Exotic Capillary Tubes

In contrast to the standard capillary tube, an exotic capillary tube is a rotationally symmetric tube of variable cross-section which if positioned correctly in a vessel of fluid possesses a continuum of equilibrium configurations. The controlling variables are the capillary constant k = ρ g/σ and the contact angle γ. Lowering the tube slightly from its natural position causes the tube to completely fill up while raising the tube slightly forces the tube to drain out. Other surprising consequences follow.     

Scherk-Type Capillary Graphs

This paper concerns the regularity of a capillary graph (the meniscus profile of liquid in a cylindrical tube) over a corner domain of angle α. By giving an explicit construction of minimal surface solutions previously shown to exist (Indiana Univ. Math. J. 50 (2001), no. 1, 411–441) we clarify two outstanding questions. Solutions are constructed in the case α = π/2 for contact angle data (γ₁, γ₂) = (γ, π − γ) with 0 < γ < π. The solutions given with |γ − π/2| < π/4 are the first known solutions that are not C² up to the corner. This shows that the best known regularity (C^{1, ∈}) is the best possible in some cases. Specific dependence of the H?lder exponent on the contact angle for our examples is given. Solutions with γ = π/4 have continuous, but horizontal, normal vector at the corners in accordance with results of Tam (Pacific J. Math. 124 (1986), 469–482). It is shown that our examples are C^{0, β} up to and including the corner for any β < 1. Solutions with |γ − π/2| > π/4 have a jump discontinuity at the corner. This kind of behavior was suggested by numerical work of Concus and Finn (Microgravity sci. technol. VII/2 (1994), 152–155) and Mittelmann and Zhu (Microgravity sci. technol. IX/1 (1996), 22–27). Our explicit construction, however, allows us to investigate the solutions quantitatively. For example, the trace of these solutions, excluding the jump discontinuity, is C^2/3.     

Extended Annular Capillary Surfaces

The height of the surface of a fluid in an annular tube is explored using a shooting method to solve a boundary value problem where the radii and the contact angles are given. The contact angles on the inner and outer tube surface need not be the same. These surfaces are then extended so that they are no longer graphs. The extended surfaces are shown to solve a boundary value problem over an annular base domain where given inclination angles are achieved at given radii.     

Comment on Capillary Coupling

Letter to the Editor

Comment on Capillary Coupling     

Generalized Solutions of the Capillary Problem

It was pointed out by Finn [2] that the capillary problem in zero gravity has not always a classical (smooth) solution in the case that the bounded domain Ω⊂ℝ² contains cusps or corners. Here, ω denotes the cross section of a given cylinder, in which a liquid is contained. If special energy terms could become infinite the variational formulation is not free of limitations as well. Therefore, the concept of generalized solutions, which can be traced back to Miranda [11], has been developed and could be a way out. We want to prove an existence result for such solutions under very weak preconditions. The proof is closely related to Giusti's paper [6], but we do not require full smoothness of the boundary. The major new difficulty is that we also want to consider domains with non-Lipschitz boundary. This excludes the application of some theorems. On the other hand, we use special geometric conditions in ℝ² and consequently, the proof cannot easily be generalized to a higher dimension. Furthermore, we construct some generalized solutions explicitly.     

Dynamic Stability of Equilibrium Capillary Drops

We investigate a model for contact angle motion of quasi-static capillary drops resting on a horizontal plane. We prove global in time existence and long time behavior (convergence to equilibrium) in a class of star-shaped initial data for which we show that topological changes of drops can be ruled out for all times. Our result applies to any drop which is initially star-shaped with respect to a small ball inside the drop, given that the volume of the drop is sufficiently large. For the analysis, we combine geometric arguments based on the moving-plane type method with energy dissipation methods based on the formal gradient flow structure of the problem.     

Petrophysical and Capillary Properties of Compacted Salt

This paper reports experimental results that demonstrate petrophysical and capillary characteristics of compacted salt. The measured data include porosity, gas permeability, pore size distribution, specific surface area, and gas-brine breakthrough and capillary pressure. Salt samples employed in the experiments were prepared by compacting sodium chloride granulates at high stresses for several hours. They represent an intermediate consolidation stage of crushed salt under in-situ conditions. The porosity and permeability of compacted salt showed similar trends to those expected in backfilled regions of waste repositories excavated in salt rock. The correlation between the measured porosity and permeability seems to be independent of the compaction parameters for the range examined in this study. The correlation also shows a different behaviour from that of rock salt. The data of all petrophysical properties show that the pore structure of compacted salt can be better characterized by fracture permeability models rather than capillary bundle ones. Simple creep tests, conducted on the fully-brine-saturated compacted salt samples, yielded similar strain rates to those obtained by a steady-state mechanical model developed from the tests on fully brine-saturated granular salt. A modified procedure is proposed for the evaluation of restored-state capillary pressure data influenced by the material creep. The characteristic parameters for the capillary behaviour of compacted salt are determined by matching the Brooks-Corey and van Genuchten models with the measured data. The Leverett functions determined with different methods agree well.     

A Capillary Microstructure of the Wetting Front

This article reports the experimental results of a study of the wetting-front microscale structure formed only by capillary forces in homogeneous and random etched glass capillary models. In the homogeneous model, water propagates through the capillary system, evenly filling the capillaries across the direction of flow. Air is trapped by the pinch-off mechanism inside the pore bodies in the form of individual bubbles. The experiments specified three consecutive steps of the pinch-off mechanism, film flow, snap-off, and interface movement. In the random model, both the bypass and pinch-off, forming bypass/cut-off mechanism, create residual air structure. Bypass traps air inside large capillary-pore aggregates which are bounded by small-diameter capillaries in where pinch-off traps air in the adjacent pores. An analysis of the residual air distribution versus depth below the surface in the homogeneous and random micromodels made it possible to identify three successive zones, namely a transition zone, a transmission zone, and a wetting-and-front zone. In the transition zone, the residual air content increases with depth from zero to the constant value in the transmission zone where it remains practically constant. The capillary processes within the wetting-and-front combined zone govern air replacement with wetting and formation of the transmission zone.     

Immiscible Displacement in the Interacting Capillary Bundle Model Part I. Development of Interacting Capillary Bundle Model

An interacting capillary bundle model is developed for analysing immiscible displacement processes in porous media. In this model, pressure equilibration among the capillaries is stipulated and capillary forces are included. This feature makes the model entirely different from the traditional tube bundle models in which fluids in different capillaries are independent of each other. In this work, displacements of a non-wetting phase by a wetting phase at different injection rates were analysed using the interacting capillary bundle model. The predicted evolutions of saturation profiles were consistent with both numerical simulation and experimental results for porous media reported in literature which cannot be re-produced with the non-interacting tube bundle models.     

Capillary flow behavior of bicomponent polymer blends

The flow behavior of bicomponent polymer blends of four types of polymers (polypropylene, polystyrene, high-density polyethylene and polymethyl-methacrylate) was examined using a capillary extrusion rheometer. The viscosity of the blend was generally less than the value calculated by the theoretical or empirical additivity rules proposed in previous reports, whereas the entrance pressure loss, which is considered to be an effect of elasticity, was larger than the estimated value. Thus the variation of the viscosity with blending ratio was inversely proportional to the variation in the elastic property. The cross-section of the material extruded in a roughly dispersed state showed an annularly stratified flow pattern in which the lower viscosity component polymer appeared to form the outer skin layer. However, the observation that the viscosity of the properly blended material at certain blending ratios was sometimes lower than that of either homopolymer could not be explained.     

Capillary rheometry for polymer melts revisited

Capillary rheometry provides an efficient access to high shear rate flow properties relevant for processing. An automated gas driven capillary rheometer developed at BASF enables accurate measurements at imposed wall shear stress, thus supplementing instruments operating at imposed flow rate. A simplified treatment of dissipative heating based on the assumption of a radially flat temperature profile is outlined and justified by means of finite element simulations. The combined treatment of dissipation and pressure dependent viscosity yields relations to treat throttling experiments at imposed flow rate. Throttle pressure coefficients from a long die and an orifice agree for LDPE but significantly differ for PMSAN. The effect is explained on the basis of identical pressure coefficients for shear and elongational flows, with regard to a constant stress, however. The effect of melt compressibility is negligible in practical capillary rheometry if the temperature and pressure coefficients of the melt density are by an order of magnitude smaller than those of the viscosity. Gas pressure driven instruments allow an effective determination of wall slip velocities from Mooney plots. This is of advantage for the investigation of the mechanism of additives or processing aids. Furthermore, imposed pressure experiments are pertinent to investigate the spurt effect of HDPE and to demonstrate that two different slip processes contribute to the apparent flow curve above spurt.

Determination of Capillary Pressure Function from Resistivity Data

A model has been derived theoretically to correlate capillary pressure and resistivity index based on the fractal scaling theory. The model is simple and predicts a power law relationship between capillary pressure and resistivity index (P _c = p _e · I ^β) in a specific range of low water saturation. To verify the model, gas-water capillary pressure and resistivity were measured simultaneously at a room temperature in 14 core samples from two formations in an oil reservoir. The permeability of the core samples ranged from 0.028 to over 3000 md. The porosity ranged from less than 8 to over 30. Capillary pressure curves were measured using a semi-permeable porous-plate technique. The model was tested against the experimental data obtained in this study. The results demonstrated that the model could match the experimental data in a specific range of low water saturation. The experimental results also support the fractal scaling theory in a low water saturation range. The new model developed in this study may be deployed to determine capillary pressure from resistivity data both in laboratories and reservoirs, especially in the case in which permeability is low or it is difficult to measure capillary pressure.     

Radial Capillary Transport from an Infinite Reservoir

Radial capillary transport occurs, for example, when wine spreads in the tablecloth ink in paper, rain drops in textiles, or dye into yarn. It is of technical relevance for propellant and other liquid transport in space. We present a theoretical and experimental study on the more basic situation when liquid spreads radially from an infinite reservoir. Our theoretical model predicts both outward and inward radial transport in a porous screen. While the outward wicking is fed by a circular wick in the center, the inward wicking is fed by a ring-like wick from the outside. For both cases, an analytical solution is obtained in terms of time versus radius as well as radius versus time aided by the Lambert W function. In the experiments, we use four different filter papers combined with three cylindrical wicks for outward wicking and one ring wick for inward wicking, respectively. The wicking process is recorded by a digital camera. Afterward, the resulting image series are evaluated with Matlab routines to detect the wicking front line. From the wetted area, we derive the mean radius versus time. Beside radially outward and inward wicking, we consider also experimental reference data from horizontal and vertical wicking in a strip.     

On the Capillary Problem for Compressible Fluids

Classical capillarity theory is based on a hypothesis that virtual motions of fluid particles distinct from those on a surface interface have no effect on the form of the interface. That hypothesis cannot be supported for a compressible fluid. A heuristic reasoning suggests that even small amounts of compressibility could have significant effect on surface behavior. In an earlier work, Finn took a partial account of compressibility, and formulated a variant of the classical capillarity equation for fluid surface height in a vertical capillary tube; he was led to a necessary condition for existence of a solution with prescribed mass in a tube closed at the bottom. For a circular tube, he proved that the condition also suffices, and that solutions are uniquely determined for any contact angle γ. Later Finn took more complete account of compressibility and obtained a new equation of highly nonlinear character but for which the same necessary condition holds. In the present work we consider that equation for circular tubes. We prove that the necessary condition again suffices for existence when 0 ≤ γ < π, and we establish uniqueness when 0 ≤ γ ≤ π/2. Our result is put into relief by the observation that for the unconstrained problem of a tube dipped into an infinite liquid bath, solutions do not in general exist when γ > π/2. Presumably an actual fluid would in that case descend to the bottom of the tube. This kind of singular behavior does not occur for the equation previously considered, nor does it occur in the present case under the presence of a mass constraint.     

Capillary Floating and the Billiard Ball Problem

In a study of capillary floating, Finn (J Math Fluid Mech 11:443?C458, 2009) described a procedure for determining cross-sections of non-circular, infinite convex cylinders that float horizontally on a liquid surface in every orientation with contact angle ??/2. Finn??s procedure yielded incomplete results for other contact angles; he raised the question as to whether an analogous construction would be feasible in that case. In the note, Finn (J Math Fluid Mech 11:464?C465, 2009) pointed out a connection with an independent problem on billiard caustics citing the unpublished work (Gutkin in Proceedings of the Workshop on Dynamics and Related Questions, PennState University, 1993) of the present author. Here we present a solution of the billiard problem in full detail, thus settling Finn??s question in a surprising way. In particular, we show that such floating cylinders exist if and only if the contact angle lies in a certain, explicitly described countably dense set. Moreover, for each element ?? in this set we exhibit a family of convex, non-circular cylinders that float in every orientation with contact angle ??. Our discussion contains other material of independent interest for the billiard ball problem.     

Capillary Adhesion Between Elastically Hard Rough Surfaces

The adhesion versus vapor pressure (p/p _s) trend between two elastically hard rough surfaces is modeled and compared with experimental results. The experimental samples were hydrophilic surface-micromachined cantilevers, in which the nanometer-scale surface roughness is on the order of the Kelvin radius. The experimental results indicated that adhesion increases exponentially from p/p _s=0.3 to 0.95, with values from 1 mJ/m² to 50 mJ/m². Using the Kelvin equation to determine the force-displacement curves, the mechanics of a wetted rough interface are treated in two ways. First, the characteristics of a surface with rigid asperities of uniform height are derived. At low p/p _s, menisci surrounding individual asperities do not interact. Beyond a transition value, [p/p _s]_tr, a given meniscus grows beyond the asperity it is associated with, and liquid fills the interface. Capillary adhesion in each realm is found according to the integrated work of adhesion. Second, a more general approach allowing an arbitrary height distribution of Hertzian asperities subject to capillary forces is justified and developed. To compare with experimental results, a Gaussian height distribution is first assumed but significantly underestimates the measured adhesion. This is because equilibrium is found far into the Gaussian tail, where asperities likely do not exist. It is shown that by bounding the tail to more likely limits, the measured adhesion trend is more closely followed but is still not satisfactorily matched by the model. The uniform summit height model fits the data very well with a single free parameter. These results can be rationalized if the upper and lower surfaces are geometrically correlated.