Flow of Newtonian and non-newtonian fluids in a concentric annulus with a rotating inner cylinder

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1 Korea-Australia Rheology Journal, 25(2), (May 2013) DOI: /s Flow of Newtonian and non-newtonian fluids in a concentric annulus with a rotating inner cylinder Young-Ju Kim 1, Sang-Mok Han 2 and Nam-Sub Woo 1, * 1 Korea Institute of Geoscience and Mineral Resources, 30, Gajeong-dong, Yuseong-gu, Daejeon, , Republic of Korea 2 Jeollabuk-do Branch Institute, Korea Marine Equipment Research Institute, Gunsan , Republic of Korea (Received August 7, 2012; final revision received December 19, 2012; accepted March 7, 2013) We examine the characteristics of helical flow in a concentric annulus with radii ratios of 0.52 and 0.9, whose outer cylinder is stationary and inner cylinder is rotating. Pressure losses and skin friction coefficients are measured for fully developed flows of water and a 0.4% aqueous solution of sodium carboxymethyl cellulose (CMC), when the inner cylinder rotates at the speed of 0~62.82 rad/s. The transitional flow has been examined by the measurement of pressure losses to reveal the relation between the Reynolds and Rossby numbers and the skin friction coefficients. The effect of rotation on the skin friction coefficient is largely changed in accordance with the axial fluid flow, from laminar to turbulent flow. In all flow regimes, the skin friction coefficient increases due to inner cylinder rotation. The change of skin friction coefficient corresponding to the variation of rotating speed is large for the laminar flow regime, becomes smaller as the Reynolds number increases for the transitional flow regime, and gradually approaches zero for the turbulent flow regime. The value of skin friction coefficient for a radii ratio of 0.52 is about two times larger than for a radii ratio of 0.9. For 0.4% CMC solution, the value of skin friction coefficient for a radii ratio of 0.52 is about four times larger than for a radii ratio of 0.9. Keywords: Non-Newtonian fluids, helical flow, concentric annulus, pressure loss, rotating cylinder 1. Introduction The flow of non-newtonian fluids in annuli is important for numerous engineering applications in chemical, petroleum, biomedical, and food processes (Wilson 2001). An understanding of the characteristics of such non-newtonian fluid flow is of considerable interest in many areas of science and engineering. A typical example from the petroleum industry is the case of drilling mud and cement slurry, which are flowing between a drill pipe and its casing in an oil well. Directional drilling has been used by the petroleum industry for decades. Knowledge of the velocity distribution and pressure loss during flow is required to obtain a desired flow rate, and to determine the rates of heat transfer accompanying such flow. This study examines the pressure loss problem associated with the flow instability of Newtonian and non-newtonian fluids in a concentric annulus with rotation of the inner cylinder for various Reynolds numbers (Re) up to that of turbulent flow. In a study of non-newtonian fluid flow in concentric annuli, Fredrickson (1960) analyzed the fully developed flow of ideal Bingham and power law fluids. The relationship between flow rate and the frictional pressure *Corresponding author: nswoo@kigam.re.kr gradient was published in the form of a set of plots. Stuart (1958) and Diprima (1960) used nonlinear theory to investigate the relation between the Taylor number and stability. Yamada (1962) and Watanabe et al. (1973) showed that the flow is relatively stable when the outer cylinder is rotating, and thus the critical (bulk flow) Reynolds number is larger and the pressure loss is smaller than would be the case for a rotating inner cylinder. The critical Reynolds number decreases as both the rotational Reynolds number and the ratio of eccentricity increases (Nakabayashi et al. 1974; Nouri et al. 1993; Nouri and Whitelaw 1994). Escudier and Gouldson (1995) showed that the influence of rotation on the axial velocity distribution is most apparent at low Reynolds numbers. Variations in annular gap, wellbore eccentricity, and shaft rotational speed have strong effects on the pressure loss of fluid flowing in the narrow annulus of a slimhole drilling wellbore (Delwiche et al. 1992). Moreover, in the case of an inclined annulus, the axial component of particle velocity plays a less important role, and to have satisfactory transport, the annular mud velocity in this case may be lower than in the case of a vertical annulus. However, the increasing radial component of particle slip velocity pushes particles toward the lower wall of the annulus; consequently, the annular mud velocity must be sufficient to avoid bed formation. Due to these factors, it is difficult to calculate 2013 The Korean Society of Rheology and Springer 77

2 Young-Ju Kim, Sang-Mok Han and Nam-Sub Woo accurately and to control the pressures in slim hole wellbores when the inner cylinder rotates with a proper rotational speed below 1000 rpm, as required for safety. In this case, the drilling fluid flow in the small clearance of the concentric annulus has the characteristics of vortex flow in the transitional regime. In this study, the pressure losses and the skin friction coefficients through rotating annuli with diameter ratios of 0.52 and 0.9 have been investigated numerically and experimentally, under fully developed flow using water and 0.4% CMC aqueous solution. In the oil and gas drilling operation, the radii ratios (between the diameters of hole and drill pipe) of conventional drilling are 0.3~0.5, and those of slim hole drilling are over 0.8. The radii ratios of 0.52 and 0.9 are selected, because they can show the best flow characteristics of the conventional and slim hole drillings, respectively. The 0.4% CMC aqueous solution exemplifies the characteristics of shear thinning fluids. The rotational speed of the inner cylinder changes from 0~62.82 rad/s (0 to 600 rpm). At this time, the range of Reynolds number is, 100 Re 12, Data Reduction Average axial velocity in a concentric annulus with rotation can be expressed in terms of the pressure loss dp/dz, as follows (Papanastasiou 1994): u dp r 2 2 = ( R 2 ) e z + { ωr}e r (1) 4ηdz where, u is the velocity vector, η=r 1 /R 2 is the ratio of radii, ω is the angular velocity of the rotating inner cylinder, and e is the unit vector. The skin friction coefficient C f is given by: D h dp C f = (2) dz 2 2ρv z where, D h =2(R 2 R 1 ) is the hydraulic diameter. Equations (1) and (2) can be combined as follows: Equations (1) and (2) are used in this study, with the following constants: D h = m; R 1 = m; R 2 = m for η =0.52 and D h = m; R 1 = m; R 2 = m for η =0.9. Also, the pressure losses have been measured by Eq. (3): dp gh sinθ ( ρ ccl4 ρ) = (3) dz z where, ρ, ρ ccl4, θ, h, and z denote the density of the fluid, the density of CCl 4, the inclined angle, the head difference, and the distance between pressure holes for each. The experimental values of the skin friction coefficients in the laminar region can be evaluated by substituting Eq. (1) and Eq. (3) into Eq. (2). Fig. 1. (Color online) Schematic diagram of experimental apparatus; all dimensions in m. 3. Experimental and Numerical Methods 3.1 Experimental method The experimental setup consists of a cylinder part, a supporting part, a fluid-providing and rotating part, and a measuring part, which measures the flow rate, pressure loss, and temperature, as shown in Fig. 1. A centrifugal pump delivers the working fluid from a supply tank to a surge tank. The fluid flows into the annular passage of 3.82 m length, with outer brass pipe of nominal inside radius of 19.2 mm and an inner stainless steel rod radius of 10 mm, for η=0.52. For η=0.9, R 2 is 19 mm and R 1 is 17 mm. To ensure fully developed flow in the measuring section, the length of the straight pipe downstream of the test section is 2.32 m, corresponding to 126 hydraulic diameters, with a uniform step at the inlet, in order to produce an artificially-thickened boundary layer. The rotating cylinder with a length of 1.5 m and its non-rotating counterpart are connected by bronze bearings. Also, in order to prevent vibration and eccentricity caused by the rotation of the cylinder, connectors are installed at three locations on the outer cylinder. Static pressure is measured by using the 0.5 mm diameter holes installed on the outer cylinder, which is distributed longitudinally along the flow direction. Static pressures are read from a calibrated manometer bank, with 78 Korea-Australia Rheology J., Vol. 25, No. 2 (2013)

3 Flow of Newtonian and non-newtonian fluids in a concentric annulus with a rotating inner cylinder Fig. 2. Viscosity of 0.1~0.4 CMC and 5% bentonite solutions using the power law model. ±1 mm resolution. The specific gravity of the CCl 4 manometer fluid is 1.88, giving a height range of 20~ 600 mm. Drilling fluids usually have non-newtonian properties. Non-Newtonian fluids are those for which the strain rate and stress curve are not linear, i.e. the viscosity of non- Newtonian fluids is not constant at a given temperature and pressure, but depends on other factors, such as the rate of shear in the fluid, the apparatus in which the fluid is contained or even the previous history of the fluid. As shown by Lauzon and Leid (1979), the power law model adequately describes most drilling fluids at shear rate normally encountered in wellbore annuli during normal drilling operation. The carrier fluid used in the experiment, 0.4% CMC solution, is a shear-thinning fluid. In the case of CMC and solution, n <1 and the power law relating the shear stress τ to the shear rate γ is given by: τ = kγ n (4) where, n is the flow behavior index and K is the consistency factor. The apparent viscosity µ a for a power law fluid may be expressed in terms of n and K as follows, µ a = kγ n 1 (5) The viscosity of 0.1%~0.4% CMC solutions and 5% bentonite solution are shown in Fig. 2. The viscosity is measured by using a Brookfield DV-III + programmable viscometer. As shown in the figure, the viscosity of 0.4% CMC solution changes from 13 cp to 10 cp at the shear rate range from 100 to 350, respectively The cylinder and the rotating parts are supported by a 4 m long H-beam made of construction steel. In practice, the inner cylinder is slightly distorted, and it proves impossible to achieve a concentric geometry over the entire length of the test section. Fig. 3. Pressure differences of water as a function of z with various Re. The flow rate is measured with a magnetic flowmeter, with accuracy within ±0.5%. The temperature of the working fluid is measured with a digital multimeter. The inner cylinder can be rotated at any speed up to a maximum of 1500 rpm, by means of an alternating current motor and gear box. A glass box with the thickness of 5 mm filled with water has been installed to prevent diffused reflection, because the visualization part is cylindrical. Particles used for visualization are Polyvinyl chloride powders (PVC, specific gravity=1.1), with a mean diameter of 150 µm. The development of the flow field is identified by the change of the axial pressure gradient. Therefore, the value of pressure losses according to the flow rate has been measured between tap 1 and taps 2 ~ 5 of Fig. 1, to check the development of the flow field. In the case of water, the pressure loss along the z direction has been illustrated in Fig. 3, with various Re. Since the measured values of P 1,2 and P 1,3 have large errors, due to the short distances between taps, experiments have been repeated several times, to minimize the errors. Errors have appeared at a maximum of ±7% for Re 1000, because the viscosity coefficient is relatively small. The measured pressure losses along the flow of z-direction at each tap are shown in Fig. 3, to confirm the development of the flow field Numerical analysis The continuity and momentum equations for threedimensional, incompressible fluid flow were used to solve the equations of motion as, follows: u = 0 u u 1 = -- p+ v 2 u (7) ρ where, ρ, u, ν, and p denote the velocity vector, the density, (6) Korea-Australia Rheology J., Vol. 25, No. 2 (2013) 79

4 Young-Ju Kim, Sang-Mok Han and Nam-Sub Woo Fig. 4. Computational grid. the kinetic viscosity, and the pressure of the fluid, respectively. In numerical study, the working fluids were water and power law fluids, and the numerical results were compared with experimental results, by using the commercial CFD code, FLUENT. The computational grid of the flow field is shown in Fig. 4. The inner cylinder radius is R 1 =10 mm, the outer cylinder radius is R 2 =19.2 mm, the axial length of the cylinder is z=1.5 m, and the radii ratio is Constant velocity and pressure boundary conditions of the fluid were imposed on the inlet and outlet of the annulus, respectively. No-slip boundary conditions were used at the inner and outer cylinders. The range of the rotational speeds of the inner cylinder was between 0 and 600 rpm. The numerical analysis was carried out in a laminar regime. The Taylor vortex, fluid velocity, and pressure loss for an annulus with various rotational speeds and Reynolds numbers were calculated, to understand the effects of rotation and flow rate on flow instability. 4. Results and Discussion 4.1. Experimental result The rotating annulus flow with axial flow and inner cylinder rotation is classified as four flow regimes laminar flow, laminar flow + Taylor vortices, turbulent flow, and turbulent flow + Taylor vortices. In this study, the annular flow of the test section in Fig. 1 shows the helical flow pattern. The relations between skin friction coefficient C f and Reynolds number Re are shown in Fig. 5, for various rotational speeds in the range of water flow rate 1~60 l/min (200 Re 12,000). Where, Reynolds number Re is defined as Eq. (8): v Re z D = h v (8) Fig. 5. Skin friction coefficients of water as a function of Re at 0~500 rpm: (a) η=0.52, and (b) η=0.9. where, v z is the fluid velocity in the z direction, D h is the hydraulic diameter, and ν is the kinematic viscosity. In the case of water, it was hard to measure the pressure loss accurately because it was so small for low values of Re. However, using 0.4% CMC solution, it was relatively easy to measure the pressure loss, since its zero shear viscosity is thirteen times greater than that of water. The measurement of pressure loss for 0.4% CMC solution was carried out for 100 Re 1,500. Fig. 6 shows the effects of rotation on the skin friction coefficient for 0.4% CMC solution. The critical Reynolds number for 0.4% CMC solution is found to be lower by 0.87%~6.3% than that of water at 0~600 rpm, because of the different viscosities at the same rotational speed. The laminar regime is confined to the range Re < Re c, where the critical Reynolds number decreased according to the increase in rotational speed. From the experimental results for the skin friction coefficient for both water and 0.4% CMC solution, we con- 80 Korea-Australia Rheology J., Vol. 25, No. 2 (2013)

5 Flow of Newtonian and non-newtonian fluids in a concentric annulus with a rotating inner cylinder Fig. 7. Pressure losses of 0.4% CMC as a function of Re at 200, 400 and 600 rpm (η=0.52). Fig. 6. Skin friction coefficients of 0.4% CMC solution as a function of Re at 0~500 rpm: (a) η=0.52, and (b) η=0.9. clude that the influence of rotational speed on the skin friction coefficient for 0.4% CMC solution is relatively less than for the influence on water, because the value of the Rossby number for 0.4% CMC solution is greater than the value for water (The Rossby number is the ratio of Re ω to Re, which represents the ratio of Coriolis to inertial forces). The variation of the average skin friction coefficient with respect to Reynolds number in a Newtonian laminar flow field is shown in Fig. 5. The correlation agrees with the equation C f = 23.8/Re, as presented by Shah and London (1978), in the radii ratio of 0.52 as shown in Fig. 5(a). When the radii ratio is 0.9 (Fig. 5(b)), the correlation between the skin friction coefficient and Reynolds number is C f = 11/Re. That is, the gradient of the skin friction coefficient with Reynolds number is the same for both radii ratios. However, the value of skin friction coefficient for a radii ratio of 0.52 is about two times larger than for a radii ratio of 0.9. In the case of 0.4% CMC solution, the value of the skin friction coefficient for a radii ratio of 0.52 is about four times larger than for a radii ratio of 0.9, as shown in Fig. 6. The pressure losses for 0.4% CMC solution are shown in Fig. 7. As previous researchers also observed, the laminar Taylor vortex exists in the range of 131 Re 927 (Wereley and Lueptow 1998). Our results show that the laminar Taylor vortex exists in the range of 0 Re Re l,t. The value of Re l,t increases as the rotational speed of the inner cylinder, N increases, and at the same time, Ro decreases. If the bulk flow increases for Re > Re l,t, the flow belongs to the laminar flow regime, where the Taylor vortex has the least influence. The relative skin friction coefficient C f * is defined by Eq. (9), where subscripts s and R represents the skin friction coefficient for rotation and non-rotation, respectively. C f * C fr, C fs, C fr, = We obtained C * f for water using Eq. (9), as shown in Fig. 8. The value of C * f increased significantly, from 40% to 80%, in the laminar regime, as N increased from 100 to 600 rpm. On the other hand, for 0.4% CMC solution, the value of C * f increased slightly, from 4% to 30%, as N increased from 100 to 600 rpm. Thus, the influence of rotational speed on pressure loss for 0.4% CMC solution is relatively weaker than the influence on water, because the value of Rossby number Ro for 0.4% CMC solution is greater than Ro for water (i.e. Ro 0.7). If C * f is used to express the influence of N (i.e. Re w ) on the skin friction coefficients, C * f is consistently unchanged * in the laminar regime, as seen in Fig. 8. However, C f tends to increase in the transition regime for Re Re c, and to decrease in the turbulent regime (i.e. Re >> Re c ). However, this tendency decreases as N increases, due to the Taylor vortex. The increased rate of skin friction coefficient due to the rotation is about 65% for water at N = (9) Korea-Australia Rheology J., Vol. 25, No. 2 (2013) 81

6 Young-Ju Kim, Sang-Mok Han and Nam-Sub Woo Fig. 8. Normalized relative skin friction coefficients of water with Re at 100, 300 and 600 rpm (η=0.52). Fig. 10. (Color online) Taylor vortices with rotation of inner cylinder: (a) 100 rpm, and (b) 150 rpm (Re=0). and 0.4% CMC solution, respectively. The C f can be correlated with Ro and Re using Eq. (10), and using Eq. (11) for water, and 0.4% CMC solution, respectively: C f Re = 45.3 Ro (10) Fig. 9. Relation of C f Re with Ro for laminar flow in (a) water, and (b) 0.4% CMC water solution (η=0.52) rpm. Fig. 9 shows the relation between C f Re and Ro for water C f Re = 77.3 Ro (11) The deviation of experimental data with the correlations is within ±9%. As the Rossby number becomes infinite, it tends to asymptotically approach C f = 23.8/Re for water, and C f = 69.3/Re for 0.4% CMC solution. For water, Taylor vortices are observed between the outer rest cylinder and the inner rotating cylinder. Fig. 10 shows Taylor vortices for Re=0 at N=100 and 150 rpm, respectively. When the flow rate is zero, the Taylor vortex cannot be observed for N < 50 rpm, due to the small effect of rotation on the flow. The wavelength λ is defined as the length of a pair of Taylor vortices. The value of λ is 20 mm at N=100 rpm (see Fig. 10). For 200 < N < 300 rpm at Re = 0, the wavelength is not clearly observed, because an increase of rotational speed is promoted in the turbulent effect. After the Taylor vortices break down, the rotating flow between the coaxial cylinders becomes turbulent flow. Thus, the Taylor vortex cannot be observed for N > 400 rpm, due to the turbulent effect. This physical phenomenon is very important in considering the rotating flow between the coaxial cylinders. When the gap between the outer rest cylinder and the inner rotating cylinder becomes wide, the wavelength of the Taylor vortices becomes longer with time. If the gap between the inner and outer cylinders is very narrow, the wavelength λ of the Taylor vortices can be predicted using the following relation (Ogawa 1993): 82 Korea-Australia Rheology J., Vol. 25, No. 2 (2013)

7 Flow of Newtonian and non-newtonian fluids in a concentric annulus with a rotating inner cylinder Fig. 12. (Color online) Velocity vectors of the flow used for particle tracking studies; note the location of vortices (η=0.52). Fig. 11. (Color online) Flow characteristics at 100 rpm. λ = R 1 (116.8/Re w ) 2/3 (mm) (12) As the rotational speed N changed from 100 to 150 rpm, the theoretical value of λ obtained by Eq. (12) decreased by 23%, but the measured λ shown in Fig. 10 decreased by approximately 17.3%. Because Eq. (12) is only valid for a narrow gap (η > 0.8), the theoretical λ is different from the experimental λ, for the case of η=0.52. The wavelength is significantly influenced by the rotation, and the size of the gap between the concentric cylinders Numerical results In order to validate the CFD calculation with FLUENT, numerical results were compared with the experimental results. The velocity vectors and pressure contours at the r-è plane are shown in Fig. 11. The maximum velocity is located at the inner part of the gap, and the tangential velocity increases as the rotational speed increases. The measurement of the axial pressure gradient, which is found to be constant with axial distance, confirms the fully developed flow field. Fig. 12 shows a pair of Taylor vortices when the inner cylinder rotated at 100 and 300 rpm for η=0.52. The magnitude of the wavelength, which is defined as the length of a pair of Taylor vortices, decreased as rotational speed increased. As shown in Fig. 13, the relationship between skin friction coefficient and Reynolds number shows the same tendency, in both experiment and numerical analysis. But in high rotational speed, the difference between experimental and numerical results increases, because of experimental errors such as vibration and eccentricity. Fig. 14 shows the relationship between pressure losses and the Reynolds numbers of water, for rotational speeds of 0~600 rpm. The pressure loss increases as the rotational speed increases, and numerical results shows the same tendency as the experimental results. Fig. 15 shows profiles of rotating flow velocities of the Newtonian and non-newtonian fluids when the inner cylinder rotated from 200 to 600 rpm, respectively, for an axial velocity of m/s. Distances were normalized to the radial distance S from the outer to inner wall. In the case of non-newtonian fluid, the location of the maximum axial velocity moves to the inner cylinder as the rotational speed increases, as shown in Fig. 15(a). In Fig. 15(b), the tangential velocity gradient of the Newtonian fluid is much greater than that of the non-newtonian fluid. Korea-Australia Rheology J., Vol. 25, No. 2 (2013) 83

8 Young-Ju Kim, Sang-Mok Han and Nam-Sub Woo Fig. 13. Skin friction coefficients of water as a function of Re in a laminar region: (a) 0 rpm, (b) 100 rpm, and (c) 300 rpm. Fig. 14. Pressure losses of water as a function of Re at 0~600 rpm (η=0.52). Fig. 15. Rotating flow velocities of Newtonian and non-newtonian fluids (η=0.52). 5. Conclusions In this study, the effects of rotational speed, flow rate, and radii ratios on the pressure loss and skin friction coefficients in the vortex flow field were investigated experimentally and numerically, for rotating flow between an outer rest cylinder and an inner rotating cylinder. It is determined that the critical Reynolds number of 0.4% CMC has a slightly lower value than that of water, due to the difference in rheological characteristics. The pressure loss increases slightly as the rotational speed increases, although the gradient of pressure losses decreases as the Reynolds number increases in the laminar and transition regimes. Also, the Rossby number of the water is smaller than that of 0.4% CMC solution. Thus, the effect of rotation on the pressure loss in water is much greater than that for 0.4% CMC solution. The value of the skin friction coefficient for a radii ratio of 0.52 is about two times greater than the value for a radii 84 Korea-Australia Rheology J., Vol. 25, No. 2 (2013)

9 Flow of Newtonian and non-newtonian fluids in a concentric annulus with a rotating inner cylinder ratio of 0.9. For 0.4% CMC solution, the value of the skin friction coefficient with a radii ratio of 0.52 is about four times greater than that of the value for a radii ratio of 0.9. In addition, the location of the maximum axial velocity of non-newtonian fluid moves to the inner cylinder as the rotational speed increases, and the tangential velocity gradient of the Newtonian fluid is much greater than the tangential velocity gradient for the non-newtonian fluid. Acknowledgement This work was supported by the Development of Drill Riser System Project ( ) of the KIGAM, funded by the Ministry of Knowledge Economy of Korea. Nomenclature C f C fr, C fs, * C f skin friction coefficient skin friction coefficient with rotation skin friction coefficient without rotation relative skin friction coefficient (see Eq. (9)) hydraulic diameter, ( ) pressure loss (Pa/m) D h 2 R 2 R 1 dp dz m ratio of the eccentricity to the difference of radii N rotational speed (rpm, rpm) difference of pressure (Pa) radius of inner cylinder (mm) radius of outer cylinder (mm) P R 1 R 2 Re Re c Re lt, bulk flow Reynolds number, v z D h v critical Reynolds number Reynolds number discriminating laminar-taylor vortex regime and pure laminar regime Re ω rotational Reynolds number, ωr 1 ( R 2 R 1 ) v Ro Rossby number, 2v z ωr 1 v z fluid velocity in the z-direction (m/s) z distance between pressure taps (mm) Greek Symbols η ratio of radii, R 1 R 2 λ wavelength (mm) µ absolute viscosity ( Pa s ) ν kinematic viscosity ( m 2 s ) ρ density of fluid ( kg m 3 ) ω angular velocity of rotating cylinder (rad/s) References Delwiche, R.A., M.W.D. Lejeune, and D.B. Stratabit, 1992, Slimhole Drilling Hydraulics, Society of Petroleum Engineers Inc. SPE 24596, Diprima, R.C., 1960, The Stability of a Viscous Fluid Between Rotating Cylinders with an Bulk Flow, Journal of Fluid Mechanics 366, Escudier, M.P., and I.W. Gouldson, 1995, Concentric annular flow with centerbody rotation of a Newtonian and a shear-thinning liquid, Journal of Fluid Mechanics 16, Fredrickson, A.G., 1960, Helical Flow of an Annular Mass of Visco-elastic Fluid, Chemical Engineering Science 11, Lauzon, R.V., and K.I.B. Reid, 1979, New Rheological Model Offers Field Alternatives, Oil and Gas Journal 77, Nakabayashi, K., K. Seo, and Y. Yamada, 1974, Rotational and Axial through the Gap between Eccentric Cylinders of which the Outer One Rotates, Bull. JSME, 17(114), Nouri, J.M., H. Umur, and J.H. Whitelaw, 1993, Flow of Newtonian and Non-Newtonian Fluids in Concentric and Eccentric Annuli, Journal of Fluid Mechanics 253, Nouri, J.M., and J.H. Whitelaw, 1994, Flow of Newtonian and Non-Newtonian Fluids in a Concentric Annulus with Rotation of the Inner Cylinder, Journal of Fluid Mechanics 116, Ogawa, A., 1993, Vortex Flow, CRC Press Inc., Papanastasiou, T.C., 1994, Applied Fluid Mechanics, Daewoong Press Inc. Shah, R.K., and A.L. London, 1978, Laminar Flow Forced Convection in Ducts, Academic Press, New York. Stuart, J.T., 1958, On the Nonlinear Mechanics of Hydrodynamic Stability, Journal of Fluid Mechanics 4, Watanabe, S., and Y. Yamada, 1973, Frictional Moment and Pressure Drop of the Flow through Co-Axial Cylinders with an Outer Rotating Cylinder, Bull. JSME 16(93), Wereley, S.T., and R.M. Lueptow, 1998, Spatio-temporal character of non-wavy and wavy Taylor-Couette flow, Journal of Fluid Mechanics 364, Wilson, C.C., 2001, Computational Rheology for Pipeline and Annular Flow, Gulf Professional Publishing. Yamada, Y., 1962, Resistance of a Flow through an Annulus with an Inner Rotating Cylinder, Bull. JSME 5(18), Korea-Australia Rheology J., Vol. 25, No. 2 (2013) 85