MIXED CONVECTION IN NON-NEWTONIAN FLUID FLOW OVER A ROTATING CIRCULAR CYLINDER

Abstract

The problem of fluid flow and heat transfer from a rotating circular cylinder placed in the free stream has always been enticing the attentiveness of researcher owing to its industrial application such as heat transfer from rotating machinery, spinning projectiles, and several others (Badr and Dennis, 1985a,b). According to a theoretical perspective, this problem configuration unveils a bountiful variety of conceivable flow regime, besides, to lift enhance ment, suppressing flow fluctuations and control boundary layer formed on rotating-part, etc. (Prandtl, 1925a,b; Badr and Dennis, 1985a,b; Badr, 1983; Ingham, 1983; Ingham and Tang, 1990; Badr et al., 1990; Tang and Ingham, 1991). Therefore, meaningful experimental and numerical investigations have been performed over the years to explore steady as well as unsteady fluid flow and heat transfer characteristics of a rotating circular cylinder in a free stream for various flow regimes. To an extent, these studies are mainly limited to Newtonian f luids (air and water). This problem configuration is also applicable in the manufacturing of polymer sheet, paper making, textile industries, plastic, and glass industries (Panda and Chhabra, 2010; Thakur et al., 2016). In such industries, the non-Newtonian fluids are being used as process fluids like slurries and suspensions, etc. These non-Newtonian fluids show the strong dependence of viscosity on the shear rates. It is doubtless to state that much more extensive researches have been done regarding the hydrodynamic characteristics and heat transfer characteristics aspects in forced convection regime as a compared to that in mixed convection regime (combined forced and free convec tion) even in a Newtonian fluid. In most viable applications, mixed convection is the mode of heat transfer, i.e., both forced and free convection occur together. While the presence of free convection might be small but it is always present and plays a vital role. To deter mine the effect and extent of mixed convection regime on the fluid flow and heat transfer characteristics, the non-dimensional parameter called Richardson number (Ri = Gr/Re2) is employed. Here, Gr and Re are the Grashof and Reynolds numbers, respectively. The experimental as well as numerical studies for rotating bluff-bodies are thoroughly sum marized in the literature (Prandtl, 1925a,b; Badr and Dennis, 1985a,b; Ingham, 1983; Ing ham and Tang, 1990; Tang and Ingham, 1991; Townsend, 1980; Stojkovi´c et al., 2002, 2003; Chew, 2013). They studied the steady as well as unsteady state of a rotating circular cylin der in the free stream of Newtonian fluids and presented the detailed flow kinematic in terms of streamlines, drag and lift coefficients. Further, most of the research on the convective heat transfer from a rotating cylinder are focused on the forced convection mode of heat transfer for unconfined/confined channel even for Newtonian fluids (Badr and Dennis, 1985a; Paramane and Sharma, 2009; Prasad et al., 2011; Sharma and Dhiman, 2012). Additionally, few studied are carried out in a mixed convection in Newtonian fluids (Chatterjee and Sinha, 2014; Paramane and Sharma, 2010a). In their study, they reported the effect of rotation of the cylinder on the onset of vortex shedding in the presence of cross buoyancy even in Newtonian fluids. To the best of my knowledge, only two available studies considers the non-Newtonian fluids as a process fluid in their study (Panda and Chhabra, 2010; Thakur et al., 2019). They numerically performed the parametric studies for the non-Newtonian shear-thinning fluids, i.e., power-law index less than one (n ≤ 1), in the forced convection mode of heat transfer for the different values of Prandtl number (Pr). This dissertation aims to fill this gap in the existing literature. In particular, considerations is given to drag, lift, and heat transfer characteristics for a rotating circular cylinder submerged in streaming cross-flow of power-law fluids over a wide range of conditions, namely, Reynolds number (Re), Prandtl number (Pr), Richardson number (Ri), rotational velocity (α) and, blockage ratio (β). The physical problems mentioned above are represented by the laws of conservation of mass, momentum, and energy with four major unknown variables, mainly pressure, x-, and y-components of the velocity and temperature. The flow is subjected to the appropriate boundary conditions and solved using a finite element method based commercial software COMSOLMultiphysics, where a numerically adequate grid is fetched for rigorous qualitative and quantitative flow analysis. The absolute convergence criteria of 10−7 for the continuity, components of velocity and 10−15 for the thermal energy equation are found to be sufficient. The post-processing analysis is carried out for various quantitative and theoretical analysis. The descriptions of the various problems studied in the thesis work are explained as follows: 1. Mixed Convection Heat Transfer in Non-Newtonian Fluids from an Uncon f ined Rotating Cylinder The detailed analysis of flow and thermal fields have been presented and discussed for the steady mixed convection heat transfer from an isothermally heated cylinder rotating in power-law fluids. The numerical results are interpreted in terms of the pressure coefficient and the local Nusselt number over the surface of the cylinder, individual and total drag and lift coefficients and the surface averaged Nusselt number. In presence of rotation and/or buoyancy, the overall downward lift force is generated which is strongly dependent on the rotational velocity than Richardson number (Ri). The magnitude of the overall lift force is found to be higher in the shear-thickening (n > 1) than Newtonian (n = 1) fluids and lower in shear-thinning (n < 1) fluids. The overall drag force reduces and it becomes negative at higher rotational velocity. This reduction in the overall drag force gets manifested as the effect of buoyancy is increased. For fixed value of Re, Pr, n, and Ri, as the value of rotational velocity is increased, the reduction in the value of average Nusselt number is observed for both Newtonian and shear-thickening fluids whereas for shear-thinning fluids at lower rotational velocity, there is an increase in the value of average Nusselt number. The reduction in the heat transfer with rotation is due to the formation of a concentric layer of f luid around the cylinder which act as a buffer zone. This buffer zone increases as the value of the power-law index (n) increase. Apart from this, the streamline patterns reveal that the unsteadiness produced in the flow due to thermal buoyancy vanishes as the rotational velocity is imposed on the cylinder. The layer of fluid around the cylinder with the single saddle point is formed, whose size increases as the fluid behavior change from shear-thinning to shear-thickening fluids. The position of the single saddle point or saddle zone move in the opposite direction of the cylinder rotation as the effect of thermal buoyancy is increased. The qualitatively similar to streamline pattern, the isotherm contours are formed which get crowded near the cylinder as the value of Prandtl number (Pr) is increased. This effect gets intensified as the degree of shear-thinning is increased. For the higher value of the rotational velocity a concentric ring like structure of isotherms is formed around the cylinder. Whose size increases as the value of rotational velocity and/or power-law index is increased. 2. Power-Law Non-Newtonian Fluid Flow Across a Channel Confined Rotating Circular Cylinder Two-dimensional steady Poiseuille flow of power-law type of non-Newtonian fluids across a channel confined rotating circular cylinder in cross-flow has been numerically. Extensive results on the individual and total drag and lift coefficients, velocity profile, shear rate polar contours, streamline and isotherm patterns have presented and discussed for all the parametric conditions. The investigation in terms of streamline patterns reveal that rotation causes the fluid to rotate with cylinder which result in formation of layer of fluid around the cylinder for all the value of Reynolds number (Re), power-law index (n) and blockage ratio (β). This influence of rotation is seen to be more prominent for higher value of blockage ratio (β). Moreover, the size of layer of fluid around the cylinder grows as the fluid behavior Abstract vpower-law fluids. The numerical results are interpreted in terms of the pressure coefficient and the local Nusselt number over the surface of the cylinder, individual and total drag and lift coefficients and the surface averaged Nusselt number. In presence of rotation and/or buoyancy, the overall downward lift force is generated which is strongly dependent on the rotational velocity than Richardson number (Ri). The magnitude of the overall lift force is found to be higher in the shear-thickening (n > 1) than Newtonian (n = 1) fluids and lower in shear-thinning (n < 1) fluids. The overall drag force reduces and it becomes negative at higher rotational velocity. This reduction in the overall drag force gets manifested as the effect of buoyancy is increased. For fixed value of Re, Pr, n, and Ri, as the value of rotational velocity is increased, the reduction in the value of average Nusselt number is observed for both Newtonian and shear-thickening fluids whereas for shear-thinning fluids at lower rotational velocity, there is an increase in the value of average Nusselt number. The reduction in the heat transfer with rotation is due to the formation of a concentric layer of f luid around the cylinder which act as a buffer zone. This buffer zone increases as the value of the power-law index (n) increase. Apart from this, the streamline patterns reveal that the unsteadiness produced in the flow due to thermal buoyancy vanishes as the rotational velocity is imposed on the cylinder. The layer of fluid around the cylinder with the single saddle point is formed, whose size increases as the fluid behavior change from shear-thinning to shear-thickening fluids. The position of the single saddle point or saddle zone move in the opposite direction of the cylinder rotation as the effect of thermal buoyancy is increased. The qualitatively similar to streamline pattern, the isotherm contours are formed which get crowded near the cylinder as the value of Prandtl number (Pr) is increased. This effect gets intensified as the degree of shear-thinning is increased. For the higher value of the rotational velocity a concentric ring like structure of isotherms is formed around the cylinder. Whose size increases as the value of rotational velocity and/or power-law index is increased. 2. Power-Law Non-Newtonian Fluid Flow Across a Channel Confined Rotating Circular Cylinder Two-dimensional steady Poiseuille flow of power-law type of non-Newtonian fluids across a channel confined rotating circular cylinder in cross-flow has been numerically. Extensive results on the individual and total drag and lift coefficients, velocity profile, shear rate polar contours, streamline and isotherm patterns have presented and discussed for all the parametric conditions. The investigation in terms of streamline patterns reveal that rotation causes the fluid to rotate with cylinder which result in formation of layer of fluid around the cylinder for all the value of Reynolds number (Re), power-law index (n) and blockage ratio (β). This influence of rotation is seen to be more prominent for higher value of blockage ratio (β). Moreover, the size of layer of fluid around the cylinder grows as the fluid behavior change from shear-thinning (n < 1) to Newtonian (n = 1) and shear thickening (n > 1). Further, it is interesting to note here that for the lower value of blockage ratio (β ≤ 4), the two stagnation points are formed on the upper half of the cylinder. These two stagnation points come closer to each other as the value of rotational velocity. This influence of rotation in conjunction with blockage ratio is more prominent for the shear-thinning (n < 1) fluids as compared to Newtonian (n = 1)andshear-thickening (n > 1) fluids. Further, it is interesting to note here that for lower value of blockage ratio i.e., β = 2, at higher value of rotational velocity (α = 2), the formation of the secondary wake takes place at the bottom wall of the channel for the Reynolds number (Re = 40), called here as the bottom-wall wake. The size of the bottom wall wake increase as the fluid behavior change from shear-thickening (n > 1) to Newtonian (n = 1) fluids and finally to shear-thinning (n < 1) fluids. Subsequently, for the shear-thickening fluids no such bottom-wall wake is formed, otherwise under identical conditions. Further, for the lower values of blockage ratio (β ≤ 4), the increase in the magnitude of lift coefficient is seen to be very small with rotation as compared to the higher blockage ratio (β), even for rotational velocity (α = 2). This effect is further accentuated as the shear-thickening (n > 1) tendency of the fluid behavior increases. On the other hand, for lower blockage ratio (β ≤ 4), the total drag coefficient is seen to increase with the increasing value of the rotational velocity (α). The increase in the total drag coefficient is seen to be higher for shear-thickening (n > 1) than Newtonian (n = 1) fluids and lower for shear-thinning (n < 1) fluids. Subsequently, for higher values of blockage ratio (β ≥ 8), the total drag coefficient is seen to decrease with the increasing value of rotational velocity (α). This effect is seen to be more pronounced for the shear-thickening behavior of fluids. The influence as mentioned above of rotation, is observed to be more prominent for lower value of Reynolds (Re). 3. Forced Convection Heat Transfer in the Power-Law Fluid Flow from a Chan nel Confined Rotating Circular Cylinder The aforementioned hydrodynamic study has also been extended to the forced convec tion heat transfer from an isothermal rotating circular cylinder to power-law type of non Newtonian fluids to ascertain the extent of rotation and wall confinement. The effects of rotation (α), and blockage ratio (β) and the dimensionless parameter (Re, n & Pr) on the temperature profile (isotherm contours), local and average Nusselt number have been pre sented. The isotherm patterns shows that for fixed value of power-law index (n), rotational velocity (α), and blockage ratio (β), the clustering of isotherms in the vicinity of the cylinder increases with the increasing value of Reynolds (Re) and Prandtl (Pr) numbers. Further, isotherm patterns get distorted in the presence of rotation. For fixed value of Reynolds num ber (Re), Prandtl number (Pr), rotational velocity, and blockage ratio (β), the isotherms is seen to be crowded near the surface of the cylinder with decreasing value of power-law index (n). It has also been seen that as the fluid behavior changes from shear-thickening (n > 1) to Newtonian (n = 1) and shear-thinning (n < 1), the isotherm contours get extended in the rare side of the cylinder. This effect happens because thermal boundary layer becomes thin and flow field decays faster. This effect of power-law fluids is more prominent for the higher value of blockage ratio (β). Further, due to rotation, the cylinder is surrounded by a concentric ring-like structure of isotherms for all the values of Reynolds number (Re), Prandtl number (Pr), power-law index (n) and blockage ratio (β). Interestingly, this influ ence of the rotation seen to be more prominent for the higher value of the blockage ratio. Subsequently, for fixed value of Re, Pr, n, and α, the clustering of isotherms is seen to be increased with the decreasing value of blockage ratio (β). For the fixed value of power-law index (n), rotational velocity (α), and blockage ratio (β), the surface average Nusselt num ber (Nuavg) is observed to increase with the increasing value of Reynolds (Re) and Prandtl (Pr) number. Further, this enhancement in the value of average Nusselt number is seen to be accentuated as the behavior of fluid changes from shear-thickening (n > 1) to Newtonian (n = 1) and finally to shear-thinning (n < 1) fluids. Subsequently, for the fixed value of Reynolds number (Re), Prandtl number (Pr), power-law index (n) and rotational velocity (α), the value of average Nusselt number increases as the wall effect is increased i.e., with the decreasing value of blockage ratio (β). This effect of the wall is further dominated as the shear-thinning behavior tendency of fluid increases. Further, it is interesting to note here that, for lower value of blockage ratio (β ≤ 4), effect of the rotation on the average Nusselt is seen to be infinitesimally small, whereas for higher value blockage ratio (β ≥ 8), the value of Nuavg decreases with the increasing value of rotational velocity (α) for all the values of Reynolds and Prandtl number. This effect of rotation on the heat transfer for higher value of blockage ratio is further get manifested as the fluid behavior changes from shear-thinning to Newtonian fluids and shear-thickening fluids. 4. Mixed Convection Heat Transfer in the Power-Law Fluid Flow from a Channel Confined Rotating Circular Cylinder Finally, the flow and heat transfer characteristics associated with a non-Newtonian fluid as it flows past a rotating circular cylinder in the presence of wall and buoyancy effect is explored. The flow encompasses a combined effect of flow cross-buoyancy (Ri), rotation (α), and blockage ratio (β). It has been noticed that at the lower value of the blockage ratio, the effect of the buoyancy is infinitesimally small as compared to unconfined flow; this effect turns out to be more pronounced with decreasing value of (n). Further, for the lower value of blockage ratio (β ≤ 4), as the value of rotational velocity (α) is increased in the presence of the buoyancy, the qualitatively very similar flow patterns is seen as it was in the case of forced convection (Ri = 0) for all the values of Reynolds number (Re), Prandtl number (Pr) and power-law index (n). Further, for the blockage ratio β = 2, and at power-law index (n = 0.4) rotational velocity (α = 2), the similar type of bottom wall wake was for in the presence of buoyancy (Ri > 0) as it was in forced convection (Ri = 0) mode of heat transfer for Reynolds number (Re = 40) except that its size grow with the increasing value of Richardson number (Ri). On the other hand, for the higher value of blockage ratio (β ≥ 8), the effect of rotation in conjunction with buoyancy is seen to be qualitatively similar to that of unconfined flow for all the value of Re, Pr, and n. Subsequently, in the presence of rotation and/or buoyancy, the isotherms patterns become asymmetric along the x-axis. It has been observed that for fixed value of power-law index (n), rotational velocity (α), blockage ratio (β), and Richardson number (Ri), the clustering of isotherms near the surface of the cylinder increases as the value Re and/or Pr is increased. This effect is further accentuated as the value of power-law index (n) decreases. Further, the effect of blockage ratio on the isotherms patterns is seen to be qualitatively similar to that of forced convection (Ri = 0) for all the value of parameters spanned herein, otherwise under identical conditions. For the higher values of blockage ratio (β ≥ 8), at fixed values of Reynolds (Re) and Prandtl number (Pr), and power-law index (n), the qualitatively similar trends in the value of total drag coefficients is seen in presence of rotation and buoyancy, as it was in the case of unconfined flow configuration. Subsequently, for the blockage ratio β = 2 and 4, as the shear-thickening tendency of flow increases, the value of the drag coefficient is seen to increase with the increasing value of rotational velocity irrespective of buoyancy effect. Whereas, the opposite trend is seen for shear-thinning fluids. For a fixed values of Reynolds number (Re), power-law index (n), and blockage ratio (β), infinitesimally small effect of Prandtl number (Pr) and cross-buoyancy (Ri > 0) have been observed on the drag coefficient for all the values of rotational velocity (α). Further, for the lower values of blockage ratio (β ≤ 4), the effect of rotation and buoyancy are found to very weak. Further, for fixed values of Reynolds and Prandtl numbers, power-law index, Richardson number, and blockage ratio, the magnitude of lift coefficient (CL) increase with an increase in the value of rotational velocity (α) for all the value of n and β. Further, for fixed values of Re, Pr, Ri, α and, β, the magnitude of lift coefficient is seen to decrease with the increasing value of power index (n), this effect is more prominent for the lower values of blockage ratio especially for blockage ratio (β = 2 and 4). On the other hand, for higher values of blockage ratio (β ≥ 8), the variation in the lift coefficient is seen to qualitatively similar to that of unconfined flow in the presence of rotation and buoyancy. Moreover, for higher blockage ratio (β), the lift coefficient is seen to increase with the rotation and buoyancy. This effect is seen to be more pronounced as the tendency of shear-thickening (n > 1) behavior of f luid increases. On the other hand, for fixed values of the Re, n, Pr, Ri, and β, the value average Nusselt number decreases with increasing value of the rotational velocity irrespective of buoyancy effect. This effect increases with the increasing value of power-law index (n) and blockage ratio (β). For a fixed value of Re, Pr, n, and Ri, the low value of blockage ratio (β ≤ 4), rate of decrement of heat transfer is found to be very small with increasing rotational velocity (α) than higher values of blockage ratio for all the values of Ri. The strong influence of rotation and cross-buoyancy is seen for higher value of blockage ratio (β).