We are interested in the amount of fluid left behind a drop moved inside a capillary tube. Long ago, Taylor showed that for very viscous liquids moved at small velocities, the film thickness is a monotonic increasing function of the capillary number. New data obtained with liquids of low viscosity are reported here and compared with Taylor’s law. Two successive effects are observed: above a threshold in capillary number, the film is thicker than a Taylor film; at a very high speed, the deposition law becomes a decreasing function of the drop velocity. Both behaviors are analyzed thanks to scaling arguments and shown to be consequences of inertia.

Quick deposition of a fluid on the wall of a tube

Multiphase monolith reactors: Chemical reaction engineering of segmented flow in microchannels

In a capillary, the two-phase pressure drop in Taylor slug flow was measured. A carefully designed inlet section for the capillary allowed the independent variation of gas bubble and liquid slug length. Gas and liquid superficial velocities were varied from 0.04 m/s to 0.3 m/s. If the slug length was smaller than 10 times the capillary diameter, the length-averaged friction factor for the liquid slug increased drastically from the single phase value (f = 16/Re) due to differences in curvature at the front and the back of the bubble. The use of different liquids allowed the independent variation of Re and Ca. The flow of elongated bubbles in capillaries was simulated using the CFD code FIDAP. It was found both numerically and experimentally that for Re ≫ 1, the extra pressure terms may be taken account using (Ca/Re) as a parameter. The numerical results agreed with the experimental data, provided that Marangoni effects of impurities are taken into account. The results allow the determination of slug length from pressure drop measurements in closed equipment where the slug length cannot otherwise be measured easily, such as monoliths and microreactors. © 2005 American Institute of Chemical Engineers AIChE J, 2005

Inertial and Interfacial Effects on Pressure Drop of Taylor Flow in Capillaries

Providing a comprehensive introduction to the fundamentals and applications of flow and heat transfer in conventional and miniature systems, this fully enhanced and updated edition covers all the topics essential for graduate courses on two-phase flow, boiling, and condensation. Beginning with a concise review of single-phase flow fundamentals and interfacial phenomena, detailed and clear discussion is provided on a range of topics, including two-phase hydrodynamics and flow regimes, mathematical modeling of gas-liquid two-phase flows, pool and flow boiling, flow and boiling in mini and microchannels, external and internal-flow condensation with and without noncondensables, condensation in small flow passages, and two-phase choked flow. Numerous solved examples and end-of-chapter problems that include many common design problems likely to be encountered by students, make this an essential text for graduate students. With up-to-date detail on the most recent research trends and practical applications, it is also an ideal reference for professionals and researchers in mechanical, nuclear, and chemical engineering.

/pdf/two-phase-flow-boiling-and-condensation-in-conventional-and-5aink3qfmk.pdf

Two-phase flow, boiling and condensation in conventional and miniature systems

On the CFD modelling of Taylor flow in microchannels

The flow induced by a long bubble steadily displacing a liquid confined by two closely located parallel plates or by a cylindrical tube of small diameter is numerically analyzed. The technique employed solves the complete set of governing equations simultaneously. The present analysis encompasses, and also extends, the whole range of Capillary values previously studied with various numerical techniques. The results shown uncover a type of recirculating flow pattern that appears to have been overlooked before. The effects of the inertial forces on the liquid flow rate are also assessed.

The axisymmetric and plane cases of a gas phase steadily displacing a Newtonian liquid—A simultaneous solution of the governing equations

In this work the interfacial shapes and the flow occurring at the trailing meniscus of a long bubble is numerically analyzed. The technique employed solves the complete set of governing equations simultaneously. The numerical results reported complete previous descriptions of the creeping flow regime; the influence of the inertia forces on the free surface shapes, interfacial undulations, and flow patterns is also analyzed.

The rear meniscus of a long bubble steadily displacing a Newtonian liquid in a capillary tube

In this work a numerical analysis of two-dimensional Faraday waves is presented. This study is based on direct numerical simulation of Navier–Stokes and continuity equations with appropriate boundary conditions. Stability maps on the (F-α) plane for viscous liquid layers with equilibrium depths between 5×10−5 m and 10−5 m are presented; comparisons are made with the linear stability predictions obtained with Benjamin and Ursell’s model for an inviscid fluid and with Kumar and Tuckerman’s model for a viscous fluid. Regions in which time-periodic solutions are no longer obtained and nonlinear effects are relevant, and are also delimited and analyzed: in these zones the disintegration of the free surface into drops may take place.

/pdf/a-numerical-analysis-of-the-influence-of-the-liquid-depth-on-4ouq0c8a55.pdf

A numerical analysis of the influence of the liquid depth on two-dimensional Faraday waves

In this work numerical solutions of the dip coating problem in the presence of a soluble surfactant are shown. Predictions of film thickening as well as thickening factors are in very good agreement with published experimental data, showing that pure hydrodynamic modeling suffices to mimic the process. Our numerical solutions provide a wealth of information on the functioning of the dip coating system; they show the appearance of a second stagnation point located in the bulk phase near the dynamic meniscus and they give clues about how the flow patterns might change as the surfactant becomes less soluble.

Numerical prediction of the film thickening due to surfactants in the Landau–Levich problem

A numerical investigation is carried out to study the effects of an insoluble surfactant on the dip coating of a flat substrate. Predictions of both the film thickness and the concentration of surfactant in the film as a function of the capillary number compare well with the solutions of a simpler asymptotic model based on the lubrication approximation. Streamline patterns confirm the existence of a stagnation point located in the bulk phase in the region of the dynamic meniscus—a conjecture postulated forty years ago. The evolution of the flow patterns and the interfacial variables shows how the classical result of Landau and Levich is recovered as the coating speed is augmented. Finally, we show that the effect of inertia forces cannot be neglected when the viscosity of the coating liquid is low.

/pdf/a-deeper-insight-into-the-dip-coating-process-in-the-1u7zmgczpj.pdf

Maria Delia Giavedoni

Papers

The axisymmetric and plane cases of a gas phase steadily displacing a Newtonian liquid—A simultaneous solution of the governing equations

The rear meniscus of a long bubble steadily displacing a Newtonian liquid in a capillary tube

A numerical analysis of the influence of the liquid depth on two-dimensional Faraday waves

Numerical prediction of the film thickening due to surfactants in the Landau–Levich problem

A deeper insight into the dip coating process in the presence of insoluble surfactants: A numerical analysis