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.     

Motion of viscous drops in tubes filled with yield stress fluid

The dynamics of deformable viscous drops in yield stress medium is investigated experimentally. Tetrachloroethylene drops settle in a cylindrical tube filled with neutralized Carbopol solution under the action of gravity. For a single fluid particle, stopping conditions, established velocities and deformation patterns are examined for a variety of the physical parameters of the continuous medium determined by Carbopol concentration and the geometry of the system. For a pair of drops, we present time evolution of separation distance and of the shapes of the drops. It was found that if the initial separation between equal size drops does not exceed some critical value, the trailing particle moves with a considerably higher velocity than that of the leading one and eventually the two drops coalesce. Afterwards, a single large fluid particle is formed and it sometimes breaks up forming one large leading drop and one or several small trailing satellites. This behavior is qualitatively similar to that obtained previously in numerical simulations.     

Dynamic modelling of the deformation of a drop in a four-roll mill

The deformation of a drop flowing along the centre streamline of a four-roll mill (4RM) has been investigated. The velocities and elongation rates along the centre streamline in the 4RM were measured using particle tracking velocimetry. The deformation and position of the deforming drops were photographed with a video camera. A dynamic, one-dimensional, analytical simulation model describing the drop deformation has been developed. The model is based on Taylor's [1964. International Congress on Applied Mechanics, vol. 11, 790-796] static conical drop shape model, but has been extended to include elliptic drops undergoing rapid deformation. The model was incorporated into a numerical scheme using Matlab and the drop deformation in the 4RM was simulated. The simulations were compared with the results of the experiments with the help of a dynamic Weber number incorporating the exact effect of the continuous phase stress on the deformation of the drop. With a dynamic Weber number of 0.42 the agreement between the experiments and the simulations along the whole deformation process was excellent for all three drop diameters studied. With this model the deformation of drops of all sizes in different elongation fields can be calculated, for example sub-micron-sized drops in a high-pressure homogeniser.     

Collisions of spheres with wet and dry porous layers on a solid wall

A theory that combines Darcy's law for flow in porous media with inelastic solid mechanics, to model collisions of solid spheres with wet or dry porous layers placed on a solid wall, is found to closely describe the trends in data collected from particle-collision experiments. An exponential-hardening, stress-strain model is used for the porous layer, validated with dynamic mechanical analyzer measurements. Low-velocity collisions were performed in the low-gravity environment afforded during parabolic flight of a KC-135 aircraft, and also under normal gravity with a pendulum-based setup. Both theory and experiments show a decrease in the dry restitution coefficient with an increase in impact velocity, mainly due to increased inelastic losses in the porous material. The wet restitution coefficient is also found to decrease with an increase in the impact velocity, in contrast to the wet restitution coefficient for collisions of a solid sphere with a wet wall without a porous layer. Moreover, a critical impact velocity (below which no rebound occurs) is observed for wet collisions without a porous layer but not with a porous layer. The wet restitution coefficient is always found to be lower than the dry restitution coefficient, due to the viscous losses associated with fluid flow in addition to the inelastic losses associated with the porous layer.     

Unsteady mass transfer in the continuous phase around axisymmetric drops of revolution

Unsteady mass transfer in the continuous phase around any axisymmetric drop of revolution at high Peclet numbers has been theoretically studied. General equations for the concentration profile, the molar flux, the concentration boundary layer thickness, and the time to reach steady state have been obtained using a similarity transformation and by the method of characteristics. Solutions for large number of problems can be immediately obtained, with the only requirements being the shape of the drop and the tangential velocity at the surface of the drop.     

Unsteady mass transfer around axisymmetric drops of revolution in variable diffusion coefficient liquids

Unsteady mass transfer in the continuous phase around axisymmetric drops of revolution at high Peclet numbers has been theoretically studied. The liquid is a binary system, having a variable diffusion coefficient, which depends on the solute concentration. The solution to the problem was obtained by extending the theory of Favelukis and Mudunuri, developed for a constant diffusion coefficient liquid. The procedure consists of transforming the differential mass balance, for a binary system, into a partial differential equation which has an analytical solution, and an ordinary differential equation that needs to be solved numerically. Solutions to a large number of problems can be immediately obtained with the only requirements being the shape of the drop, the tangential velocity at the surface of the drop and an expression for the variable diffusion coefficient liquid. An approximate analytical solution is also suggested which is in excellent agreement with the numerical results.     

Viscous flow of a Newtonian fluid in a curved channel with flexible moving porous walls

We consider flow of a viscous Newtonian fluid in a curved channel with moving porous walls; the upper wall is flexible and its position in unknown a priori. This work is motivated from a papermaking application namely roll forming. We solve the leading order terms in equations of motion using perturbation methods and present analytical expressions for the variation in channel size, pressure, and viscous shear. The stability of the solution is also examined and we report the conditions for marginal stability.     

Viscous flow in a channel partially filled with a porous medium and with wall suction

In this paper, we consider a coupled two-dimensional flow of a Newtonian fluid, both above and through a porous medium. In the fluid-only region, the two-dimensional flow field is governed by the Navier-Stokes equation. We consider the Brinkman-extended Darcy law relationship in the porous medium. Inertial terms are retained in the formulation and the interface conditions between the two domains are those as outlined by Ochoa-Tapia and Whitaker (Int. J. Heat Mass Transfer 38 (1995) 2635). It should be noted that these interface conditions are formulated with an empirical constant β that is unknown a priori. The model equations were solved using two independent methods. In the first method, we pose a similarity variable and reduce the governing equations to two, coupled, non-linear ordinary differential equations. In the second approach, the governing equations were re-posed as a one-domain problem, using the procedure outlined by Basu and Khalili (Phys. Fluids 11 (1999) 1031), so that the conditions at the interface need not be considered. The resulting equation was solved directly, in primitive variable form, using a finite volume formulation. This enabled us to determine β by comparing the resulting solutions.     

A stochastic model for breakage frequency of viscous drops in turbulent dispersions

In industrial liquid-liquid mass-transfer equipment, many a times the dispersed phases involved are highly viscous. The viscosity of dispersed drops influences the rate processes, especially their breakage rate. A new stochastic model for predicting the breakage frequency of viscous drops in a turbulent dispersion applying the random behavior of the turbulent fluctuations, has been proposed. It has been assumed that the correlation time of turbulent fluctuations across the viscous drops is so small compared to the time scale of drop deformation, that the turbulent fluctuation can be considered as a white-noise process.     

Numerical simulation and experimental study of emulsification in a narrow-gap homogenizer

The present experimental and theoretical study investigates the fragmentation of the oil phase in an emulsion on its passage through a high-pressure, axial-flow homogenizer. The considered homogenizer contains narrow annular gap(s), whereupon the initially coarse oil drops break into fine droplets. The experiments were carried out using either a facility with one or two successive gaps, varying the flow rate and the material properties of the dispersed phase. The measured drop size distributions in the final emulsion clearly illustrated that the flow rate, as well as the dispersed-phase viscosity, and the interfacial tension can significantly affect the drop size after emulsification. The larger mean and maximum drop diameters obtained for the homogenizer with one gap in comparison to those obtained with two gaps (at the same Reynolds number and material parameters of the emulsion phases), highlighted the strong relevance of the flow geometry to the emulsification process. The numerical simulation of the carrier phase flow fields evolving in the investigated homogenizer was proven to be a very reliable method for providing appropriate input to theoretical models for the maximum drop size. The predictions of the applied droplet breakup model using input values from the numerical simulations showed very good agreement with the experimental data. In particular, the effect of the flow geometry—one-gap versus two-gaps design—was captured very well. This effect associated with the geometry is missed completely when using instead the frequently adopted concept of estimating input values from very gross correlations. It was shown that applying such a mainly bulk flow dependent estimate correlation makes the drop size predictions insensitive to the observed difference between the one-gap and the two-gaps cases. This obvious deficit, as well the higher accuracy, strongly favors the present method relying on the numerical simulation of the carrier phase flow.     

Modelling the motion of cylindrical particles in a nonuniform flow

The models currently used in computational fluid dynamics codes to predict solid fuel combustion rely on a spherical shape assumption. Cylinders and disks represent a much better geometrical approximation to the shape of bio-fuels such as straws and woods chips. A sphere gives an extreme in terms of the volume-to-surface-area ratio, which impacts both motion and reaction of a particle. For a nonspherical particle, an additional lift force becomes important, and generally hydrodynamic forces introduce a torque on the particle as the centre of pressure does not coincide with the centre of mass. Therefore, rotation of a nonspherical particle needs to be considered. This paper derives a model for tracking nonspherical particles in a nonuniform flow field, which is validated by a preliminary experimental study: the calculated results agree well with measurements in both translation and rotation aspects. The model allows to take into account shape details of nonspherical particles so that both the motion and the chemical reaction of particles can be modelled more reasonably. The ultimate goal of such a study is to simulate flow and combustion in biomass-fired furnaces using nonspherical particle tracking model instead of traditional sphere assumption, and thus improve the design of biomass-fired boilers.     

Shape oscillations and path transition of bubbles rising in a model bubble column

We investigate experimentally the occurrence of shape oscillations accompanied by path transition of periodically produced air bubbles rising in water. Within the period of bubble formation, the induced velocity is measured to examine bubble-liquid and bubble-bubble interactions. The flow is produced in a small-scale bubble column with square-shaped cross section. A capillary aerator produces bubbles of size 3.4 mm at a frequency of 5 Hz. Measuring techniques employed are high-speed imaging to capture bubble shape oscillations and path geometry, and laser-Doppler anemometry (LDA) to measure the velocity in the liquid near the rising bubbles. The experimentally obtained bubble shape data are expanded in Legendre polynomials. The results show the occurrence of oscillations by the periodicity of the expansion coefficients in space. Significant shape oscillations accompanied by path transition are observed as the second-mode oscillation frequency converges to the frequency of the initial shape oscillations. The mean velocity field in the water obtained by LDA agrees well with potential theory. An analysis of the decay of the induced flow shows that there is no interaction between the flow fields of two succeeding 3.4 mm bubbles in the rectilinear path when the bubble production frequency is lower than 7.4 Hz.     

Deformation and breakup of a second-order fluid droplet in an electric field

The effects of elastic property on the deformation and breakup of an uncharged drop in a uniform electric field are investigated theoretically using the second-order fluid model as a constitutive equation. Two dimensionless numbers, the electric capillary number (C) and the Deborah number (De), the dimensionless parammeters governing the problem. The asymptotic analytic solution of the nonlinear free boundary problem is determined by utilizing the method of domain perturbation in the limit of small mathcal C and small De. The asymptotic solution provides the limiting point of C above which no steady-state drop shape exists. The linear stability theory shows that the elastic property of fluids give either stabilizing or destabilizing effect on the drop, depending on the deformation mode.     

Numerical study on gas and liquid slugs for Taylor flow in a T-junction microchannel

The rapid development of microfabrication techniques creates new opportunities for applications of microchannel reactor technology in chemical reaction engineering. The extremely large surface-to-volume ratio and the short transport path in microchannels enhance heat and mass transfer dramatically, and hence provide many potential opportunities in chemical process development and intensification. Multiphase reactions involving gas/liquid reactants with a solid as a catalyst are ubiquitous in chemical and pharmaceutical industries. The hydrodynamics of the flow affects the reactor performance significantly; therefore it plays a prominent role in reactor design. For gas/liquid two-phase flow in a microchannel, the Taylor slug flow regime is the most commonly encountered flow pattern. The present study deals with the numerical simulation of the Taylor flow in a microchannel, particularly on gas and liquid slugs. A T-junction empty microchannel with varying cross-sectional width (0.25, 0.5, 0.75, 1, 2 and 3 mm) served as the model micro-reactor, and a finite volume based commercial computational fluid dynamics (CFD) package, FLUENT, was adopted for the numerical simulation. The gas and liquid slug lengths at various operating and fluid conditions were obtained and found to be in good agreement with the literature data. Several correlations in the T-junction microchannel were developed based on the simulation results. The slug flows for other geometries and inlet conditions were also studied.     

Effect of compatibilization on the deformation and breakup of drops in step-wise increasing shear flow

Drop deformation and breakup were investigated in the presence of a block copolymer in step-wise simple shear flow using a home-made Couette cell connected to an Anton Paar MCR500 rheometer. Polyisobutylene (PIB) was used as the matrix, while five different molecular weights of polydimethylsiloxane (PDMS) were selected to provide drops with a relatively wide range of viscosity ratio. A block copolymer made of PDMS-PIB was used for interfacial modification of the drop-matrix system. The copolymer concentration was 2 wt% based on the drop phase. The experiments consisted in analyzing the drop shape and measuring the variation of the length to diameter ratio, L/D, both in steady state and in transient regimes till breakup. This allowed revising of the classical Grace curve that reports the variation of the critical capillary number for breakup as a function of viscosity ratio and providing also a new one for blends compatibilized with an interfacial active agent with a given molecular weight.     

Comparison of steric effects in the modeling of spheres and rodlike particles in field-flow fractionation

The elution of spheres and rods in field-flow fractionation (FFF) is studied using a Brownian dynamics method. The particle motions for spheres are governed by a familiar Langevin equation which models drag force and diffusion. The rods are modeled as prolate ellipsoids and the particle motions are governed by a similar but orientation dependent Langevin equation, and the Jeffrey equation with rotational diffusion. Modeling of particle elution for spheres from 10 to 1000 nm was examined. The simulation captures the steric transition, and results for mean elution time are in good agreement with the steric inversion theory of Giddings [Giddings, J.C., 2000. In: Field-Flow Fractionation Handbook, Wiley-Interscience; Giddings, J.C., 1978. Separation Science and Technology 13, 241; Giddings, J.C., Myers, M.N., 1978. Separation Science and Technology 13, 637]. The sphere simulations are compared with simulations for rods of equal diffusivity, as under “normal mode” conditions (i.e., diffusion controlled) such particles should elute at the same rate. The results for rods show that nanotube size particles elute by a normal mode mechanism up to a size of about 500 nm (based on a particle diameter of 1 nm). At larger sizes, the rods begin to deviate from normal mode theory, but less strongly and in the opposite sense as for spheres. While the steric effect for spheres causes larger spheres to elute faster than predicted by normal mode theory, an inverse steric effect occurs for rods in which larger rods move increasingly slower than predicted by theory. The difference is attributed to the fact that the speed up observed for spheres is dictated by size exclusion of the particles at the boundary, while rods slow down due to increasing alignment at the boundary. Spheres and rods of equivalent diffusivity elute at the same rate up to a sphere size of approximately 90 nm (500 nm rods), at which point there are increasingly greater differences in mean elution times. While this affects the calibration of such operations, it also indicates that length based separations for nanotubes are not bound by the same limitation as occurs for spheres due to steric inversion.     

Visualization of compound drops formation in multiphase processes for the identification of factors influencing bubble and water droplet inclusions in oil drops

An innovative methodology for visualizing and identifying some mechanisms by which complex structures such as air-in-oil-in-water (A/O/W) and water-in-oil-in-water (W/O/W) may be formed inside mixing tanks dispersing various phases is described. In the case of A/O/W inclusions, isolated inclusion events could be observed by the first time with an experimental setup designed to produce sudden turbulence in a small confined space simulating a three-phase fermentation system. It was observed that high-energy direct-collisions of the bodies are not required for inclusions to occur; rather, a gentle contact between the phases was needed. Then, by maintaining an oil drop in a fixed position while it was impacted by single air bubble, it was feasible to calculate the percentage of air-bubble inclusions into oil drops for different compositions of the continuous phase. By adding biomass as a solid phase, the inclusion occurrence reached 61%; likely this was caused by a mechanical effect of the added biomass (making the interface breakable or unstable) with a minor contribution by the decreased surface tension. In the case of W/O/W, a basic mechanism by which the inclusion of water droplets in oil drops may occur is described. This was derived from the analysis of the hydrodynamic process of the formation of a water drop inside a volume of oil where the differential pressures occurring along the water–oil interface were mapped. This is the first time that factors influencing water and air inclusions in oil drops are identified, and possible mechanisms behind their occurrence are proposed, based on visual evidence.     

Transport equation for the local residence time of a fluid

The residence time of a fluid particle (or local residence time) is the time that the particle has spent inside a domain since its entry. A transport equation that provides the local residence time field is derived by the balance on a control volume of a variable denominated ”quantity of residence time”. A similar transport equation has been derived previously, but a more general expression is obtained through the present derivation. While the earlier equation is applicable to steady state, constant density flows, the equation proposed here can be applied to transient flows where density changes occur. The derivation also clarifies how to treat boundary conditions, a matter that has raised uncertainty in previous works. Furthermore, the concept of residence time of a chemical species is introduced, and the corresponding equation is formulated for specific cases.     

Incipient motion of a small particle in the viscous boundary layer at a pipe wall

A force balance is derived for a hemispherical particle in the viscous boundary layer at the wall of a horizontal pipe conveying Newtonian fluid; the hemisphere, of radius much less than that of the pipe, rests on the bottom with its flat face against the wall. The drag on the hemisphere is calculated from the creeping flow field of Price (Q. J. Mech. Appl. Math. Pt. 1 (1993)). This yields a prediction of the maximum velocity gradient at the wall for equilibrium, with limiting friction between the hemisphere and the wall. It is shown that the flow field of Price predicts a zero lift force but the validity of this, for actual flows, is questioned. Use of a hemisphere formulates a relevant well-posed problem, capable of mathematical solution. However, the flow field around real particles, e.g. sand, is complex, because of their irregular shapes, but the hemisphere work gives a qualitative indication of the behaviour of irregular particles. For turbulent flow in a pipe it is pertinent to consider a particle wholly within the viscous sub-layer, because it is isolated from significant turbulence and therefore hard to move; for such flow, the theory gives Eq. (21) to predict the critical pipe velocity, v_C, for incipient motion of the hemisphere. For laminar flow, the wall shear rate is readily obtained from the parabolic velocity profile leading to Eq. (26) for v_C. The flow field of Price (and therefore the force acting on the hemisphere) is valid only for creeping flow (i.e. very low particle Reynolds number). Modifications to the force balance are tentatively suggested to account for inertial components to the drag force. The predictions of critical velocity are tested against our data for the incipient motion of small hemispheres at pipe walls in hydraulic conveying as well as new and previously published data for both hydraulic and pneumatic conveying. The new method of predicting incipient motion works well for both the pneumatic and hydraulic conveying of hemispheres and sand shaped particles but it overpredicts the critical velocity for more rounded particles. The dependency of critical velocity on particle shape is under-researched.