Mathematical Models and Methods in Applied Sciences

A deterministic model of polymer crystallization, derived from a previous stochastic one, is considered. The model describes the crystallization process of a rectangular sample of a material cooled at one of its sides. It is a reaction–diffusion system, composed of a PDE for the temperature and an ODE for the phase change of a polymer melt from liquid to crystal. The two equations are strongly coupled since the evolution of temperature depends on a source term, due to the latent heat developed during the phase change, the nucleation and growth rates are functions of the local (in time and space) temperature. The main difference with respect to the previous model is the introduction of a critical temperature of freezing in these functions. The paper does not contain detailed analytical aspects, that are left to subsequent investigations. A qualitative analysis of the proposed model is carried out, based on numerical simulations. An interesting feature shown by the simulations is that the solution exhibits an advancing moving band of crystallization in the mass distribution, as well as a moving boundary in the temperature field, both advancing with the same decreasing velocity. For some values of the parameters, which are typical of the physical problem, the advance takes place by jumps due to regular stops of the most advanced point of crystallization. The duration of these halts increases as the applied temperature decreases. This may indicate that the crystallization time is not a monotone function of the applied temperature. A simplified mathematical model is eventually proposed which reproduces the same patterns.

A thin micropolar fluid with new boundary conditions at the fluid-solid interface, linking the velocity and the microrotation by introducing a so-called "boundary viscosity" is presented. The existence and uniqueness of the solution is proved and, by way of asymptotic analysis, a generalized micropolar Reynolds equation is derived. Numerical results show the influence of the new boundary conditions for the load and the friction coefficient. Comparisons are made with other works retaining a no slip boundary condition.

We give a direct derivation of a theory of martensitic heterogeneous wires in the zero thickness and homogenization limit via a convergence result. We start from three-dimensional nonlinear hyperelasticity theory augmented by a term of interfacial energy of the van der Waals type. The derivation involves no a priori choice of asymptotic expansion or Ansatz. It yields a wire theory with two Cosserat vector fields. A formal derivation is given of higher theories for homogeneous wires, which yields one corrector for the deformation of the central line and two correctors for the Cosserat vector fields. Finally, we present a few numerical results.

This paper presents an optimality condition for the optimization problem of the assembly distribution in a nuclear reactor, by using the homogenization method. In this paper the reactivity of the reactor core is measured by the critical eigenvalue for both continuous and multigroup neutron transport models. In particular, we extend the spectral theory of the critical eigenvalue and prove the differentiability of the latter with respect to the design parameter, the configuration of the fuels.

For bounded smooth domains, we study how the solution of the Stokes problem is bounded in terms of the data when the domain changes. We show that standard bounds for a fixed domain hold with the same constant for wide classes of domains. This is done by first reviewing the original results of Cattabriga and then, in terms of geometric properties of the domains, by specifying when to apply Cattabriga's intermediate results. We do the same with standard strong extension and trace theorems.

We apply these results to an elegant technique of analysis due to Wahlbin to overcome the disparity between a curved domain and the domains where finite-element computations are carried out in practice without resorting to numerical quadrature.

In this short note, we discuss the basic approach to computational modeling of dynamical systems. If a dynamical system contains multiple time scales, ranging from very fast to slow, computational solution of the dynamical system can be very costly. By resolving the fast time scales in a short time simulation, a model for the effect of the small time scale variation on large time scales can be determined, making solution possible on a long time interval. This process of computational modeling can be completely automated. Two examples are presented, including a simple model problem oscillating at a time scale of 10^–9 computed over the time interval [0,100], and a lattice consisting of large and small point masses.

Under Grad angular cutoff assumption, we first establish the global-in-time classical solutions near Maxwellians to the Boltzmann equation in a periodic box. Furthermore, the exponential decay of such a solution is obtained.