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].     

An analytical method for selecting the optimal nozzle external geometry for fluid dynamic gauging

Fluid dynamic gauging (FDG) was developed to measure, in situ and in real time, the thickness of a soft deposit layer immersed in a liquid without contacting the surface of the layer. An analysis based on the lubrication assumption for the flow patterns in the space between the nozzle and the surface being gauged yielded analytical expressions for the relationships between the main flow variables and system parameters. Nozzle shapes for particular pressure, pressure gradient and shear stress profiles could then be identified. The effect of flow rate, nozzle geometry and nozzle position on the pressure beneath the nozzle and shear stress on the gauged surface showed very good agreement with computational fluid dynamics (CFD) simulations. Case studies presented include nozzle shapes for uniform pressure and shear stress profiles, which are useful for measuring the strength of soft deposit layers.     

Analytical and numerical evaluation of two-fluid model solutions for laminar fully developed bubbly two-phase flows

An analysis of the two-fluid model in the case of vertical fully developed laminar bubbly flows is conducted. Firstly the phase distribution in the central region of the pipe (where wall effects vanish) is considered. From the model equations an intrinsic length scale L is deduced such that the scaled system reduces to a single equation without parameters. With the aid of this equation some generic properties of the solutions of the model for pipes with diameter greater than about 20L (the usual case, since L is of the order of the bubble radius) are found. We prove that in all physically meaningful solutions an (almost) exact compensation of the applied pressure gradient with the hydrostatic force occurs (with ρ_eff the effective density and the gravity). This compensation implies flat void fraction and velocity profiles in the central region not affected by the wall, even when no turbulence effects are accounted for.We then turn to consider the complete problem with a numerical approach, with the effect of the wall dealt via wall forces. The previous mathematical results are confirmed and the near-wall phase distributions and velocity profiles are found. With the numerical code it is also possible to investigate the regime in which the pressure gradient is greater than the weight of the pure liquid, in which case a region of strictly zero void fraction develops surrounding the axis of the pipe (in upward flow of bubbles).Finally, the same code is used to study the effect of reducing the gravity. As decreases, so does the relative velocity between the phases, making the lift force increasingly dominant. This produces, in upward bubbly flows, narrower and sharper void fraction peaks that also appear closer to the wall.     

Axial dispersion of particles in a slugging column—the role of the laminar wake of the bubbles

Axial solid dispersion promoted by Taylor bubbles in a batch liquid column was studied. A mechanistic model was developed to predict the axial solid dispersion. The model is based on the upward transport of particles inside closed wakes of non-interacting Taylor bubbles. The model predictions are compared with experimental data. The experimental data were obtained in a test tube of internal diameter. The particle volumetric distribution was measured by several differential pressure transducers placed along the column. Two classes of glass beads, mean diameter 180 and , were suspended in aqueous glycerol solutions, with glycerol percentage ranging from 40% (v/v) to 100% (v/v). The amount of particles in the column was such that the volumetric particle fractions were 0.1, 0.2 and 0.3, supposing homogeneous liquid-solid suspension. The air flow rate ranged from 90×10⁻⁶ to at PTN conditions. The obtained experimental data are in good agreement with the model predictions for laminar wakes, i.e., closed wakes with internal recirculation and without vortex shedding. The experimental data show a higher upward particle transport for wakes in the transition laminar-turbulent regime; closed wakes with internal recirculation and vortex shedding. The upward particle transport is higher for increasing air flow rate, decreasing particle diameter and increasing amount of particles in the column.     

Modelling laminar pulsed flow for the enhancement of cleaning

A Green function method is presented which enables computation of laminar flow of an incompressible Newtonian fluid in circular and annular pipes, subject to an arbitrary forcing periodic pressure gradient, in terms of Bessel functions. The response to a step change in pressure gradient in an annular pipe is presented. The method allows direct calculation of wall shear stress and flow rates generated by pulsed flows, which are of interest in fouling mitigation and cleaning-in-place systems.     

Liquid–gas flow patterns in a narrow electrochemical channel

The flow in a narrow (3 mm wide) vertical gap of an electrochemical cell with gas evolution at one electrode is modeled by means of the two-phase Euler–Euler model. The results indicate that at certain conditions an unsteady type of flow with vortices and recirculation regions can occur. Such flow pattern has been observed experimentally, but not reported in previous modeling studies. Further analysis establishes that the presence of a sufficient amount of small (∼10 μm) bubbles is the main factor causing this type of flow at high current densities.     

Flow visualization of the liquid emptying process in scaled-up gravure grooves and cells

The liquid-emptying process in scaled-up gravure grooves and cells is studied using flow visualization in order to better understand gravure coating and printing processes. Water and two different glycerin/water mixtures serve as the test liquids, and the emptying process is initiated by moving over the groove or cell a rotating roller or a glass top with a curved surface. For the scaled-up groove, a region of recirculating flow is observed to attach to the moving glass top. When the glass top is used to drive flow in the scaled-up cell, an air bubble may appear inside the cell when the gap between the bottom of the curved surface and the top of the cell is zero. When this gap is positive, a liquid bridge is formed, dragged across the cell, and then broken, leaving some liquid inside the cell. The amount of liquid remaining in the cell, V_r, is measured for different liquids, surface speeds, and gap distances for both the glass top and the rotating roller. The effect of using a soft elastomeric covering on the glass top and roller is also explored. For each liquid, V_r increases as the speed of the glass top or roller increases. The data are correlated by multiplying V_r by a liquid-dependent shift factor, which leads to a power-law relationship between the shifted V_r and the capillary number. These experimental observations and measurements can be used to benchmark theoretical calculations, which can then be applied to design gravure grooves and cells that empty in a controlled way.     

Investigation of laminar flow in a stirred vessel at low Reynolds numbers

Many mixing applications involve viscous fluids and laminar flows where the detailed as well as overall flow structures are important. In order to understand the fluid dynamic characteristics of low Re laminar flows in mixing vessels, the flow induced by a Rushton impeller for three Re namely, 1, 10 and 28, was studied both experimentally and computationally. It was found that for the highest Re, the flow exhibited the familiar outward pumping action associated with radial impellers under turbulent flow conditions. However, as the Re decreases, the net radial flow during one impeller revolution was reduced and for the lowest Re a reciprocating motion with negligible net pumping was observed. This behaviour has not been reported in the literature in the past and represents a highly undesirable flow pattern from the standpoint of effective mixing. The CFD results successfully reproduced this behaviour. In order to elucidate the physical mechanism responsible for the observed flow pattern, the forces acting on a fluid element in the radial direction were analysed. The analysis indicated that for the lowest Re, the material derivative of radial velocity near the blade tip is small thus a balance exists between pressure and viscous forces; the defining characteristic of creeping flow. The velocity and pressure forces are in phase because the velocity is driven by the pressure field generated by the rotation of the impeller. Based on these findings, a simplified analytic model of the flow was developed that gives a good qualitative as well as quantitative representation of the flow.     

Mechanism of mixing enhancement with baffles in impeller-agitated vessel, part I: A case study based on cross-sections of streak sheet

Enhancement mechanism of mixing with baffle in agitated vessel using rotated two-bladed paddle impeller was investigated under a laminar condition. In mixing pattern, the toroidal isolated mixing region in the baffled vessel becomes distortive and much smaller than that of unbaffled vessel. From the visualization of streak cross-sections in the baffled vessel, interestingly, the renewed streak folds (streak lobes) are generated at the vicinity of baffles in both the vertical and horizontal cross-sections. These behaviors of streak are unlike the unbaffled case that the streak stretches straightforwardly. The streak lobe is known as the mixing template that its number and size are key factor for laminar mixing in agitated vessel. The results suggest that baffles can effectively transform the circumferential flow to vertical and/or radial flows. Consequently, in the baffled vessels, not only the vicinity of vessel wall but also the tip of baffles can become the origination of streak lobe formation, and folds of streak in the vertical and circumferential directions are further enhanced with baffles.     

Measurement of liquid film thickness and the determination of spin-up radius on a rotating disc using an electrical resistance technique

A method of rapid measurement of liquid film thickness on a spinning disc surface is presented. The concepts used are based on the analysis of the electrical resistance of the liquid film and its relationship to film thickness when high frequency voltages are applied. A comparison with the Nusselt model is made for a wide range of operating conditions. Experimental data are presented and calculations of mean radial flow velocity based on film thickness measurements are made and shown to compare well with a simplified two-dimensional model. An empirical model for the extent of the spin-up radius is developed to provide a guide to the zone at which the Nusselt model breaks down.     

Influence of Drying Method on Enzymatic Stain and Contact Angle of Maple

Lumber enzymatically stained has a significantly reduced value. The process involved in producing this degradation has not been well characterized. The influence of temperature and fluid flow transport of sap solutes were probed using contact angle analysis. Temperature effects on surface chemistry, as detected by polar components, was identified but was found not to attribute to staining. Dispersal components, used to indicate fluid flow transport of solutes, pointed to a positive influence on the production of stain. A more detailed layer-by-layer analysis was discussed to provide a more definitive conclusion of attributing the degree of staining to fluid flow transport of sap solute.     

The experimental observation and modelling of microdroplet formation within a plastic microcapillary array

This paper reports an experimental study of the formation of a two-phase liquid mixture in a circular capillary tube of 0.74 mm diameter. Organic liquid, the continuous phase, flowed through the capillary. Aqueous liquid, the dispersed phase, was injected through a hypodermic entering the side of the capillary and a stream of aqueous droplets was formed in the flowing organic liquid. The observed droplet diameters depended strongly on the ratio of the flow-rates between the dispersed and continuous phases: droplet diameters ranged between 480 and 64 μm. A simple model gave good predictions, matching the data and showing how the droplet diameter is dependant on the flow rates of the two phases. The flow geometry was similar to the T-junction configuration used for emulsion formation in microfluidic devices and was fabricated from an extruded plastic capillary array termed a microcapillary film (MCF).     

Reynolds number effects in a simple planetary mixer

Planetary mixers are widely used in a diverse range of industrial applications. This paper presents an experimental investigation of mixing in a planetary mixer, and a comparison with numerical simulations based on a simple mathematical model of the flow. The model allows an exact expression for the velocity field in the Stokes flow regime, apparently the first for a mixer with genuinely moving parts, which permits accurate numerical tracking of material interfaces. Experiments performed at low Reynolds number (Re?1) show good agreement with corresponding numerical simulations, but as the Reynolds number is increased, the agreement between experiments and Stokes-flow numerics worsens, in a manner that reflects improving experimental mixing quality. Specifically, we find that islands of poor mixing shrink as Re increases. Our results suggest that, while numerical simulations in the Stokes flow regime may be used as a ‘sieve’ to select good mixing protocols at small Re, experiments or computational fluid dynamics simulations are required properly to evaluate mixing protocols operated at finite Reynolds numbers.     

Simulations of microfluidic droplet formation using the two-phase level set method

Microdroplet formation is an emerging area of research due to its wide-ranging applications within microfluidic based lab-on-a-chip devices. Our goal is to understand the dynamics of droplet formation in a microfluidic T-junction in order to optimize the operation of the microfluidic device. Understanding of this process forms the basis of many potential applications: synthesis of new materials, formulation of products in pharmaceutical, cosmetics and food industries. The two-phase level set method, which is ideally suited for tracking the interfaces between two immiscible fluids, has been used to perform numerical simulations of droplet formation in a T-junction. Numerical predictions compare well with experimental observations. The influence of parameters such as flow rate ratio, capillary number, viscosity ratio and the interfacial tension between the two immiscible fluids is known to affect the physical processes of droplet generation. In this study the effects of surface wettability, which can be controlled by altering the contact angle, are investigated systematically. As competitive wetting between liquids in a two-phase flow can give rise to erratic flow patterns, it is often desirable to minimize this phenomenon as it can lead to a disruption of the regular production of uniform droplets. The numerical simulations predicted that wettability effects on droplet length are more prominent when the viscosity ratio λ (the quotient of the viscosity of the dispersed phase with the viscosity of the continuous phase) is O(1), compared to the situation when λ is O(0.1). The droplet size becomes independent of contact angle in the superhydrophobic regime for all capillary numbers. At a given value of interfacial tension, the droplet length is greater when λ is O(1) compared to the case when λ is O(0.1). The increase in droplet length with interfacial tension, σ, is a function of with the coefficients of the regression curves depending on the viscosity ratio.     

Flow distribution and pressure drop in 2D meshed channel circuits

The present paper investigates some families of simple 2D circuits or grids of intersecting tubes or channels (thus forming meshes) from the point of view of the fluid distribution and hydraulic characteristics in slow laminar flow. Square grids with a single inlet and a single outlet, and comprising from 4 to 100 meshes are considered, either with all channels identical, or with different channels along the borders of the circuit. Analytical resolution of the mesh and node relations to obtain the steady flow distributions are convenient up to 5×5 meshes, and formal computation furnishes numerical solutions for higher order circuits. This paper discusses some details of the flow and pressure distributions, in particular the symmetries, and also the behavior of some global properties such as overall or internal resistance and flow heterogeneity, as a function of the ratio of resistances of the two types of channels.     

Centrifugal-force driven flow in cylindrical micro-channel

Micro-channel flow driven by the centrifugal force was studied by analytical and experimental methods. The analytical expression of flow rate as a function of angular revolution speed was applied to the flow under the condition that the increase in frictional loss due to Coriolis force could be neglected. The experimental verification was made by measuring the outlet volume of water through capillaries that were radially mounted. The experimental results show that cavitation can occur in micro-channel flow driven solely by centrifugal force.     

Mixing efficiency of a multilamination micromixer with consecutive recirculation zones

Adding recirculation zones to a mixer for a microplant is proposed for enhanced mixing efficiency. A multilamination interdigital micromixer has been widely used in microchemical plants for precision or small scale chemical process. The mixing efficiency of this micromixer is relatively low as the mixing of two fluids is executed by the laminar diffusion process. To assist the mixing by fluid action, a series of recirculation zones were added to the mixing chamber. The effectiveness of the recirculation zones on mixing was estimated through a numerical simulation which indicated the dependence on Reynolds number. Mixing efficiency increased at Reynolds number that is relevant to the condition that is prevalent in a microchemical plant. The proposed micromixer was fabricated by the lithography process on the photosensitive glass wafers. The mixing qualities of the fabricated micromixer were measured by two methods; the flow visualization of dilution type experiments and the reactivity measurement. The measurement of color intensity of the mixed fluid followed the predictions by the simulation. For a Reynolds number greater than 400 that was relevant in mixers for microchemical plant, a mixing efficiency higher than 90% was obtained by adding the recirculation zones.     

Low Reynolds number flow in a channel with oscillating wavy-walls: An analytical study

An analytical study using perturbation methods has been investigated on the flow profiles and power requirements in an oscillating flow field with a wavy-walled boundary. The velocity field is found for a fluid in between two wavy-walled plates that oscillate in phase. The wavy-walled boundary causes the flow to produce regions of recirculations within the recesses of the boundary. Once the velocity profiles are identified, the power required to drive the oscillatory fluid motion is then calculated and compared to a flat plate geometry without recesses. As expected, the analytical results show that more power is needed to drive a wavy-walled system than a flat plate configuration.     

Forces on the surface of small ellipsoidal particles immersed in a linear flow field

Knowledge of the forces which a fluid motion exerts on the surface of suspended material is important for many processes in which the particles are broken apart by the hydrodynamic forces. In this paper, we examine the stresses on a small ellipsoidal particle which is immersed in either a constant, simple shear, two-dimensional straining or axisymmetric straining flow. Calculations have been performed using Oberbeck's and Jeffery's models and have been appropriately visualized. Furthermore, the motion and orientation of the ellipsoid have been examined and the extreme values of stresses have been analyzed. A simple criterion for the break-up of particles is proposed. The analysis shows that the straining flows are particular important for the load on particles. In contrast, simple shear flows are less crucial as the presence of vorticity leads to a rotation of the particles.