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This is the author’s version of a work that was submitted/accepted for

pub-lication in the following source:

Sojoudi, Atta,

Saha, Suvash C.

,

Gu, YuanTong

, & Hossain, Md. Anwar

(2013)

Steady natural convection of non-Newtonian power-law fluid in a

trape-zoidal enclosure.

Advances in Mechanical Engineering, 2013(653108).

This file was downloaded from:

http://eprints.qut.edu.au/63377/

c

Copyright 2013 Atta Sojoudi et al.

This is an open access article distributed under the Creative Commons

Attribution License, which permits unrestricted use, distribution, and

re-production in any medium, provided the original work is properly cited.

Notice: Changes introduced as a result of publishing processes such as

copy-editing and formatting may not be reflected in this document. For a

definitive version of this work, please refer to the published source:

(2)

Volume 2013, Article ID 653108,8pages

http://dx.doi.org/10.1155/2013/653108

Research Article

Steady Natural Convection of Non-Newtonian

Power-Law Fluid in a Trapezoidal Enclosure

Atta Sojoudi,

1

Suvash C. Saha,

2

Y. T. Gu,

2

and M. A. Hossain

3

1Mechanical Engineering Department of Sharif University of Technology, Tehran 11365-9567, Iran

2School of Chemistry, Physics & Mechanical Engineering, Queensland University of Technology, 2 George Street,

GPO Box 2434, Brisbane QLD 4001, Australia

3Bangladesh Academy of Science, University of Dhaka, Dhaka 1207, Bangladesh

Correspondence should be addressed to Suvash C. Saha; s c saha@yahoo.com

Received 4 May 2013; Accepted 13 October 2013

Academic Editor: Waqar Khan

Copyright © 2013 Atta Sojoudi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Numerical investigation of free convection heat transfer in a differentially heated trapezoidal cavity filled with non-Newtonian Power-law fluid has been performed in this study. The left inclined surface is uniformly heated whereas the right inclined surface is maintained as uniformly cooled. The top and bottom surfaces are kept adiabatic with initially quiescent fluid inside the enclosure. Finite-volume-based commercial software FLUENT 14.5 is used to solve the governing equations. Dependency of various flow parameters of fluid flow and heat transfer is analyzed including Rayleigh number (Ra) ranging from 105to 107, Prandtl number (Pr) from 100 to 10,000, and power-law index (𝑛) from 0.6 to 1.4. Outcomes have been reported in terms of isotherms, streamlines, and local Nusselt number for various Ra, Pr,𝑛, and inclined angles. Grid sensitivity analysis is performed and numerically obtained results have been compared with those results available in the literature and were in good agreement.

1. Introduction

Rectangular enclosures with differentially heated vertical sidewalls are of great importance to many fields of studies in heat transfer phenomena such as natural convection. It is one of the most widely investigated configurations because of its prime importance as a benchmark geometry to study convec-tion effects and compare numerical techniques. Addiconvec-tionally, the geometry has many applications in different industrial techniques and equipments such as solar collectors, food pre-servation, compact heat exchangers, and electronic cooling systems among other practical applications. As a consequ-ence of these applications, a thorough literature exists in this

field of study especially in the case of Newtonian fluids [1–4].

Natural convection laminar flow of non-Newtonian Power-law fluids performs an important role in various engi-neering applications which are related to pseudoplastic fluids. It should be noted that the pseudoplastic fluid is character-ized by apparent viscosity or that consistency decreases in-stantaneously with an increase in shear rate. The study of fluid

flow and heat transfer related to Power-law non-Newtonian fluids has attracted many researchers in the past half-century. An excellent research on pseudoplastic fluid was conducted

by Boger [5]. At first, boundary-layer flows for such

non-Newtonian fluids were investigated by Acrivos [6]. Since then,

a large number of literatures are created due to their wide relevance to pseudoplastic fluids like chemicals, foods, poly-mers, molten plastics, and petroleum production and various natural phenomena.

It is important to be noted that most of fluids employed in chemical and petrochemical processes or many other indus-tries seems to show non-Newtonian behavior. The natural convection of a non-Newtonian fluid over enclosures such as a cylindrical enclosure or a heated plate has received more

attention [7–14]. Several methodologies including analytical

[7], numerical [8], and experimental [9] approaches have

been employed in most of these studies, and the results indi-cated that the free convection features are considerably affected by the rheological properties of the fluid. However, the crucial issue of the buoyant convective process in various

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2 Advances in Mechanical Engineering g H H 𝜕T/𝜕y = 0 𝜕T/𝜕y = 0 T = TC T = TH 𝜑 y,  x, u

Figure 1: Schematic of the cavity and the coordinate systems.

other geometries/enclosures of a non-Newtonian fluid has remained largely unexplored.

Kim et al. [15] studied unsteady buoyant convection of a

non-Newtonian Power-law fluid within a square enclosure. The authors used the finite volume technique realizing that the rheological properties have a considerable effect on the transient process. Additionally, the numerical solutions had an extensive qualitative agreement with the descriptions obtained from the scale analysis. Following their study, steady-state analysis is performed in this study for a trape-zoidal configuration of Non-Newtonian fluid. We have per-formed parametric studies by varying angle of the inclined surfaces, Rayleigh number, Prandtl number, and Power-law index.

2. Mathematical Formulation

Consider a two-dimensional trapezoidal enclosure of length

(base) and height𝐻, which is filled with an incompressible

Power-law non-Newtonian fluid.Figure 1displays the

enclo-sure with top and bottom insulated walls. The left inclined wall is heated and the right inclined wall is cooled with con-stant temperature. With invocation of Boussinesq’s approxi-mation, governing equations take the form as below:

𝜕𝑢 𝜕𝑥+ 𝜕V 𝜕𝑦 = 0, (1) 𝑢𝜕𝑢 𝜕𝑥+ V 𝜕𝑢 𝜕𝑦 = − 1 𝜌0 𝜕𝑝 𝜕𝑥+ 1 𝜌0( 𝜕𝜏𝑥𝑥 𝜕𝑥 + 𝜕𝜏𝑥𝑦 𝜕𝑦 ) , (2) 𝑢𝜕V 𝜕𝑥+ V 𝜕V 𝜕𝑦 = − 1 𝜌0 𝜕𝑝 𝜕𝑦+ 1 𝜌0( 𝜕𝜏𝑥𝑦 𝜕𝑥 + 𝜕𝜏𝑦𝑦 𝜕𝑦 ) + 𝑔𝛽 (𝑇 − 𝑇0) , (3) 𝑢𝜕𝑇 𝜕𝑥+ V 𝜕𝑇 𝜕𝑦 = 𝛼 ( 𝜕2𝑇 𝜕𝑥2 +𝜕 2𝑇 𝜕𝑦2) , (4)

where(𝑢, V) represent velocity components in the horizontal

𝑥 and vertical 𝑦 directions; 𝑇 represents the temperature;

𝑝 represents the pressure; 𝑔 represents the gravitational

accel-eration; and𝜌, 𝛽, and 𝛼 represent the density, thermal

expan-sion coefficient, and thermal diffusivity of the fluid at

ref-erence temperature𝑇0. The related boundary conditions are

𝑢 = V = 0 at 𝑥 cos (𝜑) + 𝑦 sin (𝜑) = 0, 𝑥 cos (𝜑) − 𝑦 sin (𝜑) = 𝐻 cos (𝜑) ,

0 ≤ 𝑦 ≤ 𝐻, 0 ≤ 𝑥 ≤ 𝐻,

(5a)

𝜕𝑇

𝜕𝑦 = 0 at 𝑦 = 0, 𝐻, (5b)

𝑇 = 𝑇𝐻 at𝑥 cos (𝜑) + 𝑦 sin (𝜑) = 0, 0 ≤ 𝑦 ≤ 𝐻, (5c)

𝑇 = 𝑇𝐶 at𝑥 cos (𝜑) − 𝑦 sin (𝜑) = 𝐻 cos (𝜑) , 0 ≤ 𝑦 ≤ 𝐻.

(5d)

Dimensionless forms of (1)–(4) can be obtained in the

fol-lowing fashion: (𝑋, 𝑌) =(𝑥, 𝑦)𝐻 , (𝑈, 𝑉) = (𝑢, V)𝛼/𝐻, 𝑃 = 𝜌 𝑝 0𝛼2/𝐻, 𝜃 = 𝑇 − 𝑇0 Δ𝑇 . (6)

The crucial part of the formulation is to assign a suitable fun-damental equation, which relates definite components of stress tensor to the relevant kinematics variables. For this pur-pose, a purely viscous Power-law non-Newtonian fluid is

assumed, which follows the Ostwald-De Waele Power-law [7–

9]:

𝜏𝑖𝑗= 2𝜇𝑎𝐷𝑖𝑗= 2𝐾(2𝐷𝑘𝑙𝐷𝑘𝑙)(𝑛−1)/2𝐷𝑖𝑗. (7)

In the above, two material parameters are involved, that is,𝐾,

the consistency factor and𝑛, the Power-law index, and 𝐷𝑖𝑗

represents the rate of deformation tensor. Apparently,𝑛 = 1

corresponds to those fluids of Newtonian behavior with the

coefficient of viscosity𝐾, whereas 𝑛 > 1 indicates the dilatant

(or shear thickening) behavior and𝑛 < 1 shows pseudoplastic

(or shear thinning) behavior of a non-Newtonian fluid. The pseudoplastic fluids have generally a high viscosity, and mal variation of viscosity has also a direct effect on the ther-mal and flow fields. In the present setup, the dependency of 𝐾 on temperature is not assumed; a small temperature

differ-ence,Δ𝑇, is assumed.

𝐷𝑖𝑗is simplified to the following equation for the

two-dimensional Cartesian coordinates:

𝐷𝑖𝑗= 1 2( 𝜕𝑢𝑖 𝜕𝑥𝑗 + 𝜕𝑢𝑗 𝜕𝑥𝑖) . (8)

From (7) and (8), we get (9) for apparent viscosity [11]:

𝜇𝑎 = 𝐾{2 [ (𝜕𝑢 𝜕𝑥) 2 + (𝜕V 𝜕𝑦) 2 ] + (𝜕V 𝜕𝑥+ 𝜕𝑢 𝜕𝑦) 2 } (𝑛−1)/2 . (9)

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Obviously, for𝑛 = 1, 𝐾 represents the conventional viscosity.

However, for nonunit𝑛, non-Newtonian behavior, complex

dependence of viscosity on fluid’s property, and velocity com-ponents gradients are diagnosed. Based on the physical rationalizations and trial-and-error efforts, a grouping, which

consists of the consistency coefficient𝐾, the Power-law index

𝑛, the fluid density 𝜌0, and the cavity height𝐻, emerges to be

appropriate [9]:

]󸀠= (𝐾

𝜌0)

1/(2−𝑛)

𝐻2(1−𝑛)/(2−𝑛). (10)

It is important to be noted that application of]󸀠, which is in

dimension of m2s−1, is analogous to that of kinematics

vis-cosity of Newtonian fluids. Using (10), Prandtl number and

Rayleigh number are defined, respectively, as below [12]:

Pr= (𝐾/𝜌0) 1/(2−𝑛)𝐻2(1−𝑛)/(2−𝑛) 𝛼 , Ra= 𝑔𝛽Δ𝑇𝐻 3 𝛼(𝐾/𝜌0)1/(2−𝑛)𝐻2(1−𝑛)/(2−𝑛) . (11)

It is of great interest for many researchers to investigate local Nu of hot wall in many thermal systems. Similarly, local Nu is studied for left hot inclined wall which is defined as follows:

Nu= −𝜕𝜃

𝜕𝑛, (12)

where𝑛 denotes the normal direction on left-side plane.

3. Numerical Procedure

Finite-volume-based code is used to discretize and solve the

coupled set of equations (1)–(4) employing commercial

soft-ware Ansys FLUENT 14.5. In this framework, QUICK scheme was used for convective terms and SIMPLE algorithm was employed for the coupling of the pressure and velocity.

Con-vergence criteria were set to 10−5for all relative residuals.

A grid of 81× 81 has been required for obtaining

accept-able results, as shown inTable 1; a refinement to 101× 101 leads

to a maximum difference of 2.04% and 0.35% in terms of

maximum stream function(𝜓Max) and average (Nuavg) for

Pr = 100 of a square enclosure. As an additional check of the results’ accuracy, the present solution has been validated against the Benchmark solutions obtained, in the case of the classical Newtonian fluids and non-Newtonian fluids in a square enclosure. Nusselt numbers of some certain cases are

compared inTable 2.

4. Results and Discussion

In this section, the results correspond to the influence of

important parameters, namely, inclination angle(0 ≤ 𝜑 ≤

60), Power-law index (0.6 ≤ 𝑛 ≤ 1.4), Rayleigh number (104

Ra≤ 106), and Prandtl number (100≤ Pr ≤ 10,000), on heat

transfer and fluid flow. The results are presented in the form of local Nusselt number, isotherm, and stream function for the

Table 1: Preliminary tests on the grid size effect (Ra= 105, Pr= 100, and𝜑 = 0).

Grid sizes

𝑛 61 × 61 81 × 81 101 × 101

|𝜓Max| Nuavg |𝜓Max| Nuavg |𝜓Max| Nuavg

0.6 0.1225 7.085 0.1223 7.045 0.1223 7.020

1 0.0049 4.771 0.0049 4.752 0.0048 4.741

1.4 4.954𝑒 − 6 3.785 4.947𝑒 − 6 3.775 4.940𝑒 − 6 3.770 Table 2: Validation of the numerical code for a square enclosure.

Ra Pr 𝑛 Present

Nuavg Nuavg[15] Error (%)

107 100 0.6 40.91 41.02 0.26 106 100 0.6 17.46 17.51 0.28 107 100 0.8 24.87 24.97 0.40 107 100 1 17.45 17.52 0.39 107 104 0.6 21.89 22.05 0.72 105 100 0.6 6.12 6.15 0.48

above parameters.Figure 2illustrates isotherms and

stream-lines of various angles for trapezoidal enclosure of Ra = 105,

Pr = 100, and𝑛 = 1. As expected due to presence of hot

and cold walls, fluid rises up from bottom horizontal edge, adjacent to the hot inclined wall and flows up along it reach-ing the top horizontal edge. Then, the fluid flows down beside the oblique cold wall forming a roll with clockwise rotation inside the cavity. By the increase of angle, horizontal iso-therms occupy much area of the enclosure. Also the formed roll is elongated toward the side walls by the increment of

trapezoidal angle.Figure 3indicates local Nu of hot wall for

three angles of Pr = 100, Ra = 105, and𝑛 = 1. For square

en-closure (𝜑 = 0), local Nu has a maximum value of nearly 14 at

the top end of hot inclined side wall. By tilting the angle to 30∘,

maximum Nu is reduced to almost 8 and its position is near top end again. By further increase in angle value (𝜑 = 60), Nu value is reduced more and many positions may be regarded to have maximum Nu of nearly 2. It is concluded that by the increment of trapezoidal angle, average Nu is reduced and this may be attributed to the increase of mean distance bet-ween two differentially heated inclined side walls.

Figure 4 displays isotherms and streamlines of distinct

Power-law index𝑛, from 0.6 to 1.4 for Pr = 100, Ra = 105, and

𝜑 = 30. When shear thinning behavior is converted to the shear thickening behavior by the increment of Power-law index, maximum stream function is reduced from nearly

0.3 kg/s to 9.3× 10−6kg/s. This reveals that for𝑛 > 1, fluid

gradually rises up to the top edge and we would expect lower

Nu for𝑛 > 1 with respect to those 𝑛 < 1. Isotherm lines show

that the intrusion of fluid at top and bottom edges is thickened by the increment of Power-law index. This fact shows that much part of fluid inside the enclosure is expressed for the

case of higher𝑛. Local Nu is displayed inFigure 5for different

𝑛 of Pr = 100, Ra = 106, and𝜑 = 30. Shear thinning fluid has a

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4 Advances in Mechanical Engineering 290.2 290.4 290.5 290.6 290.8 𝜑 = 0∘ (a) 𝜑 = 0∘ 2.4E − 04 7.3E − 04 1.7E − 032.9E − 03 4.4E − 03 (b) 290.2 290.4 290.5 290.7 290.8 𝜑 = 30∘ (c) 𝜑 = 30∘ 2.6E − 04 3.4E − 03 1.8E − 03 4.9E − 03 7.8E − 04 (d) 290.2 290.4 290.5 290.7 290.9 290.8 𝜑 = 60∘ (e) 𝜑 = 60∘ 3.6E − 03 1.5E − 03 5.7E − 03 3.0E − 04 (f)

Figure 2: Isotherms (left) and streamlines (right) for different inclination when Ra = 105, Pr = 100, and𝑛 = 1.

0 0.5 1 1.5 2 0 2 4 6 8 10 12 14

Hot wall curve length (m)

Loc al N u 𝜑 = 0∘ 𝜑 = 30∘ 𝜑 = 60∘

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290.2 n = 0.6 290.9 290.7 290.6 290.5 290.3 (a) 𝜓max= 0.3159 (b) 290.2 n = 0.8 290.9 290.8 290.6 290.5 290.5 (c) 𝜓max= 0.0669 (d) 290.3 n = 1.0 290.7 290.8 290.5 290.5 (e) 𝜓max= 0.0101 (f) 290.3 290.3 n = 1.2 290.7 290.8 290.5 (g) 𝜓max= 0.00038 (h) 290.3 290.2 n = 1.4 290.7 290.8 290.6 290.5 (i) 𝜓max= 9.33e − 06 (j)

Figure 4: : Isotherms (left) and streamlines (right) for different𝑛 when Ra = 106, Pr = 100, and𝜑 = 30∘.

fluid. This maximum value is located near top edge of sloped

hot wall for all𝑛.

Figure 6 represents isotherms and stream functions of

different Ra for Pr = 100,𝑛 = 1, and 𝜑 = 30. Stream lines reveal

that for Ra = 104a clockwise roll is formed within the

enclo-sure and with the increase of Ra; this roll is extruded and

elon-gated toward the side walls generating two small rolls. Also maximum stream function is increased from 0.0022 kg/s for

Ra= 104to 0.01 kg/s for Ra = 106. It is clear that larger value of

Ra results in higher Nu due to the higher rate of heat transfer

from hot wall to the cold wall andFigure 7reveals this fact for

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6 Advances in Mechanical Engineering 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60

Hot wall curve length (m)

Loc al N u n = 0.6 n = 0.8 n = 1.2n = 1.4 n = 1.0

Figure 5: Local Nu for different𝑛 when Pr = 100, 𝜑 = 30, and Ra = 106.

Ra = 1.0 × 104 (a) 𝜓max= 0.00222 (b) Ra = 1.0 × 105 (c) 𝜓max= 0.00521 (d) Ra = 1.0 × 106 (e) 𝜓max= 0.01014 (f)

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0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1

Hot wall curve length (m)

Loc al N u Ra = 104 Ra = 105 Ra = 106

Figure 7: Local Nu for different Ra when Pr = 100,𝜑 = 30, and 𝑛 = 1.

Pr = 100 290.75 290.60 290.45 290.25 (a) 𝜓max= 5.2 × 10−3 (b) Pr = 1000 290.70 290.60 290.30 290.50 (c) 𝜓max= 5.2 × 10−4 (d) Pr = 10000 290.75 290.90 290.45 290.25 290.55 (e) 𝜓max= 5.2 × 10−5 (f)

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8 Advances in Mechanical Engineering also investigated. Note that the values of Pr are much larger

than unity for non-Newtonian fluids and it has been shown that an increase of this parameter makes the contribution of

convective terms in (4) negligible [15], but to have better

in-sight into the fluid flow, we present numerical results of

dis-tinct Pr.Figure 8displays isotherms and streamlines of fluid

for different Pr of Ra = 105,𝑛 = 1, and 𝜑 = 30. There exists

negligible difference of isotherms and stream functions. Stream function for Pr = 100 is reduced from 0.0052 kg/s to

5.2× 10−5kg/s for Pr = 10,000. Note that Nu never changes as

the Power-law index is unity for different Pr [15].

5. Conclusions

A numerical study has been performed on steady natural con-vection of non-Newtonian fluids within a trapezoidal cavity with differentially heated walls. The main objective of the pre-sent work was to observe the influence of parameters, namely, Power-law index, trapezoidal angle, and Pr and Ra in terms of isotherms, streamlines, and local Nusselt number. Main out-comes of the study are as follows.

(i) By the increase of trapezoidal angle, the formed roll within the enclosure is elongated and extruded to-ward the side walls. Additionally maximum Nu on left hot wall is reduced by the increase of trapezoidal angle.

(ii) Shear thinning behavior of the working fluid has higher Nu value than that of shear thickening. This may be attributed to the lower maximum stream func-tion of higher Pr.

(iii) Increment of Ra enhances local Nu and makes the generated roll at the core of enclosure extruded to the side walls.

(iv) Pr variation has not significant effect on Nu as most non-Newtonian fluids contain higher values of Pr. Also maximum stream function is reduced by the increase of Pr.

References

[1] G. de Vahl Davis, “Natural convection of air in a square cavity: a bench mark numerical solution,” International Journal for Num-erical Methods in Fluids, vol. 3, no. 3, pp. 249–264, 1983. [2] A. F. Emery and J. W. Lee, “The effects of property variations on

natural convection in a square enclosure,” Journal of Heat Trans-fer, vol. 121, no. 1, pp. 57–62, 1999.

[3] O. Aydin, A. ¨Unal, and T. Ayhan, “Natural convection in rec-tangular enclosures heated from one side and cooled from the ceiling,” International Journal of Heat and Mass Transfer, vol. 42, no. 13, pp. 2345–2355, 1999.

[4] S. Ostrach, “Natural convection in enclosures,” Advances in Heat Transfer, vol. 8, pp. 161–227, 1972.

[5] D. V. Boger, “Demonstration of upper and lower Newtonian fluid behaviour in a pseudoplastic fluid,” Nature, vol. 265, no. 5590, pp. 126–128, 1977.

[6] A. Acrivos, “A theoretical analysis of laminar natural convection heat transfer to non-Newtonian fluids,” AIChE Journal, vol. 6, no. 4, pp. 584–590, 1960.

[7] S. Haq, U. USA, C. Kleinstreuer, and J. C. Mulligan, “Transient free convection of a non-Newtonian fluid along a vertical wall,” Journal of Heat Transfer, vol. 110, no. 3, pp. 604–607, 1988. [8] J. F. T. Pittman, J. F. Richardson, and C. P. Sherrard, “An

experi-mental study of heat transfer by laminar natural convection bet-ween an electrically-heated vertical plate and both Newtonian and non-Newtonian fluids,” International Journal of Heat and Mass Transfer, vol. 42, no. 4, pp. 657–671, 1999.

[9] A. F. Emery, H. W. Chi, and J. D. Dale, “Free convection through vertical plane layers of non-Newtonian Power-law fluids,” Jour-nal of Heat Transfer, vol. 93, no. 2, pp. 164–171, 1971.

[10] T. Y. W. Chen and D. E. Wollersheim, “Free convection at a vert-ical plate with uniform flux condition in non-Newtonian Power-law fluids,” Journal of Heat Transfer, vol. 95, no. 1, pp. 123– 124, 1973.

[11] S. W. Churchill and H. H. S. Chu, “Correlating equations for laminar and turbulent free convection from a vertical plate,” International Journal of Heat and Mass Transfer, vol. 18, no. 11, pp. 1323–1329, 1975.

[12] Z. P. Shulman, V. I. Baikov, and E. A. Zaltsgendler, “An approach to prediction of free convection in non-newtonian fluids,” Inter-national Journal of Heat and Mass Transfer, vol. 19, no. 9, pp. 1003–1007, 1976.

[13] M. Kaddiri, M. Na¨ımi, A. Raji, and M. Hasnaoui, “Rayleigh-B´enard convection of non-Newtonian Power-law fluids with temperature-dependent viscosity,” ISRN Thermodynamics, vol. 2012, Article ID 614712, 10 pages, 2012.

[14] A. Sojoudi and S. C. Saha, “Shear thinning and shear thickening non-Newtonian confined fluid flow over rotating cylinder,” American Journal of Fluid Dynamics, vol. 2, no. 6, pp. 117–121, 2012.

[15] G. B. Kim, J. M. Hyun, and H. S. Kwak, “Transient buoyant con-vection of a power-law non-Newtonian fluid in an enclosure,” International Journal of Heat and Mass Transfer, vol. 46, no. 19, pp. 3605–3617, 2003.

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