Drag on two coaxial rigid spheres moving along the axis of a cylinder filled with Carreau fluid

The boundary effect on the drag on two identical, rigid spheres moving along the axis of a long cylinder filled with a Carreau fluid for Reynolds number ranges from 0.1 to 40 is investigated. The influences of the key parameters of the problem under consideration, including the separation distance between two spheres, the relaxation time constant and the power-law index of a Carreau fluid, the Reynolds number, and the ratios (radius of sphere/radius of cylinder), on the drag acting on two spheres are investigated. We show that the boundary effect for the present case is more significant than that for the corresponding Newtonian fluid. The presence of the cylinder has the effect of enhancing the convective motion in the rear part of a sphere, thereby forming wakes and a reverse flow field, and this phenomenon is enhanced by the shear-thinning nature of a fluid. If the boundary effect is insignificant, the shear-thinning nature of a fluid has the effect of reducing the deviation of the ln(drag coefficient)-ln(Reynolds number) curve from a Stokes'-law-like relation. On the other hand, if it is significant, this deviation has a local minimum as the shear-thinning nature of a fluid varies.     

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.     

Drag on non-spherical particles in non-newtonian fluids

Extensive experimental results on the free fall of a range of non-spherical particles such as square bars, cylinders, spheres and crushed rock chips in Newtonian, inelastic, viscoelastic and Boger fluids are presented. It is demonstrated that the use of a volume equivalent sphere diameter in addition to a shape factor provides an adequate approximation for the non-sphericity of particles used in this study. The applicability of two rheological models, namely, the power-law and the Carreau viscosity model has been examined in representing the drag coefficient results. Appropriate predictive expressions of the drag coefficient as a function of the particle Reynolds number and the Deborah number, encompassing wide ranges of rheological and kinematic conditions, are presented.     

THE SLOW MOTION OF A SPHERICAL PARTICLE IN A CARREAU FLUID

The drag on a spherical particle is studied for two limiting cases, namely for the rigid sphere and for the bubble. An approximate solution is found for creeping flow around a particle suspended in a shear-thinning fluid. The three parameter Carreau model is used to represent the suspending liquid. The drag force on the particle for both cases is calculated by a perturbation method around the Newtonian solution in the limit of small Carreau number. The resulting expressions are found to be dependent on the Carreau number and on the power-law index.     

Wall effects in flow past a circular cylinder in a plane channel: a numerical study

This paper describes a numerical study on the steady flow of an incompressible Newtonian fluid past a circular cylinder confined in a plane rectangular channel. Using FLUENT (version 6), two-dimensional steady state computations were carried out for an uniform inlet velocity and for different values of the Reynolds numbers in the range between 0.1 and 200 and blockage ratios (ratio of the channel width to the cylinder diameter) in the range between 1.54 and 20. The flow parameters such as drag coefficient, length of the recirculation zone, and the angle of separation are presented as functions of the Reynolds number and blockage ratio. The total drag coefficient (C_D) was found to decrease with an increase in the blockage ratio (λ) for a fixed value of the Reynolds number (Re) and to decrease with increasing Reynolds number for a fixed value of λ. Similarly, for a fixed value of λ, both the angle of separation and the length of the recirculation zone increase with the increasing Reynolds number.     

大长径比细长颗粒的沉降实验和曳力系数的关联

引　言对球状颗粒的研究已取得丰硕的成果[1] ,但对其他形状颗粒的研究则较少 ,特别是细长颗粒 .形状的特殊性使细长颗粒的曳力系数与球状颗粒明显不同 ,仅有的文献[2～ 4] 将研究重点放在静止液体中颗粒沉降时的取向和终端沉降速度 ,且一般采用球形度表示曳力系数 ,颗粒长径比     

MOTION OF AND MASS TRANSFER FROM AN ASSEMBLAGE OF SOLID SPHERES MOVING IN A NON-NEWTONIAN FLUID AT HIGH REYNOLDS NUMBERS

An approximate solution for the motion of an assemblage of solid spheres moving in a power-law fluid in the high Reynolds number region is obtained using a combination of Happel's free-surface cell model and the boundary layer theory. It is theoretically predicted that the drag coefficient will decrease with the increase of the shear-thinning anomaly. The results of the present analysis are in reasonably good agreement with the available experimental data for fixed and fluidized beds. The influence of the non-Newtonian behavior on the mass transfer rate from an assemblage of solid spheres is also discussed.     

MOTION OF AND MASS TRANSFER FROM AN ASSEMBLAGE OF SOLID SPHERES MOVING IN A NON-NEWTONIAN FLUID AT HIGH REYNOLDS NUMBERS

An approximate solution for the motion of an assemblage of solid spheres moving in a power-law fluid in the high Reynolds number region is obtained using a combination of Happel's free-surface cell model and the boundary layer theory. It is theoretically predicted that the drag coefficient will decrease with the increase of the shear-thinning anomaly. The results of the present analysis are in reasonably good agreement with the available experimental data for fixed and fluidized beds. The influence of the non-Newtonian behavior on the mass transfer rate from an assemblage of solid spheres is also discussed.     

低粘附功管壁的内流减阻研究

以乙烯基三乙氧基硅烷 无水乙醚溶液处理玻璃管道内壁 ,降低流体水与管壁间的粘附功 ,研究了低粘附功内壁管道的内流减阻性能。结果表明 ,在管壁与液体间的粘附功小于液体的内聚功条件下 ,当雷诺数大于临界雷诺数时 ,低粘附功内壁管道呈现明显的内流减阻效果 ,临界雷诺数的值与一定的临界壁面剪力值或层流底层临界厚度值相对应 ;在减阻区内 ,减阻率随雷诺数的增加及管道内径的减小而增大 ,阻力系数的无因次关联式可表示为λ=f (Re ,X) ,其中X为无因次数群 [W / (du2 ρ) ]。     

Sedimentation of a cylindrical particle along the axis of a cylindrical tube filled with Carreau fluid

The boundary effect on the movement of a particle in a Carreau fluid is investigated theoretically by considering the sedimentation of a cylindrical particle along the axis of a cylindrical tube. The influences of the key parameters of the system under consideration on the drag coefficient and the associated flow field are discussed. These include the relaxation time constant and the power-law index of a Carreau fluid, the length of a particle, and the diameter of a cylindrical tube. We show that the flow field and the drag coefficient are affected more significantly by the boundary effect, measured by the ratio (particle diameter/tube diameter) than by the size of a particle and the properties of the fluid. In general, the terminal velocity of a particle correlates nonlinearly with the ratio (particle diameter/tube diameter). The problem of a particle in an unbounded fluid can be recovered as a limiting case of the present one.     

Experimental measurements of the effect of viscosity on drag for liquid drops

The drag coefficients of drops of various liquids falling in air were measured experimentally. The drag coefficient was linearly related to the viscosity in the Reynolds number and viscosity range measured. Measurements also suggested there is no difference between Newtonian and non-Newtonian liquids.     

Electrophoresis of a non-conducting Newtonian drop of low electrical potential normal to a plane

The electrophoretic behavior of a non-conducting, Newtonian drop of low surface potential normal to a plane is investigated theoretically under the conditions of weak applied electric field and arbitrary thickness of double layer. The governing equations and the associated boundary conditions expressed in terms of bipolar spherical coordinates are solved by an orthogonal collocation method. In general, the thinner the double layer surrounding a drop and/or the longer its distance from a planar surface, the larger its mobility, and if a drop is sufficiently close to a plane, its mobility may change sign. These results are similar to the case of a rigid particle. The mobility of a drop decreases with the increase in the ratio (viscosity of drop fluid/viscosity of dispersion medium). Under the conditions assumed, a drop can be treated as a rigid sphere if the viscosity ratio exceeds about 100, and as a bubble if it is smaller than about 0.01.     

Drag on ensembles of fluid spheres translating in a power-law liquid at moderate Reynolds numbers

This work elucidates the role of power-law rheology on the sedimentation velocity of an ensemble of mono-size spherical Newtonian droplets (free from surfactants) translating in a power-law continuous phase numerically by solving the momentum equations of both phases. A simple sphere-in-sphere cell model has been used to account for inter-drop interactions. In particular, in this study, the effects of the Reynolds number (Re_o), the internal to external fluid characteristic viscosity ratio (k), the volume fraction of the dispersed phase (Φ) and the power-law index of the continuous phase (n_o) on the external flow field, pressure drag (C_dp), friction drag (C_df) and total drag (C_d) coefficients have been analyzed over wide ranges of parameters as follows: 1 ≤ Re_o ≤ 200, 0.1 ≤ k ≤ 50, 0.2 ≤ Φ ≤ 0.6 and 0.6 ≤ n_o ≤ 1.6. Based on the extensive numerical results obtained in this work, a simple predictive correlation has been proposed for the total drag coefficient, which can be used to predict the rate of sedimentation of ensembles of Newtonian fluid spheres in power-law liquids in a new application.     

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.     

Power‐Law Fluid Flow Over a Sphere: Average Shear Rate and Drag Coefficient

Based on the consideration of the rate of mechanical energy dissipation, an expression for the average shear rate for a sphere falling in a power‐law fluid in the creeping flow regime has been deduced. The average shear rate in a power‐law fluid (n<1) appears to be higher than that in an equivalent Newtonian fluid. This in turn has been combined with the numerical predictions of drag coefficient (up to Reynolds number of 100) of a sphere to develop a generalized drag correlation for power‐law liquids encompassing both n > 1 and n < 1 which appears to apply up to much higher values of the Reynolds number. The available experimental data have been used to demonstrate the reliability and accuracy of the new correlation for shearthinning liquids. Also, in the limit of n = 1, this expression reproduces the standard drag curve with a very high accuracy.     

Effect of blockage on drag and heat transfer from a single sphere and an in-line array of three spheres

The effect of blockage ratio on the steady flow and heat transfer characteristics of incompressible fluid over a sphere and an in-line array of three spheres placed at the axis of a tube has been investigated numerically. The Navier-Stokes and thermal energy equations have been solved numerically using FLUENT for the following ranges of parameters: for a single sphere, 2 ≤ β ≤ 10; 1 ≤ Re ≤ 100; for the three-sphere system, for two values of sphere-to-sphere distance, namely s = 2 and 4. All computations were carried out for two values of the Prandtl number, i.e., 0.74 and 7, corresponding to the flow of air and water respectively. Extensive results on streamline patterns, wake characteristics (angle of separation and recirculation length), drag coefficient and Nusselt number are presented to elucidate the interplay between the blockage and the Reynolds number and their influence on drag and Nusselt number.     

Hydrodynamic drag forces on two porous spheres moving along their centerline

This paper numerically evaluates the hydrodynamic drag force exerted on two highly porous spheres moving steadily along their centerline (sphere #1 and sphere #2) through a quiescent Newtonian fluid over a Reynolds number ranging from 0.1 to 40. At creeping flow limit, the drag forces exerted on both spheres were identical. At higher Reynolds numbers the drag force on sphere #1 was higher than sphere #2, revealing the shading effects produced by sphere #1 on sphere #2. At dimensionless diameter (β, =d_f/2k^0.5, d_f and k are floc diameter and interior permeability, respectively) >20, the spheres can be regarded nonporous. At β<20, the drag forces dropped. At β<2, the drag forces approached “no-spheres” limit. An increased size ratio of two spheres (d_f1/d_f2) would increase the drag force on sphere #1 and reduce that on sphere #2. At increasing β for both spheres, the drag force on sphere #2 was increased because of the more difficult advective flow through its interior, and at the same time the drag was reduced owing to the stronger wake flow produced by the denser sphere #1. The competition between these two effects leads to complicated dependence of drag force on sphere #2 on β value. These effects were minimal when β became low. Two identical spheres could move steadily along their centerline. At higher Reynolds number, the two spheres would move closer because of the incorporation of inertia force. For spheres of different diameters, the sphere # 2 would move faster than sphere #1 regardless of their size ratio and β value. This occurrence yielded efficient coagulation when two porous spheres were moving in-line.     

SLOW NON-NEWTONIAN FLOW PAST AN ASSEMBLAGE OF RIGID SPHERES

Previous work on slow flow of non-Newtonian fluids past particles assemblages has been reviewed. Using a combination of Happel's free surface model and variational principles, bounds on the drag have been obtained for the creeping flow of a Carreau Model fluid past an assemblage of rigid spheres. The bounds are related to friction factor for flow through fixed beds of spherical particles. Numerical results covering a wide range of model parameters and bed voidages are presented.

Theoretical predictions are validated by comparing with experimental results reported in the literature that involve viscoelastic fluids. Arithmetic averages of the two bounds compare well for 182 data points with an average error of 12%. It is demonstrated that the present analysis, though based on a purely viscous model, can predict creeping flow behaviour in rigid particles assemblage for both inelastic and viscoelastic fluids.     

Lift and drag forces on a sphere attached to a wall in a Blasius boundary layer

Lift and drag forces on a sphere attached to a planar wall, over which a laminar flat plat boundary layer flows, are examined numerically in this study. Particle Reynolds number ranged from 0.1–250, which represents steady, laminar flow about the sphere, and the plate Reynolds number was held constant at 32 400. A finite-volume computational fluid dynamics program was utilised. Simulation results were validated against analytical results for drag and lift in creeping flow and against experimental results available in the literature for lift at higher particle Reynolds number. The model results were curve-fitted and interpolating drag and lift coefficient functions are reported. The lift and drag results are shown to be weakly dependent upon plate Reynolds number. The resulting correlations are expected to be useful in the development of particle impending motion and aerosol entrainment predictions of particles adhering to planar walls.     

A STUDY OF SPINLINE DYNAMICS USING ORTHOGONAL COLLOCATION

Previous work on slow flow of non-Newtonian fluids past particles assemblages has been reviewed. Using a combination of Happel's free surface model and variational principles, bounds on the drag have been obtained for the creeping flow of a Carreau Model fluid past an assemblage of rigid spheres. The bounds are related to friction factor for flow through fixed beds of spherical particles. Numerical results covering a wide range of model parameters and bed voidages are presented.

Theoretical predictions are validated by comparing with experimental results reported in the literature that involve viscoelastic fluids. Arithmetic averages of the two bounds compare well for 182 data points with an average error of 12%. It is demonstrated that the present analysis, though based on a purely viscous model, can predict creeping flow behaviour in rigid particles assemblage for both inelastic and viscoelastic fluids.