Wide Range Of Capillary Numbers Research Articles

Free surface Stokes flow over topography

We study two-dimensional viscous flow over topographical features under the action of an external body force. The Stokes equations are written as a set of harmonic and biharmonic equations for vorticity and stream function. A direct biharmonic boundary integral equation method is used to transform these equations to a pair of integral equations on the boundary of the domain. These equations are solved with a preassumed free surface profile to obtain values of the flow variables on the boundary. The location of the free boundary is then updated by considering the normal stress condition and using an iteration technique. We have studied the flow over steps and trenches of different depths and for a wide range of capillary numbers, Ca. Our computation shows that for small Ca, the free surface develops a ridge before the entrance to a step down and a depression region right before a step up. The magnitude and location of these features depend on the capillary number and the step depth. For large capillary numbers the free surface nearly follows the topography and the ridge and depression are found to be exponentially small in the capillary number. On the other hand, our results agree well with lubrication theory for small capillary numbers on the order of 10−2 or less, even for steep features.

Film Drainage between Colliding Drops at Constant Approach Velocity: Experiments and Modeling

Experiments and modeling of the drainage of the thin liquid film between two deformable spherical drops approaching each other at constant velocity in another liquid are being presented. Two numerical models based on the lubrication theory have been developed considering the cases of immobile or mobile drop interfaces. The absolute film thickness and the thinning rate have been measured using laser interferometry for a wide range of capillary numbers. In all studied cases, the model with immobile interfaces was found to give the best predictions of the experimental time evolution of the film thickness and radial expansion. These results made it possible to derive a typical time scale of the drainage process.

The motion of long bubbles through viscoelastic fluids in capillary tubes

The penetration of long gas bubble through a viscoelastic fluid in a capillary tube has been studied in order to investigate the influence of viscoelastic material properties on the hydrodynamic coating thickness and local flow kinematics. Experiments are conducted for three tailored ideal elastic (Boger) fluids, designed to exhibit similar steady shear properties but substantially different elastic material functions. This allows for the isolation of elastic and extensional material effects on the bubble penetration process. The shear and extensional rheology of the fluid is characterized using rotational and filament stretching rheometers (FSR). The fluids are designed such that the steady-state extensional viscosity measured by the FSR at a Deborah number (De) greater than 1 differs over three orders of magnitude (Trouton ratio = 103–106). The experiment set up to measure the hydrodynamic coating thickness is designed to provide accurate data over a wide range of capillary numbers (0.01 < Ca < 100). The results indicate that the coating thickness in this process increases with an increase in the extensionally thickening nature of the fluid. Experiments are also conducted using several different capillary tube diameters (0.1 < D < 1 cm), in order to compare responses at similar Ca but different flow De. Suitable scaling methods and nonlinear viscoelastic constitutive equations are explored to characterize the displacement process for polymeric fluids. Bubble tip shapes at different De are recorded using a CCD camera, and measured using an edge detection algorithm. The influence of the mixed flow field on the bubble tip shape is examined. Particle tracking velocimetry experiments are conducted to compare the influence of viscoelastic properties on the velocity field in the vicinity of the bubble tip. Local shear and extension rates are calculated in the vicinity of the bubble tip from the velocity data. The results provide quantitative information on the influence of elastic and extensional properties on the bubble penetration process in gas-assisted injection molding. The bubble shape and velocity field information provides a basis for evaluating the performance of constitutive equations in mixed flow.

FINITE DIFFERENCE SIMULATION OF A STRATIFIED TWO-PHASE SYSTEM

SUMMARY We have analyzed several different approaches for simulating the fluid motion of a stratified two-phase system. The flow regime is transformed to a regular domain for numerical integration and the standard finite difference formulas are applied to discretize the governing and mapping equations. Five different interface iteration schemes that are based on the kinematic condition and the normal stress balance have been derived to update the position of the liquid-liquid interface. We have found that the iterative scheme based on the normal stress balance is more stable and is applicable to a wide range of capillary numbers, and that these appear to be the only sensitive parameter in the stratified two-phase system under consideration.

Coating the inside of a capillary of square cross section

Coating the walls of a capillary is usually done in two steps: (1) wetting or filling the capillary with a viscous fluid, and (2) clearing the capillary by forcing air through it. Clearing amounts to driving the wetting viscous fluid out of the capillary by another immiscible fluid. When air is blown into one end of a tube containing the viscous fluid, a “bubble” forms with the back end open. The bubble travels down the tube forcing some of the liquid out and leaving behind a fraction, in the form of a uniform film covering the walls. This paper reports flow visualization in capillaries of square cross secton. Visualization of the bubble was done by viewing both side and diagonal planes of the capillary in order to analyze the effect of the square cross section. Measurements from the captured images were used to characterize the shape of the fluid—fluid interface. The coating thickness of liquid deposited on the capillary walls was measured as the fraction of the capillary volume occupied by the liquid. The results for a wide range of capillary numbers show that this fraction increased monotonically with increasing capillary number, approaching an asymptotic limit of 0.64. Sequential particle tracking techniques were used to determine the flow field in different regions surrounding the bubble. A semi-quantitative comparison of the flow behavior, in capillaries of square and circular cross section, has been carried out.

Some mechanisms of immiscible fluid displacement in small networks

We report experimental observations on immiscible displacement in two small networks using three different pairs of fluids, air-oil, air-water, and oil-water, to vary the wettability. The experiments were run for a wide range of capillary number, from 10 −7 to 10 −3. Various mechanisms are observed. These are film spreading and drainage, Haines' jump, free slip and stick-slip meniscus motion, contact angle hysteresis, snap-off, coalescence, and blocking of film and bubble. For the air-oil case, oil is perfectly wetting in the network. In imbibition, the displacement occurs first via thin film spreading, followed by snap-off of menisci, and then by piston-like displacement at low flow rates. As the flow rate increases, piston-like displacement dominates because film spreading is comparatively slow. Snap-off of menisci in the throats is a necessary condition for air trapping. In drainage, meniscus snap-off and coalescence are observed in one network. For both imbibition and drainage, during each snap-off or piston-like displacement event, all menisci move freely along the channels to adjust their curvatures, due to the lubrication of the wetting film. For the other two fluid pairs at low flow rates, this curvature readjustment through free slipping of meniscus is not observed, presumably due to the absence of wetting film during the displacement. At high flow rate, oscillation of menisci due to volumetric competition is observed. Neither wetting film spreading nor throat snap-off is observed. Stick and slip motion of meniscus is observed, probably due to the roughness and/or heterogeneous wettability of the solid surface. For the oil-water system the wettability seems to be time dependent. Coalescence between two menisci can occur in the throat, in the pore, or at the pore-throat boundary during displacement. Trapping of the displaced phase is due to its being bypassed or snapped off in the throat.

ANALYSIS OF FREE COATING ONTO A ROTATING ROLL

An oblique flat surface model is used lo simulate the rotating roll. The flow regions proposed by Landau and Levich is utilized in the analysis. The inertia force terms are included and a more realistic 2-dimensional boundary conditions for the free surface is used in this paper. Also, the solution is obtained by utilizing a refinement of integral method. The results are compared with available experimental data. It is shown that the prediction is sufficiently accurate over a wide range of capillary number, and better than the other existing results. The film thickness increase with respect to increasing capillary number up to a certain higher values, from then this tendency slows down and even decreases due to inertia effects

Velocity effects in emulsion flow through porous media

Velocity plays an important role in the flow behavior of dispersed drops in underground media. Its magnitude designates whether a drop is captured, breaks up, or percolates through the medium. A quantitative description of these effects is lacking. This work investigates the possible roles of velocity in the transport of dilute, stable oil-in-water emulsions through underground porous media from the viewpoint of deep-bed filtration. A wide range of capillary number is considered. Theoretical analyses are presented for droplets having sizes corresponding to the two capture regimes of straining and interception. For the straining capture regime, the effect of velocity is important when capillary number is on the order of 10 −4 or above. For the interception capture regime, velocity effects are important when capture is in the secondary-minimum energy well and the velocity is close to a critical re-entrainment value. In both capture regimes drop breakup occurs only when the capillary number is about unity. The theoretical findings are verified by new experimental studies on the flow of oil-in-water emulsions at velocities of 0.005 to 0.08 cm/s in sandpacks with permeabilities of 0.7 and 2 μm 2.

Uniqueness of film thickness and meniscus profiles in vertical withdrawal

The authors' new one-dimensional free coating theory is considered further, and compared with previous theories. It is shown that a unique value of the dimensionless film thickness, T 0, exists over a wide range of Capillary Number, Ca. The shape of the upper meniscus is predicted from the new one-dimensional theory and compared with previous one- and two-dimensional creeping flow solutions. The superiority of the new one-dimensional theory over the previous theory is demonstrated, and the shape of the upper meniscus is shown to provide a more sensitive test of withdrawal theories. Comparison is also made with recently presented data.