Axisymmetric Laplace Equation Research Articles

Numerical algorithm with fifth‐order accuracy for axisymmetric Laplace equation with linear boundary value problem

AbstractIn order to obtain the numerical solutions of the axisymmetric Laplace equation with linear boundary problem in three dimensions, we have developed a quadrature method to solve the problem. Firstly, the problem can be transformed to a integral equation with weakly singular operator by using the Green's formula. Secondly, A quadrature method is constructed by combing the mid‐rectangle formula with a singular integral formula to solve the integral equation, which has the accuracy of and low computational complexity. Thirdly, the convergence of the numerical solutions is proved based on the theory of compact operators and the single parameter asymptotic expansion of errors with odd power is got. From the expansion, we construct an extrapolation algorithm (EA) to further improve the accuracy of the numerical solutions. After one extrapolation, the accuracy of the numerical solutions can reach the accuracy of . Finally, two numerical examples are presented to demonstrate the efficiency of the method.

Volume of a liquid drop detaching from a sphere

AbstractA theoretical and experimental study is conducted to investigate the detached volume from a pendant drop on the surface of a sphere. Observation of drop detachment by high‐speed video camera reveals that the movement of the upper part of the neck of the drop is quite slow compared to that of the detaching lower part. The surface profile of the upper part was calculated approximately as a static problem using the axisymmetric Laplace equation. Using the drop profile, the system energy, including the work done by the solid–liquid wetting behavior, was calculated. Based on the condition of minimum energy, the volume of the detached part V was calculated. The volume V increases with the sphere diameter and approaches the value for the pendant drop attached to a plate. In addition, V is strongly dependent on the wettability between the sphere and the liquid and decreases with the receding contact angle. The detached volume of the water drop was measured for spheres of porous brick of various diameters. The experimental and theoretical results were found to be in good agreement. © 2010 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/htj.20305

Point Collocation Mesh-Free Method Using FMLSRKM for Solving Axisymmetric Laplace Equation

In this paper, the fast moving least square reproducing kernel method (FMLSRKM) for solving the axisymmetric Laplace problem is proposed. To maximize the advantages of FMLSRKM, a point collocation scheme is used. The accuracy of the proposed method is verified by applying it to a test electrostatic model with analytic solution, and the electric potential distribution in the arc region of a gas circuit breaker is also analyzed as a practical application.

Numerical simulation of non‐viscous liquid pinch off using a coupled level set boundary integral method

AbstractThe pinch off of an inviscid fluid column is described using a potential flow model with capillary forces. The interface velocity is obtained via a Galerkin boundary integral method for the 3D axisymmetric Laplace equation, whereas the interface location and the velocity potential on the free boundary are both approximated using level set techniques on a fixed domain. The algorithm is validated computing the Raleigh‐Taylor instability for liquid columns which provides an analytical solution for short times. The simulations show the time evolution of the fluid tube and the algorithm is capable of handling pinch‐off and after pinch‐off events. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Deformation and adhesive contact of elastomeric membranes

AbstractWe have developed a highly sensitive inflation technique for probing adhesive interactions between soft materials. In this method, an elastomeric membrane is placed across a cylindrical glass tube and is pressurized into contact with a substrate. The adhesive interaction between the membrane and the substrate is obtained from the angle of contact, which is determined indirectly from the measured contact radius and applied pressure. An analysis of the membrane profile based on the numerical solution of the axisymmetric Laplace equation is developed. In a nonadhesive situation, where the membrane contact angle is known to be zero, the analysis is used to obtain the relationship between the membrane deformation and the biaxial membrane strain. Linearization of the governing equations is used to develop an analytic solution for the contact angle and energy release rate, making a connection to existing, more restrictive analyses of this problem. Results from three different types of membranes are presented as illustrative applications of the method. The substrate in these experiments is the gold electrode surface of a quartz crystal resonator, and the effect of membrane contact on the crystal resonance is summarized briefly. Adhesive interactions between the membranes are minimized in most cases by adding a grafted polyethylene glycol brush to one or both of the contacting surfaces. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 3361–3374, 2007

The convergence of V-cycle multigrid algorithms for axisymmetric Laplace and Maxwell equations

We investigate some simple finite element discretizations for the axisymmetric Laplace equation and the azimuthal component of the axisymmetric Maxwell equations as well as multigrid algorithms for these discretizations. Our analysis is targeted at simple model problems and our main result is that the standard V-cycle with point smoothing converges at a rate independent of the number of unknowns. This is contrary to suggestions in the existing literature that line relaxations and semicoarsening are needed in multigrid algorithms to overcome difficulties caused by the singularities in the axisymmetric Maxwell problems. Our multigrid analysis proceeds by applying the well known regularity based multigrid theory. In order to apply this theory, we prove regularity results for the axisymmetric Laplace and Maxwell equations in certain weighted Sobolev spaces. These, together with some new finite element error estimates in certain weighted Sobolev norms, are the main ingredients of our analysis.

Adhesive Contact of a Membrane with a Hemispherical Indenter: Theoretical Analysis and Model Liquid System

The axisymmetric Laplace equation is solved numerically to extract contact-angle data for a flat liquid/vapor interface contacting a submerged hemispherical solid. The liquid/vapor interface is treated as a membrane, with a membrane tension equal to the surface energy of the liquid. By measuring the vertical displacement of the membrane and the projected contact area the membrane makes with the hemisphere, the contact angle and correspondingly the driving force for motion of the contact line can be measured. We show that characteristic receding and advancing contact angles can be obtained by measuring the contact radii formed upon initial contact between the interface and hemisphere and final contact just prior to detachment of the interface, respectively. Use of the technique is illustrated with a model experiment involving the contact of an air/water interface with a poly(methyl methacrylate) surface.

Galerkin boundary integral analysis for the axisymmetric Laplace equation

AbstractThe boundary integral equation for the axisymmetric Laplace equation is solved by employing modified Galerkin weight functions. The alternative weights smooth out the singularity of the Green's function at the symmetry axis, and restore symmetry to the formulation. As a consequence, special treatment of the axis equations is avoided, and a symmetric‐Galerkin formulation would be possible. For the singular integration, the integrals containing a logarithmic singularity are converted to a non‐singular form and evaluated partially analytically and partially numerically. The modified weight functions, together with a boundary limit definition, also result in a simple algorithm for the post‐processing of the surface gradient. Published in 2005 by John Wiley & Sons, Ltd.

Optimisation of a periodic solenoidal focusing system

An optimisation study of a periodic structure of superconducting solenoids used as a focusing system for a muon cooling channel is presented. In order to optimise the field quality of these focusing solenoids, the general solution of the axi-symmetric Laplace equation with periodic boundary conditions in the longitudinal direction is calculated. The objective function for the field optimisation has been derived from the equation of motion of a charged particle inside the periodic decay channel. A coil cross-section has been found that reduces the first order non-linearity of the focusing system by 27% (with respect to a coil with rectangular cross-section) while keeping the linear focusing power constant.

Gravitational fields of rotating disks and black holes

AbstractThe two known exact solutions of Einstein's field equations describing rotating objects of physical significance – a black hole and a rigidly rotating disk of dust – are discussed using a single mathematical framework related to Jacobi's inversion problem. Both solutions can be represented in such a form that they differ in the choice of a complex parameter and a real solution of the axisymmetric Laplace equation only.A recently found family of solutions describing differentially rotating disks of dust fits into the same scheme.

Gravitational fields of rotating disks and black holes

The two known exact solutions of Einstein's field equations describing rotating objects of physical significance - a black hole and a rigidly rotating disk of dust - are discussed using a single mathematical framework related to Jacobi's inversion problem. Both solutions can be represented in such a form that they differ in the choice of a complex parameter and a real solution of the axisymmetric Laplace equation only. A recently found family of solutions describing differentially rotating disks of dust fits into the same scheme.

Boundary element analysis of axisymmetric resonant cavities

Two numerical techniques are applied to boundary element analysis of axisymmetric modes of a cylindrical resonant cavity. One of the techniques is based on a novel expression for the fundamental solution of the axisymmetric Helmholtz equation with double exponential formulas. The other is an integral equation method in which the fundamental solution of the Laplace equation is used by considering the axisymmetric Helmholtz equation as the axisymmetric Laplace equation with an inhomogeneous term. Axisymmetric modes of a cyclindrical resonant cavity are analyzed by these techniques, and the resonant wave numbers are obtained with high accuracy. These results show that the axisymmetric resonant cavities can be analyzed by the boundary element method. >