Mixed Convection From a Circular Cylinder to Power Law Fluids

Ind. Eng. Chem. Res. 2009, 48, 8219–8231

8219

Mixed Convection From a Circular Cylinder to Power Law Fluids A. A. Soares,† J. Anacleto,†,‡ L. Caramelo,† J. M. Ferreira,† and R. P. Chhabra*,§ Departamento de Fı´sica, UniVersidade de Tra´s-os-Montes e Alto Douro, Apartado 1013, 5001-801 Vila Real, Portugal, IFIMUP and INsInstitute of Nanoscience and Nanotechnology Departamento de Fı´sica da Faculdade de Cieˆncias da UniVersidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal, and Department of Chemical Engineering, Indian Institute of Technology, Kanpur, India 208016

The heat transfer characteristics from a circular cylinder immersed in power law fluids have been studied in the mixed convection regime when the imposed flow is oriented normal to the direction of gravity. The continuity, momentum, and thermal energy equations have been solved numerically using a second-order finite difference method to obtain the streamline, surface viscosity, and vorticity patterns, to map the temperature field near the cylinder and to determine the local and surface-averaged values of the Nusselt number. Overall, mixed convection distorts streamline and isotherm patterns and increases the drag coefficient as well as the rate of heat transfer from the circular cylinder. New results showing the complex dependence of all these parameters on power law index (n ) 0.6, 0.8, 1, 1.6), Prandtl number () 1,100), Reynolds number (1-30), and the Richardson number (0, 1, and 3) are presented herein. Over this range of conditions, the flow is assumed to be steady, as is the case for Newtonian fluids. 1. Introduction Due to its fundamental and pragmatic significance, momentum and heat transfer characteristics of a circular cylinder immersed in moving fluids have been studied extensively for more than 100 years. Typical examples where this type of flow occurs include tubular and pin-type heat exchangers, membrane modules for separation, and use of bluff-bodies as flow dividers in polymer processing and in thermal treatment of foodstuffs. Also, the recent exponential growth in the performance of modern electronic equipment, and thus increased power consumption and heat generation, has provided impetus for renewed interest in developing methods of enhancing the rate of heat removal from such equipment. Consequently, a significant body of information is now available pertaining to various aspects of the flow and heat transfer from a cylinder in Newtonian fluids like air and water. It is, however, fair to say that the flow phenomena (drag and lift coefficients and wake characteristics, for instance) have been studied much more extensively than the corresponding heat transfer phenomena. Within the context of heat transfer, indeed very limited information is available on mixed convection from a cylinder, even in Newtonian fluids.1,2 Suffice it to say that adequate information is now available on the prediction of engineering design parameters over a wide range of interest. In most practical situations, free convection, how so ever small, is always present and thus heat transfer occurs in the mixed convection regime. In a given situation, the importance of mixed convection is gauged by the value of the so-called Richardson number, Ri, which is defined as the ratio of the Grashof number to the square of the Reynolds number. Thus, a small value of the Richardson number (Ri f 0) indicates that heat transfer occurs primarily by forced convection or, conversely, Ri ∼ O(1) corresponds to the case when the imposed * To whom correspondence should be addressed. E-mail: chhabra@ iitk.ac.in. † Universidade de Tra´s-os-Montes e Alto Douro. ‡ IFIMUP and INsInstitute of Nanoscience and Nanotechnology Departamento de Fı´sica da Faculdade de Cieˆncias da Universidade do Porto. § Indian Institute of Technology.

velocity and that induced by buoyancy are of comparable magnitudes. Further complications arise depending upon the orientation of the cylinder with respect to the direction of flow. Thus, for instance, when the imposed flow is upward over a heated cylinder, the rate of heat transfer is enhanced due to the aiding buoyancy whereas the rate of heat transfer will deteriorate in case of the downward flow over a heated cylinder (opposing flow). Similarly, there are situations when the buoyancy induced velocity is oriented normal to the imposed flow, thereby resulting in the so-called crossflow configuration. The present work is concerned with the crossflow configuration. However, a terse review of the previous literature is instructive prior to the presentation of the present study. As noted earlier, the literature is limited on mixed convection from a circular cylinder even in Newtonian fluids like air and water.3-6 Chang and Sa7 examined numerically the effects of mixed convection heat transfer on vortex shedding in the near wake of a heated/cooled circular cylinder, and their findings are consistent with the experimental results of Noto et al.8 and the subsequent numerical study of Hatanaka and Kawahara.9 The influence of buoyancy on heat transfer and wake structure at low Reynolds numbers (Re ) 20-40) has been investigated numerically by Patnaik et al.10 for a circular cylinder placed in a vertical stream. Kieft et al.11 have studied the effect of mixed convection from a heated cylinder in horizontal crossflow configuration and found that this configuration leads to asymmetrical flow patterns. More recently, the effects of mixed convection on the wake instability of a heated cylinder in contraflow have been investigated experimentally12 and numerically.13 It is thus abundantly clear that, over the years, mixed convection from a heated circular cylinder to Newtonian fluids has attracted a fair bit of attention from the experimental, analytical, and numerical standpoints, e.g., see refs 1, 2, and 14 and references therein, albeit most of these studies relate to air as the working fluid, i.e., Pr ) 0.7. Furthermore, an examination of these survey articles shows that buoyancy forces enhance the heat transfer rate when they aid the forced flow and decrease the same when they oppose it. Such aiding and opposing flow conditions have received a great deal of attention (e.g., see refs 1 and 2). In contrast, limited work has been

10.1021/ie801187k CCC: $40.75  2009 American Chemical Society Published on Web 11/21/2008

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Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009

Figure 1. Real (x, y) and computational (ε, θ) plane. Variables include gravitational acceleration (g), the free stream fluid temperature (T∞), and uniform approach velocity (U∞). Table 1. Comparison of Present Results with Literature Values of Surface Averaged Nusselt Number (Nu) for Newtonian Fluids (n ) 1), for Prandtl Number Pr ) 0.7 Nu Re

Ri

1

0.0 1.0 2.0 4.0 0.0 1.2 2.4 4.0 0.0 1.25 2.5 5.0 0.0 0.25 2.0 4.0

5

20

40

a

3,a

Hatton et al. 0.965 1.04 1.088 1.149 1.562 1.679 1.750 1.821 2.548 2.698 2.795 2.939 3.318 3.364 3.552 3.701

Badr

5

0.800 0.865 0.902 0.955 1.45 1.51 1.65 1.82 2.54 2.65 2.85 3.10 3.48 3.49 3.76 4.17

Chen et al.34

present

2.5806 2.690 2.911 3.240 3.470 3.481 3.780 4.228

0.95 0.97 1.01 1.10 1.48 1.53 1.60 1.70 2.43 2.57 2.67 3.01 3.20 3.21 3.57 3.84

Experimental correlation from ref 3.

reported for mixed convection for a circular cylinder in crossflow configuration, when the mean flow is oriented normal to the direction of gravity. Aside from the aforementioned studies based on the application of the complete field equations, some results have also been obtained by employing the standard boundary layer flow approximation; e.g., see ref 15. On the other hand, many multiphase and macromolecular systems encountered in industrial practice (polymer, food and pharmaceutical processing applications, mineral slurries, and mine-tailings, for instance) exhibit a variety of non-Newtonian flow characteristics such as shear-dependent viscosity, yield stress, and viscoelasticity, etc.16 More recently, both shearthinning behavior17 and heat transfer18 have been investigated for multiwalled carbon nanotubes. In spite of the wide occurrence of non-Newtonian characteristics in a broad spectrum of applications, no prior information is available on mixed convection heat transfer from a circular cylinder submerged in power law type media. The present work aims to fill this gap in the current literature. It is, however, instructive to review the pertinent studies briefly here which, in turn, facilitates the subsequent presentation of the new results obtained in this work. Over the past 10 years or so, reliable numerical results on the drag and wake characteristics for the flow of power law fluids over a circular cylinder have accrued, albeit these predictions are restricted to the so-called two-dimensional steady flow regime.19-25 The corresponding scant heat transfer results

also suggest that heat transfer is facilitated by shear-thinning behavior, and it is somewhat impeded in shear-thickening fluids,26-28 for both a confined as well as an unconfined cylinder. This observation is also applicable for cylinders of square and elliptical cross sections in the two-dimensional steady flow regime.29-31 As far as is known to us, there have been only two prior studies of heat transfer from long cylinders of square cross section to power law fluids in the mixed convection regime; both of which demonstrated varying levels of enhancement in the value of the Nusselt number.32,33 It is thus safe to conclude that no prior results are available on heat transfer from a circular cylinder to power law fluids in the mixed convection regime. The present work aims to fill this gap in the current literature and to investigate the effect of the buoyancy on heat transfer from a long isothermal circular cylinder to power law fluids in the crossflow configuration over a moderate range of Reynolds and Prandtl numbers and of power law indices. 2. Problem Statement and Mathematical Formulation Consider the steady and incompressible flow of an inelastic power law fluid normal to a circular heated cylinder with a constant surface temperature, Ts. The constant free-stream velocity and temperature are U∞ and T∞, respectively. The imposed flow is assumed to be normal to the direction of gravity. The effect of temperature variation on thermophysical fluid properties (density F, specific heat at constant pressure cp, thermal conductivity k, and power law parameters K and n) is considered negligible except for the body force term in the momentum equation (Boussinesq approximation). Due to the infinite length of the cylinder axis along the z-direction, the flow is two-dimensional, i.e., no flow variable depends upon the z coordinate and thus Vz ) 0. The relevant governing equations (continuity, momentum, and thermal energy) can be expressed in their dimensionless form in terms of the polar coordinates (ε, θ) with ε ) ln(r/a), (e.g., see ref 26) where a is the radius of cylinder, giving ∂ ∂ψ 1 ∂ ε ∂ψ e ψ+ )0 ∂θ ∂θ ∂ε eε ∂ε

(

)

(

)

(1)

ε-component

)(

)

∂ψ ∂2ψ ∂ψ ∂2ψ ∂ψ +ψ +ψ ) + ∂θ ∂ε ∂ θ ∂ε ∂θ2 ∂ε ∂τrθ Gr 2n -ε ∂ ε 1 ∂p + (e τrr) + - τθθ (2a) T sin θ e 2 2 ∂ε 2Re Re ∂ε ∂ε

(

[

]


where η is the dimensionless viscosity and εij are the dimensionless components of the rate-of-deformation tensor. The equation for the dimensionless power law viscosity is

θ -component

(

)

1 ∂p ∂ψ ∂2ψ ∂ψ ∂2ψ ∂ψ +ψ )+ + + 2 ∂θ ∂ε ∂ε ∂ε ∂ε ∂ θ 2 ∂θ ∂τθθ Gr 2n -2ε ∂ 2ε (e τrθ) + T cos θ e 2 Re ∂ε ∂θ 2Re energy equation -

(

)

(

[

)

]

)) (

(

(2b)

)

F(2a)nU∞2-n (4) Re ) K where K denotes the power law consistency index and n is the power law index. The power law model predicts shear-thinning behavior for n < 1 and shear-thickening for n > 1. Evidently, n ) 1 corresponds to the standard Newtonian fluid behavior. The Prandtl number is defined as

( )

cpK U∞ n-1 k 2a The Grashof number is defined as Pr )

Gr )

η ) I2n-1

(6)

where g is the gravitational acceleration and β is the coefficient of volumetric expansion of the power law fluid. The dimensionless components of the extra stress tensor for a power law fluid are written as τij ) -ηεij

(7)

) ( )]

[(

∂2ψ 2 ∂2ψ ∂2ψ 2 + + 4 ∂θ ∂ ε ∂ε2 ∂θ2 The vorticity in its scaled form is given as follows: I2 ) e-2ε ψ -

(9)

∂2ψ ∂2ψ ∂ψ +ψ+ω)0 (10) + 2 +2 2 ∂ε ∂ε ∂θ Eliminating the pressure in eqs 2a and 2b by the method of cross-differentiation and introducing the vorticity ω, with some rearrangement, leads to η

(

)

∂ω ∂2ω ∂2ω ∂ω + 2µ + γω ) F + M (11a) + + 2λ ∂ε ∂θ ∂ε2 ∂θ2

where λ)

∂η Re eε ∂ψ - η - n+1 ∂ε ∂θ 2

µ)

∂η Re eε ∂ψ + +ψ ∂θ 2n+1 ∂ε

(

γ ) -2

(

F) ψ-

)(

(11b)

)

(11c)

∂η Re eε ∂ψ +η+ n ∂ε 2 ∂θ

)

(11d)

∂2ψ ∂2ψ ∂2η ∂2η ∂η + 2 - 2 +2 2 2 ∂ε ∂ε ∂θ ∂θ ∂ε ∂2η ∂2ψ ∂η 4 (11e) ∂θ ∂ ε ∂θ ∂θ ∂ ε

(

)

∂T Gr ε ∂T e sin θ - T cos θ (11f) ∂ε 2n+1Re ∂θ Due to the two-dimensional nature of the flow here (xy plane) and since the oncoming flow is aligned with the x direction, we need only consider the region x2 + y2 g 1. Thus, the corresponding region in the (ε, θ) plane is defined by ε g 0 and 0 e θ e 2π (Figure 1). It needs to be emphasized here that the viscous dissipation term has been neglected in the energy equation used here because for the range of conditions of 1 e Re e 30 and 0 e Ri e 3, the effective viscosity and/or shear rate close to the surface

[

M)

K2U∞2(n-1)

(8)

where I2 is the dimensionless second invariant of the rate-ofdeformation tensor given as follows:

(5)

βF2g(Ts - T∞)(2a)2n+1

)/2

(

eεRePr ∂ψ ∂2T ∂2T ∂T + 2 + + 2 ∂θ ∂ε2 ∂θ2 ∂ε ∂T eεRePr ∂ψ eεRePr ∂ψ ψ+ +T 1+ ) 0 (3) ∂θ 2 ∂ε 2 ∂θ where the stream function ψ, vorticity ω, and pressure p have been scaled using eεU∞a, e-εU∞/a, and U∞2F/2, respectively. The components of the extra stress tensor τij are scaled using K(U∞/ a)n. The thermal boundary condition at ε ) 0 is the constant temperature (Ts) condition, and consequently, the temperature T is scaled as e-ε/(Ts - T∞) in the energy equation (eq 3). This scaling procedure, used by D’Alessio and Pascal20 and Chhabra et al.21 for the momentum transfer and by Soares et al.26 for both momentum and heat transfer, is also employed in the present study because it suppressed the numerical instabilities and thus enabled convergent solutions in the range of conditions studied here. The Reynolds number appearing in eqs 2 and 3 is defined as

(

8221

(

) ]

Table 2. Effect of the Richardson Number on the Drag Coefficient (Cd) Pr ) 1 Ri

Re ) 1

Re ) 5

0 1 3

12.87 12.95 14.02

4.26 4.66 4.89

0 1 3

10.45 10.86 13.19

3.93 4.34 5.14

0 1 3

7.43 8.06 13.30

3.70 4.35 7.72

Re ) 10

Pr ) 100 Re ) 20

Re ) 30

Re ) 1

Re ) 5

2.01 2.34 3.33

1.67 1.97 2.81

12.87 12.86 13.39

4.26 4.34 4.62

1.99 2.33 3.06

1.66 1.98 3.14

10.45 10.42 10.80

3.93 4.00 4.20

2.14 2.58 4.02

1.86 2.29 3.65

7.43 7.44 8.51

3.70 3.73 3.98

n ) 0.8 2.83 3.21 3.95

Re ) 30

2.85 2.93 3.10

2.01 2.04 2.19

1.67 1.69 1.83

1.99 2.02 2.15

1.66 1.69 1.81

2.14 2.17 2.28

1.86 1.89 1.99

n ) 1.0

n ) 1.6 2.74 3.27 4.17

Re ) 20

n ) 0.8

n ) 1.0 2.76 3.10 4.23

Re ) 10

2.76 2.82 2.98 n ) 1.6 2.74 2.78 2.93

8222


Figure 2. Effect of Richardson number Ri on the streamline patterns for Re ) 1 and Pr ) 1 and 100 at different values of n (flow direction ) left to right).

cylinder is not expected to be excessively high. Similarly, the assumption that the flow remains 2D is probably a reasonable approximation under such conditions. The momentum and energy equations are coupled through eq. 11f, which incorporates the buoyant forces in the present problem. The realistic physical boundary conditions for this type of flow are expressed as follows: On the cylinder surface, i.e. at ε ) 0, the usual no-slip condition is applied, i.e. ∂ψ ∂ψ ) )0 ∂ε ∂θ which together with eq 10 gives ψ ) 0 and ω ) -

(12a)

∂2ψ ∂ε2

(12b)

The isothermal boundary condition at the surface of the solid cylinder is given by T)1

(12c)

Far away from the cylinder surface, for ε∞ ) 4, we use the asymptotic approximation for stream function and vorticity given by Chhabra et al.21 ψ ≈ sin(θ) + ω≈-

Cd -ε θ e - erf(Q) 2 π

(

CdReI2(1-n)⁄2 2

√π

n+1

Qe-Q

2

)

(12d) (12e)

where Cd is the drag coefficient,

Re (1-n)⁄4 θ sin I2 (12f) 2 2n and erf(Q) is the standard error function. The far away stream temperature boundary conditions are written as Q ) eε/2

()

∂T )0 (12g) ∂ε It needs to be emphasized here that the aforementioned boundary condition of a constant temperature at the far field (∂T/∂ε ) 0) does not depend on the type of fluid and is therefore valid for both Newtonian and non-Newtonian conditions. It is, however, appropriate to mention here that, although the vorticity and stream function boundary conditions usually applied in the far-field mimic the well-known asymptotic solution for the Newtonian case, the non-Newtonian viscosity value was incorporated21,26 in the aforementioned solution with the aim of extending its validity to non-Newtonian fluids. In terms of the velocity boundary conditions, there is no ambiguity as the free stream velocity condition is valid for any type of fluid. However, in developing the corresponding boundary conditions in terms of vorticity and stream function, any possible loss of accuracy in such far-field asymptotic solution for non-Newtonian fluids will become less significant as one approaches the surface of the cylinder where the no-slip boundary condition is clearly valid for both Newtonian and non-Newtonian conditions. T ) 0 and


8223

Figure 3. Effect of Richardson number Ri on the streamline patterns for Re ) 30 and Pr ) 1 and 100 at different values of n (flow direction ) left to right).

Therefore, the far field boundary conditions for power law fluids developed in our previous works21,26 are believed to be quite adequate for this purpose. The fact that the use of this boundary condition led to reliable values of drag coefficient and Nusselt number (for forced convection case) inspires confidence in the use of these conditions in the present case also. The coupled set of elliptic governing equations given by eqs 3, 10, and 11a-11f together with the corresponding boundary conditions (eqs 12a-12g), have been discretized using the finite difference method. The resulting system of algebraic equations has been solved using an iterative Gauss Seidel relaxation method (e.g., see refs 21 and 26). Once the values of ω, ψ, and T are computed in the flow domain 0 e ε e 4 and 0 e θ e 2π, the local Nusselt number Nu(θ) and average Nusselt number Nu, can be determined using the following expressions: The local Nusselt number at a point on the surface of the isothermal cylinder is defined by ∂T h(2a) ) -2 -T k ∂ε ε)0 The surface averaged Nusselt number is given by

(

Nu(θ) )

∫

)

(13)

1 2π Nu(θ) dθ (14) 2π 0 Thus, in summary, once the values of the stream function, vorticity, and temperature fields are known, these can be postprocessed to obtain the values of drag coefficient, local and surface-averaged Nusselt numbers as functions of the physical (n) and kinematic variables (Re, Ri, Pr). These results elucidate Nu )

the interplay between the fluid rheology and the characteristic conditions of flow when the imposed velocity is oriented normal to the direction of gravity vector. 3. Numerical Method The finite difference method with a second order upwind differencing scheme was applied to the convective terms of temperature and vorticity, as developed in previous studies (e.g., see refs 21 and 26), has been used here to discretize and solve the set of coupled equations formed by the governing stream function, vorticity and energy eqs 10, 11a-11f, and 3, respectively, together with the power law constitutive relationship (eq 8) and the boundary conditions outlined in eqs 12a-12g. The numerical solutions were obtained for the computational domain shown in Figure 1. For an (N +1) × (M + 1) computational mesh, the spacing in the ε and θ directions are ε∞/N and 2π/M, respectively. All results reported herein have been checked for grid independence. In general, it is somewhat easier to meet the convergence criterion for shear-thickening fluids (n > 1) than that for shear-thinning fluids (n < 1), although in both cases the nonlinearity of the system of equations increases as the value of the power law index deviates increasingly from unity. This makes convergence for the governing equations difficult for small and high values of the power law index. In addition, for Ri * 0, the difficulty to meet the convergence criterion increases as the value of Pr decreases. The steady values of ψ, ω, and T as function of ε and θ were obtained using the Gauss Seidel

8224


Figure 4. Effect of Richardson number Ri on the dimensionless surface power law viscosity at n ) 0.8 and 1.6 for (a) Re ) 1 and Pr ) 100, (b) Re ) 1 and Pr ) 1, (c) Re ) 30 and Pr ) 100, (d) Re ) 30 and Pr ) 1.

relaxation iterative algorithm.21,26 The solutions of the field equations were used to calculate the Nusselt numbers via the use of eqs 13 and 14. To obtain convergence, it was also necessary to apply an under-relaxation factor between 0.5 and 1 to the vorticity and temperature variables. The iteration procedure was repeated until the variation of Nu per iteration was less than a preset value of 10-7. The outer boundary was positioned at ε∞ ) 4, corresponding to asymptotic boundary conditions at a distance of ∼54.6 radii away from the cylinder. The numerical solutions for each value of the Reynolds number were obtained using a rectangular computational mesh. The mesh sizes used for Re e 5 were 101 × 101, but for Re > 5, the sizes used were 201 × 201. These meshes were found to be adequate to account for all possibilities of flow and heat transfer phenomena.26 As mentioned previously, in the present work the results for non-Newtonian flow patterns are presented for Reynolds numbers between 1 and 30, and we assume that the flow remains 2D for the range of conditions (0 e Ri e 3, 0.6 e n e 1.6, and Pr ) 1,100) embraced by the present study. 4. Results and Discussion In the present study, the problem of 2D steady heat transfer in the mixed convection regime from an isothermal circular

cylinder to a power law fluid has been studied over the range of conditions as follows: Reynolds numbers 1 e Re e 30, power law indices 0.6 e n e 1.6, Prandtl numbers Pr ) 1 and 100, and Richardson numbers Ri ) 0, 1, and 3. Extensive results on the streamline patterns, surface vorticity and viscosity, isotherm patterns, and local (Nu(θ)) and surface-averaged (Nu) Nusselt numbers, have been obtained to elucidate the influence of buoyancy effects on the rate of heat transfer from a circular cylinder to power law fluids. In addition, the effect of the Richardson number on the flow field was also studied by calculating the values of the drag coefficient. 4.1. Validation of Results. As mentioned previously, no prior numerical or experimental results are available in the literature for mixed convection from a horizontal heated circular cylinder in power law fluids. However, the present results for Newtonian fluids (n ) 1) for forced convection (Ri ) 0), described in detail in our previous work,26 were found to be in excellent agreement with the literature values and are therefore not repeated here. On the other hand, the present results for mixed convection (Ri * 0) in Newtonian values are compared with the literature values3,5,34 in terms of the surface-averaged values of the Nusselt number (Nu), for Re ) 1, 5, 20, and 40 at Pr ) 0.7. Under these conditions, as expected, our results always showed an increase of Nu with Ri for Newtonian fluids,


8225

Figure 5. Effect of Richardson number Ri on the surface vorticity profiles at Pr )1 and 100 for (a) Re ) 1 and n ) 0.6 (b) Re ) 1 and n ) 1.0, (c) Re ) 1 and n ) 1.6, (d) Re ) 30 and n ) 0.8, (e) Re ) 30 and n ) 1.0, (f) Re ) 30 and n ) 1.6.

consistent with previous experimental3 and numerical5,34 results (Table 1). Such increase was found to range from 15% to 24%, which is roughly in line with the values from the literature shown also in Table 1 (range 12-26%). From Table 1, it is clearly seen that the present results deviate at most by 8.8%, 12%, and 9.2% from those of Hatton et al.,3 Badr,5 and Chen et al.,34 respectively. Similar differences were also observed for Ri ) 0, except for Hatton et al.3 where this discrepancy decreased to 1.6% for Re ) 1. The discrepancy between the present values and the experimental correlation of Hatton et al.1 is perhaps not surprising, owing to the uncertainty in the experimental measurements and to the assumptions inherent in our analysis, such as temperature-independent fluid properties and neglecting the cylinder-end effects on velocity and temperature fields leading to the two-dimensional flow approximation. Likewise, the discrepancy between the present results and those of Badr5 and Chen et al.34 must be due to the differences inherent to the problem formulation, grids and/or domain sizes, discretization schemes, numerical methods, etc. Finally, attention is drawn to the fact that the present results for Ri ) 0 are in excellent agreement with the literature values,26,35 where the discrepancy between the numerical results for Ri ) 0 given by

the different authors did not exceed 5%. However, discrepancies of this magnitude are not at all uncommon in such studies.36 4.2. Drag Coefficient and Streamline Patterns. The hydrodynamic drag force component along the x direction (Figure 1) can be expressed in terms of the corresponding drag coefficient Cd, and the present study showed that an increase in Richardson number from Ri ) 0 to 3 resulted in increased Cd value for shear-thinning (n ) 0.8), shear-thickening (n ) 1.6), and Newtonian (n ) 1) fluids, with Prandtl numbers Pr ) 1 and 100 and Reynolds numbers Re ) 1-30 (Table 2). This trend can safely be ascribed to the increasing distortion of streamlines (thereby sharpening of the velocity gradients) near the surface of the cylinder with the increasing value of the Richardson number. For all the aforementioned values of Ri and Pr, the drag coefficient was a decreasing function of both Re and n. The increase in Cd with mixed convection parameter Ri became more pronounced with increased shear-thickening at lower Prandtl number (Pr ) 1). At higher Prandtl number (Pr ) 100), the opposite behavior was observed for higher Reynolds number (Re ) 10-30) whereas for lower Reynolds number (Re ) 1-5), the increase in Cd with Ri was much less

8226


Figure 6. Effect of Richardson number Ri on the isothermal patterns for Re ) 1 and Pr ) 1 and 100 at different values of n (flow direction ) left to right).

pronounced at n ) 1. For instance, at Pr ) 100 and Re ) 30, an increase in Ri from 0 to 3 resulted in an increase in Cd of 10% at n ) 0.8, 9% at n ) 1, and 7% at n ) 1.6, whereas at Pr ) 100 and Re ) 1 the increase in Cd was 4% at n ) 0.8, 3.4% at n ) 1, and 14% at n ) 1.6. This study also showed that, for shear-thinning (n ) 0.8), shear-thickening (n ) 1.6), and Newtonian (n ) 1) conditions, the increase in Cd with the increasing level of mixed convection became more pronounced as Re increased from Re ) 1 to 30, except for Pr ) 100 under highly shear-thickening conditions (n ) 1.6) where the opposite effect was observed. For instance at Pr ) 100 and n ) 0.8, an increase in Ri from 0 to 3 resulted in a 4% increase in Cd for Re ) 1 whereas for Re ) 30 it was 10%. However, for Pr ) 100 and n ) 1.6, similar increase in Ri led to an increase of 14% and 7%, respectively, at Re ) 1 and 30. Representative streamline patterns close to the cylinder for Reynolds numbers Re ) 1 and 30, Richardson numbers Ri ) 0, 1, and 3, Prandtl number Pr ) 1 and 100, and power law index n ) 0.6-0.8, 1.0, and 1.6 are shown in Figures 2 and 3. The main effect of mixed convection term (Ri * 0) was a loss of symmetry of the streamline patterns, which are known to be symmetrical in the absence of mixed convection, as shown in Figures 2 and 3. Such symmetry-breaking was in part characterized by the presence, in the streamline patterns, of a positive slope which increased with the increasing values of the Richardson number Ri (Figures 2 and 3). While such upward slope is not seen in pure forced convection, i.e. at Ri ) 0, once the mixed convection term was introduced (Ri * 0), the positive slope was found to increase with the value of the power law index (Figures 2 and 3). The present study also showed that

such positive slope became less pronounced at higher values of Re and Pr (Figures 2 and 3), so that at Re ) 30 and Pr ) 100, it was no longer observed (Figure 3). It is so in part due to the thinning of the thermal boundary layer at high Prandtl number and due to the increasing contribution of convection with Reynolds number. For Re ) 30 (Figure 3), in the absence of buoyancy effects (Ri ) 0), the flow separates from the trailing edge of the cylinder to form two symmetrical vortices behind it. However, when mixed convection is present (Ri * 0), the reverse flow in the wake of the cylinder is strongly influenced by the increase in Ri and consequently these twin vortices break down and the flow becomes asymmetrical, an observation which is consistent with the streamline patterns obtained by Dhiman et al.32 for power law fluid flow past a square cylinder and is also in line with the streamline patterns around a circular cylinder in a Newtonian flow given by Chen et al.34 at Re ) 20 and 40 and Pr ) 0.7. Our results thus show that for higher values of Re ) 30 and Pr ) 100, mixed convection again results in the loss of symmetry (Figure 3) and that although such breakup is no longer characterized by a positive slope in the streamline patterns, the asymmetrical flow is now characterized by the formation of a vortex which is detached from the rear of the cylinder in the wake region, an effect which becomes more visible in shear-thickening fluids (Figure 3). For a fixed value of Ri, the size of the recirculation zone grows and shows a complex behavior as the fluid behavior changes from shearthinning to shear-thickening. This observation is consistent with the growth of the wake region for pure forced convection (Ri ) 0). Also, it is clear from these figures that the buoyancy forces are aiding the flow close to the cylinder surface and consequently


the velocity gradient increases significantly with the increase of Ri, an effect which is more pronounced at Pr ) 1 when the boundary layer is relatively thick. 4.3. Variation of Power Law Viscosity and Vorticity on Cylinder Surface. For low Reynolds numbers (Re ) 1) and in the absence of mixed convection (Ri ) 0), the viscosity was characterized under shear-thinning conditions by the presence of two local maxima located respectively at the front (θ ) 180°) and rear (θ ) 0°) stagnation points, whereas under shearthickening conditions (n ) 1.6) such stagnation points corresponded to two local minima (Figure 4a and b). The introduction of mixed convection (Ri * 0) always resulted in a shift in the position of these local maxima/minima in the direction of increasing θ, i.e., anticlockwise (Figure 4a and b). This effect gets accentuated as the Prandtl number is decreased from Pr ) 100 (Figure 4a) to Pr ) 1 (Figure 4b), i.e. for lower Prandtl number the variation of viscosity profile with Ri was always more pronounced. This trend is also consistent with the anticlockwise shift in streamline patterns with increasing Ri, which was also more pronounced for low Prandtl numbers (Figure 2). Moreover, at higher Reynolds number (Re ) 30), for both shear-thinning and shear-thickening fluid behaviors, a decrease in Prandtl number from Pr ) 100 (Figure 4c) to Pr ) 1 (Figure 4d) again resulted in a more pronounced variation of the viscosity profile with mixed convection, i.e. with increasing Ri, an effect which is again consistent with streamline patterns (Figure 3). Also, at such high Reynolds number value of Re ) 30, in the absence of mixed convection (Ri ) 0) the viscosity profile was characterized under shear-thinning condition (n ) 0.8) by the presence of four local minima located respectively at the front stagnation point (θ ) 180°), rear stagnation point (θ ) 0°), and separation points of the fluid from the cylinder surface (θ ) 45°, 315°), whereas under shear-thickening conditions (n ) 1.6), these four points corresponded to local minima (Figure 4c and d). In this case (Re ) 30), an increase in the value of the Richardson number resulted in a decrease in the height of the separation point maxima under shear-thinning conditions n ) 0.8 and an increase in the height of separation point minima under shear-thickening conditions n ) 1.6 (Figure 4c and d). These effects can qualitatively be explained by the fact that the effect of buoyancy forces becomes progressively more significant with the increasing value of the Richardson number, leading to an increase in velocity and temperature gradients. Such increase in velocity gradients produces different types of viscosity behavior depending on the value of power law index, i.e., decreases the effective viscosity for shearthinning fluids (n ) 0.8) and increases the effective viscosity for shear-thickening fluids (n ) 1.6). An inspection of the surface vorticity profiles with and without the introduction of mixed convection effect, i.e. Ri ) 0 and Ri * 0, respectively (Figure 5), showed that the vorticity profiles also get modified significantly in the presence of cross buoyancy forces. For power law index in the range 0.6 e n e 1.6, such changes in surface vorticity profile with Ri became more pronounced as Prandtl number decreased from Pr ) 100 to 1 (Figure 5), an observation which is consistent with the more pronounced variation of viscosity profile with mixed convection at lower Pr shown in Figure 4. Moreover, at high Prandtl number, Pr ) 100, the dependence of surface vorticity profile on mixed convection parameter Ri became less significant upstream of the cylinder, i.e. for θ ) 90°-270° (Figure 5), an observation that is again consistent with the viscosity profiles shown in Figure 4. In contrast, at low Prandtl number, Pr ) 1 (at Re ) 1), the maximum variation in surface vorticity profiles

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(Figure 5) became less significant downstream of the cylinder. It was also found that, for power law fluids, the locations of the separation points where surface vorticity changes sign (Figure 5) correspond to the viscosity peaks in Figure 4. For a fixed value of Ri, a decrease in power law index from n ) 1.6 to 0.8 always resulted in an increase in the magnitude of both maximum and minimum vorticity peaks (Figure 5), an observation which is in line with the previous results for forced convection.21,35 4.4. Isotherm Patterns. This study also showed asymmetry of the isotherm patterns for mixed convection (Figures 6 and 7) and that such asymmetry exhibited a complex dependence on the Reynolds number (Re ) 1 and 30) and Prandtl number (Pr ) 1 and 100), as well as on the power law index (n ) 0.6, 1, 1.6). The extent of such symmetry-breaking is characterized by a distortion in the isotherms which increased anticlockwise as mixed convection increased, i.e. with greater Ri (Figures 6 and 7), and did not occur in the absence of mixed convection, i.e. at Ri ) 0. This effect becomes less pronounced with the decreasing value of the power law index, an observation that can qualitatively be explained by the fact that the values of effective fluid viscosity decrease with the increasing degree of shear-thinning behavior, which increases the flow rate close to the cylinder surface, consequently reducing the effect of buoyancy forces on the isotherms. The present study also showed that in the presence of mixed convection (and also in its absence), a decrease in power law index from n ) 1.6 to 0.6 resulted in slight overall increase in compactness of the isothermals, which slightly increased the temperature gradient overall (Figures 6 and 7). A similar trend was observed as Prandtl number increased from Pr ) 1 to 100, although in this case the increase in temperature gradient was much steeper, presumably due to the thinning of the thermal boundary layer (Figures 6 and 7). The increase in the overall temperature gradient with shear-thinning, which is reported here for a circular cylinder when mixed convection is considered (Ri * 0), is consistent with similar observations in the absence of mixed convection (Ri ) 0) for both circular26,27 and square31 cylinders, all of which, in turn, are in line with the previously predicted37 and experimentally observed38 decrease in boundary layer thickness with increased levels of shear-thinning. In general (Figures 6 and 7), a higher temperature gradient close to the cylinder surface is visible from the isotherm contours. Temperature contours are denser around the front of the cylinder than in its rear, indicating that convective cooling of the cylinder surface is more intense upstream. 4.5. Heat Transfer Characteristics. The study of local Nusselt number Nu(θ) profiles without and with the introduction of buoyancy effects, i.e. Ri ) 0 and Ri * 0, respectively (Figure 8), showed that an increase from Ri ) 0 to 3 generally resulted in an increase in local Nusselt number Nu(θ) irrespective of the type of fluid behavior, albeit the extent of enhancement is somewhat dependent on the value of the power law index. For both Re ) 1 and 30 at low Prandtl number (Pr ) 1), the maximum value of Nu(θ) was located at the front stagnation point (θ ) 180°), and the minimum value occurred at the rear stagnation point of the cylinder at θ ) 0°, for Ri ) 0. As the Richardson number is progressively increased from Ri ) 0 to 3, the maximum value of Nu(θ) still occurs close to θ ) 180°; however, the location of the minimum shifted in anticlockwise direction of increasing θ (Figure 8). This trend is qualitatively similar to that observed by Badr4 and by Chen et al.34 in air. At higher Prandtl numbers, such as Pr ) 100, for both Re ) 1 and 30, for n g 1 the maximum value of Nu(θ) was seen to

8228


Figure 7. Effect of Richardson number Ri on the isothermal patterns for Re ) 30 and Pr ) 1 and 100 at different values of n (flow direction ) left to right).

occur near θ ) 180° (Figure 8). However, for n < 1, a local minimum of Nu(θ) appeared near θ ) 180° in between the two symmetrical maxima (Figure 8), which is also consistent with the results of Soares et al.26 in the absence of buoyancy effects. Both the location of the maximum (for n g 1) and that of the two symmetrical maxima and in-between local minima (for n < 1), showed virtually no variation as Ri increased from 0 to 3 (Figure 8). At such high Prandtl numbers (Pr ) 100), at Re ) 1, the minimum of Nu(θ) was again located in the rear of the cylinder at θ ) 0° and again shifted its position anticlockwise as Ri increased from 0 to 3, whereas at Re ) 30 the variation of the local Nusselt number in the rear of the cylinder with Ri showed a more complex behavior (Figure 8). Finally, a general examination of these figures clearly shows that for fixed values of Ri and n, as the value of the Peclet number (Pe ) RePr) increases, the contribution of convection gradually increases and consequently the overall values of Nu(θ) increase. Moreover, an increase in power law index from 0.6-0.8 to 1.6 resulted overall in a decrease in Nu(θ) and a change in its profile. This trend is more pronounced for higher values of Pe and can qualitatively be explained by the fact that for n > 1, the effective viscosity increases with the shear rate, lowering the rate of heat transfer, and of course the reverse is true for n < 1. The variation in local Nusselt number Nu(θ) profiles before and after the introduction of buoyancy effects was thus determined by a complex interplay between the kinematic (Ri, Re, Pr) and physical (n) characteristics of the system, and in order to better understand the effect of these characteristics on the system, it is both useful and convenient to use the mean value of the Nusselt number (averaged over the cylinder surface), Nu, thereby eliminating one variable from the process. The present results

showed that, in general, the value of the surface averaged Nusselt number increased (by up to about 10-11%) with the Richardson number while keeping the values of Re, Pr, and n fixed (Table 3), and this is shown in Figure 8. It was also found that the rate of increase of Nu with Ri became more pronounced for higher Re values (Table 3). However, for Re ) 30, as n was increased from 0.8 to 1.6, the increase in Nu with Ri became less significant so that under high shear-thickening condition of n ) 1.6, the increase of Nu with mixed convection was only observed above Ri ) 1 (Table 3), i.e., for n ) 1.6 and Ri < 1, increased mixed convection resulted in decreased Nu. Such dependence of the variation of Nu with Ri on degree of shearthickening may be partly due to the decrease in Nu with increasing power law index (Table 3), although this decrease was present for all values of the Reynolds number. Moreover, such a decrease in Nu with increasing n, which was observed for Ri ) 0-3 (Table 3), is consistent with the previous observations in the absence of mixed convection.26 Finally, both in the presence and absence of mixed convection, an increase in the surface-averaged Nusselt number Nu with Peclet number Pe ) RePr can be inferred from Table 3. Such an effect is in line with the increase in local Nusselt number Nu(θ) with Pe which can be inferred from Figure 8 and is also consistent with previous results26 for forced convection. Before closing this discussion, it needs to be emphasized here that while the assumption of temperature-independent viscosity affords a great simplification, it also restricts the applicability of the present results to situations where either the temperature difference between the fluid and the cylinder is small and/or the viscosity of the substance in use is not too sensitive to the temperature variation in the domain. Qualitatively, since an increase in


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Figure 8. Effect of Richardson number Ri on local Nusselt number profiles Nu(θ) at Pr )1 and 100 for (a) Re ) 1 and n ) 0.6 (b) Re ) 1 and n ) 1.0, (c) Re ) 1 and n ) 1.6, (d) Re ) 30 and n ) 0.8, (e) Re ) 30 and n ) 1.0, (f) Re ) 30 and n ) 1.6 (downstream θ ) 0o, upstream θ ) 180°). Table 3. Effect of Mixed Convection Parameter Ri on the Variation of Surface-Averaged Nusselt (Nu) with n, Re, and Pr Pr ) 1 Ri

Re ) 1

Re ) 5

0 1 3

1.03 1.04 1.14

1.66 1.70 1.88

0 1 3

1.02 1.03 1.14

1.63 1.68 1.85

0 1 3

1.01 1.02 1.12

1.57 1.62 1.78

Re ) 10

Pr ) 100 Re ) 20

Re ) 30

Re ) 1

Re ) 5

2.81 2.90 3.33

3.31 3.46 4.00

3.37 3.37 3.43

6.65 6.76 7.09

2.72 2.84 3.13

3.19 3.33 3.84

3.21 3.21 3.29

6.26 6.40 6.72

2.50 2.63 2.96

2.91 3.04 3.46

3.07 3.07 3.18

5.60 5.83 6.12

n ) 0.8 2.13 2.19 2.52

Re ) 30

9.19 9.50 10.17

12.77 13.31 14.20

16.00 16.27 17.43

12.02 12.42 13.21

15.05 15.11 16.13

10.83 10.92 11.72

13.42 13.09 14.04

n ) 1.0

n ) 1.6 1.96 2.03 2.32

Re ) 20

n ) 0.8

n ) 1.0 2.07 2.14 2.45

Re ) 10

8.60 8.96 9.60 n ) 1.6

temperature leads to a decrease in viscosity, e.g. see the work of Sun et al.,39 and a decrease in power law index results in a similar trend, it is expected that an increase in temperature would affect our results in a similar manner to an increase in shear rate for shear-thinning fluids. Therefore, should it become

7.76 8.06 8.62

necessary to account for the temperature-dependent viscosity, it is perhaps adequate (at least as a first-order approximation) to use the same correction factor as that used for mixed convection in Newtonian fluids.40 However, it should be borne in mind that these are the very first set of results which

8230


incorporate the effects of mixed convection for a circular cylinder under non-Newtonian conditions and, naturally, these will get superseded by future studies which would consider the role of temperature-dependent thermophysical properties. Finally, it is clearly demonstrated here that depending upon the values of n, Ri, Re, and Pr, free convection can enhance the value of the convective heat transfer coefficient by up to 10-15% which will translate into improved thermal efficiency. Similarly, the buoyancy effects can significantly alter the temperature field close to the surface of the cylinder which directly influences the product quality during the thermal processing of temperature-sensitive substances. It is therefore not always justified to ignore the role of free convection. 5. Conclusions The non-Newtonian flow across an isothermally heated long circular cylinder was investigated numerically to determine the effect of buoyancy on flow and heat transfer, for three values of the Richardson number, i.e., Ri ) 0, 1, 3; Reynolds number Re ) 1-30; Prandtl number Pr ) 1 and 100; and power law index n ) 0.6-1.6. The drag coefficient Cd was found to be a decreasing function of Re and n for all values of Ri, and an increase in the degree of mixed convection always led to an increase in the value of Cd. The increased degree of mixed convection was also found to distort streamline patterns anticlockwise, to shift anticlockwise the position of power law viscosity maxima and minima, and to change the vorticity profile at the cylinder surface. The dependence of all these parameters on the Richardson number Ri is seen to be rather pronounced for low Prandtl numbers, such as Pr ) 1. An increase in the value of the power law index increased the effect of mixed convection on streamline pattern distortion and decreased the magnitude of surface vorticity maxima. The effects of increased mixed convection on heat transfer manifest in the form of increased anticlockwise distortion in isothermal patterns as well as increased values of both local and surface-averaged Nusselt numbers by varying amounts. The effects of mixed convection on heat transfer characteristics were generally more significant at lower values of power law index (shear-thinning fluids), and such a decrease in n was found to increase the magnitude of surface-averaged Nusselt number. A decrease in n and/or increase in Pr also led to the crowding of isotherms thereby the sharpening of the temperature gradients. The effects of power law index and/or Prandtl number on the growth of the recirculation zone, on temperature gradient, and on Nusselt number, in the mixed convection regime, were found to be qualitatively similar to those in the case of forced convection for the range of physical conditions considered here. Nomenclature a ) radius of the cylinder (m) cp ) specific heat at pressure constant (J/(kg K)) F ) dimensionless function, eq 11e Gr ) Grashof number (βF2g(Ts - T∞)(2a)2n+1)/(K2U∞2(n-1)) (-) h ) heat transfer coefficient, (W/(m2 K)) H ) distance from the external boundary to the cylinder surface (m) g ) gravitational acceleration (m/s2) I2 ) dimensionless second invariant of the rate-of-deformation tensor (-) K ) power law consistency index, (Pa sn) k ) thermal conductivity of the fluid, (W/(m K)) M ) dimensionless function, eq 11f n ) power law index (-)

Nu ) average Nusselt number (-) Nu(θ) ) local Nusselt number (-) p ) dimensionless pressure (-) Pe ) Peclet number (-) Pr ) Prandtl number (-) r ) cylindrical coordinate (m) Re ) Reynolds number, (F(2a)nU∞2-n/K) (-) Ri ) Richardson number (Gr/Re2) (-) U∞ ) uniform approach velocity (m/s) T ) dimensionless temperature (-) Ts ) temperature on the surface of the cylinder (K) T∞ ) free stream fluid temperature (K) Vz ) z-component of velocity (m/s) Greek Letters β ) coefficient of volumetric expansion (1/K) γ ) dimensionless function, eq 11d (-) θ ) angle (radians) λ ) dimensionless function, eq 11b (-) µ) dimensionless function, eq 11c (-) ψ ) dimensionless stream function (-) η ) dimensionless viscosity (-) ω ) dimensionless vorticity (-) F ) fluid density, (kg/m3) ε ) dimensionless polar coordinate, ln(r/a) (-) ε∞) position of external boundary εij ) components of rate-of-deformation tensor, (1/s) τij ) dimensionless components of extra stress tensor (-) Subscripts θ ) angular component r ) radial component

Literature Cited (1) Zdravkovich, M. M. Flow around circular cylinders: Fundamentals; Oxford University Press: Oxford, U. K., 1997; Vol. 1. (2) Zdravkovich, M. M. Flow around circular cylinders: Applications; Oxford University Press: Oxford, U. K., 2003; Vol. 2. (3) Hatton, A. P.; James, D. D.; Swire, H. V. Combined forced and natural convection with low-speed air flow over horizontal cylinders. J. Fluid Mech. 1970, 42, 17–31. (4) Badr, H. M. A theoretical study of laminar mixed convection from a horizontal cylinder in a cross stream. Int. J. Heat Mass Transfer 1983, 26, 639–653. (5) Badr, H. M. Laminar combined convection from a horizontal cylinder in cross stream. Int. J. Heat Mass Transfer 1984, 27, 15–27. (6) Ahmad, R. A.; Qureshi, Z. H. Laminar mixed convection from a uniform heat flux horizontal cylinder in a crossflow. J. Thermophys. Heat Transfer 1992, 6, 277–287. (7) Chang, K.-S.; Sa, J.-Y. The effect of buoyancy on vortex shedding in the near wake of a circular cylinder. J. Fluid Mech. 1990, 220, 253– 266. (8) Noto, K.; Ishida, H.; Matsumoto, R. A breakdown of the Karman vortex street due to the natural convection. Flow Visualization III; Springer: New York, 1985; pp 348-352. (9) Hatanaka, K.; Kawahara, M. A numerical study of vortex shedding around a heated/cooled circular cylinder by the three-step Taylor-Galerkin method. Int. J. Numer. Methods Fluids 1995, 21, 857–867. (10) Patnaik, B. S. V.; Narayana, P. S. A.; Seetharamu, K. N. Numerical simulation of vortex shedding past a circular cylinder under the influence of buoyancy. Int. J. Heat Mass Transfer 1999, 42, 3495–3507. (11) Kieft, R. N.; Rindt, C. C. M.; Steenhoven, A. A. V. The wake behaviour behind a heated horizontal cylinder. Exp. Therm. Fluid Sci. 1999, 19, 183–193. (12) Hu, H.; Kochesfahani M. The wake behind a heated cylinder in forced and mixed convection regimes. 2005 ASME Summer Heat Transfer Conference, San Francisco, CA, 2005. (13) Khyati, B. V.; Hui, H.; Wang, Z. J. Numerical investigation of effect of buoyancy on the wake instability of a heated cylinder in contra flow. AIAA 2007, 0801, 1–19.

Ind. Eng. Chem. Res., Vol. 48, No. 17, 2009 (14) Zukaukas, A.; Zingza A. Heat Transfer of Cylinder in Cross Flow; Hemisphere: Washington, D.C., 1985. (15) Nazar, R.; Amin, N.; Pop, I. Mixed convection boundary-layer flow from a horizontal circular cylinder with a constant surface heat flux. Heat Mass Transfer 2004, 40, 219–227. (16) Chhabra, R. P.; Richardson, J. F. Non-Newtonian Flow and Applied Rheology, 2nd ed.; Butterworth-Heinemann: Oxford, U. K., 2008. (17) Kinloch, I. A.; Roberts, S. A.; Windle, A. H. A rheological study of concentrated aqueous nanotube dispersions. Polymer 2002, 43, 7483– 7491. (18) Ding, Y.; Alias, H.; Wen, D.; Williams, R. A. Heat transfer of aqueous suspensions of carbon nanotubes (CNT nanofluids). Int. J. Heat Mass Transfer 2006, 49, 240–250. (19) Tanner, R. I. Stokes paradox for power -law fluids around a cylinder. J. Non-Newt. Fluid Mech. 1993, 50, 217–224. (20) D’ Alessio, S. J. D.; Pascal, J. P. Steady flow of a power law fluid past a cylinder. Acta Mech. 1996, 117, 87–100. (21) Chhabra, R. P.; Soares, A. A.; Ferreira, J. M. Steady non-Newtonian flow past a circular cylinder: a numerical study. Acta Mech. 2004, 172, 1–16. (22) Bharti, R. P.; Chhabra, R. P.; Eswaran, V. Steady flow of power law fluids across a circular cylinder. Can. J. Chem. Eng. 2006, 84, 1–15. (23) Sivakumar, P.; Bharti, R. P.; Chhabra, R. P. Effect of power law index on critical parameters for power law flow across an unconfined circular cylinder. Chem. Eng. Sci. 2006, 61, 6035–6046. (24) Khan, W. A.; Culham, J. R.; Yovanovich, M. M. Fluid flow and heat transfer in power law fluids across circular cylinders-analytical study. J. Heat Transfer 2006, 128, 870–878. (25) Bharti, R. P.; Chhabra, R. P.; Eswaran, V. Two-dimensional steady Poiseuille flow of power law fluids across a circular cylinder in a plane confined channel:wall effects and drag coefficients. Ind. Eng. Chem. Res. 2007, 46, 3820–3840. (26) Soares, A. A.; Ferreira, J. M.; Chhabra, R. P. Flow and forced convection heat transfer in cross flow of non-Newtonian fluids over a circular cylinder. Ind. Eng. Chem. Res. 2005, 44, 5815–5827. (27) Bharti, R. P.; Chhabra, R. P.; Eswaran, V. Steady forced convection heat transfer from a heated cylinder to power law fluids. Int. J. Heat Mass Transfer 2007, 50, 977–990. (28) Bharti, R. P.; Chhabra, R. P.; Eswaran, V. Effect of blockage on heat transfer from a cylinder to power law fluids. Chem. Eng. Sci. 2007, 62, 4729–4741.

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(29) Sivakumar, P.; Bharti, R. P.; Chhabra, R. P. Steady flow of power law fluids across an unconfined elliptical cylinder. Chem. Eng. Sci. 2007, 62, 1682–1702. (30) Bharti, R. P.; Sivakumar, P.; Chhabra, R. P. Forced convection heat transfer from an elliptical cylinder to power law fluids. Int. J. Heat Mass Transfer 2008, 51, 1838–1853. (31) Dhiman, A. K.; Chhabra, R. P.; Eswaran, V. Heat transfer to power law fluids from a heated square cylinder. Numerical Heat Transfer 2007, 52, 185–201. (32) Dhiman, A. K.; Anjaiah, N.; Chhabra, R. P.; Eswaran, V. Mixed convection from a heated square cylinder to Newtonian and power law fluids. J. Fluids Eng. 2007, 129, 506–513. (33) Dhiman, A. K.; Chhabra, R. P.; Eswaran, V. Steady mixed convection from a confined square cylinder. Int. Commun. Heat Mass Transfer 2008, 35, 47–55. (34) Chen, C.-K.; Yang, Y.-T.; Wu, S.-R. Laminar mixed convection from a circular cylinder using a body-fitted coordinate system. J. Thermophys. Heat Transfer 1994, 8, 695–701. (35) Bharti, R. P.; Chhabra, R. P.; Eswaran, V. Numerical study of the steady forced convection heat transfer from an unconfined circular cylinder. Heat Mass Transfer 2007, 43, 639–648. (36) Roache, P. J. Verification and Validation in computational Science and Engineering; Hermosa Publishers: Albuquerque, NM, 1998. (37) Chhabra, R. P. Bubbles, drops and particles in non-Newtonian fluids, 2nd ed.; CRC Press: Boca Raton, FL, 2006. (38) Coelho, P. M.; Pinho, F. T. Vortex shedding in cylinder flow of shear-thinning fluids. II. Flow characteristics. J. Non-Newt. Fluid Mech. 2003, 110, 177–193. (39) Sun, K. H.; Pyle, D. L.; Fitt, A. D.; Please, C. P.; Baines, M. J.; Hall-Taylor, N. Numerical study of 2D heat transfer in a scraped surface heat exchanger. Comput. Fluids 2004, 33, 869–880. (40) Kreith, F. The CRC Handbook of Thermal Engineering; CRC Press: Boca Raton, FL, 2000.

ReceiVed for reView August 1, 2008 ReVised manuscript receiVed September 23, 2008 Accepted September 24, 2008 IE801187K

Mixed Convection From a Circular Cylinder to Power Law Fluids

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