A numerical study of the rheological properties of suspensions of ...

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166 M. B. Mackaplow and E. S. G. Shaqfehwhere each term in the series represents the effect of a certain class of fibre-fibreinteractions. C” is an 0(1) constant that is a function of fibre shape and orientationdistribution. Comparing (22) to (19) yields x/b - [(l/4)ln(l/4)]1’2 for all orientationdistributions, assuming randomness in the centre-of-mass. By neglecting the[In( 1/4)] 1/2 part of this expression, we see that this analysis predicts that the screeninglength is approximately equal to the average interfibre spacing. This is in contrast tothe prediction of Dinh & Armstrong (1984).As discussed by Mackaplow et al. (1994), some of the values of C” shown inShaqfeh & Fredrickson( 1990) are incorrect. This results from Shaqfeh & Fredrickson(1990) assuming that the suspension volume fraction, 4, is 2nn13/A2, regardless of fibreshape, even though this is only true for cylinders. The corrected values for spheroidalfibres are 1.034 and 0.202 for aligned and isotropic suspensions, respectively. Thecorresponding values for cylindrical fibres are 0.1585 and -0.6634.Vyx) =i,1 0 00 -1 0 ‘X.0 0 05.1.1. Aligned suspensionsFigures l(a) and l(b) show plots of Q vs. n13 for fibres with an aspect ratio of 100that are all aligned with the principal axis of extension of the imposed flow field. Foreach set of simulations we have shown the mean and its 95% confidence interval.In figure l(a) we see that for n13 > 1,but quantitatively lies 25%-30% below it. However, by adjusting the constant, C”,
Rheological properties of suspensions of rigid, non-Brownian jibres 167Volume fraction of fibres, q10-5 5 2 40.140i """ F0 Slmulatlan averages- Dilute theoryQ0.1360.130 k$01 011 1.0n13Volume fraction of fibres, y0 0.001 0.002 0.003 0.004 0.005 0.006I I I I I Ir'Q0. 1 2 -0.10-____..-.-,,/'___._..-.. .__..-./' ___._.-*--___..---/..-/--,,~+,.'. /.: ..':,n13FIGURE 1. Normalized extra particle stress, Q, of a suspension of aligned, spheroidal fibres withaspect ratio, A = 100, as a function of the suspension volume fraction. Shown are the predictions ofnumerical simulations, the dilute theory (with and without two-body corrections), and the semidilutetheory of Shaqfeh & Fredrickson (1990) (both the original and modified versions).in the semi-dilute theory from the value given by Shaqfeh & Fredrickson (1990),1.034, to -1.25, and assigning a coefficient of 2.4 to the O(l/ln(l/4)) term, a muchbetter quantitative agreement with the simulations is achieved. This is shown infigure l(b). As will be subsequently shown in figures 2, 3, and 4, by utilizing theseparticular adjusted coefficients the semi-dilute theory is in an excellent quantitativeagreement with the simulations for suspensions having a wide range of concentrationsand particle aspect ratios. The need to adjust C" and keep the O(l/ln(l/4)) termcorresponds to keeping classes of interparticle interactions that are neglected in theasymptotic theory. Thus, it appears the scaling predicted by Shaqfeh & Fredrickson(1990) is correct, but more terms in the series must be kept to get good quantitativeaccuracy.Of particular interest is the fact that the dilute theory with two-body correctionspredicts, with small error, the behaviour of the suspension at concentrations up ton13 = 15. This is interesting because this theory was developed to be valid only inthe dilute region. One indicator of the breakdown of the two-body theory is whenthe correcting term to the dilute theory become of the same order as the term it iscorrecting. This does not occur until n13 - O(ln2(2A)). For an aspect ratio of 100,
Page 1 and 2: J. Fluid Mech. (1996), uol. 329, pp
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