Maxwell fluid, n a m e l y by introduction of the dependence of quantities ~a, q~ on the flow c h a r a c t e r i s t i c s and by use of m o r e i n t r i c a t e " e f f e c t i v e " s t r a i n r a t e t e n s o r s Bfi in

474

Eqs. (17') and (17"). Into account is also taken here that the tensors Bfi can depend on the flow c h a r a c t e r i s tics. LITERATURE I. 2.

3. 4. 5.

6. 7. 8. 9. 10. 11. 12.

CITED

C. Truesdell and W. Noll, "Nonlinear field theories of mechanics," in: Handbook of Physics, Vol. Ill/3, Springer- Verlag, Berlin ( 1965). C. Truesdell (ed.), Continuum Mechanics, Vol. 2, Rational Mechanics of Materials, Gordon and Breach

(1965). G. Astarita and G. Marucci, Basic Hydromechanies of Newtonian Fluid [Russian translation], Mir, Moscow (1978)o A.S. Lodge, Body Tensor Fields in Continuum Mechanics, Academic Press, New York (1974). R.R. Bird, R. C. Armstrong, and O. Hassager, Dynamics of Polymer Fluids, Vol. 1, Wiley, New York (1977), G.V. Vinogradov and A. Ya. Malkin, Rheology of Polymers [in Russian], Khimiya, Moscow (1977). Chaig Dei Khan, Rheology of Polymer Reprocessiug [in Russian], Khimiya, Moscow (1979). A.S. Lodge, "Constitutive equations from molecular network theories for polymer solutions," Rheol. Acta, 7_, No. 4, 379-392 (1968). F . R . Gantmakher, Theory of Matrices, Chelsea Publ. T . D . Goddard, " P o l y m e r i c fluid mechanics," Adv. Appl. Mech., 19__,143-219 (1979). C. Truesdell and R. A. Toupin, "Classical field t h e o r i e s , " Handbook of Physics, Vol. III/1, SpringerVerlag, Berlin (1960), pp. 226-793. L . I . Sedov, Introduction to Mechanics of Continuous Media [in Russian], Fizmatgiz, Moscow (1962).

INTERRELATIONSHIP BIOLOGICAL

OF

RHEOLOGICAL

CHARACTERISTICS

IN

E. G. Tutova, E. V. and I. V. Zhavnerko

Ivashkevich,

AND COMPLEX

SYSTEMS UDC

532.535

The interrelationship of rheologieal properties and quantitative indices of fluids of biological origin is analyzed. Possible variants of the use of the viscosity for estimating the state of the system are presented. In studying labile systems, whose properties depend greatly on the parameters of the external medium (t, P, q~)or the state of the system itself (t, W), it is necessary to choose a characteristic physical indicator, which reflects to a certain extent the state of the substance, as well as the kinetics and dynamics of its variation. Typical representatives of such materials are heterogeneous systems of biological origin- microbe biomasses. It is well known that the presently existing methods of microbiological analysis are imperfect and are distinguished by their long duration., measured in days, and high degree of error. The effect of the error can be eliminated by multiple repetition of the experiment and statistical analysis of the results obtained, as is customary in studying probabilistic processes. However, inthis case, the duration of the analysis increases even more, which can be eliminated only by developing and applying new improved methods, based on the interrelationship of physical and biological properties of the system. It is well known that microbiological materials of different nature are characterized by a wide range of rheological properties from Newtonian to plastic [i], which can serve as qualitative and quantitative indices of heat and mass transfer in bioengineering and biotechnology processes. Thus, the viscosity of the starting feed media exceeds by not more than a factor of 1.5-2 the viscosity of water, while during the growth of life of microorganisms, this quantity increases by one to two orders of magnitude. The increase in the viscosity of the

A. V. Lykov Institute of Heat and Mass Transfer, Academy of Sciences of the Belorussian SSR, Minsk. Translated from Inzherno-Fizicheskii Zhurnal, Vol. 42, No. 4, pp. 677-681, April, 1982. Original article submitted February 5, 1981. 0022-0841/82/4204-0475507.50

9 1982 Plenum Publishing Corporation

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