We study the asymptotic tail behavior of the first-passage time over a moving boundary for a random walk conditioned to return to zero, where the increments of the random walk have finite variance. Typically, the asymptotic tail behavior may be described through a regularly varying function with exponent -½, where the impact of the boundary is captured by the slowly varying function. Yet, the moving boundary may have a stronger effect when the tail is considered at a time close to the return point of the random walk bridge, leading to a possible phase transition depending on the order of the distance between zero and the moving boundary.

In this work, the hybrid solution reconstruction formulation proposed by Luo et al. [H. Luo, H. Dai, P. F. de Sousa and B. Yin, On the numerical oscillation of the direct-forcing immersed-boundary method for moving boundaries, Computers & Fluids, 56 (2012), pp. 61–76] for the finite-difference discretization on Cartesian meshes is implemented in the finite-element framework of the local domain-free discretization (DFD) method to reduce the numerical oscillations in the simulation of moving-boundary flows. The reconstruction formulation is applied at fluid nodes in the immediate vicinity of the immersed boundary, which combines weightly the local DFD solution with the specific values obtained via an approximation of quadratic polynomial in the normal direction to the wall. The quadratic approximation is associated with the no-slip boundary condition and the local simplified momentum equation. The weighted factor suitable for unstructured triangular and tetrahedral meshes is constructed, which is related to the local mesh intervals near the immersed boundary and the distances from exterior dependent nodes to the boundary. Therefore, the reconstructed solution can account for the smooth movement of the immersed boundary. Several numerical experiments have been conducted for two- and three-dimensional moving-boundary flows. It is shown that the hybrid reconstruction approach can work well in the finite-element context and effectively reduce the numerical oscillations with little additional computational cost, and the spatial accuracy of the original local DFD method can also be preserved.

Enhancement of a heat transfer rate of mixed convection flow in a three-dimensional vertical channel with insertion of a moving slender block is investigated numerically. A slender block is installed along the direction of the channel flow, and the movement of the slender block is in periodic motion and transverse to the channel flow. The interaction between the moving block and the channel flow destroys and suppresses the velocity and thermal boundary layers on the heat surface periodically. Various ratios of the Richardson numbers (Gr/Re2) are simulated. The results show that under a higher velocity of the channel flow and a lower magnitude of Gr/Re2, the enhancement of heat transfer rate is better. Oppositely, under a lower velocity of the channel flow and a higher magnitude of Gr/Re2, the effect of natural convection driven by the buoyancy force is stronger and it is unfavorable to the heat transfer. A counter effect of the heat transfer rate is observed. These phenomena which are seldom analyzed before by numerical simulation are carried out in this study.

We study the problem of breakup of an air bubble in a Hele-Shaw cell. In particular, we propose some sufficient conditions of breakup of the bubble, and ways to find the contraction points of its parts. We also study regulated contraction of a pair of bubbles (in which the rates of air extraction from the bubbles are controlled) and study various asymptotic questions (such as the asymptotics of contraction of a bubble to a degenerate critical point, and asymptotics of contraction of a small bubble in the presence of a big bubble)

We consider the equation of a one-dimensional viscous heat-conducting compressible gas in the variable domain with the appropriate boundary conditions. We study the large-time behaviour of the solution in the particular case where the displacement of the variable boundary is given by $L(t)=L_0(1+at)^\alpha$ with $0lt\alphalt1$, where $a$ is a positive constant and $L_0$ is the initial amplitude of our domain.

AMS 2000 Mathematics subject classification: Primary 35B40; 76N15

An innovative computation procedure is developed to solve the external flow problems for viscous fluids. The method is able to handle the infinite domain so that it is convenient for the external flows. The code is based on the projection method of the Navier-Stokes equations. We use the three-step explicit finite element method to solve the momentum equation by extracting the boundary effects from the finite computation domain. The pressure Poisson equation for the external field is treated by the boundary element method. The arbitrary Lagrangian-Eulerian (ALE) scheme is employed to incorporate the present algorithm to deal with the moving boundary, such as the motion of an impulsively moving circular cylinder in a viscous fluid. The model demonstrates that drag force is well predicted for a circular cylinder moving in a still viscous fluid starting from rest, to a constant acceleration, and then maintaining at a uniform velocity. In the constant acceleration phase, the drag force is closed to the added mass effect from the ideal flow theory. On the other hand, the drag force is equal to viscous flow theory in the constant velocity phase.

The problem of estimating an unknown compact convex set K in the plane, from a sample (X1,···,Xn) of points independently and uniformly distributed over K, is considered. Let Kn be the convex hull of the sample, Δ be the Hausdorff distance, and Δn := Δ (K, Kn). Under mild conditions, limit laws for Δn are obtained. We find sequences (an), (bn) such that

(Δn - bn)/an → Λ (n → ∞), where Λ is the Gumbel (double-exponential) law from extreme-value theory. As expected, the directions of maximum curvature play a decisive role. Our results apply, for instance, to discs and to the interiors of ellipses, although for eccentricity e < 1 the first case cannot be obtained from the second by continuity. The polygonal case is also considered.

Let be the sequence of partial sums of independent N(μ, 1) random variables. The boundary in the {(n, Sn)} plane which minimizes the expected number of times n that Sn will be below a boundary when μ = θ > 0 subject to a given expected number of visits that Sn will be above the same boundary when μ = 0 is shown to be linear and is also characterized. Similar results are shown for Brownian motion.