Low-Reynolds-number motion of a heavy sphere between two parallel plane walls

A novel boundary-integral algorithm [Staben, M.E., Zinchenko, A.Z., Davis, R.H., 2003. Motion of a particle between two parallel plane walls in low-Reynolds-number Poiseuille flow. Physics of Fluids 15, 1711-1733; Erratum: Phys. Fluids 16, 4206] is used to obtain O(1)-nonsingular terms that are combined with two-wall lubrication asymptotic terms to give resistance coefficients for near-contact or contact motion of a heavy sphere translating and rotating between two parallel plane walls in a Poiseuille flow. These resistance coefficients are used to describe the sphere's motion for two cases: a heavy sphere driven by a Poiseuille flow in a horizontal channel and a heavy sphere settling due to gravity through a quiescent fluid in an inclined channel. When the heavy sphere contacts a wall in either system, which occurs when the gap between the sphere and the wall becomes equal to the surface roughness of the sphere (or plane), a contact-force model using the two-wall resistance coefficients is employed. For a heavy sphere in a Poiseuille flow, experiments were performed using polystyrene particles with diameters 10%-60% of the channel depth, driven through a glass microchannel using a syringe pump. The measured translational velocities for these particles show good agreement with theoretical results. The predicted translational velocity increases for increasing particle diameter, as the spheres extend further into the Poiseuille flow, except for particles that are so large (diameters of 80%-85% of the channel depth) that the upper wall has a dominant influence on the particle velocity. For a heavy sphere settling in a quiescent fluid in an inclined channel, the transition from the no-slip regime to slipping motion occurs for a larger inclination angle of the channel with respect to the horizontal for an increase in particle diameter, since the larger particles are more slowed by the second wall. Limited experiments were performed for Teflon spheres with diameters 64%-95% of the channel depth settling in a very viscous fluid along the lower wall of an inclined acrylic channel. The measured translational velocities, which are only about 15%-25% of the tangential component of the undisturbed Stokes settling velocity, are in close agreement with theory using physical parameters obtained from similar experiments with a single wall [Galvin, K.P., Zhao, Y., Davis, R.H., 2001. Time-averaged hydrodynamic roughness of a noncolloidal sphere in low Reynolds number motion down an inclined plane. Physics of Fluids 13, 3108-3119].     

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.     

The flow of suspensions through tubes: V. Inertial effects

The behavior of particles undergoing Couette and Poiseuille flows at rates when inertial effects become significant was investigated. The rotation of rigid particles was similar to that in the Stokes flow regime, except for a drift of cylinders to limiting rotational orbits corresponding to the maximum energy dissipation. In Poiseuille flow, rigid particles migrated to an equilibrium radial position which depended on the density difference of two phases, the directions of sedimentation velocity and flow, and the ratio of particle to tube radius. Neutrally buoyant deformable particles always migrated to the tube axis. In concentrated suspensions a plasmatic layer developed near the tube wall as a consequence of radial migration. The formation of this layer modified the velocity profile and caused a reduction in the apparent viscosity coefficient.     

Lateral migration of spherical particles in porous flow channels: application to membrane filtration

Lateral migration of spherical rigid neutrally buoyant particles moving in a laminar flow field in a porous channel is induced by an inertial lift force (tubular-pinch effect) and by a permeation drag force due to convection into the porous walls. The analysis of Cox and Brenner [7], for the particle motion in a nonporous duct is extended to include the effect of the wall porosity. Criteria are established under which the inertial and permeation drag force in the lateral direction can be vectorially added. Particle trajectories and concentrations profiles are calculated for a plane Poiseuille flow with one porous wall. For particles with radius of 1 μm, inertial and permeation drag forces are of comparable size under flow conditions often met in ultra- and hyperfiltration of dilute suspensions. For smaller particles the permeation drag force dominates.     

Investigation of the Effect of a Dividing Wall in a Moving Bed

Moving beds are widely used in many scientific fields due to their advantages, but their operation is affected by the phenomena of cavity and pinning, which should be avoided. In this article, a perforated dividing wall was located in a two‐dimensional rectangular moving bed with gas cross‐flow and the effect of the wall on some properties of the moving bed was investigated. Experimental results show that the location of the perforated dividing wall can improve the upper limit of the gas velocity of the moving bed since it prohibits the existence of a cavity and pinning. The dividing wall separates the apparatus into two channels. A cavity forms in the channel close to the upstream face while pinning occurs in the other channel. In comparison to a moving bed without a dividing wall, the cavity size and pinning width in a moving bed with a dividing wall are smaller at the same gas velocity.     

EXTRUSION AND LUBRICATION FLOWS OF VISCOPLASTIC FLUIDS WITH WALL SLIP

The simplest model flow which approximates the extrusion (shallow screw channels) and lubrication flow is the steady, laminar flow occurring between two infinitely long parallel plates i.e., the generalized plane Couette flow. Here we develop an analytical model of the generalized plane Couette flow of viscoplastic fluids. The deformation and flow behavior of viscoplastic fluids can be realistically represented with the Herschel-Bulkley constitutive equation, which we have utilized as the basis for the development of our analytical model. Furthermore, as also demonstrated here, the deformation behavior of viscoplastic fluids is generally complicated by the presence of wall slip at solid walls, which occurs as a function of the wall shear stress. The wall slip versus the wall shear stress behavior of viscoplastic fluids can be experimentally characterized using viscomelric flows, including steady torsional and capillary flows. Thus determined Navier's wall slip coefficient can then be utilized in modeling of processing flows. In our analytical model of the generalized plane Couette flow of viscoplastic fluids the Navier's wall slip boundary condition was included. This model should be an important engineering tool, which provides design expressions for the extrusion and lubrication flows of viscoplastic fluids, with or without wall slip occurring at the walls. @KEYWORDS:Extrusion, lubrication, flow, viscoplastic, slip.     

SHEAR AND INTERFACIAL INSTABILITIES OF OIL-WATER FLOW IN AN INCLINED CHANNEL

A study of the linear stability of a laminar flow of an oil-water system in an inclined channel is presented. A novel shear-mode of instability, which is necessarily decaying in plane Poiseuille flow, is found to be the primary instability in certain situations. When the channel is sufficiently inclined, the long-wave mode can become unstable, regardless of the total volumetric flow rate of the fluids The consequences to oil transport are discussed.     

VOLUMETRIC FLUX RATE ENHANCEMENT AND REDUCTION IN CONICAL VISCOUS FLOWS WITH MULTIPLY CONNECTED CROSS SECTIONS

Axisymmetric viscous flow in a conical channel is considered for geometrical systems in which the cross section is multiply-connected. The volumetric flux rates are compared for a 2-flow region and a 1 -flow region of equal cross sectional area, and same mean radial pressure gradient. It is found that for various configurations of the 2-flow system there can be both flux rate enhancement and reduction. As part of an introduction some similar results are presented for Poiseuille flow, and flow between non parallel plane walls the former of which satisfies the exact Navier-Stokes equations.     

Slot die coating window for a uniform fuel cell ink dispersion

The continuous roll-to-roll slot die coating technique has shown great potential for fuel cell electrode fabrication. It is essential to determine the particulate coating window to obtain a defect-free film of uniform thickness and uniform particle dispersion. For this dual-purpose, an additional upper operating limit has been proposed via a “flow field analysis scheme” that combines analysis and simulation. In this analysis, the fundamentals of flow pattern transition (Poiseuille–Couette–bubbly flow) behind coating limits are clarified. The Couette flow is advantageous for uniform shear rate and dispersion. Furthermore, phase-field simulations systematically investigate the effect of fluid and geometrical parameters and quantitatively present the flow limit with explicit expression. Results reveal that the slope of the upper limit is inversely correlated with shear-thinning index n while proportional to the coating gap H. Combining the bubble entrainment boundary, lower n and H are favorable for a larger particulate coating window.     

Eccentric laminar couette flow of long cylindrical capsules

The equation of motion for laminar Couette flow of an incompressible Newtonian fluid, induced by the eccentric longitudinal motion of a long circular cylinder in a circular pipe, is solved analytically. Expressions are obtained and typical plots presented for local velocity distribution, local shear stress distributions on both the surface of the pipe wall and the surface of the cylindrical core, total drag force on both pipe wall and moving core, twisting moment on the core, and volumetric flow rate of the liquid. As the eccentricity approaches unity, the solution for flow rate is shown to coincide exactly with a practical formula developed by Kruyer and Ellis to represent the Couette flow induced by the motion of cylindrical capsules.     

A novel unified wall slip model for Poiseuille flow of polymer melt in the circular tube

In this study, to better reflect the slip effect of Poiseuille flow for polymer melt extruded through a circular tube, a novel unified wall slip model and flow equation based on two phase fluid system were proposed via a purely phenomenological approach. According to the different combinations of boundary conditions and flow parameters, the novel slip model was transformed into other models, such as adsorption–desorption model, entanglement–disentanglement model, lubrication layer model, Z–W model, and no‐slip model. The numerical simulation based on computed fluid dynamics was performed to verify the feasibility of the novel slip model. In the simulations, the radial flow velocity profile, shear rate, and viscosity distribution were obtained for six different models. Moreover, the effect of different slip coefficient combinations for the novel slip model on the radial flow velocity, slip velocity, volumetric flow rate error, and viscosity distribution of melt were also investigated and discussed. Results showed that the novel unified slip model not only incorporated the characteristics of other five models above mentioned, but also well interpreted the reason of simultaneously occurring the sharkskin surface defect and gross melt fracture phenomenon when flow rate of melt was extremely large. POLYM. ENG. SCI., 56:328–341, 2016. © 2015 Society of Plastics Engineers     

二维环形Couette设备中剪切引起的二维圆形固粒迁移的动态数值模拟

The shear-induced migration of neutrally-buoyant non-colloidal circular particles in a two-dimensional circular Couette flow is investigated numerically with a distributed Lagrange multiplier based fictitious domain method.The effects of inertia and volume fraction on the particle migration are examined.The results indicate that inertia has a negative effect on the particle migration.In consistence with the experimental observations,the rapid migration of particles near the inner cylinder at the early stage is observed in the simulation,which is believed to be related to the chain-like clustering of particles.The migration of circular particles in a plane Poiseuille flow is also examined in order to further confirm the effect of such clustering on the particle migration at early stage.There is tendency for the particles in the vicinity of outer cylinder in the Couette device to pack into concentric rings at late stage in case of high particle concentration.     

A mechanism for extrusion instabilities in polymer melts

A mechanism for explaining some of the instabilities observed during the extrusion of polymer melts is further explored. This is based on the combination of non-monotonic slip and elasticity, which permits the existence of periodic solutions in viscometric flows. The time-dependent, incompressible, one-dimensional plane Poiseuille flow of an Oldroyd-B fluid with slip along the wall is studied using a non-monotonic slip equation relating the shear stress to the velocity at the wall. The stability of the steady-state solutions to one-dimensional perturbations at fixed volumetric flow rateis analyzed by means of a linear stability analysis and finite element calculations. Self-sustained periodic oscillations of the pressure gradient are obtained when an unstable steady-state is perturbed, in direct analogy with experimental observations.     

Lattice Boltzmann modelling of unsteady-state 2D concentration profiles in adsorption bed

This study describes a lattice Boltzmann model (LBM) developed to simulate two-dimensional (2D) unsteady-state concentration profiles, including breakthrough curves, in a tubular column packed with adsorbents. The model using d3q19 (three dimensions and 19 speeds) lattice solves the 3D time-dependent convection-diffusion-adsorption equation for an ideal binary gaseous mixture assuming different velocity profiles in the column, including radially flat (plug flow) and non-uniform across the column's cross-section. The simulation results show significant concentration gradient across the cross-section depending upon the d/d_p ratio. The model results corroborate the experimental measurements made in the adsorption bed that the concentration due to breakthrough may be larger near the wall than at the core of the column due to the relatively larger local velocity in the vicinity of the wall. The LBM results have significance from the perspective of the physical understanding of the concentration profiles prevalent in the adsorption bed as well as effective design of a large-scale column. The model results are validated with the analytical solution to 1D axial dispersion problem, and to a few simple flow problems, such as Poiseuille and Couette flows.     

Lattice Boltzmann simulation of 2D and 3D non-Brownian suspensions in Couette flow

In this study, the Lattice Boltzmann (LB) method is applied for computer simulation of suspension flow in Couette systems. Typical aspects of Couette flow such as wall effects and non-zero Reynolds numbers can be studied well with the LB method because of its time-dependent character. Couette flow of single, two and multi-particle systems was studied, where two-dimensional (2D) systems were compared with three-dimensional (3D) systems.Computations on multi-particle 3D suspensions, for instance to assess the viscosity or shear-induced diffusivity, were found to be very intensive. This was only partly a consequence of the 3D system size. The critical particle grid size, necessary for accurate results, was found to be relatively large, increasing the system to impractical sizes.It is however demonstrated that it is possible to carry out computer simulations on 2D suspensions and use relatively simple, linear scaling relations to translate these results to 3D suspensions, in this way avoiding intensive computations. By doing so, the LB method is shown to be well-suited for study of suspension flow in Couette systems, particularly for aspects as particle layering near solid walls, hydrodynamic particle interactions and viscous stresses at non-zero Reynolds numbers, which cannot be easily solved with alternative methods. It also opens the way to employ the LB method for other unexplored aspects, such as particle polydispersity and high Reynolds number flow, with large relevance to practical processing of suspensions.     

TECHNICAL NOTE: THE EFFECT OF SLIP ON THE FLOW IN A ROTATING CHANNEL

Slip flow occurs in a wide variety of practical chemical engineering processes. This work models, for the first time, the interaction of surface slip and system rotation normal to the flow direction. It is found that the flow fields are affected considerably. For Poiseuille flow, slip increases the longitudinal flow rate at low rotation, but decreases it at high rotation. For Couette flow, slip decreases the longitudinal drag. The solutions are also exact solutions of the Navier-Stokes equations.     

Effect of wall velocities on the determination of optimal separation times in electrical field flow fractionation (EFFF)

Electrical field flow fractionation (EFFF) has two perpendicular driving forces that help to produce an optimal separation of solute in a mixture [Giddings, Science 1993; 260:1456–1465]. For Couette flow based devices, the ratio of the velocity of the capillary walls offers an extra parameter that can be exploited to enhance the efficiency of EFFF applications. The analysis of the effects of this parameter on optimal times of separation is the subject matter of this contribution. The use of this additional parameter increases flexibility in the design of new devices for the improvement of the separation of solutes, such as proteins, DNA, and pharmaceuticals, as it will be illustrated with the results of this analysis (Jaroszeski et al., 2000 ; Trinh et al., 1999 ). The analysis has been illustrated by selecting parameter values that represent a number of potential useful applications. A set of five parameters (i.e., z, the valence; µ, electrophoretic mobility; Pe, Peclet number; Ω, the orthogonal applied electrical field; and R, the ratio of channel wall velocities) has been combined to obtain the best operating conditions for optimal separation of solutes. Results indicate that R, the ratio of the channel wall velocities, is actually the most important driving parameter.     

Toward Understanding of Two‐Phase Eccentric Helical Reactor Performance

Mass transfer investigations in a two‐phase gas‐liquid Couette‐Taylor flow (CTF) reactor and a numerical flow simulation are reported. The CTF reactor is characterized by high values of the mass transfer parameters. Previous mass transfer investigations have yielded high values of the volumetric mass transfer coefficients (of the order of 10^–1 s^–1) and the specific interfacial area, compared to those obtained in a stirred tank (10³ m² m^–3). In order to intensify mass transfer in the CTF reactor, an eccentric rotor (rotating inner cylinder) was used. In the eccentric annulus with rotating inner cylinder, due to frequent variation of the hydrodynamic flow field parameters, nonlinear hydrodynamic conditions occurred. These conditions can influence the rate of mass transfer. The experimental results of benzaldehyde oxidation in an eccentric CTF reactor confirmed an increase in mass transfer, as against a concentric CTF reactor. Numerical simulation of the Couette‐Taylor (helical) flow was performed in a concentric and in an eccentric annulus. Calculation of parameters such as velocity, static pressure, kinetic energy and energy dissipation rate revealed a significant effect of gap eccentricity on the flow behavior.     

Penetration of Shear Flow Into an Array of Rods Aligned With the Flow

Shear flow over a square array of widely‐spaced rods aligned with the flow is investigated using singularity methods to solve Stokes equation. The flow field is determined for various arrays occupying a fraction of a Couette channel, for solid volume fractions from 0.001 to 0.1. Flow penetration into an array is quantified by the slip velocity at the array edge. This velocity is much greater than when the flow is across the array but still less than the value predicted by the use of Brinkman's equation.     

A study of the evaporation of water on a wet mobile plane surface in a hot draft

The authors study the case of the evaporation of water from a wet plane surface moving with a constant velocity in the same direction as a laminar hot draft. They numerically solve the unsteady transfer equations formulated with vorticity, stream function and constant physical properties, using a finite-difference scheme with an implicit alternating directions method. They determine the space–time distributions of temperatures, humidities and velocity components as well as the local values of their gradients according to the system parameters. In this study, the main results can be summarized as follows: The emission of vapor does not significantly disturb air flow. The velocity of the wall does not affect the hydrodynamic entrance length, that is correctly given by the Schlichting formula, but it increases the mean temperature in a section of the tunnel and the temperature and humidity wall gradients in proportion to the Lewis number. A privileged point exists in a section of the tunnel, at which the longitudinal velocity component is independent of the mobile plane velocity. Finally, the authors study the influence of inner drying with the help of the Luikov model.