4. Findings and discussion

In order to make a comparison and to compare the findings given in the literature in connection with settling of small particles for inelastic fluids with the behaviour of viscoelastic non-Newtonian fluids having variable material coefficient that show shear-thinning due to being mostly polymeric and viscoelastic nature of aforementioned materials; based on an example given by Rameshwaran et al. [9] for generalized Newtonian fluid, results of a typical example (CEF fluid) discussed in this study are compared visually with stream function, velocities, pressures and stresses with contour curves, and numerically with extrema tables. For both fluids the contour curves are shown in Figure 8 and in Figures 10–13 and in Figures 15–27, and the contour levels are shown numerically in Table 3 and 6. The contours are drawn for the ψ stream function, vr, vz velocities, p pressure, τrr, τθθ, τzz, and τrz stresses, η viscosity coefficient, υ1 normal stress coefficient (Figures 10–13, 15–27). These contours and their extrema are compared with the generalized Newtonian fluid given in the literature [9].

As explained above, although the transition from system we used to a simpler example in the literature taking υ1 as equal to 0 is not completely possible, this comparison was very useful in general terms and allowed for new findings.

This comparison results can be explained as follows:

3.4.3. Numerical solution

88 Polymer Rheology

F, Euclidean and Infinite norms are shown in Table 9.

Figure 27. Equation numbers producing error are shown on the vertical axis.

Obtained mesh configuration contains (linear and quadratic) 755 nodes in total (Figure 14). Totally, 1715 equations are used. Extrema contour levels are given in Table 6. Error computation, Standard deviation, Drag coefficient in Table 7 and Histogram results about equation errors are given in Table 8. Minimum and maximum values of the minimized equation vector

Figure 26. Normal stress coefficient (υ1) [0.0029 0.1135 0.2241 0.3346 0.4452 0.5558 0.6663 0.7769 0.8875 0.9981].

Stream function ψ, radial velocity vr, axial velocity vz, pressure p, normal stress τrr, axial stress τzz, axisymmetric stress τθθ, shear stress τrz, viscosity coefficient η, normal stress coefficient v1 contours and error histogram are given in Figures 10–13 and in Figures 15–27.

ψ stream function contours are completely similar for each two fluids. The extreme values match up with each other. vr radial velocity contours are also in complete harmony to each other. Although max values being 0.2171 and 0.2183 are approximately equal, there is small difference between minimums, 0.1933 and 0.1772. At vz contours, harmony is perfect, extreme values are respectively 0.00/0.00 and 1.1249/1.0215. Because vz = 1 must be at cylinder wall as a boundary condition, the value we found is more realistic. The courses of p contours are similar. The contour given in the literature is not very detailed, however our contour is completely detailed and also it gives a better idea as it is colorful. What attracts attention is that max and min values of each contour are at the same place at source and sink points of the sphere surface. Extreme values are 4.9296/3.4962 and 7.8010/5.9995. The pressure drops for the fluids between sphere's front and back faces caused by sphere motion, which settles along the cylinder axis are 7.8 4.93 = 2.87 and 5.9995 3.4962 ≈ 2.50, respectively. The small difference between the pressure drops can be explained by the elastic effect.

The viscosity coefficient (η) contours are harmonic. The contour given by us is colorful and detailed, and it continues throughout the cylinder. Shear-thinning in this contour obviously manifests itself. In the vicinity of the sphere where the shear-rate γ\_ is very large, η coefficient becomes smaller and approaches to zero. Extreme values are respectively 0.244/0.0845 and 1.4672/1.0000. For both fluids Carreau equation and shear-thinning curve is used, and according to this extreme values given for the generalized Newtonian fluid seem to be exaggerated. The contour of the first normal stress coefficient υ1 properly illustrates the formula it was derived from and shows a course parallel to contour η. In conformity to the relaxation time curve <sup>υ</sup><sup>1</sup> <sup>η</sup> given by Barnes et al. [8], this ratio approaches to 1 for small values of γ\_ and goes


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Table 10. Comparison of the normal and shear stresses.

to zero for large values. Extrema values are 0.0029/0.9981. Likewise ratio of the υ1 and η extreme values is 0.9981/1.0000 ≈ 1.0 for small γ\_ and 0.0029/0.0845 ≈ 0.034 for large γ\_.

The main differences are at the stresses. These support our expectations. The magnitudes of normal stresses τrr, τθθ, τzz in absolute values are increased considerably. Thus, elastic effect appeared by inclusion of υ1 in our equations. In spite of this, the shear stress τrz in absolute value decreased.

So, the shear effect decreased relatively, while the extensional effect increased. The category of the flow slightly changed from shear flow toward extensional flow (Table 10).

## 5. Conclusions

Since the materials under consideration are rather polymeric and consequently viscoelastic, it is compulsory to include the elastic behaviour in the analysis by taking the effects of the normal stress into account in addition to those of the shear stresses and to use a constitutive equation appropriate to this. In this study, this point of view is tried to be ensured, and in order not to neglect the viscoelastic effect, a constitutive equation such as CEF, which includes normal stress coefficients, υ1, υ2, is used.

The results of the analysis, as expressed above in the findings and discussion section, confirms this idea. Comparison of the normal and shear stresses given in Table 10 exposed an increase which reaches 3 times in absolute value in the normal stresses and in parallel a decrease which reaches 4 times in the shear stresses. This table reveals that elastic effect in polymeric fluids cannot be neglected. In this manner a fact has been emphasized, and an important point in fluid literature has been clarified.

## Author details

Şule Celasun

Address all correspondence to: celasuns@itu.edu.tr

Department of Mechanics, Mechanical Engineering Faculty, Istanbul Technical University, Taksim, Istanbul, Turkey
