Laminar convection to rotating disks in non-Newtonian power-law fluids

A theoretical result has been obtained for heat transfer to a rotating disk in a non-Newtonian fluid which includes the convective transport in the radial direction. The transient heat transfer problem with step change in disk temperature is studied by instant-local similarity method and the approximate solution is very accurate for small time.     

Forced convection heat transfer from an elliptical cylinder to power-law fluids

Forced convection heat transfer to incompressible power-law fluids from a heated elliptical cylinder in the steady, laminar cross-flow regime has been studied numerically. In particular, the effects of the power-law index (0.2 ? n ? 1.8), Reynolds number (0.01 ? Re ? 40), Prandtl number (1 ? Pr ? 100) and the aspect ratio of the elliptic cylinder (0.2 ? E ? 5) on the average Nusselt number (Nu) have been studied. The average Nusselt number for an elliptic cylinder shows a dependence on the Reynolds and Prandtl numbers and power-law index, which is qualitatively similar to that for a circular cylinder. Thus, heat transfer is facilitated by the shear-thinning tendency of the fluid, while it is generally impeded in shear-thickening fluids. The average Nusselt number values have also been interpreted in terms of the usual Colburn heat transfer factor (j). The functional dependence of the average Nusselt number on the dimensionless parameters (Re, n, Pr, E) has been presented by empirically fitting the numerical results for their easy use in process design calculations.     

Steady forced convection heat transfer from a heated circular cylinder to power-law fluids

Forced convection heat transfer to incompressible power-law fluids from a heated circular cylinder in the steady cross-flow regime has been investigated numerically by solving the momentum and thermal energy equations using a finite volume method and the QUICK scheme on a non-uniform Cartesian grid. The dependence of the average Nusselt number on the Reynolds number (5 ⩽ Re ⩽ 40), power-law index (0.6 ⩽ n ⩽ 2) and Prandtl number (1 ⩽ Pr ⩽ 1000) has been studied in detail. The numerical results are used to develop simple correlations as functions of the pertinent dimensionless variables. In addition to the average Nusselt number, the effects of Re, Pr and n on the local Nusselt number distribution have also been studied to provide further physical insights. The role of the two types of thermal boundary conditions, namely, constant temperature and uniform heat flux on the surface of the cylinder has also been presented.     

Laminar forced convection from a rotating horizontal cylinder in cross flow

The influence of non-dimensional rotational velocity, flow Reynolds number and Prandtl number of the fluid on laminar forced convection from a rotating horizontal cylinder subject to constant heat flux boundary condition is numerically investigated. The numerical simulations have been conducted using commercial Computational Fluid Dynamics package CFX available in ANSYS Workbench 14. Results are presented for the non-dimensional rotational velocity α ranging from 0 to 4, flow Reynolds number from 25 to 40 and Prandtl number of the fluid from 0.7 to 5.4. The rotational effects results in reduction in heat transfer compared to heat transfer from stationary heated cylinder due to thickening of boundary layer as consequence of the rotation of the cylinder. Heat transfer rate increases with increase in Prandtl number of the fluid.     

Two-dimensional unsteady forced convection heat transfer in power-law fluids from a cylinder

Forced convection heat transfer characteristics of a cylinder (maintained at a constant temperature) immersed in a streaming power-law fluids have been studied numerically in the two-dimensional (2-D), unsteady flow regime. The governing equations, namely, continuity, momentum and thermal energy, have been solved using a finite volume method based solver (FLUENT 6.3) over wide ranges of conditions (power law index, 0.4 ? n ? 1.8; Reynolds number, 40 ? Re ? 140; Prandtl number, 1 ? Pr ? 100). In particular, extensive numerical results elucidating the influence of Reynolds number, Prandtl number and power-law index on the isotherm patterns, local and average Nusselt numbers and their evolution with time are discussed in detail. Over the ranges of conditions considered herein, the nature of flow is fully periodic in time. The heat transfer characteristics are seen to be influenced in an intricate manner by the value of the Reynolds number (Re), Prandtl number (Pr) and the power-law index (n). Depending upon the value of the power-law index (n), though the flow transits from being steady to unsteady somewhere in the range ～33 < Re < 50, the fully periodic behavior is seen only beyond the critical value of the Reynolds number (Re). As expected, the average Nusselt number increases with an increase in the values of Reynolds and/or Prandtl numbers, irrespective of the value of the flow behavior index. A strong influence of the power-law index on both local and time-averaged Nusselt numbers was observed. Broadly, all else being equal, shear-thinning behavior (n < 1) promotes heat transfer whereas shear-thickening behavior (n > 1) impedes it. Furthermore, this effect is much more pronounced in shear-thinning fluids than that in shear-thickening fluids.     

Flow over and forced convection heat transfer in Newtonian fluids from a semi-circular cylinder

Momentum and heat transfer characteristics of a semi-circular cylinder immersed in unconfined flowing Newtonian fluids have been investigated numerically. The governing equations, namely, continuity, Navier–Stokes and energy, have been solved in the steady flow regime over wide ranges of the Reynolds number (0.01 ? Re ? 39.5) and Prandtl number (Pr ? 100). Prior to the investigation of drag and heat transfer phenomena, the critical values of the Reynolds number for wake formation (0.55 < Re^c < 0.6) and for the onset of vortex shedding (39.5 < Re_c < 40) have been identified. The corresponding values of the lift coefficient, drag coefficient, and Strouhal number are also presented. After establishing the limit of the steady flow regime, the influence of the Reynolds number (0.01 ? Re ? 39.5) and Prandtl number (Pr = 0.72, 1, 10, 50 and 100) on the global flow and heat transfer characteristics have been elucidated. Detailed kinematics of the flow is investigated in terms of the streamline and vorticity profiles and the variation of pressure coefficient in the vicinity of the cylinder. The functional dependence of the individual and total drag coefficients on the Reynolds number is explored. The Nusselt number shows an additional dependence on the Prandtl number. In addition, the isotherm profiles, local Nusselt number (Nu_L) and average Nusselt number (Nu) are also presented to analyze the heat transfer characteristic of a semi-circular cylinder in Newtonian media.     

Laminar natural convection of non-Newtonian power-law fluids between concentric circular cylinders

The two-dimensional steady-state natural convection of power-law fluids is studied numerically between two concentric horizontal cylinders with different constant temperatures. The governing equations are discretized using finite volume technique based on second order upwind and are solved using the SIMPLE algorithm. The effects of Rayleigh number (10³ ≤ Ra ≤ 10⁵) and Prandtl number (10 ≤ Pr ≤ 10³) on the dimensionless velocity and temperature are investigated for both pseudoplastic and dilatant fluids. Also the mean Nusselt number for various values of governing parameters is obtained and discussed. The results indicate that with increasing the power-law index from 0.6 to 1.4, the mean Nusselt number decreases. In the best case among the range of parameters considered here the heat transfer rate for pseudo-plastic fluid (n = 0.6) is 170% higher than the Newtonian one and for dilatant fluid (n = 1.4) the heat transfer rate is 43% lower than the Newtonian fluid. So the pseudoplastic and dilatant fluids are more efficient than Newtonian fluids for cooling and insulating purposes, respectively. It is shown that as the Rayleigh number increases the cooling effect of pseudoplastic fluid and the insulating effect of dilatant fluid become more pronounced.     

Laminar free convection underneath a hot horizontal infinite flat strip

Analyses were conducted to study the problem of natural convection underneath a hot and isothermal horizontal infinite flat strip. It included a numerical and an analytical investigation of the problem. The work offers simple closed form solutions for the critical flow depth, boundary layer thickness and heat transfer coefficient. The governing equations were solved by the integral method. The justification for applying self-similar boundary layer profiles was demonstrated both numerically and from analyses of published experimental results. The solution was improved through the use of numerical analyses as well as from analytical inspection of limiting cases. The results were successfully tested against published experimental data. Furthermore, the work offers an explanation for the discrepancy that exists amongst the various heat transfer correlations found in the literature.     

Laminar mixed convection in a channel with a built-in semi-circular cylinder under the effect of cross-buoyancy

Effects of cross-buoyancy mixed convection on flow and heat transfer characteristics of a long semi-circular cylinder (long in neutral direction) in a confined channel have been investigated in the laminar regime. The numerical results have been presented and discussed for the range of conditions as Reynolds number (Re) = 1–40, Richardson number (Ri) = 0–4, Prandtl number (Pr) = 0.71–50 and blockage ratio (β) = 16.67%–50%. The drag coefficient increases with increasing Richardson number and/or blockage ratio. The average Nusselt number is showing a maximum relative enhancement of approximately 45% for Ri = 4 with respect to corresponding forced convection value (Ri = 0). The average Nusselt number increases with increase in Prandtl number and shows a maximum relative enhancement of approximately 1136% for Pr = 50 with respect to corresponding value at Pr = 0.71. On the other hand, the maximum relative variation of the total drag coefficient is found to be approximately 55% for Ri = 4 with respect to corresponding value at Ri = 0. Finally, the simple heat transfer correlation is obtained for the proceeding range of control parameters.     

Effect of aspect ratio on natural convection in power-law liquids from a heated horizontal elliptic cylinder

The influence of aspect ratio and shear-dependent viscosity on free convection heat transfer from a horizontal heated elliptic cylinder in power-law fluids has been investigated. In particular, the coupled momentum and energy equations have been solved numerically over the following ranges of conditions: Grashof number, 10 ? Gr ? 10⁵; Prandtl number, 0.72 ? Pr ? 100; power-law index, 0.3 ? n ? 1.5 and aspect ratio, 0.2 ? E ? 5. The new extensive results demonstrate the influence of the Grashof number (Gr), Prandtl number (Pr), power-law index (n) and aspect ratio (E) on the macroscopic heat and momentum transfer characteristics like local and average values of Nusselt number (Nu) and drag coefficients (C_D). Further insights are developed by examining the structure of the flow and temperature fields adjacent to the cylinder. Broadly speaking, all else being equal, shear- thinning fluid behaviour promotes heat transfer whereas shear-thickening viscosity has a deleterious effect on it with reference to that in Newtonian fluids. Also, the rate of heat transfer gradually increases as the cylinder shape passes from blunt to slender with respect to the direction of gravity. Finally, the present numerical values of the Nusselt number are correlated using a simple analytical form which facilitates interpolation of the present results for the intermediate values of the governing parameters. The paper is concluded by presenting detailed comparisons with the previous numerical and experimental results available in the literature, especially in Newtonian fluids.     

Effect of temperature-dependent viscosity on forced convection heat transfer from a cylinder in crossflow of power-law fluids

The steady, two-dimensional and incompressible flow of power-law fluids across an unconfined isothermal heated circular cylinder is investigated numerically to ascertain the effect of temperature-dependent viscosity on the flow and forced convection heat transfer phenomena. Extensive numerical results elucidating the variation of the heat transfer characteristics and drag coefficient on the severity of temperature dependence of viscosity (0 ? b ? 0.5), power law index (0.6 ? n ? 1.6), Prandtl number (1 ? Pr ? 100) and Reynolds number (1 ? Re ? 30) are presented. The coupled momentum and energy equations are expressed in the stream function/vorticity formulation and solved using a second-order accurate finite difference method to determine the local and surface-averaged Nusselt numbers, the drag coefficient, and to map the flow domain in terms of the temperature and flow fields near the cylinder. The variation of viscosity with temperature is shown to have a substantial effect on both the local and surface-averaged values of the Nusselt number. As expected, the results also suggest that the rate of heat transfer shows positive dependence on the Reynolds number and Prandtl number. Furthermore, stronger the dependence of viscosity on the temperature, the greater is the enhancement in the rate of heat transfer. Finally, all else being equal, shear-thinning fluid behaviour facilitates heat transfer while the shear-thickening behaviour has deleterious effect on heat transfer.     

Momentum and heat transfer characteristics of a semi-circular cylinder immersed in power-law fluids in the steady flow regime

The continuity, momentum and energy equations describing the flow and heat transfer of power-law fluids over a semi-circular cylinder have been solved numerically in the two-dimensional steady flow regime. The influence of the Reynolds number (Re), Prandtl number (Pr) and power-law index (n) on the local and global flow and heat characteristics have been studied over wide ranges of conditions as follows: 0.01 ? Re ? 30, 1 ? Pr ? 100 and 0.2 ? n ? 1.8. The variation of drag coefficient and Nusselt number with the Reynolds number, Prandtl number and power-law index is shown over the aforementioned ranges of conditions. In addition, streamline and isotherm profiles along with the recirculation length and distribution of pressure coefficient and Nusselt number over the surface of the semi-circular cylinder are also presented to gain further insights into the nature of the underlying kinematics. The wake size (recirculation length) shows almost linear dependence on the Reynolds number (Re ? 1) for all values of power-law index studied herein. The drag values show the classical inverse variation with the Reynolds number, especially for shear-thinning fluids at low Reynolds numbers. The point of maximum pressure coefficient is found slightly displaced from the front stagnation point for highly shear-thinning fluids, whereas for shear-thickening and Newtonian fluids, it coincides with the front stagnation point. For fixed values of the Prandtl number and Reynolds number, the rate of heat transfer decreases with the gradual increase in power-law index; this effect is particularly striking at high Prandtl numbers due to the thinning of the thermal boundary layer. Conversely, as expected, shear-thinning behavior facilitates heat transfer and shear-thickening impedes it. The effect of power-law index on both momentum and heat-transfer characteristics is seen to be appreciable at low Reynolds numbers and it gradually diminishes with the increasing Reynolds number.     

Momentum and heat transfer from a square cylinder in power-law fluids

Numerical solutions are sought, using FLUENT, to the mass, momentum and thermal energy equations for the 2-D flow of power-law fluids over a cylinder of square cross-section. The major thrust of this work is to delineate the values of the Reynolds number denoting the onset of flow separation and the limits of the steady flow regime for both shear-thinning and shear-thickening type fluids. Extensive results are reported on streamline and vorticity contours over wide ranges of power-law index (0.2–1.4) corroborating the occurrence of these two transitions. Having established the limits of the steady flow regime, drag and Nusselt number results are obtained in this regime as functions of the Reynolds number (0.1–40), of Prandtl number (0.7–100) for highly shear-thinning fluids (power-law index < 0.5) thereby extending the range of currently available results to that encountered in practical applications. The Nusselt number shows positive dependence on both the Reynolds and Prandtl numbers. Also, shear-thinning characteristics can augment the rate of heat transfer by up to 100% under appropriate conditions.     

Mixed convection heat transfer from a horizontal plate to non-newtonian fluids

Steady laminar mixed convection of non-Newtonian fluids over a horizontal plate has been analyzed. After a suitable coordinate transformation to reduce the complexity of the governing boundary-layer equations, the resulting nonlinear coupled differential equations were solved with an implicit finite difference scheme. Of particular interest are the effects of the power-law viscosity index, the generalized Prandtl number and the buoyancy parameter on fluid flow and heat transfer characteristics. It was found that both the dimensionless skin friction group and the dimensionless heat transfer group increase with higher buoyancy effects for any non-Newtonian fluid. Dilatant fluids exhibit a distinctively different behavior with respect to dimensionless heat transfer group when compares to pseudoplastics in the leading edge of the flat plate. Furthermore, higher generalized Prandtl numbers generate lower skin friction and larger heat transfer coefficients.     

Velocity measurements in the free convection flow above a heated horizontal cylinder

The fluid dynamics of the free convection water flow above a heated horizontal cylinder were investigated using particle image velocimetry. Velocity measurements were obtained in three orthogonal planes, with each plane acquired independently in time. The experiment was performed for a Rayleigh number of 1.33 × 10⁶ and for water heights above the cylinder surface of H/D = 1, 2, 4, 6, 8, and 12. The top water surface was open to room air, and the height of the free surface introduced different effects of vertical confinement. For all water heights studied, the plume above the cylinder and its interaction with the free surface were observed. For water heights of H/D = 4 and 6, the appearance of a swaying motion was captured in the plane perpendicular to the cylinder axis, and a meandering motion was detected in the other two orthogonal planes. A vortex was observed to form alongside of the plume and rise to the free surface, where it detached from the plume structure and moved along the free surface. Vortex dynamics and the physical characteristics of the swaying and meandering motions appeared to be interdependent, providing a possible relation between vortex formation, swaying, and meandering. Comparisons with previous analytical and experimental results highlight the complicated three-dimensional flow structure that governs heat transfer from the cylinder.     

Laminar natural convection of power‐law fluids over a horizontal heated flat plate

Natural convection of power‐law fluids over a horizontal flat plate with constant heat flux is studied. The stretching transformations relating the similarity forms of the boundary layer velocity, pressure, and temperature profiles are applied to the governing boundary layer equations. The resultant set of coupled ordinary differential equations are solved analytically and numerically using the integral method and the finite difference method, respectively. The results are presented for the details of the velocity and temperature fields for various values of the non‐Newtonian power‐law viscosity index (n) and the generalized Prandtl number (Pr*). At a fixed value of the viscosity index, increasing the Prandtl number increases the skin friction and wall temperature. For Pr* > 1, a lower viscosity index results in larger wall skin friction, temperature scale, and thermal boundary layer thickness, and thus lower Nusselt number. The reverse trend is observed for Pr* < 1. By using an integral solution and the numerical results, a semi‐analytical correlation for the Nusselt number is obtained, valid for and .     

Laminar natural convection above a horizontal laser beam

The steady laminar natural convective plume above a horizontal laser beam has been studied. The plume, which is caused by absorption of thermal energy from the beam, is three-dimensional. The three-dimensionality is a consequence of mass continuity and the variation in the thermal energy absorption in the propagation direction. Flow visualizations have verified the three-dimensionality in that a significant velocity component, in a direction opposite to that of laser beam propagation, was observed.The problem is reduced by similarity analysis to a system of ordinary differential equations which are solved numerically for the Prandtl number, Pr = 1.0. Integral approximations are also presented for Pr = 0.7, 1.0, 10.0 and 100.0. The effect of Pr on the velocities, and temperature boundary layer thicknesses is discussed in detail.