Two-dimensional Time-dependent Problems Research Articles

On the modelling of joint formation in dissolutive brazing processes

In this paper, earlier dissolutive wetting models describing the dynamics of an axisymmetric alloy drop spreading on pure metal substrate are extended to describe reactive wetting and subsequent joint formation in brazing processes. A two-dimensional time-dependent problem is formulated, and the model equations are nondimensionalized, revealing the possibilities for asymptotic model reduction. Whilst the numerical solution of the time-dependent problem, which contains two moving contact lines and would not in general be amenable to lubrication theory, is relegated to future work, the steady-state problem is analyzed in detail. The analysis offers an arguably more transparent alternative to an earlier energy minimization approach for finding the location of the meniscus, which ultimately constitutes the joint. The results of the present model are found to compare favourably to those of earlier experimental and theoretical work.

Unsteady Rarefied Gas Flow with Shock Wave in a Channel

The two-dimensional time-dependent problem of rarefied gas flow in a plane channel, formed by parallel plates of finite length and closed at one end, is solved on the basis of the kinetic S-model. The flow develops as a result of rupture of a diaphragm which separates the gas at rest in the channel and the gas at rest in a reservoir of infinite volume. The effect of gas deceleration at the channel walls under the conditions of diffuse molecular reflection from the channel walls and end face is studied. Decay of a shock wave and disappearance of a homogeneous flow zone behind the shock wave is traced for three variants of conditions at the channel inlet: (1) gas enters the channel from a reservoir of infinite length and width (as the basic variant), the simultaneous motion in the reservoir and channel being studied; (2) the high-pressure reservoir represents a usual channel section; and (3) the motion of the gas in the reservoir is not considered at all, instead of this, the boundary conditions of the evaporation-condensation type under the conditions of gas at rest in the reservoir are imposed in the inlet cross-section.

Core loading pattern optimization based on simulated annealing and successive linear programming

An algorithm is developed to determine optimum core loading patterns for a pressurized water reactor burnup cycle. The best locations of fuel assemblies are searched by simulated annealing. The optimum placement of burnable absorbers for any stochastically generated fuel loading pattern is determined using successive linear programming. Two-dimensional time-dependent problems are considered for numerical study.

Finite element method for one-dimensional and two-dimensional time dependent problems with B-splines

One-dimensional and two-dimensional time dependent problems have been solved by the Galerkin method with cubic B-splines as basis functions. The redefining of basis functions has been done for two-dimensional problems, for the Dirichlet type boundary conditions, resulting in a non-homogeneous part in the approximation. The equidistribution of the error principle, given by Carl de Boor for one-dimensional problems, has been extended to two-dimensional problems. The solutions for nonlinear problems are obtained as the limit of solutions of a sequence of linear problems generated by the quasilinearization technique. The method developed with these features compares favourably with the methods available in literature.