Non-Monotonous Behaviour of Shear Viscosity - Empirical Modelling

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1 Non-Monotonous Behaviour of Shear Visosity - Empirial Modelling PETR FILIP, JIRI DAVID Institute of Hydrodynamis, Aad.Si.Czeh Rep. Pod Patankou 5, Prague 6 CZECH REPUBLIC filip@ih.as.z Abstrat: - Pratially all hitherto used empirial models used for haraterisation of shear visosity of liquids desribe only its monotonous ourse. However, the onset of new materials is aompanied by more ompliated harateristis of their behaviour inluding non-monotonous ourse of shear visosity. This feature is refleted not only in an existene of one extreme point (maximum or minimum) but there an appear both extreme points; i.e. that shear visosity initially exhibits shear thinning, after attaining a loal minimum onverts to shear thikening, and again after reahing a loal maximum has a shear-thinning harater. It is lear that for an empirial desription of this omplex behaviour a hitherto used number of s (four, five) in lassial monotonous models (suh as Cross or Carreau-Yasuda) is no longer tenable. If more s are applied, there should be given an emphasis on a relatively simple algebrai form of the proposed models, unambiguity of the involved s and their sound interpretation in the whole modelling. This ontribution provides an overview of the existing non-monotonous models and proposes a new 1- model inluding a demonstration of its flexibility using various experimental data. Key-Words: - Empirial modelling; Non-monotonous visosity; Flow urves 1 Introdution Constitutive modelling represents an inevitable step in ompleting balane equations desribing flow behaviour of non-newtonian fluids. In priniple, there are approximately two ways how to ope with this problem. The first one is oriented to a desription of one harateristi only (suh as e.g. shear visosity) but a suffiiently preise approximation is expeted. The seond approah onsists in a proposal of differential or integral onstitutive equation simultaneously desribing more rheologial harateristis (suh as shear and elongational visosities, first and possibly seond normal stress differenes, et.). However, aurateness of the individual harateristi (suh as e.g. shear visosity) is muh lower ompared to the empirial modelling. It depends on the omplexity of the given problem and user s requirements whih approah is more suitable to apply. Empirial modelling of purely visous and visoplasti materials started in the 2s with the proposals of 2- onstitutive models (power-law, Bingham [1,2]) followed by the 3- ones: Hershel-Bulkley [2] as a ombination of the preeding two models and the Williamson one [3] initiating an algebrai struture appearing in now-a-days frequently used models suh as the 4- Cross [4] and 5- Carreau-Yasuda [5,6] ones. A detailed analysis with respet to the number of empirial s is given e.g. in Filip et al. [7]. Reently, the empirial models desribing elastovisoplasti behaviour (Saramito [8], de Souza Mendes and Thompson [9]) have been published. With the onset of a new larger lass of materials, shear visosity no longer exhibits a monotonous ourse only. In this ase the lassial models fail and there is a neessity to apply models enabling a desription of non-monotonous ourse of behaviour of this quantity, i.e. to desribe simultaneously shear thinning and shear thikenning behaviour in their various ombinations. Unfortunately, a number of models devoted to this phenomenon is less than sare. These models an be lassified into two groups: either the proposals of algebraially new forms or modifiations of the lassial models. Ranging to the first group, the 6- model proposed in David and Filip [1] enables to desribe the situations when one loal extreme (either maximum or minimum) appears in-between the initial and terminal Newtonian plateaux (with the possibility to inlude an intermediate one) ( f ( ɺ p q) ) ( f ( ɺ p q) ) ( ( ɺ γ )) ( ( ɺ γ )) η exp γ ;,, + η exp γ ;,, η= b+ exp f ;, p, q + exp f ;, p, q (1) ISBN:

2 ( ) ( ) p ( ) ( ) p q f ɺ γ ;, p, q = sign log ɺ γ log ɺ γ, (2) the s η and η determine the initial and terminal Newtonian plateaux, respetively. The s, p, q are supposed to be positive and the b > -2. Nevertheless, due to a low number of s this model exhibits some restritions summarised in Filip et al. [7], the presented 8- model omposed of two similar terms simplified from rel. (1) exp( f) ( ) exp( ) η exp( f ) exp( ) exp( ) η η= + b + exp f + f + b + f + f ( ɺ γ) p ( ɺ γ ;, ) = log ; f f p f f ( ɺ γ ;, p ) = log ( ɺ γ) p (3), (4) enables to desribe onseutively shear thinning, shear thikenning, and again shear thinning. In both models the s inluded partiipate in a desription of the flow urve within the whole range of shear rates. Ranging to the seond group, Zatloukal et al. [11] modified the simplified 4-P Carreau-Yasuda model (with η ) by an additional 3-P multipliative funtion ensuring non-monotonous behaviour of visosity log( η). 1+ ( kɺ γ) tanh( β ) (5) ξ η tanh( αγɺ + β ) = log a (1 n) / a Galindo-Rosales et al. [12,13] have reently published the 11- model desribing onseutively shear thinning, shear thikenning, and shear thinning behaviour eah monotonous part was desribed by its orresponding Cross model in suh a way that at loal extremes the individual Cross models were oupled. The problems an arise from a determination of loal extremes and onsequently the ranges of shear rates the individual sets of s should be applied, restrited smoothness of a omposed visosity funtion, and modelling some points on the visosity urve lose to the extreme points from one side only (and negleting behaviour from the other side). The inreasing number of s appearing in these non-monotonous models is not surprising. Even for only shear thinning materials not exhibiting a symmetry, i.e. the drop-off in visosity from the initial Newtonian plateau is not mirrored by its rapid levelling-off towards the terminal Newtonian plateau, Roberts et al. [14] introdued the 8- model. Every empirial model desribing a nonmonotonous ourse of shear visosity should fulfil the following attributes - dependene on a limited number of independent model s; - the s should partiipate throughout the whole range of shear rate (stress) as the strit determination of the individual monotonous subintervals an be evaluated only approximately due to both non-ontinuous harater of experimental data and their inauray; - the s should be readily and unambiguously evaluated from experimental visosity data; - the s should represent sound physial or mathematial interpretation onneted with the measured visosity funtion; - every proposed model should exhibit good mathematial struture to enable suffiiently aurate and relatively simple fit to the measured data; - appliability to a larger amount of materials. The aim of this ontribution is to present a 1- empirial model desribing sequential hanging of shear thinning, thikening, and again thinning regimes. The effiieny of the model is illustrated using data of other authors. 2 Proposal of the 1-Parameter Model As introdued above, the 6- and 8- nonmonotonous models are not able to evaluate more ompliated flow behaviour of shear visosity. To widen the lass of materials for whih an evaluation is possible, the following 1- model is proposed 1ɺ γ + 2 / ɺ γ 5 exp( f ) η= f + f ( ɺ γ 1/ ɺ γ) ( exp( ) exp( )) (6) ISBN:

3 9 9 ( ) ( ) 1 8γ log 8γ f= sign log ɺ ɺ. (7) Analogously to the 6-P model, the proposed 1-P model is a sum of two terms. The resaling of shear rate, i.e. the ombination of the exponential and logarithmi funtions was motivated by the model presented in Chynoweth and Mihopoulos [15]. 3 Comparison with the Experimental Data An appliability of the proposed 1-P model is doumented in omparison with the experimental data by Zhang et al. [16] and White et al. [17]. Fig.1 depits visosity as a funtion of shear rate for different solid frations of fumed silia in ethylene glyol. Fig.2 shows a omparison between the model and the experimental data relating shear rate with visosity of a suspension of fumed silia in ethylene glyol at 3% (w/w) and arbonyl iron partiles for various intensity of magneti field. The appliation of the model to the data in White et al. [17] desribing shear thikening behaviour of a suspension of ornstarh in water at 55 wt.% is presented in Fig.3. The lists of the values attained by the model s orresponding to Figs.1-3 are given in Tables 1-3. In all ases the agreement between empirial fitting and the experimental data is very good. Table 1 The model s orresponding to Fig.1. solid fration [%] *E *E *E Fig.2 Appliation of the 1-P model to the experimental data presented by Zhang et al. [16]. Table 2 The model s orresponding to Fig.2. Fig.1 Comparison of the 1-P model with the experimental data presented by Zhang et al. [12]. magneti field intensity [ka/m] , *E-4 4.4*E-4 5.4*E ISBN:

4 Aknowledgement The authors wish to aknowledge GA CR for the finanial support of Grant No. 13/9/266. Fig.3 Comparison of the 1-P model with the experimental data presented by White et al. [17]. Table 1 The model s orresponding to Fig E Conlusion The 1- model was proposed for the desription of non-monotonous behaviour of visosity in dependene on shear rate. Analogous form an be proposed in dependene on shear stress as for some materials this relation seems to be more orresponding to the physial reality, see Lomellini and Ferri [18]. Seemingly large number of s is adequate when ompared with the other models. The omparisons with the experimental data of other authors showed very good appliability of the proposed model. Referenes: [1] R.B.Bird, G.C.Dai, B.J.Yarusso, The rheology and flow of visoplasti materials, Reviews in Chem. Engng., Vol.1, 1983, pp [2] C.W.Maosko, Rheology Priniples, Measurements, and Appliations, VCH, New York, [3] R.V.Williamson, The flow of pseudoplasti materials, Ind.Eng.Chem., Vol.21, 1929, pp [4] M.M.Cross, Rheology of non-newtonian fluids: A new flow equation for pseudoplasti systems, J.Colloid Si., Vol.2, 1965, pp [5] P.J.Carreau, Ph.D.Thesis, University of Wisonsin, Madison, [6] K.Yasuda, R.C.Armstrong, R.E.Cohen, Shearflow properties of onentrated-solutions of linear and star branhed polystyrenes, Rheol.Ata, Vol.2, 1981, pp [7] P.Filip, J.David, R.Pivokonsky, Modelling of non-monotonous flow urves using empirial onstitutive equations, Ata Tehnia CSAV, vol.51, 26, pp [8] P.Saramito - A new elastovisoplasti model based on the Hershel Bulkley visoplasti model, J. Non-Newton. Fluid Meh., Vol.158, 29, pp [9] P. de Souza Mendes, R.L.Thompson, A ritial overview of elasto-visoplasti thixotropi modeling, J. Non-Newton. Fluid Meh., Vol , 212, [1] J.David, P.Filip, Phenomenologial modelling of non-monotonous shear visosity funtions, Appl.Rheol., Vol.14, 24, pp [11] M.Zatloukal, J.Vlek, C.Tzoganakis, P.Saha, Improvement in tehniques to determine extensional rheology from entrane flows, In: Pro. 6th Europ. Conf. on Rheology, Eds.: Münstedt, H., Kashta, J., Merten, A., 1-6 Sep.22, Erlangen, Germany 22, pp [12] F.J.Galindo-Rosales, F.J.Rubio-Hernández, A.Sevilla, An apparent visosity funtion for shear thikening fluids, J. Non-Newton.Fluid Meh., Vol.166, 211, pp [13] F.J. Galindo-Rosales, F.J. Rubio-Hernández, A. Sevilla, R.H. Ewoldt - How Dr. Malom M. Cross may have takled the development of An apparent visosity funtion for shear ISBN:

5 thikening fluids, J. Non-Newton. Fluid Meh., Vol.166, 211, pp [14] G.P.Roberts, H.A.Barnes, P.Carew, Modelling the flow behaviour of very shear-thinning liquids. Chem. Eng. Si., Vol.56, 21, pp [15] S.Chynoweth, Y.Mihopoulos, Generi properties of rheologial flow urves, J.Non- Newton. Fluid Meh., Vol.69, 1997, pp [16] X. Zhang, L. Weihua, X. Gong, Study on magnetorheologial shear thikening fluid, Smart Mater. Strut., Vol.17, 28, [17] E.White, M.Chellamuthu, J.Rothstein, Extensional rheology of a shear-thikening ornstarh and water suspension, Rheol. Ata, Vol.49, 21, pp [18] P.Lomellini, D.Ferri, Polymer melt rheology versus shear rate or versus shear stress? In: Pro.13th Int. Congress on Rheology, Eds.: Binding, D.M., Hudson, N.E., Mewis, J., Piau, J.-M., Petrie, C.J.S., Townsend, P., Wagner, M.H., Walters, K., 2-25 Aug.2, Cambridge, UK 2, pp ISBN: