212 research outputs found
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
Measurement of Critical Contact Angle in a Microgravity Space Experiment
Mathematical theory predicts that small changes in container shape or in contact angle can give rise to large shifts of liquid in a microgravity environment. This phenomenon was investigated in the Interface Configuration Experiment on board the USML-2 Space Shuttle flight. The experiment's "double proboscis" containers were designed to strike a balance between conflicting requirements of sizable volume of liquid shift (for ease of observation) and abruptness of the shift (for accurate determination of critical contact angle). The experimental results support the classical concept of macroscopic contact angle and demonstrate the role of hysteresis in impeding orientation toward equilibrium
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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
The prescribed mean curvature equation in weakly regular domains
We show that the characterization of existence and uniqueness up to vertical
translations of solutions to the prescribed mean curvature equation, originally
proved by Giusti in the smooth case, holds true for domains satisfying very
mild regularity assumptions. Our results apply in particular to the
non-parametric solutions of the capillary problem for perfectly wetting fluids
in zero gravity. Among the essential tools used in the proofs, we mention a
\textit{generalized Gauss-Green theorem} based on the construction of the weak
normal trace of a vector field with bounded divergence, in the spirit of
classical results due to Anzellotti, and a \textit{weak Young's law} for
-minimizers of the perimeter.Comment: 23 pages, 1 figure --- The results on the weak normal trace of vector
fields have been now extended and moved in a self-contained paper available
at: arXiv:1708.0139
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Numerical solution of the multidimensional Buckley--Leverett equation by a sampling method
A method developed earlier for solving numerically the one-dimensional Buckley--Leverett equation for two phase immiscible flow in a porous medium is extended to the case of non-uniform flow in two space dimensions. The method has the feature of tracking solution discontinuities sharply for purely hyperbolic problems, without requiring devices such as the introduction of artificial dissipation. It is found that the method is computationally efficient for solving a numerical example for the five-spot configuration of water flooding of a petroleum reservoir
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
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