Small amplitude lateral sloshing in a cylindrical tank with a hemispherical bottom under low gravitational conditions Summary report

Small amplitude lateral sloshing in cylindrical tank with hemispherical bottom under low gravitational condition

Mathematical and computational studies of equilibrium capillary free surfaces

The results of several independent studies are presented. The general question is considered of whether a wetting liquid always rises higher in a small capillary tube than in a larger one, when both are dipped vertically into an infinite reservoir. An analytical investigation is initiated to determine the qualitative behavior of the family of solutions of the equilibrium capillary free-surface equation that correspond to rotationally symmetric pendent liquid drops and the relationship of these solutions to the singular solution, which corresponds to an infinite spike of liquid extending downward to infinity. The block successive overrelaxation-Newton method and the generalized conjugate gradient method are investigated for solving the capillary equation on a uniform square mesh in a square domain, including the case for which the solution is unbounded at the corners. Capillary surfaces are calculated on the ellipse, on a circle with reentrant notches, and on other irregularly shaped domains using JASON, a general purpose program for solving nonlinear elliptic equations on a nonuniform quadrilaterial mesh. Analytical estimates for the nonexistence of solutions of the equilibrium capillary free-surface equation on the ellipse in zero gravity are evaluated

Equilibrium liquid free-surface configurations: Mathematical theory and space experiments

Small changes in container shape or in contact angle can give rise to large shifts of liquid in a microgravity environment. We describe some of our mathematical results that predict such behavior and that form the basis for physical experiments in space. The results include cases of discontinuous dependence on data and symmetry-breaking type of behavior. 23 refs., 9 figs

Universality for 2D Wedge Wetting

We study 2D wedge wetting using a continuum interfacial Hamiltonian model which is solved by transfer-matrix methods. For arbitrary binding potentials, we are able to exactly calculate the wedge free-energy and interface height distribution function and, thus, can completely classify all types of critical behaviour. We show that critical filling is characterized by strongly universal fluctuation dominated critical exponents, whilst complete filling is determined by the geometry rather than fluctuation effects. Related phenomena for interface depinning from defect lines in the bulk are also considered.Comment: 4 pages, 1 figur

Numerical solution of nonlinear elliptic partial differential equations by a generalized conjugate gradient method

We have studied previously a generalized conjugate gradient method for solving sparse positive-definite systems of linear equations arising from the discretization of elliptic partial-differential boundary-value problems. Here, extensions to the nonlinear case are considered. We split the original discretized operator into the sum of two operators, one of which corresponds to a more easily solvable system of equations, and accelerate the associated iteration based on this splitting by (nonlinear) conjugate gradients. The behavior of the method is illustrated for the minimal surface equation with splittings corresponding to nonlinear SSOR, to approximate factorization of the Jacobian matrix, and to elliptic operators suitable for use with fast direct methods. The results of numerical experiments are given as well for a mildy nonlinear example, for which, in the corresponding linear case, the finite termination property of the conjugate gradient algorithm is crucial. Wir haben früher eine verallgemeinerte Methode der konjugierten Gradienten studiert, um dünnbesetzte positiv definite Systeme von linearen Gleichungen zu lösen, die von der Diskretisierung von elliptischen partiellen Differential-Randwertproblemen herrühren. Wir betrachten hier die Verallgemeinerung auf den nichtlinearen Fall: Wir spalten den ursprünglichen diskretisierten Operator auf in eine Summe von zwei Operatoren. Einer von diesen Operatoren entspricht einem leicht lösbaren System von Gleichungen, und wir beschleunigen die aus dieser Spaltung hervorgehende Iteration mit (nichtlinearen) konjugierten Gradienten. Das Verhalten der Methode wird illustriert durch Anwendung auf die Minimalflächen-Gleichung, mit Spaltungen entsprechend dem nichtlinearen SSOR-Verfahren, der angenäherten Faktorisierung der Jacobi-Matrix, oder den elliptischen Operatoren, die sich für schnelle direkte Methoden eignen. Die Resultate von numerischen Experimenten für ein nur schwach nichtlineares Beispiel sind ebenfalls angegeben. Für den entsprechenden linearen Fall ist in diesem Fall die Konvergenz des konjugierten Gradienten-Algorithmus in einer endlichen Anzahl von Schritten wesentlich.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41643/1/607_2005_Article_BF02252030.pd

Capillary filling with wall corrugations] Capillary filling in microchannels with wall corrugations: A comparative study of the Concus-Finn criterion by continuum, kinetic and atomistic approaches

We study the impact of wall corrugations in microchannels on the process of capillary filling by means of three broadly used methods - Computational Fluid Dynamics (CFD), Lattice-Boltzmann Equations (LBE) and Molecular Dynamics (MD). The numerical results of these approaches are compared and tested against the Concus-Finn (CF) criterion, which predicts pinning of the contact line at rectangular ridges perpendicular to flow for contact angles theta > 45. While for theta = 30, theta = 40 (no flow) and theta = 60 (flow) all methods are found to produce data consistent with the CF criterion, at theta = 50 the numerical experiments provide different results. Whilst pinning of the liquid front is observed both in the LB and CFD simulations, MD simulations show that molecular fluctuations allow front propagation even above the critical value predicted by the deterministic CF criterion, thereby introducing a sensitivity to the obstacle heigth.Comment: 25 pages, 8 figures, Langmuir in pres

Droplet shapes on structured substrates and conformal invariance

We consider the finite-size scaling of equilibrium droplet shapes for fluid adsorption (at bulk two-phase co-existence) on heterogeneous substrates and also in wedge geometries in which only a finite domain $\Lambda_{A}$ of the substrate is completely wet. For three-dimensional systems with short-ranged forces we use renormalization group ideas to establish that both the shape of the droplet height and the height-height correlations can be understood from the conformal invariance of an appropriate operator. This allows us to predict the explicit scaling form of the droplet height for a number of different domain shapes. For systems with long-ranged forces, conformal invariance is not obeyed but the droplet shape is still shown to exhibit strong scaling behaviour. We argue that droplet formation in heterogeneous wedge geometries also shows a number of different scaling regimes depending on the range of the forces. The conformal invariance of the wedge droplet shape for short-ranged forces is shown explicitly.Comment: 20 pages, 7 figures. (Submitted to J.Phys.:Cond.Mat.

Geometry dominated fluid adsorption on sculptured substrates

Experimental methods allow the shape and chemical composition of solid surfaces to be controlled at a mesoscopic level. Exposing such structured substrates to a gas close to coexistence with its liquid can produce quite distinct adsorption characteristics compared to that occuring for planar systems, which may well play an important role in developing technologies such as super-repellent surfaces or micro-fluidics. Recent studies have concentrated on adsorption of liquids at rough and heterogeneous substrates and the characterisation of nanoscopic liquid films. However, the fundamental effect of geometry has hardly been addressed. Here we show that varying the shape of the substrate can exert a profound influence on the adsorption isotherms allowing us to smoothly connect wetting and capillary condensation through a number of novel and distinct examples of fluid interfacial phenomena. This opens the possibility of tailoring the adsorption properties of solid substrates by sculpturing their surface shape.Comment: 6 pages, 4 figure

On cylindrical container sections for a capillary free-surface experiment

Small changes in container shape or in contact angle can give rise to large shifts of liquid in a microgravity environment. These shifts can be used as a basis for accurate determination of contact angle. The authors describe container shapes, designed for a forthcoming USML-2 experiment, in the form of a circular cylinder with two diametrically opposed ``canonical proboscis`` protrusions. Computational studies indicate that these containers can be designed to have the desirable properties that sufficient liquid will participate in the shift to permit easy observation, but that the change will be abrupt enough to allow precise contact angle determination