an equation of state for non-newtonian fluids

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An equation of state for non-Newtonian fluids

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1966 Br. J. Appl. Phys. 17 803

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BRIT. J. APPL. PWYS., 1966, VOL. 17

ation of state for non-NewtonianA. KAYEDepartment of Mathematics, Manchester College of Science and Technology

M S. received 3vd February 1966Abstract, A n equation of state for non-Newtonian fluids is suggested which shouldapply to polymer solutions and melts. It assumes that the polymer molecules formtemporary crosslinks in solution and that the lifetimes of these crosslinks are stress-dependent. Irrespective of the form of the stress-dependence it is shown that aconsequence of the equation of state is that

PU2 X P I I -P22where p u ( i , j=1 , 2 , 3) are the components of the stress tensor in simple shear flowreferred to the usual axes.

l.:IntroductionmaterialLodge (1956, 1964) has suggested the following equation of state for an incompressible

where pij are the rectangular Cartesian coordinates of the stress tensor, p is the magnitudeof the isotropic stress, xi and xi are the rectangular Cartesian coordinates of a typicalparticle at times t and t respectively, Tis the absolute temperature and k is Boltzmanmsconstant. S is a numerical constant approximately equal to 2, and N ( t - , T ) s afunction of t - and T which depends on the material in question. This equation ofstate was derived from tha t of a material which obeys the kinetic theory of rubber elasticity(Treloar 1958) by assuming, inter alia, that the cross links of tha t theory are time-dependentin the sense th at they break a nd reform but the total number remains constant-thus th eequation of state should describe a material which contains long molecular chains withtemporary crosslinks. A comparison with the rheory of rcibber elasticity enables a physicalinterpretation of S and N( t - , T ) o be given: N( t- , T )dt is the nnmber of crosslinksand S N ( t - , T ) t the number of chain segments per unitvolume which are formed orjoin the network at the time t in a n interval of time dt an d still remain joined to the networkat time t .This equation of state gives a good qualitative description of the properties of poly-mer solutions and melts b ut does no t give a quantitative description; for example, it pre-dicts a stress-independent viscosity. One assumption made in Lodges theory is that therate of breakage of crosslinks does not depend on the stress. Not only is this physicallyunlikely but leads directly to a viscosity independent of shear rate.It is the purpose of this paper to devise an equation of state, similar to that of Lodge, inwhich the rate of breakage of crosslinks is stress-dependent.2. The equation of state

We consider a special form of Lodges equation of state. Consider a number of cross-links A,dt which are formed at time t in an interval of time dt. At time t let the num berof these crosslinks which remain unbroken be Adt. The rate of breakage of these cross-803


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804 A . Kayelinks, (dA/d t) dt, will in Lodges theory be a function of t - and A ; we only considera special case in which the rate of breakage is proportional to Adt, that is

This is, of course, the physically reasonable assumption that the rate of breakage is pro-portional to the number left. Noting tha t when t = , A =A,, we find

A ( t )=Aoe-t@-t) (3)where A( t )dt is the number of crosslinks still unbro ken at time t of the group A,dt whichwere formed in the interval of time dt at time t . Tha t is, we may takeif A refers to unit volume. A ( t )=N ( t - , T ) (4)Now, if the to tal number of crosslinks in the network at time t is N o , his w ill be given by

( 5 )

(6 )

Ak o= / N ( t - , T ) t = A,,-k(t- t) dt =0f = - 3 t f = - 0 3Hence N ( t - , T )=kNoe-k(t-t)where of course k and N o may be functions of T.Let us now assume th at the rate of breakage depends not only on A but also on thestress. Since the rat e of breakage is a scalar we must take dA/dt as a function of theinvariants of the stress. Moreover, since the material we are considering may be assumedto be incompressible these invariants should be independent of the magnitude of the iso-tropic stress. It is shown in the Appendix that the two such invariants are

Q 1 = PI2- 3Pz (7 )where Q2 = ?,S - ?l?2 f 273,

PI=c Paa3a = lP,=det pap.Hence we may take

( 9 )

where g is some function of Q 1 nd Q z which specifies how the rate of breakage of cross-links depends on stress and g* ( t ) is that function oft which is equal to g ( Ql, 0,)when thestress invariants are evaluated a t time I. Integrating we findt = t

A ( t )=A . exp (- j g*(t) dt).f tAt time t the rate of breakage of crosslinks per unit volume is g*(t)No since by hypothesisthe rate of breakage is independent of the time of formation and is proportional to thenumber of crosslinks. Since we also assume that the rate of breakage is equal to the rateof formation, the number formed at time t in an interval dt, A&, is given by g*(t)N,dt.Hencet

~ ( t , , T )=g* ( ( t ) ~ ,xp (- g*(t> di t ) . (14)I


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An equation of state for non-Newtonian fluids 805N ( t , t ' , T ) t' is the number of crosslinks which were formed in time dt' at time t' and arestill unbroken at time t. N ( t , t ' , T ) s now not necessarily a function of t - '. By anargument similar to tha t of Lodge we now arrive at an equation of state for a materia1 withstress-dependent crosslinks

t t

The consequences of this equation in steady shear flow are now examined.

3. Steady shear flawplanes, the state of flow in steady shear flow may be defined by the equationsChoosing the xl axis parallel to the streamlines and the x2 xis normal to the shearing

X I=I' +G (t- ') xZ'xz=Xg'x3=x;

where G is the shear rate. NOW , ince the flow is steady, the components of the stress pijwill be constant and hence g will be independent of time. Thus, inserting equations (16)(17) and (18) into (15) we find thatPI1 - 22 =(SkT2No/g2)G2 (19)Pnn - 33=0Y13= 2 8 =0

= SkT Nolg) G

where g is a function of the invariants of pij.The invariants of pi j , Q1 nd Q 2 ,are given in the present case by

where g =g ( Q i , (2 2 ) .If g were known as a function of Ql nd Q 2 hen these equations would enable g to befound as a function of G, and hence the variation of p l l - z 2 and plzwith G could befound. Now i t is physically reasonable th at the rat e of breakage of crosslinks shouldincrease with stress. Hence the fo rm of g ( Q 1 , Q ,) should be such that g does not fallwith Q, and Q2 hatever values the pij take. Ql is easily proved to be a non-negativefunction of the stress components ;henceg = g + P Q i

where a and /3 are positive constants, certainly meets this requirement and gives a viscosity(7= 1 2 / G )which falls with increasing shear rate.Again, f rom equations (19)-(22)) we see a t once that

S~kTNOCpl,- zz)=P l 2and hence, irrespective of the form of the stress-dependence of the crosslink lifetime for agiven material, p I z 2 hould be proportional to p l , - n 2 .


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806 A . Kayequantity s whereThis conjecture has been discussed in the literature from a different point of view. The

P I 1 - 2 2P12

S =

has been called the 'recoverable shear' and the variation of s with p1 2has been considered.Various workers have reported a linear relation between s and p1 2 and hence pI1- z zproportional to pl:. Philippoff, Brodnyan and Gaskins (1957) have found such a relationfor low concentrations of polyisobutylene in dekalin, and Kotaka, Kurata and Tamura(1959, 1462) for polystyrene in toluene and dekalin.Yamomoto (1958) has considered the dependence of crosslink lifetimes on chain lengthand number of chain segments and has also, by a semi-quantitative argument, arrived atthe conclusion that pll - 2 ? hould be proportional to p l r 2 or Gaussian chains.Acknowledgmentswork was carried out at the College of Aeronautics, Cranfield.

The author wishes to thank Dr. A. B. Tayler for many useful discussions. Pa rt of this

AppendixLetand let Plr:P2' ,P3' be the invariants of pij', defined in a manner similar to the definitionsgiven in equations (9 ) , (10) an d (11). Consider the second-order invariant

pij' = 6i j f ~ i j (A l l

and hence, ifthen

Similarly by considering the third-order invariantwe find tha t

3a' f p' =0Q1=(P: - P2) s independent of p .

02 - a P 3 - rPz 'Pl ' f YIP,'Q2=2P13- PLP2t 7P3

and is also independent of p .ReferencesKOTAKA,. , KURATA, ., and TAMU RA,., 1959, J . Appl. Phys., 30, 1705-12.- 962, Rheol. Acta., 2, 179-86.LODGE,. S. , 1956, Trans. Faraday, Soc., 52, 120-30.- 964, Elastic Liquids (London:Academic Press).~ I P P O F F ,w.,BRODhrYAN, J. G., and GASKINS,. H., 1957, Trans. Soc. Rheol., 1, 109-18.TRELOAR,. R. G., 1958, The Physics of Rubber Elasticity, 2nd edn (Oxford: Clarendon Press).YAMOMOTO,., 1958,J.Phys. Soc. Japan, 13 , 1200-11.

an equation of state for non-newtonian fluids

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