Characterisation of inelastic poweriaw fluids using falling sphere data

Two simple methods are presented for the characterization of inelastic power law fluids from falling sphere data. The methods involve the application of shear rate or shear stress correction factors which have been derived theoretically using Slattery's solution for creeping flow about spheres. Flow curves obtained using these methods are in excellent agreement with those measured on a Weissenberg rheogoniometer for 0.83 > n > 1.0. The experimentally determined drag coefficients are found to be in good agreement with the predictions of Slattery's creeping flow first approximation solution. The wall correction factors of Faxen and Francis appear to be valid for inelastic non-Newtonian fluids up to a diameter ratio of at least 0.08.     

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.     

Boundary-layer analysis of power-law fluids

An exact numerical solution to the 2-D(Two-Dimensional) laminar boundary-layer equations of power-law non-newtonian fluids is obtained using a finite difference technique. No limitation has been imposed on the flow behavior index(n) or generalized Prandli number As a test case, velocity and temperature fields around a circular cylinder in crossflow were calculated The result clearly indicated that heat transfer of power-law materials is governed by shear dependent viscosity.     

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.     

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).     

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.     

Bubble drag and mass transfer in non-newtonian fluids: Creeping flow with power-law fluids

The drag and mass-transfer characteristics of a gas bubble moving in power-law non-Newtonian fluids are examined analytically in terms of the rheological properties of the system. An approximate solution for the case of creeping flow around circulating bubbles shows that the mass-transfer coefficient is enhanced for pseudoplastics and depressed for dilatants compared to the situation for Newtonian fluids. Some preliminary experimental results in support of the analysis are presented.     

Direct numerical simulations of a freely falling sphere using fictitious domain method: Breaking of axisymmetric wake

In the present paper, numerical simulations of the wake generated by a freely falling sphere, under the action of gravity, are performed. Simulations have been carried out in the range of Reynolds numbers from 1 to 210 for understanding the formation, growth and breakup of the axisymmetric wake. The in-house code used is based on a non-Lagrange multiplier fictitious-domain method, which has been developed and validated by Veeramani et al. (2007). The onset of instability in the wake and its growth along with the dynamic behavior of a settling sphere is examined at Reynolds number (Re) of 210. It is found that at the onset of instability the sphere starts to rotate and gives rise to a lift force due to the break of the axisymmetry in the wake which in turns triggers a lateral migration of the sphere. The lift coefficient of a freely falling sphere is 1.8 times that of a fixed sphere at a given sphere density of 4000 kg m^?3 and sphere to fluid density ratio of 4. This is attributed to the Robin's force which arises due to the rotation of the sphere. At this Reynolds number (Re=210) a double threaded wake is observed, which resembles the experimental observations of Magarvey and MacLatchy (1965).     

A study of viscous dissipation in the calendering of power-law fluids

A finite difference procedure was used to solve the equation of conservation of energy with the lubrication approximation. The temperature profiles due to viscous dissipation in the gap formed by a pair of equal sized rolls rotating at the same speed exhibit two maxima in the vicinity of the roll surfaces. The maximum temperatures also exhibit two local maxima and a minimum in the direction of flow. The calculated temperature distributions across and along the flow field reveal the possibility of excessive local heating which might be detrimental for certain temperature sensitive materials such as poly(vinyl chloride) and rubbers.     

鼓泡反应器中幂律型流体流动与传质特性

很多废水处理装置涉及非牛顿型流体中的多相流动和传质问题,研究其中的气液传质过程有助于实现装置的优化设计和高效节能运行。以鼓泡反应器内清水和不同质量分数的羧甲基纤维素钠（CMC）水溶液为实验对象,分别研究气相表观气速和液相流变特性对气泡尺寸分布、全局气含率和体积氧传质系数的影响。实验结果表明,液相的流变特性对其传质特性参数均有较大影响。与清水相比,CMC水溶液中气泡平均直径和分布范围更大;清水和CMC水溶液的全局气含率均随表观气速的增加而增大;CMC水溶液的体积氧传质系数随CMC水溶液质量分数的增加而减小。基于实验研究,得出修正的体积氧传质系数公式和适用于幂律型非牛顿流体流动体系氧传递过程的无量纲关联式,可很好地实现非牛顿流体流动的废水处理装置中气液传质参数的计算。     

A new velocity profile model for turbulent pipe-flow of power-law fluids

A new velocity profile model for turbulent flow of non-Newtonian power-law fluids in smooth pipes has been proposed, following the approach of Stein et al. (1980) for Newtonian fluids. The merit of the model, as different from all earlier works, lies in its ability to predict a zero velocity gradient at the centreline for all values of the pseudoplasticity index n. The predictions of the model when compared with the work of Bogue and Metzner (1963) have shown good agreement.     

3-D Non-isothermal die flow of power-law fluids with viscous heating

The nonisothermal flow of incompressible viscous non-Newtonian fiuids is analyzed numerically. A three-dimensional finite element code is developed for the flow simulation purpose. The energy equation is decoupled from the equations of motion and both the flow field and the thermal field are solved iteratively. Two dimesionless groups, the Peclet number and the Brinkman number, are introduced to represent the characteristics of the thermal field. Results are presented for velocity, pressure and temperature distributions in the entrance region, and comparisons made with various mesh layouts. The results provide new insight into the temperature regulation in the extrusion process.     

Laminar flow of power-law fluids in the entrance region of a pipe

The study is concerned with developing laminar flow of a power-law fluid in a circular tube. The analysis extends the filled-region concept, used previously to study Newtonian fluid flow, to the more general class of power-law fluids. Flow is analyzed in both the inlet and filled regions using an integral boundary layer method. Results obtained provide the lengths of the entrance, inlet and filled regions as a function of the generalized Reynolds number and the power-law fluid index. In addition, the variations of the local friction factor, the pressure drop and the centerline velocity along the axial coordinate are also provided. The available models are compared with the present one on the basis of experimental data. The present results are found to reach asymptotically the fully developed values, and also to be in good agreement with all available experimental data.     

A simplified couette flow solution of non-newtonian power-law fluids in eccentric annuli

Flow of non-Newtonian fluids in both the concentric and eccentric annuli is of great importance in extruders for molten plastics and wellbore fluid circulation for the removal of drilling cuttings. The steady laminar couette flow of non-Newtonian power-law fluids in eccentric annulus is employed in this study to analyze the problems of surge or swab pressures encountered when running or pulling tubular goods (pipes) in a liquid filled borehole. This is similar to the annular space created by two long co-axial cylinders with the inner cylinder in motion at a steady velocity, and a stationary outer cylinder. The solutions of the equations of motion are presented in both dimensionless form and as a family of curves for different pipe/borehole eccentricity ratios and power-law fluid index values for a more general application. The expected error in surge computation for concentric annulus as a result of eccentricity is evaluated.     

Computational analysis of the flow of pseudoplastic power-law fluids in a microchannel plate

The flow of pseudoplastic power-law fluids with different flow indexes at a microchannel plate was studied using computational fluid dynamic simulation.The velocity distribution along the microchannel plate and especially in the microchannel slits,flow pattern along the outlet arc and the pressure drop through the whole of microchannel plate were investigated at different power-law flow indexes.The results showed that the velocity profile in the microchannel slits for low flow index fluids was similar to the plug flow and had uniform pattern.Also the power-law fluids with lower flow indexes had lower stagnation zones near the outlet of the microchannel plate.The pressure drop through the microchannel plate showed huge differences between the fluids.The most interesting result was that the pressure drops for power-law fluids were very smaller than that of Newtonian fluids.In addition,the heat transfer of the fluids through the microchannel with different channel numbers in a wide range of Reynolds number was investigated.For power-law fluid with flow index (n =0.4),the Nusselt number increases continuously as the number of channels increases.The results highlight the potential use of using pseudoplastic fluids in the microheat exchangers which can lower the pressure drop and increase the heat transfer efflciency.     

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.     

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.