Steady flow of power-law fluids across an unconfined elliptical cylinder

The momentum transfer characteristics of the power-law fluid flow past an unconfined elliptic cylinder is investigated numerically by solving continuity and momentum equations using FLUENT (version 6.2) in the two-dimensional steady cross-flow regime. The influence of the power-law index (0.2?n?1.8), Reynolds number (0.01?Re?40) and the aspect ratio of the elliptic cylinder (0.2?E?5) on the local and global flow characteristics has been studied. In addition, flow patterns showing streamline and vorticity profiles, and the pressure distribution on the surface of the cylinder have also been presented to provide further physical insights into the detailed flow kinematics. For shear-thinning (n<1) behaviour and the aspect ratio E>1, flow separation is somewhat delayed and the resulting wake is also shorter; on the other hand, for shear-thickening (n>1) fluid behaviour and for E<1, the opposite behaviour is obtained. The pressure coefficient and drag coefficient show a complex dependence on the Reynolds number and power-law index. The decrease in the degree of shear-thinning behaviour increases the drag coefficient, especially at low Reynolds numbers. While the aspect ratio of the cylinder exerts significant influence on the detailed flow characteristics, the total drag coefficient is only weakly dependent on the aspect ratio in shear-thickening fluids. The effect of the flow behaviour index, however, diminishes gradually with the increasing Reynolds number. The numerical results have also been presented in terms of closure relations for easy use in a new application.     

Two-dimensional unsteady flow of power-law fluids over a cylinder

The unsteady flow of incompressible power-law fluids over an unconfined circular cylinder in cross-flow arrangement has been studied numerically. The two-dimensional (2-D) field equations have been solved using a finite volume method based solver (FLUENT 6.3). In particular, the effects of the power-law index (0.4?n?1.8) and Reynolds number (40?Re?140) on the detailed kinematics of the flow (streamline, surface pressure and vorticity patterns) and on the macroscopic parameters (drag and lift coefficients, Strouhal number) are presented in detail. The periodic vortex shedding and the evolution of detailed kinematics with time are also presented to provide insights into the nature of flow. The two-dimensional flow transits from steady to unsteady behaviour at a critical value of the Reynolds number Re∼(40-50) and the von-Karman vortex street is observed beyond the critical Reynolds number (Re). Obviously, both the lift coefficient and Strouhal number values are zero for the steady flow, but their values increase with the increasing Reynolds number (Re) in the unsteady flow regime. For highly shear-thickening fluids (n=1.8), the flow becomes unsteady at Re=40 and unsteadiness in the flow appears at Re=50 for all values of power-law index (n). As expected, the evolution of the kinematics and vortex shedding show a complex dependence on the flow parameters near the transition in the flow. For a fixed value of the Reynolds number (Re), the drag coefficient increases and lift coefficient decreases with increasing value of the power-law index (n). For a fixed value of the power-law index (n), the drag coefficient gradually increases with the Reynolds number (Re). Similar to the drag coefficient, lift coefficient also shows a complex dependence on the power-law index (n) near the transition zone. The value of the Strouhal number (St) decreases with the increasing value of the power-law index (n) at a fixed value of the Reynolds number (Re).     

Non-Newtonian power-law flow across a confined triangular bluff body in a channel

Wall effects on the flow of incompressible non-Newtonian power-law fluids across an equilateral triangular cylinder confined in a horizontal plane channel have been investigated for the range of conditions: Reynolds number, Re=1–40, power-law index, n=0.4–1.8 (covering shear-thinning, Newtonian and shear-thickening behaviors) and blockage ratio=0.125–0.5. Extensive numerical results on flow pattern, wake/recirculation length, individual and overall drag coefficients, variation of pressure coefficient on the surface of the triangular cylinder and so forth are reported to elucidate the combined effect of power-law index, blockage ratio and Reynolds number. The size of vortices decreases with an increase in the value of the blockage ratio and/or power-law index. For a fixed value of the Reynolds number, individual and overall drags decrease with decrease in power-law index and/or blockage ratio in steady confined flow regime. Simple correlations of wake length and drag are also obtained for the range of settings considered.     

Free convection in power-law fluids from a heated sphere

In this work, the governing field equations describing heat transfer from a heated sphere immersed in quiescent power-law fluids have been solved numerically. In particular, consideration has been given to elucidate the role of Grashof number (Gr), Prandtl number (Pr) and power-law index (n), on the value of the Nusselt number (Nu) for a sphere in the natural convection regime. Further insights are provided by presenting streamline and constant temperature contours. The results presented herein encompass the following ranges of conditions: 10≤Gr≤10⁷; 0.72≤Pr≤100 and 0.4≤n≤1.8 thereby covering both shear-thinning and shear-thickening types of fluid behaviours. Broadly, all else being equal, shear-thinning behaviour can enhance the rate of heat transfer by up to three-fold where as shear-thickening can impede it up to ～30?40% with reference to that in Newtonian fluids. The paper is concluded by presenting detailed comparisons with the scant experimental data and the other approximate treatments of this problem available in the literature.     

Effect of blockage on heat transfer from a cylinder to power law liquids

The influence of planar confining walls on the steady forced convection heat transfer from a cylinder to power-law fluids has been investigated numerically by solving the field equations using FLUENT (version 6.2). Extensive results highlighting the effects of the Reynolds number (1?Re?40), power-law index (0.2?n?1.8), Prandtl number (1?Pr?100) and the blockage ratio (β=4 and 1.6) on the average Nusselt number have been presented. For a fixed value of the blockage ratio, the heat transfer is enhanced with the increasing degree of shear-thinning behaviour of the fluid, while an opposite trend was observed in shear-thickening fluids. Due to the modifications of the flow and temperature fields close to the cylinder, the closely placed walls (i.e., decreasing value of the blockage ratio) further enhance the rate of heat transfer as the fluid behaviour changes from Newtonian to shear-thickening fluids (n>1), the opposite influence is seen with the decreasing value of the flow behaviour index (n) in shear-thinning (n<1) fluids. Finally, the functional dependence of the present numerical results on the relevant dimensionless parameters has been presented in the form of closure relationships for their easy use in a new application.     

Mass transfer from ensembles of fluid spheres to a power-law liquid at moderate Reynolds and Peclet numbers

The steady convective mass transfer from ensembles of mono-size Newtonian fluid spheres to power-law liquids has been studied at moderate Reynolds and Peclet numbers. The species continuity equation segregated from momentum equations has been solved numerically using a finite difference method. A simple cell model has been used to account for the modification of the flow field due to the neighbouring droplets. Extensive numerical results have been obtained which elucidate effects of the Reynolds number (Re_o), Schmidt number (Sc), power-law index (n_o), internal to external fluid characteristic viscosity ratio (k) and the volume fraction of the dispersed phase (Φ) on the rate of mass transfer. The ranges of parameters considered herein are: 1?Re_o?200, 1?Sc?10000, 0.6?n_o?1.6, 0.1?k?50 and 0.2?Φ?0.6. For shear-thinning fluids (n_o<1), the rate of mass transfer is somewhat enhanced whereas for shear-thickening fluids (n_o>1), it decreased as compared to that in Newtonian fluids (n_o=1). A simple predictive correlation has been proposed which can be used to estimate the rate of mass transfer in liquid-liquid systems in a new application involving power-law continuous phase.     

Mass transfer from a single fluid sphere to power-law liquids at moderate Reynolds numbers

The steady-state convective inter-phase mass transfer from a single Newtonian fluid sphere (free from surfactants) to a continuous phase with power-law viscosity has been studied at moderate Reynolds and Schmidt numbers under the conditions when the resistance to mass transfer in the dispersed phase is negligible. The species continuity equation, segregated from the momentum equations of both phases, has been numerically solved using a finite difference method. The effects of the Reynolds number (Re_o), power-law index (n_o), internal to external fluid characteristic viscosity ratio (k) and Schmidt number (Sc) on the local and average Sherwood number (Sh) have been analysed over the following ranges of conditions: 5?Re_o?200, 0.6?n_o?1.6, 0.1?k?50 and 1?Sc?1000. It has been observed that irrespective of the values of the Reynolds number and of the power-law index, as the value of k increases the average Sherwood number decreases for intermediate to large values of the Peclet number. As the value of the power-law index increases, the rate of mass transfer decreases for all values of the Reynolds number and the characteristic viscosity ratio thereby suggesting that shear-thinning behaviour facilitates mass transfer, whereas shear-thickening behaviour impedes it. Based on the present numerical results, a simple predictive correlation is proposed which can be used to estimate the rate of inter-phase mass transfer of a fluid sphere sedimenting in power-law liquids.     

Drag on a single fluid sphere translating in power-law liquids at moderate Reynolds numbers

A numerical investigation has been carried out to obtain the steady state drag coefficients and flow patterns of a single Newtonian fluid sphere sedimenting in power-law liquids. A finite difference method based simplified marker and cell (SMAC) algorithm has been implemented on a staggered grid arrangement to solve the continuity and momentum equations. For both phases, the convective terms have been discretized using the quadratic upstream interpolation for convective kinematics (QUICK) scheme, and diffusive and non-Newtonian terms with central differencing scheme. An exponential transformation has been applied in the radial direction for the continuous phase computational domain. In order to ensure the accuracy of the solver, extensive validation has been carried out by comparing the present results with the existing literature values for a few limiting cases. Further, in this study the effects of the Reynolds number (Re_o), internal to external fluid characteristic viscosity ratio (k) and power-law index (n_o) on the continuous phase flow field, pressure drag (C_dp), friction drag (C_df) and total drag (C_D) coefficients have been analyzed over the range of parameters: 5?Re_o?500, 0.1?k?50 and 0.6?n_o?1.6. Based on numerical results obtained in this work, a simple correlation has been proposed for the total drag coefficient, which can be used to predict the rate of sedimentation of a fluid sphere in power-law liquids.     

A numerical study of the accelerating motion of a dense rigid sphere in non-newtonian power law fluids

The equations of motion of an accelerating sphere falling through non-Newtonian fluids with power law index n in the range 0.2 ≤ n ≤ 1.8 were integrated numerically using the assumption that the drag on the sphere was a function of both power law index and terminal Reynolds number, Re_t For 10^?2 ≤ Re_t ≤ 10³ both dimensionless time and distance travelled by the sphere under transient conditions showed a much stronger dependence on the flow behaviour index, n, for shear-thinning than for shear-thickening fluids. The form of this dependence is investigated here. Furthermore, results in four typical shear-thinning fluids suggested a strong correlation between the distance and time travelled by the sphere under transient conditions and the value of the fluid consistency index. The analysis reported herein is, however, restricted to dense spheres falling in less dense fluids, when additional effects arising from the Basset forces can be neelected.     

Forced convection in cross flow of power law fluids over a tube bank

The forced convective heat transfer characteristics for incompressible power-law fluids past a bundle of circular cylinders have been investigated numerically. The cylinder-to-cylinder hydrodynamic interactions have been approximated via a simple cell model. The momentum and energy equations have been solved using a finite difference based numerical method for a range of physical and kinematic conditions. The role of the two commonly used thermal boundary conditions, namely, constant temperature or constant heat flux, on heat transfer characteristics has also been studied. Extensive numerical results elucidating the effect of shear-thinning viscosity on the values of Nusselt number have been obtained for Peclet numbers ranging from 1 to 5000, Reynolds number in the range 1-500, flow behaviour index 1?n?0.5 and three values of voidages, namely, 0.4, 0.5 and 0.6, typical of tubular heat exchangers and tube banks. Under all conditions, varying levels of enhancement in Nusselt number are observed due to shear-thinning behaviour. The surface averaged Nusselt number shows strong dependence on the values of voidage, power-law index, Reynolds and Peclet numbers. The paper is concluded by presenting comparisons with the scant experimental results available in the literature.     

Steady Flow of Power Law Fluids across a Circular Cylinder

The momentum equations describing the steady cross‐flow of power law fluids past an unconfined circular cylinder have been solved numerically using a semi‐implicit finite volume method. The numerical results highlighting the roles of Reynolds number and power law index on the global and detailed flow characteristics have been presented over wide ranges of conditions as 5 ≤ Re ≤ 40 and 0.6 ≤ n ≤ 2. The shear‐thinning behaviour (n < 1) of the fluid decreases the size of recirculation zone and also delays the separation; on the other hand, the shear‐thickening fluids (n > 1) show the opposite behaviour. Furthermore, while the wake size shows non‐monotonous variation with the power law index, but it does not seem to influence the values of drag coefficient. The stagnation pressure coefficient and drag coefficient also show a complex dependence on the power law index and Reynolds number. In addition, the pressure coefficient, vorticity and viscosity distributions on the surface of the cylinder have also been presented to gain further physical insights into the detailed flow kinematics.     

Falling ball method for determining zero shear and shear-dependent viscosity of polymeric systems: Solutions,melts, and composites

The falling ball method (FBM) is one of the well-established techniques for measuring the viscosity of Newtonian liquids at the room as well as at elevated temperatures and pressures. Owing to its simplicity and low cost, the possibility of extending its range of application to non-Newtonian systems including virgin and filled polymer melts, composites, polymer-solutions, and so forth, is explored here, In this work, theoretical results for the flow of power-law fluids past a sphere have been used to extract the values of the zero-shear viscosity and shear-dependent viscosity in the low-shear rate limit. The theoretical scheme outlined here has been validated by presenting comparisons with experimental results for scores of polymer solutions for which both falling sphere and rheological data are available in the literature. Indeed, the good correspondence obtained between these two independent data is encouraging and it is thus possible to use the FBM for shear-thinning systems when the resulting Reynolds numbers are such that the flow is viscosity-dominated, and the inertial effects are negligible. This implies that the Reynolds number should be ≤ ~1 for shear-thinning fluids and ≤ ~10⁻⁵ for shear-thickening fluids.     

Stability of fiber spinning of power-law fluids

A stability analysis of fiber spinning of isothermal power-law fluids has been carried out. The analysis for purely viscous fluids indicates that the critical extension ratio increases with an increase in power-law constant q above 1. For q greater than approximately 1.5, very high values of critical extension ratio are obtained. A stability analysis in the presence of viscous and inertial forces indicates that for q > 1 critical extension ratio can be correlated to a quantity Rq = q ? 1 + 3Re, wherein Re is the Reynolds number. For the values of Rq greather than approximately 0.5, very high values of critical extension ratios are obtained.     

Dynamics of self-organizing single-line particle trains in the channel flow of a power-law fluid

The formation of self-organizing single-line particle train in a channel flow of a power-law fluid is studied using the lattice Boltzmann method with power-law index 0.6 ≤ n ≤ 1.2,particle volume concentration 0.8% ≤Φ≤ 6.4%,Reynolds number 10 ≤ Re ≤ 100,and blockage ratio 0.2 ≤ k ≤ 0.4. The numerical method is validated by comparing the present results with the previous ones.The effect n,Φ,Re and k on the interparticle spacing and parallelism of particle train is discussed.The results showed that the randomly distributed particles would migrate towards the vicinity of the equilibrium position and form the ordered particle train in the power-law fluid.The equilibrium position of particles is closer to the channel centerline in the shear-thickening fluid than that in the Newtonian fluid and shear-thinning fluid.The particles are not perfectly parallel in the equilibrium position,hence IH is used to describe the inclination of the line linking the equilibrium position of each particle.When self-organizing single-line particle train is formed,the particle train has a better parallelism and hence benefit for particle focusing in the shearthickening fluid at high Φ,low Re and small k.Meanwhile,the interparticle spacing is the largest and hence benefit for particle separation in the shear-thinning fluid at low Φ,low Re and small k.     

Drag on a sphere in a spherical dispersion containing Carreau fluid

The drag on a rigid sphere in a spherical dispersion containing Carreau fluid is investigated theoretically based on a free surface cell model for Reynolds number in the range [0.1,100], Carreau number in the range [0,10], the power-law index in the range [0.3,1], and the void fraction in the range [0.271,0.999]. The influences of the particle concentration, the nature of the Carreau fluid, and Reynolds number, on the drag coefficient are examined. We show that the drag coefficient declines with the decreasing particle concentration, and the reversal of the flow field in the rear region of a sphere is enhanced by the shear-thinning nature of the fluid. An empirical relation, which correlates the drag coefficient with the void fraction (= 1 − particle concentration), the nature of the Carreau fluid, and Reynolds number, is proposed.     

Pressure drop during the flow of stokesian fluids through granular beds

The resistance to flow of Stokesian fluids (i.e. time—independent fluids with no yield stress) through granular beds is discussed. A definition of friction factor λ and generalized Reynolds number Re′_BK is proposed for fluids obeying the “power-law” shear stress—shear rate relation.The generalized Ergun equation, derived in this paper, gives the dependence of the friction factor on the generalized Reynolds number and flow behaviour index n. The validity of the generalized Ergun equation was proved experimentally. In the case of Newtonian fluid (for n = 1·0) a more exact form of the classical Ergun equation is obtained.     

Low Reynolds number flow of power-law fluids over two square cylinders in tandem

The governing partial differential equations have been solved numerically for the 2-D and steady powerlaw fluid flow over two square cylinders in tandem arrangement. Extensive numerical results spanning wide ranges of the governing parameters as Reynolds number (0.1≤Re≤40), power-law index (0.2≤n≤1) and inter-cylinder spacing (2≤L/d≤6) are presented herein; limited results for L/d=24 are also obtained to approach the single cylinder behavior. The detailed flow visualization is done by means of the streamline and vorticity contours in the vicinity of two cylinders. The global characteristics are analyzed in terms of the surface pressure distribution and pressure drag coefficient. The drag coefficient shows the classical inverse dependence on the Reynolds number irrespective of the value of the powerlaw index; the drag on the upstream cylinder is always greater than that for the downstream cylinder.     

The drag on a single bubble accompanied by a periodic wake

The dynamics of a single spheroidal bubble accompanied by an open periodic wake were studied in water-glycerol solutions (1 < η < 37 cP) by photographic techniques. The stability of the helical vortex, the frequency of the bubble rocking and the drag coefficient were found to be closely related. These characteristics exhibit two modes of behaviour: they are very sensitive to viscosity changes at lower Reynolds numbers Re_b; whereas at higher Re_b the viscosity does not appreaciably influence the drag coefficient C_Db and the Strouhal number S_b, while the length of the helical vortex here has reached its minimum . It has been concluded threefore that, in the transition region between the oblate spheroid and spherical cap bubbles presently investigated, the induced drag plays an important role in the overall bubble dynamics.     

Three-dimensional finite element analysis of polymeric fluid flow in an extrusion die. Part I: Entrance effect

The Galerkin finite element method has been applied to study the three-dimensional flow field of power-law fluids inside an extrusion die. Two inlet designs, i.e., center-fed and end-fed, have been considered. The effects of inertial force as represented by the Reynolds number Re, inlet geometry, and the power-law index n on lateral flow uniformity and vortex formation in the entrance region have been examined. A flow visualization technique has been carried out to experimentally verify the theoretical prediction of the three-dimensional flow field inside a die. It has been found that increasing Re or decreasing n will deteriorate flow uniformity. Depending on the direction of the inlet jet stream, the inertial force may create a flow peak in the central region of a center-fed die, or the maximum flow rate will appear close to the end of the die for an end-fed die. For highly shear-thinning fluids, lower flow rates are always observed close to the end of the dies. It is concluded that creating a plug flow in the inlet tube of the extrusion die is advantageous for both center-fed and end-fed designs.