Flow pattern variations of viscoelastic fluid flows in three‐dimensional branching channel

Transport phenomena in three‐dimensional branching channel are important because of their relevance in polymer processing. In this article, an experimental study on viscoelastic flow in a three‐dimensional cylindrical branching channel is carried out to investigate variations of flow pattern. Flow visualization in representative symmetric planes is made both for the viscoelastic fluid and Newtonian flow. From the results of the present investigation, the flow field in the three‐dimensional cylindrical branching channel is clarified within the range of laminar flow. It is confirmed that corner vortex, shedding vortex, and secondary vortex flow are all obviously changed with the fluid concentration and the Reynolds number, which are much more three‐dimensional and complex than the Newtonian fluid, and the flow pattern of the viscoelastic fliud flow largely depends on the Reynolds number and fluid concentration. Even for the viscoelastic flow at the low Reynolds number, shedding vortex and secondary vortex and complex three‐dimensional flow occur in the cylinder. The flow field is not symmetric space for the viscoelastic flow and however is fairly symmetric for the Newtonian fluid. The above reasons explain why the flow deflection happens even at the low Reynolds number flow. POLYM. ENG. SCI., 2010. © 2009 Society of Plastics Engineers     

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.     

Motion of a sphere parallel to plane walls in a Poiseuille flow. Application to field-flow fractionation and hydrodynamic chromatography

The motion of a solid spherical particle entrained in a Poiseuille flow between parallel plane walls has various applications to separation methods, like field-flow fractionation and hydrodynamic chromatography. Various handy formulae are presented here to describe the particle motion, with these applications in mind. Based on the assumption of a low Reynolds number, the multipole expansion method coupled to a Cartesian representation is applied to provide accurate results for various friction factors in the motion of a solid spherical particle embedded in a viscous fluid between parallel planes. Accurate results for the velocity of a freely moving solid spherical particle are then obtained. These data are fitted so as to obtain handy formulae, providing e.g. the velocity of the freely moving sphere with a 1% error. For cases where the interaction with a single wall is sufficient, simpler fitting formulae are proposed, based on earlier results using the bispherical coordinates method. It appears that the formulae considering only the interaction with a nearest wall are applicable for a surprisingly wide range of particle positions and channel widths. As an example of application, it is shown how in hydrodynamic chromatography earlier models ignoring the particle-wall hydrodynamic interactions fail to predict the proper choice of channel width for a selective separation. The presented formulae may also be used for modeling the transport of macromolecular or colloidal objects in microfluidic systems.     

Interaction of two touching spheres in a viscous fluid

The nature and effects of contacts between suspended particles were studied through a process in which a heavy sphere falls past a light sphere in a viscous fluid at low Reynolds number. Teflon and nylon spheres were used for the heavy and light spheres, respectively, with natural surface roughness and with the nylon sphere artificially roughened. Because of the existence of microscopic roughness on the sphere surfaces, the particles are able to make physical contact, breaking the symmetry of the trajectory predicted by hydrodynamic theory for smooth spheres. The experimental results are compared with numerical results calculated according to the theory of Davis (Phys. Fluids A 4 (1992) 2607), with a particular focus on the rotational velocities of the spheres. The numerical results from the roll/slip model provide the best fit of the experimental data. Instead of locking together like a rigid body and rotating together, two spheres initially roll without slipping and then roll with slipping after the maximum friction force is reached.     

Hydrodynamic and transport properties of packed beds in small tube-to-sphere diameter ratio: pore scale simulation using an Eulerian and a Lagrangian approach

In fixed bed catalytic reactors radial heterogeneities of the granular structure are present owing to topologic constraints imposed by the reactor wall. In order to analyse the influence of this structure on the fluid flow and the radial mass transfer properties, the study of sphere packings in cylindrical container and flow simulations at the pore scale are carried out. A collocated finite volume is used to solve the 3D Navier-Stokes equations. The Reynolds number ranges from 7 to 200 allowing to use the direct numerical simulation method in stationary flow regime combined to the no-slip condition at the interface solid/fluid. Therefore no correlation are used in this study. Furthermore, representative fixed beds, composed of several hundreds of spheres, are used for a diameter ratio of 5.96 and 7.8. Radial profile of the longitudinal velocity and the probability density function of the velocity components agree with the experimental data found in the literature. At low Reynolds number, the computation of current lines reveals the presence, at the reactor wall, of a layer whose width is around one-fourth of the sphere diameter. In this layer, the fluid flow is longitudinal and tangential. Particle tracking reveals also the existence of a second layer all along the spheres in contact with the reactor wall. Mass transfer in these two regions is controlled by the diffusive mechanism at low Reynolds number. The flow structure at high Reynolds number contains lots of eddies distributed homogeneously in the fixed beds. These structures are not recirculating zones (particle traps). On contrary, they accelerate the radial mass transfer so that the layers found at low Reynolds number tend to disappear at high Reynolds.     

Mass transfer from a contaminated fluid sphere

The unsteady mass transfer from a contaminated fluid sphere moving in an unbounded fluid is examined numerically for unsteady‐state transfer. The effect of the interface contamination and the flow regime on the concentration profiles, inside and outside a fluid sphere, is investigated for different ranges of Reynolds number (0 < Re < 200) and Peclet number (0 < Pe < 10⁵), viscosity ratio between the dispersed phase and the continuous phase (0 < κ < 10), and the stagnant‐cap angle (0° < θ_cap < 180°). It was found that the stagnant‐cap angle significantly influences the mass transfer from the sphere to a surrounding medium. For all Peclet and Reynolds numbers and κ, the contamination reduces the mass transfer flux. The average Sherwood number increases with an increase of stagnant‐cap angle and reaches a maximum equal to the average one for a clean fluid sphere at low viscosity ratio and large Peclet numbers. A predictive equation for the Sherwood number is derived from these numerical results. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

Effect of the viscosity ratio on mass transfer from a fluid sphere at low to very high Peclet numbers

The effect of the viscosity ratio on mass transfer from a fluid sphere is examined in this paper. Numerical solutions of the Navier-Stokes equations off motion and the equations of mass transfer have been obtained for the unsteady state transfer from a fluid sphere moving in an unbounded fluid medium of different viscosity. The effects of the viscosity ratio and the flow on the concentration profiles were investigated for Reynolds number, viscosity ratio and Péclet number ranges of 0?Re?400, 0?κ?1000 and , respectively. The local and average Sherwood numbers are also presented graphically. The steady state results show that the average Sherwood number is increasing as Peclet number increases for a fixed viscosity ratio. However, for a fixed Peclet number, the average Sherwood number is decreasing as the viscosity ratio increases and reaches a limit value corresponding to the average Sherwood number for a solid spherical particle. From the numerical results, a predictive equation for the Sherwood number in terms of the Peclet number, the Reynolds number and the viscosity ratio is derived.     

Numerical Investigations of Fluid Flow and Lateral Fluid Dispersion in Bounded Granular Beds in a Cylindrical Coordinates System

Results are presented from a numerical study examining the flow of a viscous, incompressible fluid through a random packing of non‐overlapping spheres at moderate Reynolds numbers, spanning a wide range of flow conditions for porous media. By using a laminar model including inertial terms and assuming rough walls, numerical solutions of the Navier‐Stokes equations in three‐dimensional porous packed beds resulted in dimensionless pressure drops in excellent agreement with those reported in a previous study. This observation suggests that no transition to turbulence could occur in the range of the Reynolds number studied. For flows in the Forchheimer regime, numerical results are presented of the lateral dispersivity of solute continuously injected into a three‐dimensional bounded granular bed at moderate Peclet numbers. In addition to numerical calculations, to describe the concentration profile of solute, an approximate solution for the mass transport equation in a bounded granular bed in a cylindrical coordinates system is proposed. Lateral fluid dispersion coefficients are then calculated by fitting the concentration profiles obtained from numerical and analytical methods. Comparing the present numerical results with data available in the literature, no evidence has been found to support the speculations by others for a transition from laminar to turbulent regimes in porous media at a critical Reynolds number.     

Creeping motions of a composite sphere in a concentric spherical cavity

An analytical study is presented for the quasisteady translation and steady rotation of a spherically symmetric composite particle composed of a solid core and a surrounding porous shell located at the center of a spherical cavity filled with an incompressible Newtonian fluid. In the fluid-permeable porous shell, idealized hydrodynamic frictional segments are assumed to distribute uniformly. In the limit of small Reynolds number, the Stokes and Brinkman equations are solved for the flow field of the system, and the hydrodynamic drag force and torque exerted by the fluid on the particle which is proportional to the translational and angular velocities, respectively, are obtained in closed forms. For a given geometry, the normalized wall-corrected translational and rotational mobilities of the particle decrease monotonically with a decrease in the permeability of its porous shell. The boundary effects of the cavity wall on the creeping motions of a composite sphere can be quite significant in appropriate situations. In the limiting cases, the analytical solutions describing the drag force and torque or mobilities for a composite sphere in the cavity reduce to those for a solid sphere and for a porous sphere.     

Direct simulations of spherical particle motion in Bingham liquids

The present work deals with the development of a direct simulation strategy for solving the motion of spherical particles in a Bingham liquid. The simulating strategy is based on a lattice-Boltzmann flow solver and the dual-viscosity Bingham model. Validation of the strategy is first performed for single phase (lid-driven cavity flow) and then for two phase flows. Lid-driven cavity flow results illustrate the flow's response to an increase of the yield stress. We show how the settling velocity of a single sphere sedimenting in a Bingham liquid is influenced by the yield stress of the liquid. The hydrodynamic interactions between two spheres are studied at low and moderate Reynolds number. At low Reynolds number, two spheres settle with equal velocity. At moderate Reynolds number, the yield effects are softened and the trailing sphere approaches the leading sphere until collision occurs.     

Numerical study on the hydrodynamics of agglomerates at intermediate Reynolds numbers

The flow pattern and hydrodynamics of a heterogeneous permeable agglomerate in a uniform upward flow at intermediate Reynolds numbers(1–40) are analyzed from three-dimensional(3 D) computational fluid dynamics simulations. Different from the homogeneous or stepwise-varying permeability models used in previous papers, a continuously radially varying permeability model is used in the present study. The effects of two dimensionless parameters, the Reynolds number and the permeability ratio, on the flow field and the hydrodynamics were investigated in detail. The results reveal that unlike the solid sphere, a small recirculating wake initially forms inside the agglomerate. The critical Reynolds number for the formation of the recirculating wake is lower than that of the solid sphere and it decreases with the increase of permeability ratio. A correlation of drag coefficient as a function of the Reynolds number and permeability ratio is proposed. Comparisons of drag coefficients obtained by different permeability models show that at intermediate Reynolds numbers(1–40),the effect of radially varying permeability on the drag coefficient must be considered.     

The heat and mass transfer from a fluid sphere at large reynolds and peclet number

Analytic expressions have been derived for the steady rate of heat or mass transfer from a fluid sphere in uniform motion at large Reynolds and Peclet number. The solution is applicable to cases where both phase resistances are of the same order of magnitude, or when the resistance of one phase is negligible. The analysis has shown that if finite values of fluid viscosity and density are considered the transfer rates are considerably less than those obtained from the ideal fluid model. The maximum flux for a fluid sphere occurs near the equatorial plane and in this respect its behavior differs significantly from a solid sphere.     

Numerical Simulation for Particle Penetration Depth Distribution in Deep Bed Filtration

A two‐dimensional model has been developed to simulate particle penetration through porous media. The particle penetration depends on many parameters including the Reynolds number, particle drag coefficient, the ratio of the diameter of injected to filtered particles, fluid velocity, and pore size, etc. The numerical model for separation efficiency in periodic porous media was studied. Previous work has described the effects of injected particle size, Reynolds number and particle drag coefficient. In this study, the porous media flow is modeled (solution of the Navier‐Stokes equations) by using the finite element method, and the analysis is restricted to the case of two‐dimensional periodic porous media. The effects of these factors and particle depth distribution in porous media are investigated. It is noted that the results for the three Reynolds numbers 1, 16.56, and 100, are qualitatively similar, and about 40 % of particles are trapped in the top part of the filter.     

Numerical study on steady and transient mass/heat transfer involving a liquid sphere in simple shear creeping flow

A numerical method is utilized to examine the steady and transient mass/heat transfer processes that involve a neutrally buoyant liquid sphere suspended in simple shear flow at low Reynolds numbers is described. By making use of the known Stokes velocity field, the convection‐diffusion equations are solved in the three‐dimensional spherical coordinates system. For the mass transfer either outside or inside a liquid sphere, Sherwood number Sh approaches an asymptotic value for a given viscosity ratio at sufficiently high Peclet number Pe. In terms of the numerical results obtained in this work, two new correlations are derived to predict Sh at finite Pe for various viscosity ratios. © 2013 American Institute of Chemical Engineers AIChE J, 60: 343–352, 2014     

微纳米颗粒受自然对流影响运动沉积特性

引言固体颗粒在流体内的运动及沉积是大气环境、河床、水文地质等自然界中普遍存在的现象,也在煤燃烧、化工制药、建筑、冶金等许多工业领域中广泛存在,因此,很早就受到关注[1-3]。一般情况下,对不同颗粒度的颗粒运动与沉积应有不同的研究方法,如微纳米颗粒可能需要考虑布朗力,而大颗粒可以忽略,大颗粒可能需要考虑颗粒的形态,     

Simulations of solid–liquid scalar transfer for a spherical particle in laminar and turbulent flow

Scalar transfer from a solid sphere to a surrounding liquid has been studied numerically. The simulation procedure involves full hydrodynamic resolution of the solid–liquid interaction and the flow (laminar and turbulent) of the carrier fluid by means of the lattice‐Boltzmann method. Scalar transport is solved with a finite volume method on coupled overlapping domains (COD): an outer domain discretized with a cubic grid and a shell around the solid sphere with a spherical grid with fine spacing in the radial direction. The shell is needed given the thin scalar boundary layer around the sphere that is the result of high Schmidt numbers (up to Sc = 1000). After assessing the COD approach for laminar benchmark cases, it is applied to a sphere moving through homogeneous isotropic turbulence with the sphere radius larger (by typically a factor of 10) than the Kolmogorov length scale so that it experiences an inhomogeneous hydrodynamic environment. This translates in pronounced scalar concentration variations and transfer rates over the sphere's surface. Overall scalar‐transfer coefficients are compared to those derived from classical Sherwood number correlations. © 2014 American Institute of Chemical Engineers AIChE J, 60: 1202–1215, 2014     

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.     

Numerical simulation of steady flow past a liquid sphere immersed in simple shear flow at low and moderate Re

This work presents a numerical investigation on steady internal, external and surface flows of a liquid sphere im-mersed in a simple shear flow at low and intermediate Reynolds numbers. The control vol...     

On the unsteady forces during the motion of a sedimenting particle

We consider the unsteady motion of a sedimenting rigid spherical particle in order to examine the relative strength of the hydrodynamical forces acting on particles in fluid flows. The relative strength of the forces on all stages of the particle motion is a major concern for closing constitutive equations describing the more complex motion of particulate flows such as fluidised beds. The formulation results in a first order nonlinear integro-differential equation in terms of the instantaneous velocity of the sphere. This equation is made dimensionless and the particle Reynolds number and the fluid-particle density ratio are identified as the relevant physical parameters describing the particle motion. We obtain analytical solutions for the limits of small density ratios and small Reynolds number. In addition, a numerical solution is used for arbitrary values of the density ratio. The results show that the motion of spherical particles is significantly affected by the unsteady drag dominated by the memory Basset force on the early stages of the motion and on the approach to the steady state (terminal velocity). The present calculations indicate that the unsteady hydrodynamic drags might become of the same order of magnitude of the dominant viscous drag for flows with moderate particle-fluid density ratio. Therefore, unsteady drags should be taken into account on modelling multiphase particulate flows with moderate density ratio.     

Drag on two co-axial rigid spheres moving normal to a plane: Newtonian and Carreau fluids

The boundary effect and the presence of a nearby entity on the drag of a rigid entity is investigated by considering the movement of two identical, rigid, coaxial spheres normal to a plane in both a Newtonian and a Carreau fluid at a low to medium large Reynolds number. The parameters key to the phenomenon under consideration, including the nature of the fluid, the separation distance between two spheres, the distance between the near sphere and the plane, and the Reynolds number, on the drag coefficient are discussed. We show that the influence of a boundary on the drag coefficient is more important than that of the nature of a fluid and that of the separation distance between two spheres. The variation of the drag coefficient as a function of Reynolds number for a Carreau fluid is similar to that for a Newtonian fluid. Due to the shear-thinning nature of the former the drag coefficient in the former is smaller than that in the latter. The influence of the index parameter of a Carreau fluid becomes appreciable only if the Carreau number is sufficiently large. Correlations between the drag coefficient and the key parameters of a system are developed for the case when the Reynolds number is smaller than l.