Non-Newtonian Natural Convection Flow along an Isothermal Horizontal Circular Cylinder Using Modified Power-law Model

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1 American Journal of Fluid ynamics 3, 3(): -3 OI:.593/j.ajfd.33. Non-Newtonian Natural Convection Flow along an Isothermal Horizontal Circular Cylinder sing Modified Power-law Model Sidhartha Bhowmick, Md. Mamun Molla, Suvash C. Saha 3,* epartment of Mathematics, Jagannath niversity, haka, Bangladesh epartment of Electrical Engineering and Computer Science, North South niversity, haka, 9, Bangladesh 3 Institute of Future Environments, School of Chemistry, Physics and Mechanical Engineering, Queensland niversity of Technology, GPO Box 3, Brisbane, QL, Australia Abstract La minar t wo-dimensional natural convection boundary-layer flow of non-newtonian fluids along an isothermal horizontal circular cylinder has been studied using a modified power-law viscosity model. In this model, there are no unrealistic limits of zero or infinite viscosity. Therefore, the boundary-layer equations can be solved numerically by using marching order implicit finite difference method with double sweep technique. Numerical results are presented for the case of shear-thinning as well as shear thickening fluids in terms of the fluid velocity and temperature distributions, shear stresses and rate of heat transfer in terms of the local skin-friction and local Nusselt number respectively. Keywords Non-Newtonian, Power-law, Shear-thinning, Shear-thickening, Finite ifference. Introduction Natural convection laminar flow of non-newtonian power-law fluids from an isothermal horizontal circular cylinder p lays an important role in numerous engineering applications those are related with pseudo-plastic fluids. The pseudo-plastic flu id is characterized by a constant viscosity at very low shear rates, a viscosity, which decreases with shear rate, at intermediate shear rates and an apparently constant viscosity at very high shear rates. The interest in heat transfer problems involving power-law non-newtonian fluids has grown in the past half century. An excellent research on non-newtonian fluids was performed by Boger[]. Acrivos[] was the first who considered boundary-layer flows for such non-newtonian fluids. Since then, a large number of papers have been published, due to their wide relevance in pseudo-plastic fluids like chemicals, foods, polymers, molten plastics and petroleum production and various natural phenomena. It should be noted that a complete survey of these literatures was impractical; however, selected papers are listed here to provide starting points for a broader literature search [3-5]. In the boundary-layer study, the previous researchers used the traditional power-law viscosity * Corresponding author: s_c_saha@yahoo.com (Suvash C. Saha) Published online at Copyright 3 Scientific & Academic Publishing. All Rights Reserved correlation that viscosity becomes infinite for small shear rates or vanishes for the limits of large shear rates, which are giving the unrealistic physical results. Because an infinite viscosity corresponds to solids and no frictionless fluid has ever been found, a partial set of measured viscosity shear relations are not sufficient for a boundary-layer study. The recently proposed modified power-law correlat ion is sketched for various values of power index n, which has been shown in Fig.. The present model is formulated based on the available experimental data for the non-newtonian fluids (see Boger[]). It is clear that the new correlation does not contain the physically unrealistic limits of zero and infinite viscosity displayed by traditional power-law correlations[]. The modified power-law, in fact, fits measured viscosity data well. The constants in the proposed model can be fixed with available measurements and are described in detail in ao and Molla[6]. The boundary-layer formulation on a flat plate is described and numerically solved for non-newtonian fluid in ao and Molla[6, 7] and the associated heat transfer for two different heating conditions is reported in Molla and ao[8, 9] for shear-thinning fluid. In this investigation, the behavior of both shear-thinning and shear-thickening fluids on the natural convection laminar flow from a uniformly heated horizontal circular cylinder are studied by choosing the power-law index as n (=.6,.8,.,.,.) to fully demonstrate the performance of various non-newtonian fluids.. Formulation of the Problem

2 American Journal of Fluid ynamics 3, 3(): -3 A steady two-dimensional laminar natural convection boundary-layer of a non-newtonian fluid over an isothermal horizontal circular cylinder of radius a with uniform surface temperature and a distributed heat source of the form gβ ( T T ) has been considered. The viscosity depends on shear rate and is correlated by a modified power-law. We consider shear-thinning and shear-thickening situations of non-newtonian fluids. It is assumed that the surface temperature of the cylinder is T w ( > T ), where, T is the amb ient temperature of the fluid and T is the temperature of the fluid. The configuration considered is as shown in Fig.. nder the above assumptions, the boundary-layer equations governing the flow and heat transfer are v + = () x x ρ u + v = µ + ρgβ ( T T ) sin () x a T T k T u + v = (3) x ρc p Where u and v are velocity components along the x and y a xes, ρ is the fluid density, µ is the dynamic viscosity of the fluid in the boundary-layer region, g is the acceleration due to gravity, β is the coefficient of thermal expansion, k is the thermal conductivity and C p is the specific heat at constant pressure. µ The kinematic viscosity ν = is correlated by a ρ modified power-law, which is K ν = ρ n for γ γ () The constants γ and γ are threshold shear rates, which are given according to the model of Boger[], K is the dimensional constant, for which dimension depends on the power-law index n. The values of these constants can be determined by matching with measurements. Outside of the preceding range, viscosity is assumed to be constant; its value can be fixed with data given in Fig.. The boundary conditions for the present problems are u =, v =, T = Tw at y = (5a) u, T T as y (5b) We introduce non-dimensional dependent and independent variables according to, x / y a / x =, y = Gr, u = u Gr, a a ν a / v = v Gr υ T T gβ =, Gr = Tw T ν ν ρc p =, Pr = ν k ( T T ) w ν a 3, µ ν =, ρ (6) Where, ν is the reference viscosity at γ, is the non-dimensional temperature of the fluid, Gr is the Grashof number and Pr is the Prandtl number. sing equation (6) in equations (-) we get the following non-dimensional equations: v + = (7) x u + v = + sin x x y y y = u + v = x K ρν n ν a Gr Pr / ( T T ) n ( n ) n 3 K ν gβ w a = ρν a ν y () n = C The length scale associated with the non-newtonian power-law is ( n) Cρ a = (3 5) / [ ( )] () n Kν gβ Tw T The corresponding boundary conditions are u =, v =, = at y = u, as y Now we introduce the parabolic transformation: u = x, = y, =, V = v, = (3) x Substituting variable (3) into equations (7)-() leads to the following equations: V + + = () sin + V + = + + (5) + V = (6) Pr, γ γ n γ =, γ γ γ where, γ = (7) γ n γ, γ γ γ The correlation (7) is a modified power-law correlat ion first presented by ao and Molla[6]. This correlation (8) (9)

3 Sidhartha Bhowmick et al.: Non-Newtonian Natural Convection Flow along an Isothermal Horizontal Circular Cylinder using Modified Power-law Model describes that if the shear rate γ γ and γ lies between the threshold shear rates, then the non-newtonian viscosity,, varies with the power-law of γ. On the other hand, if the shear rate γ does not lie within this range, then the non-newtonian viscosities are different constants, as shown in Fig.. This is a property of many measured viscosities. Equation (-6) can be solved by marching downstream with the leading edge condition satisfying the following differential equations, which are the limits of equations (-6) as. V + = (8) V + = + + (9) V = () Pr The corresponding boundary conditions are =, V =, = at =, as Equations (-6) and (8-) are discretized by a central-difference scheme for the diffusion term and a backward-difference scheme for the convection terms. Finally, we get an implicit tri-diagonal algebraic system of equations, which can be solved by a double-sweep technique. The normal velocity is directly solved from the continuity equation. The computation is started at = and marches to downstream to =3.6. After several test runs, converged results are obtained by using =.5 and =.5. In practical applications, the physical quantities of principle interest are the local skin-friction coefficients C f and the local Nusselt number Nu, which are T g ( / ) / = ( / ) / = = C f Gr = Nu Gr Figure. The flow model and coordinate system a _ x T w _ y (3) () n =. n =. n =. n =.8 n = γ 5 Figure. Modified power-law correlation for the power-law index n (=.6,.8,.,.,.) whileγ =.and γ =

4 American Journal of Fluid ynamics 3, 3(): Results and iscussion The numerical results are presented for the non-newtonian power-law of shear-thinning fluids ( n =.6 and.8) and the shear-thickening fluids (n =. and.) as well as the Newtonian case (n = ) while the Prandtl number, Pr = and 5. Based on the experimental data of Boger[] the thresholds shears γ and γ have been chosen as. and 5, respectively. The obtained results include the viscosity, velocity and temperature distribution, velocity gradient and the wall shear stress in terms of the local skin-friction coefficient, C ( Gr / ) / f and the rate of heat transfer as a form of the local Nusselt number, / Nu ( Gr / ) for the wide range of the power-law index n (=.6,.8,.,.,.). Figs. 3a and 3b show the viscosity distribution, as a function of at selected (=,, 3) locations for Pr = and 5, respectively and n =.6. Fro m Fig.3a, it is found for Pr = that there is one region of variable viscosity at = and 3, but there are two such regions at =; the primary region lies fro m. to. and the secondary variable viscosity region lies between.37 to.9. On the other hand, only one variable viscosity region was found in the case of Pr =5 in Fig. 3b. Again, Figs. -(f) show the viscosity distribution, as a function of at =,, 3 respectively with n =.6,.8,.,.,. for Pr = and 5. It is clearly seen in Fig. c that there are two viscosity distribution regions for n=.6 at = of Pr =; all other viscosity distributions are in one region. The velocity distribution as a function of at the selected locations ( =,, 3) for the different power-law indices (n =.6,.8,.,.,.) are presented in Figs. 5(a-c) fo r Pr = and 5(d-f) for Pr =5, respectively. Fig. 5 shows that for shear-thinning fluids (n=.6 and.8), the velocity increases due to the decrease of viscosities at the downstream region; consequently, the boundary layer is thinned. On the other hand, for shear-thickening fluids (n =. and.), the velocity decreases slowly and the boundary-layer is thickened as the fluid becomes more viscous. We may conclude that for Pr =5, the fluid velocity is smaller than that for Pr = and the boundary-layer thickness is larger for Pr =5 than that for Pr =. The corresponding temperature distribution are plotted for Pr = and 5 in Figs. 6(a-c) and 6(d-f), respectively. For both of these Prandtl numbers, at the downstream region, in the case of shear-thinning fluids, the variation of temperature in the boundary-layer is smaller than that of the shear-thickening non-newtonian fluids. As expected, the thermal boundary-layer is thinner for larger Prandtl numbers. Figures 7(a-c) and 7(d-f) show the corresponding velocity gradient for Pr = and 5, respectively. For the shear-thinning fluids (n=.6 and.8), the boundary-layer thickness decreases more at the downstream region than for the shear-thickening fluids (n =. and.). The boundary-layer thickness for Pr =5 is almost half of the boundary-layer for Pr =..75 = = =3.8 = = = Fi gure 3. Viscosity distribution for different values of at n =.6 for Pr =, Pr = 5

5 Sidhartha Bhowmick et al.: Non-Newtonian Natural Convection Flow along an Isothermal Horizontal Circular Cylinder using Modified Power-law Model.7 n= n= (c) n=.6.5. (d) n= (e).3 3. n=.6. (f) n= Fi gure. Viscosity distribution for different n at =, Pr =, =, Pr = 5, (c) =, Pr =, (d) =, Pr = 5, (e) = 3, Pr =, (f) = 3, Pr = 5

6 American Journal of Fluid ynamics 3, 3(): n= n= (c) (d) 8 6 (e). n= n= (f).5 n= n= Fi gure 5. Velocity distribution for different n at =, =, (c) = 3; Pr = and (d) =, (e) =, (f) = 3; Pr = 5

7 6 Sidhartha Bhowmick et al.: Non-Newtonian Natural Convection Flow along an Isothermal Horizontal Circular Cylinder using Modified Power-law Model n=.6.8 n= (c) 3 n=.6.8 (d) 3 n= (e) 3 5 n=.6.8 (f).5.5 n= Fi gure 6. Temperature distribution for different n at =, =, (c) = 3; Pr = and (d) =, (e) =, (f) = 3; Pr = 5

8 American Journal of Fluid ynamics 3, 3(): n=.6.6. n= (c) (d). n= n= (e) (f).8 n=.6.6. n= Fi gure 7. Velocity gradient for different n at =, =, (c) = 3; Pr = and (d) =, (e) =, (f) = 3; Pr = 5

9 8 Sidhartha Bhowmick et al.: Non-Newtonian Natural Convection Flow along an Isothermal Horizontal Circular Cylinder using Modified Power-law Model The effects of the non-newtonian power-law index n (=.6,.8,.,.,.) on the variation of the wall shear stress ( Gr / ) / C f are shown in Fig. 8a for Pr = and in Fig. 8b for Pr =5. The results from these figures clearly show that at the leading edge of non-newtonian fluids, whose effects start from >. 8 for Pr = and >. for Pr =5, the wall shear stress decreases for the shear-thinning fluids (n=.6 and.8) and increases for the shear-thickening fluids ( and.). At the downstream region, there is a similarity solution at = 3 and at = π, the boundary-layer of shear-thinning fluids is greater than that of shear-thickening fluids. As expected, the boundary-layer is thinner for larger Prandtl number. / Figs. 9 and 9 represent the local-rate of heat transfer in terms of the local Nusselt number ( ) Nu Gr / for Pr = and Pr =5, respectively. The local Nusselt number increases for n < and decreases for n > at the leading edge of non-newtonian fluids, whose effects start from >. for Pr = and >. 9 for Pr =5. At the downstream region, heat transfer is similar at = π. C f [Gr/x] /.6 n= C f [Gr/x] /. n= π Fi gure 8. Wall shear stress for different values of n: Pr =, Pr = 5 n= π n=.6 Nu[Gr/x] -/ Nu[Gr/x] -/ π 3 Fi gure 9. Local Nusselt number for different values of n: Pr =, Pr = 5. π

10 American Journal of Fluid ynamics 3, 3(): Conclusions This study deals with the la minar t wo-dimensional natural convection boundary-layer flow of non-newtonian fluids along an isothermal horizontal circular cylinder using a modified power-law viscosity model. The proposed modified power-law correlation agrees well with the actual measurements for non-newtonian fluids; consequently, it is a physically realistic model. The problem associated with the non-removal singularity introduced by the traditional power-law correlations do not exists for the modified power-law correlation proposed in this paper. Therefore, we may conclude from the above numerical simulations that the proposed modified power-law correlations can be used to investigate other heat transfer related problems for shear-thinning or shear-thickening non-newtonian fluids in boundary-layers. It is revealed that the effect of non-newtonian fluids eventually becomes dominant when shear rate increases within the threshold shear limits. We may summarise our results from above simulations as follows: It is seen from the numerical simulations that the velocity increases due to the decrease of viscosities at the downstream region for shear-thinning fluids. However, the velocity decreases slowly as the fluid becomes more viscous for the case of shear-thickening fluids. At the downstream region of the boundary layer, the variation of the temperature inside the boundary-layer is smaller for the case of shear-thinning fluids than that of the shear-thickening non-newtonian fluids for all Prandtl numbers considered here. The boundary-layer thickness decreases more at the downstream region for the shear-thinning fluids than that for the shear-thickening fluids. It is revealed that the boundary-layer thickness for Pr =5 is almost half of the boundary-layer for Pr =. It is observed that the local Nusselt number increases when n < and decreases when n > at the leading edge of non-newtonian fluids for both Prandtl number considered here. Nomenclature C f C a g n k C p Gr K Nu Pr T T w T Local skin-frict ion Constant Non-dimensional viscosity of the fluid Radius of the circular cylinder Acceleration due to gravity Non-Newtonian power-law index Thermal conductivity of the fluid Specific heat at constant pressure Grashof number imensional constant Local Nusselt number Prandt l number imensional temperature of the fluid Surface temperature of the cylinder Ambient t emperat ure u, v Velocity components along the x, y axes, respectively x, y Cartesian coordinate measured along the surface of the cylinder and normal to it respectively, V imensionless fluid velocities in the, direct ions, respect ively Axial direction along the circular cylinder Pseudo-similarity variable Greek symbols α Thermal diffusivity β Thermal expansion coefficient ρ Fluid density imensionless temperature of the fluid µ ynamic viscosity ν ( µ / ρ ) kinemat ic viscosity ν Reference viscosity of the fluid γ Shear rate ACKNOWLEGEMENTS The second author wishes to thank Professor L. S. ao, Arizona State niversity, SA, for the idea of the mod ified power-law viscosity model. REFERENCES [] Boger,. V., emonstration of upper and lower Newtonian fluid behavior in a pseudo plastic fluid, Nature Vol. 65 January 3 977, pp 6-8. [] Acrivos, A., A Theoretical Analysis of Laminar Natural Convection Heat Transfer to Non-Newtonian Fluids, AIChE Journal, Vol. 6, No., 96, pp [3] M.Kaddiri, M. Naïmi, A. Raji and M. Hasnaoui, Rayleigh-Bénard Convection of Non-Newtonian Power-Law Fluids with Temperature-ependent Viscosity, ISRN Thermodynamics, Vol.,, Article I 67, pages [] A. Sojoudi, S. C. Saha, Shear thinning and shear thickening non- Newtonian confined fluid flow over rotating cylinder, American Journal of Fluid ynamics, (6),, pp.7- [5] ale, J.., and Emery, A. F., The Free Convection of Heat from a Vertical Plate to Several Non-Newtonian Pseudo Plastic Fluids, Journal of Heat Transfer, Vol. 9, 97, pp [6] Chen, T. V. W., and Wollersheim,. E., Free Convection at a Vertical Plate with niform Flux Conditions in Non-Newtonian Power Law Fluids, Journal of Heat Transfer, Vol. 95, 973, pp. 3-. [7] Shulman, Z. P., Baikov, V. I., and Zaltsgendler, E. A., An Approach to Prediction of Free Convection in Non-Newtonian Fluids, International Journal of Heat and Mass Transfer, Vol. 9, No. 9, 976, pp. 3-7.

11 3 Sidhartha Bhowmick et al.: Non-Newtonian Natural Convection Flow along an Isothermal Horizontal Circular Cylinder using Modified Power-law Model [8] Som, A., and Chen, J. L. S., Free Convection of Non-Newtonian Fluids over Nonisothermal Two-imension al Bodies International Journal of Heat and Mass Transfer, Vol. 7, No. 5, 98, pp [9] Haq, S., Kleinstreuer, C., and Mulligan, J. C., Transient Free Convection of a Non-Newtonian Fluid Along a Vertical Wall, Journal of Heat Transfer, Vol., 988, pp [] Huang, M. J., Huang J. S., Chou,. L., and Cheng, C. K., Effects of Prandtl Number on Free Convection Heat Transfer from a Vertical Plate to a Non-Newtonian Fluid, Journal of Heat Transfer, Vol., Feb 989, pp [] Huang, M. J., and Chen, C. K., Local Similarity Solutions of Free-Convection Heat Transfer From a Vertical Plate to Non-Newtonian Power Law Fluids, International Journal of Heat and Mass Transfer, Vol. 33, No., 99, pp [] Kim, E., Natural Convection Along a Wavy Vertical Plate to Non-Newtonian Fluids, International Journal of Heat and Mass Transfer, Vol., No. 3, 997, pp [3] Khan, W. A., Culham, J. R., and ovanovich, M. M., Fluid Flow and Heat Transfer in power Law Fluids Across Circular Cylinders: Analytical Study, Journal of Heat Transfer, Vol. 8, Sept. 6, pp [] enier, J. P., and Hewitt, R. E., Asymptotic Matching Constraints for a Boundary-Layer Flow of a Power Law Fluid, Journal of Fluid Mechanics, Vol. 58,, pp [5] enier, J. P., and abrowski, P. P., On the Boundary-Layer Equations for Power-Law Fluids, Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences, Vol. 6, No. 5,, pp [6] ao, L. S., and Molla, M. M., Flow of a Non-Newtonian Fluids on a Flat Plate, : Boundary Layer, Journal of Thermophysics and Heat Transfer, Vol., No., October-ecember 8, pp [7] ao, L. S., and Molla, M. M., Forced convection of Non-Newtonian Fluids on a Heated Flat Plate, International Journal of Heat and Mass Transfer, Vol. 5, 8, pp [8] Molla, M. M., and ao, L. S., Flow of a Non-Newtonian Fluids on a Flat Plate, : Heat Transfer, Journal of Thermophysics and Heat Transfer, Vol., No., October-ecember 8, pp [9] Molla, M. M., and ao, L. S., The Flow of Non-Newtonian Fluids on a Flat Plate With a niform Heat Flux, ASME J. Heat Transfer, Vol. 3, 9, 7

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