3.3.2. The attempt of getting υ<sup>1</sup> ¼ υ<sup>2</sup> ¼ 0 condition as a special case

The reasons why the results obtained by Townsend [9] cannot be achieved for the generalized fluid by replacing υ<sup>1</sup> ¼ υ<sup>2</sup> ¼ 0 in our equations given above for the generalized condition are explained by Kaloni [21] in the literature.

Nonetheless, we think that is beneficial to make a comparison with Townsend [9] solutions.

#### 3.3.3. Mesh-independent results

The model used in this study, which is called as Mesh#1, was obtained as a result of a three-stage optimization. In the "Initmesh" stage, an "adaptive mesh" was formed. In the second stage number of triangles was increased with the "refinemesh" function to obtain an improvement in this first mesh. Finally, at the third stage, an optimization was achieved with the "jigglemesh" function. Therefore, Mesh#1 is an optimal mesh, and it can be considered sufficient in terms of "mesh independence" alone. Mesh#2 is an adaptive mesh, and the number of triangles in this mesh, was increased considerably with respect to Mesh#1 (Figure 9 and Table 4).

3.4. Explanatory flow characteristics remarks and results of our the theoretical problem solved numerically, cited before the sections findings and discussion and conclusions to

Figure 8. Contours of streamfunction (ψ), radial velocity (vr), axial velocity (vz), pressure (p), stress components (τrr, τzz,

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The starting example of "Settling of Small Particles in Fluid Medium" problem in the literature which has important applications in many fields such as the shelf life of foodstuffs etc., is given for "generalized Newtonian fluid" medium [9]. As is known, the motion equation is Navier-Stokes in this case, and viscosity coefficient η is function of the shear-rate γ\_. Although this paper contains useful information, especially for velocity and stress contours, the elastic effects should not be neglected because in practice the fluids encountered in this field are non-Newtonian. This study is to cover this difference. Thus, CEF model was taken as fluid, and for emphasizing the elastic effect, the normal stress coefficients υ<sup>1</sup> and υ<sup>2</sup> was included. The simulation model is in

better find out the problem studied

τθθ, τrz), and viscosity (η); a/R = 0.2, We = 2.5, and n = 0.5.

3.4.1. Explanatory remarks

Because the subject of this study is based on the fact that the elastic effect in polymeric fluids cannot be neglected, Mesh#1 and Mesh#2 results in Table 5 were presented as comparison of the normal stresses. In this table, the average difference between each mesh is approximately 2%. Because the results of two very different meshes mostly overlap, the results obtained are proven to be independent from mesh configuration, i.e., "mesh independence" is achieved. The generated mesh is improved according to the initmesh, refine mesh and jigglemesh MATLAB programs and the mesh independence is tested and ensured.

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Figure 8. Contours of streamfunction (ψ), radial velocity (vr), axial velocity (vz), pressure (p), stress components (τrr, τzz, τθθ, τrz), and viscosity (η); a/R = 0.2, We = 2.5, and n = 0.5.

#### 3.4. Explanatory flow characteristics remarks and results of our the theoretical problem solved numerically, cited before the sections findings and discussion and conclusions to better find out the problem studied

#### 3.4.1. Explanatory remarks

3.3.2. The attempt of getting υ<sup>1</sup> ¼ υ<sup>2</sup> ¼ 0 condition as a special case

explained by Kaloni [21] in the literature.

Figure 7. Mesh patterns around sphere, a/R = 0.2.

76 Polymer Rheology

3.3.3. Mesh-independent results

The reasons why the results obtained by Townsend [9] cannot be achieved for the generalized fluid by replacing υ<sup>1</sup> ¼ υ<sup>2</sup> ¼ 0 in our equations given above for the generalized condition are

Nonetheless, we think that is beneficial to make a comparison with Townsend [9] solutions.

The model used in this study, which is called as Mesh#1, was obtained as a result of a three-stage optimization. In the "Initmesh" stage, an "adaptive mesh" was formed. In the second stage number of triangles was increased with the "refinemesh" function to obtain an improvement in this first mesh. Finally, at the third stage, an optimization was achieved with the "jigglemesh" function. Therefore, Mesh#1 is an optimal mesh, and it can be considered sufficient in terms of "mesh independence" alone. Mesh#2 is an adaptive mesh, and the number of triangles in this

Because the subject of this study is based on the fact that the elastic effect in polymeric fluids cannot be neglected, Mesh#1 and Mesh#2 results in Table 5 were presented as comparison of the normal stresses. In this table, the average difference between each mesh is approximately 2%. Because the results of two very different meshes mostly overlap, the results obtained are proven to be independent from mesh configuration, i.e., "mesh independence" is achieved. The generated mesh is improved according to the initmesh, refine mesh and jigglemesh

mesh, was increased considerably with respect to Mesh#1 (Figure 9 and Table 4).

MATLAB programs and the mesh independence is tested and ensured.

The starting example of "Settling of Small Particles in Fluid Medium" problem in the literature which has important applications in many fields such as the shelf life of foodstuffs etc., is given for "generalized Newtonian fluid" medium [9]. As is known, the motion equation is Navier-Stokes in this case, and viscosity coefficient η is function of the shear-rate γ\_. Although this paper contains useful information, especially for velocity and stress contours, the elastic effects should not be neglected because in practice the fluids encountered in this field are non-Newtonian. This study is to cover this difference. Thus, CEF model was taken as fluid, and for emphasizing the elastic effect, the normal stress coefficients υ<sup>1</sup> and υ<sup>2</sup> was included. The simulation model is in


Table 3. Extrema contour levels.

the form of the flow of a non-Newtonian fluid around a sphere falling along the centerline of a cylindrical tube. The problem was tried to be solved numerically by writing continuity and motion equations and using FEM. The partial differential equation system obtained was transformed into a set of simultaneous, nonlinear, algebraic equations. Here, we chose linear triangular elements for pressures and quadratic triangular elements for velocities.

Because of the boundary conditions, the equations are "overdetermined." The problem is axisymmetric, the global coordinates r, z were transformed into local coordinates L1, L2, L3; and Gauss quadrature was used for integrating numerically over the effective area around each node. In the solution of the algebraic equation system, naturally optimized formulation with the aid of (MATLAB-Optim set f-solve) function was used instead of the methods like "weak form."

It is useful to give following explanatory remarks:

Although the differential equation is third order, the shape functions are chosen as linear for pressures and quadratic (2nd degree) polynomials for velocities. The degree difference between pressures and velocities is compulsory for stability. These are thought to be sufficient in the literature examples. Using third order polynomial would extend the problem extremely, and make the solution unreachable. Considering the elastic effect (υ1, υ<sup>2</sup> coefficients) made the problem already harder compared to the generalized Newtonian fluid example [9]. Zienkiewicz especially stresses that FEM application of very refined and sophisticated models does not always yield better solutions [19].

Because the motion is relative, the sphere was taken constant in the problem; however, a motion from bottom to top with a constant Vs velocity was introduced to the cylinder, as done in the 1st Stokes problem. In this case, the inlet is through the bottom surface, while the outlet is through the top surface of the cylinder.

As an example to viscoelastic fluid, Carreau type Separan 30 was chosen [7]. The coefficient λ = 8.04 s was taken from the table given by Tanner [7]. This fluid was used frequently in experiments, and detailed information about its properties is given in the literature [4].

The boundary conditions are generally the conditions at the inlet and outlet, and—for velocities—the no slip conditions at the sphere surface and cylinder wall. It was assumed that vr = 0 vz = 0 at sphere surface and vr = 0 vz = 1 (i.e., Vs) at cylinder wall. These values are observed

Matrices for triangles

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Vector of nodal values

V

T

Matrices for edges

Mesh#1 2 / 205 7 / 62 4/346 1/1715 Mesh#2 2 / 311 7 /84 4/536 1/2625

E

clearly on the contours (Figures 10 and 11).

Definition Matrices for points P

Figure 9. Mesh#1 and Mesh#2.

Table 4. Mesh comparison.

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Figure 9. Mesh#1 and Mesh#2.

the form of the flow of a non-Newtonian fluid around a sphere falling along the centerline of a cylindrical tube. The problem was tried to be solved numerically by writing continuity and motion equations and using FEM. The partial differential equation system obtained was transformed into a set of simultaneous, nonlinear, algebraic equations. Here, we chose linear

Flow domain Minimum Maximum ψ 0.0000 3.1250 ν<sup>r</sup> 0.1933 0.2171 νθ — ν<sup>z</sup> 0.0000 1.1249 P 4.9296 7.8010 τrr 2.6545 3.1418 τθθ 1.4139 1.8356 τrz 1.3276 2.9703 τzz 3.6693 2.8291 η 0.244 1.4672

Because of the boundary conditions, the equations are "overdetermined." The problem is axisymmetric, the global coordinates r, z were transformed into local coordinates L1, L2, L3; and Gauss quadrature was used for integrating numerically over the effective area around each node. In the solution of the algebraic equation system, naturally optimized formulation with the aid of (MATLAB-Optim set f-solve) function was used instead of the methods like "weak form."

Although the differential equation is third order, the shape functions are chosen as linear for pressures and quadratic (2nd degree) polynomials for velocities. The degree difference between pressures and velocities is compulsory for stability. These are thought to be sufficient in the literature examples. Using third order polynomial would extend the problem extremely, and make the solution unreachable. Considering the elastic effect (υ1, υ<sup>2</sup> coefficients) made the problem already harder compared to the generalized Newtonian fluid example [9]. Zienkiewicz especially stresses that FEM application of very refined and sophisticated models

Because the motion is relative, the sphere was taken constant in the problem; however, a motion from bottom to top with a constant Vs velocity was introduced to the cylinder, as done in the 1st Stokes problem. In this case, the inlet is through the bottom surface, while the outlet

As an example to viscoelastic fluid, Carreau type Separan 30 was chosen [7]. The coefficient λ = 8.04 s was taken from the table given by Tanner [7]. This fluid was used frequently in experiments, and detailed information about its properties is given in the literature [4].

triangular elements for pressures and quadratic triangular elements for velocities.

It is useful to give following explanatory remarks:

Table 3. Extrema contour levels.

78 Polymer Rheology

does not always yield better solutions [19].

is through the top surface of the cylinder.


Table 4. Mesh comparison.

The boundary conditions are generally the conditions at the inlet and outlet, and—for velocities—the no slip conditions at the sphere surface and cylinder wall. It was assumed that vr = 0 vz = 0 at sphere surface and vr = 0 vz = 1 (i.e., Vs) at cylinder wall. These values are observed clearly on the contours (Figures 10 and 11).


Table 5. Comparison of normal stresses.

Figure 10. Radial velocity (vr) [0.1772 0.1333 0.0893 0.0454 0.0014 0.0425 0.0865 0.1304 0.1743 0.2183].

Contour values were found by dividing the space between the max and min contours by ten, and were written on the figures without covering them up. That is why contour labels are generally fractional. If an integer contour (e.g., zero) is to be drawn, it can be shown with linear interpolation on the figure.

As seen in pressure (p) contours (Figure 12), similarly to the example Rameshwaran, Townsend [9], max and min values are on the "stagnation" points on the sphere (at the vicinity of the vertical axis). Here vr = vz = 0; since the total energy is constant (Bernoulli at inviscid flow), if the kinetic energy is zero, potential energy is an extremum, i.e., the pressure is at its extremum value. This is what is seen perspectively in the 3-D surf (surface) drawing (Figure 13).

The extreme values concerning the contours are gathered in Table 6. The flow is not uniform: the existence of the sphere at the center, the wall effect of cylinder (drag) and the boundary conditions prevent the flow from being uniform. Thus, max and min values are obtained.

been increased extremely, we could have got curves instead of the zigzag course. This is not a drawback in terms of the results. 10 contours were drawn by dividing space between the max and min values into 10 fragments. Contours are the geometric milieu of the nodes which have the same value of the related independent variable. Horizontal projections of the curves

Figure 12. Pressure (p) [3.4962 2.4411 1.3860 0.3310 0.7241 1.7792 2.8343 3.8893 4.9444 5.9995].

Figure 11. Axial velocity (vz) [0.0000 0.1802 0.2737 0.3671 0.4606 0.5541 0.6476 0.7411 0.8345 0.9280 1.0215].

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In the adaptive "mesh" drawn, there are 755 nodes. At these nodes vr, vz, p values are taken as unknowns and they were determined as a result of calculations. Because the contours followed the nodes, zigzag course was obtained instead of a smooth curve. If the number of nodes had Particle Settling in a Non-Newtonian Fluid Medium Processed by Using the CEF Model http://dx.doi.org/10.5772/intechopen.75977 81

Figure 11. Axial velocity (vz) [0.0000 0.1802 0.2737 0.3671 0.4606 0.5541 0.6476 0.7411 0.8345 0.9280 1.0215].

Figure 12. Pressure (p) [3.4962 2.4411 1.3860 0.3310 0.7241 1.7792 2.8343 3.8893 4.9444 5.9995].

Contour values were found by dividing the space between the max and min contours by ten, and were written on the figures without covering them up. That is why contour labels are generally fractional. If an integer contour (e.g., zero) is to be drawn, it can be shown with linear

Figure 10. Radial velocity (vr) [0.1772 0.1333 0.0893 0.0454 0.0014 0.0425 0.0865 0.1304 0.1743 0.2183].

Stresses τrr τθθ τzz

Extreme values Min Max Min Max Min Max Mesh#1 6.0019 3.3792 6.0115 3.4000 6.1339 3.7165 Mesh#2 5.9810 3.2325 5.8722 3.4022 6.3396 3.8825

As seen in pressure (p) contours (Figure 12), similarly to the example Rameshwaran, Townsend [9], max and min values are on the "stagnation" points on the sphere (at the vicinity of the vertical axis). Here vr = vz = 0; since the total energy is constant (Bernoulli at inviscid flow), if the kinetic energy is zero, potential energy is an extremum, i.e., the pressure is at its extremum

The extreme values concerning the contours are gathered in Table 6. The flow is not uniform: the existence of the sphere at the center, the wall effect of cylinder (drag) and the boundary conditions prevent the flow from being uniform. Thus, max and min values are obtained.

In the adaptive "mesh" drawn, there are 755 nodes. At these nodes vr, vz, p values are taken as unknowns and they were determined as a result of calculations. Because the contours followed the nodes, zigzag course was obtained instead of a smooth curve. If the number of nodes had

value. This is what is seen perspectively in the 3-D surf (surface) drawing (Figure 13).

interpolation on the figure.

Table 5. Comparison of normal stresses.

80 Polymer Rheology

been increased extremely, we could have got curves instead of the zigzag course. This is not a drawback in terms of the results. 10 contours were drawn by dividing space between the max and min values into 10 fragments. Contours are the geometric milieu of the nodes which have the same value of the related independent variable. Horizontal projections of the curves

Figure 13. 3-D surf drawing of pressure.


Table 6. Extrema contour levels.

(level curves) that are obtained by intersecting the surface shown in 3D in the surf drawing with the horizontal planes yield the contours. Contours were drawn with the pdecont, pdemesh, pdesurf commands at MATLAB. These 10 values, below the contours are given separately.

#### 3.4.2. Technical details for the solution of the equation system

We transformed the partial equation system with the aid of FEM to nonlinear overdetermined algebraic equation system. As we have numerous simultaneous and highly nonlinear equations, which moreover are overdetermined due to the existence of the boundary conditions, we were compelled to resort to the optimization techniques to resolve the equation

Table 7. Error computation, standard deviation, the average of η values at sphere surface (η\_avg) and drag coefficient (C).

Error spaces 0–0.5 0.5–1.0 1.0–1.5 1.5–2.0 2.0–2.5 Equation number 1637 66 11 1 0 Percentage (%) 95.4519 3.8484 0.6414 0.0583 0

Min., Max. and Norms: Min(F) Max(F) ||F||2 ||F||<sup>∞</sup>

Table 9. Euclidean and infinite norms and minimum and maximum values of the minimized equation vector F.

1.3449 1.6251 8.6075 1.6251

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Table 8. Histogram results concerning equation errors; 1715 equations were used in total.

Average error 0.1004 Standard Deviation 0.1820 η\_avg 0.1104 C 1.6502

system [22].

Figure 14. Mesh scheme for a/R = 0.2.


Table 7. Error computation, standard deviation, the average of η values at sphere surface (η\_avg) and drag coefficient (C).


Table 8. Histogram results concerning equation errors; 1715 equations were used in total.


Table 9. Euclidean and infinite norms and minimum and maximum values of the minimized equation vector F.

Figure 14. Mesh scheme for a/R = 0.2.

(level curves) that are obtained by intersecting the surface shown in 3D in the surf drawing with the horizontal planes yield the contours. Contours were drawn with the pdecont, pdemesh, pdesurf commands at MATLAB. These 10 values, below the contours are given separately.

Flow domain Minimum Maximum ψ 0.0000 3.1416 ν<sup>r</sup> 0.1772 0.2183 ν<sup>z</sup> 0.0000 1.0215 p 3.4962 5.9995 τrr 6.0019 3.3792 τzz 6.1339 3.7165 τθθ 6.0115 3.4000 τrz 0.3701 0.8390 η 0.0845 1.0000 υ<sup>1</sup> 0.0029 0.9981

We transformed the partial equation system with the aid of FEM to nonlinear overdetermined algebraic equation system. As we have numerous simultaneous and highly nonlinear

3.4.2. Technical details for the solution of the equation system

Figure 13. 3-D surf drawing of pressure.

82 Polymer Rheology

Table 6. Extrema contour levels.

equations, which moreover are overdetermined due to the existence of the boundary conditions, we were compelled to resort to the optimization techniques to resolve the equation system [22].

Figure 17. 3-D surf drawing of radial velocity.

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Figure 18. 3-D mesh drawing of axial velocity.

Figure 19. 3-D surf drawing of axial velocity.

Figure 15. Stream function (Ψ) [0.00 0.01 0.05 0.20 0.44 0.77 1.25 1.88 2.67 3.14].

Figure 16. 3-D mesh drawing of radial velocity.

The end of the iterative optimization process is determined according to the fulfillment of the Error computation, Standard deviation, Histogram and minimum and maximum values of the minimized equation vector F, Euclidean norm and Infinite norm (Tables 7– 9). We determined that the results are in the acceptable order. We used the (Optim set fsolve) function in MATLAB for iterative optimization. The basis of the method is based on "least squares method."

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Figure 17. 3-D surf drawing of radial velocity.

Figure 18. 3-D mesh drawing of axial velocity.

Figure 19. 3-D surf drawing of axial velocity.

The end of the iterative optimization process is determined according to the fulfillment of the Error computation, Standard deviation, Histogram and minimum and maximum values of the minimized equation vector F, Euclidean norm and Infinite norm (Tables 7– 9). We determined that the results are in the acceptable order. We used the (Optim set fsolve) function in MATLAB for iterative optimization. The basis of the method is based on

Figure 15. Stream function (Ψ) [0.00 0.01 0.05 0.20 0.44 0.77 1.25 1.88 2.67 3.14].

"least squares method."

84 Polymer Rheology

Figure 16. 3-D mesh drawing of radial velocity.

Figure 23. Azimuthal stress (τθθ) [6.0115 4.9658 3.9200 2.8743 1.8286 0.7829 0.2628 1.3086 2.3543 3.4000].

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Figure 24. Shear stress (τrz) [0.3701 0.2358 0.1014 0.0329 0.1673 0.3016 0.4360 0.5703 0.7047 0.8390].

Figure 25. Viscosity coefficient (η) [0.0845 0.1862 0.2879 0.3896 0.4914 0.5931 0.6948 0.7965 0.8982 1.0000].

Figure 20. 3-D mesh drawing of pressure.

Figure 21. Normal stress (τrr) [6.0019 4.9595 3.9172 2.8748 1.8325 0.7902 0.2522 1.2945 2.3369 3.3792].

Figure 22. Axial stress (τzz) [6.1339 5.0394 3.9449 2.8504 1.7560 0.6615 0.4330 1.5275 2.6220 3.7165].

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Figure 23. Azimuthal stress (τθθ) [6.0115 4.9658 3.9200 2.8743 1.8286 0.7829 0.2628 1.3086 2.3543 3.4000].

Figure 20. 3-D mesh drawing of pressure.

86 Polymer Rheology

Figure 21. Normal stress (τrr) [6.0019 4.9595 3.9172 2.8748 1.8325 0.7902 0.2522 1.2945 2.3369 3.3792].

Figure 22. Axial stress (τzz) [6.1339 5.0394 3.9449 2.8504 1.7560 0.6615 0.4330 1.5275 2.6220 3.7165].

Figure 24. Shear stress (τrz) [0.3701 0.2358 0.1014 0.0329 0.1673 0.3016 0.4360 0.5703 0.7047 0.8390].

Figure 25. Viscosity coefficient (η) [0.0845 0.1862 0.2879 0.3896 0.4914 0.5931 0.6948 0.7965 0.8982 1.0000].

4. Findings and discussion

general terms and allowed for new findings.

time curve <sup>υ</sup><sup>1</sup>

This comparison results can be explained as follows:

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

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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

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

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

<sup>η</sup> given by Barnes et al. [8], this ratio approaches to 1 for small values of γ\_ and goes

compared with the generalized Newtonian fluid given in the literature [9].

difference between the pressure drops can be explained by the elastic effect.

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].

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

#### 3.4.3. Numerical solution

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

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.
