An Approximate Computational Method for the Fluid Stiction Problem of Two Separating Parallel Plates With Cavitation

Stiction forces exerted by a fluid in a thin, quickly widening gap to its boundaries can become a strongly limiting factor of the performance of technical devices, like compressor valves or hydraulic on–off valves. In design optimization, such forces need to be properly and efficiently modeled. Cavitation during parts of a stiction process plays a strong role and needs to be taken into account to achieve a meaningful model. The paper presents an approximate calculation method which uses qualitative solution properties of the non cavitating stiction problem, in particular of its level curves and gradient lines. In this method, the formation of the cavitation boundaries is approximated by an elliptic domain. The pressure distribution along its principle axis is described by a directly integrable differential equation, the evolutions of its boundaries is guided just by pressure boundary conditions when the cavitation zone expands and by a nonlinear differential equation when it shrinks. The results of this approximate model agree quite well with the solutions of a finite volume (FV) model for the fluid stiction problem with cavitation.

High orders of Weyl series for the heat content

This article concerns the Weyl series of spectral functions associated with the Dirichlet Laplacian in a d -dimensional domain with a smooth boundary. In the case of the heat kernel, Berry and Howls predicted the asymptotic form of the Weyl series characterized by a set of parameters. Here, we concentrate on another spectral function, the (normalized) heat content. We show on several exactly solvable examples that, for even d , the same asymptotic formula is valid with different values of the parameters. The considered domains are d -dimensional balls and two limiting cases of the elliptic domain with eccentricity ε : a slightly deformed disk ( ε →0) and an extremely prolonged ellipse ( ε →1). These cases include two-dimensional domains with circular symmetry and those with only one shortest periodic orbit for the classical billiard. We also analyse the heat content for the balls in odd dimensions d for which the asymptotic form of the Weyl series changes significantly.

Bending of Simply-Supported Elliptic Plates: B.P.M. Solutions With Second-Order Derivative Boundary Conditions

A higher-order boundary perturbation method (B.P.M.) is formulated to treat a class of problems defined in an elliptic domain with associated boundary conditions expressed in terms of second-order derivatives. The method is applied to study a simply-supported elliptic plate subjected to a central lateral point load. The accuracy is investigated and the B.P.M. solution is found to yield highly accurate results for moderately elliptic domains.