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 Cie ˆncias 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, momen- tum 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 con- sumption 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, con- versely, Ri ∼ O(1) corresponds to the case when the imposed 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 Sa 7 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 asym- metrical flow patterns. More recently, the effects of mixed convection on the wake instability of a heated cylinder in contraflow have been investigated experimentally 12 and numeri- cally. 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 * 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 Cie ˆncias da Universidade do Porto. § Indian Institute of Technology. Ind. Eng. Chem. Res. 2009, 48, 8219–8231 8219 10.1021/ie801187k CCC: $40.75  2009 American Chemical Society Published on Web 11/21/2008