RESUMEN

The dynamics of hydraulic fracture, described by a system of nonlinear integrodifferential equations, is studied through the development and application of a multiparameter singular perturbation analysis. We present a new single expansion framework which describes the interaction between several physical processes, namely viscosity, toughness, and leak-off. The problem has nonlocal and nonlinear effects which give a complex solution structure involving transitions on small scales near the tip of the fracture. Detailed solutions obtained in the crack tip region vary with the dominant physical processes. The parameters quantifying these processes can be identified from critical scaling relationships, which are then used to construct a smooth solution for the fracture depending on all three processes. Our work focuses on plane strain hydraulic fractures on long time scales, and this methodology shows promise for related models with additional time scales, fluid lag, or different geometries, such as radial (penny-shaped) fractures and the classical Perkins–Kern–Nordgren (PKN) model.

RESUMEN

We consider processes in which a focused laser beam is used to induce the melting of silicium. The first goal of this paper is to propose a simple three-dimensional (3D) model of this melting process. Our model is partly based on an energy balance equation. This model leads to a nontrivial ODE describing the evolution in time of the dimension of the melt region. The second goal of this paper is to obtain approximate analytical solutions of this ODE. After using basic solution methods, we propose an original geometrical method to derive asymptotic solutions for $\textrm{time} \rightarrow \infty$. These solutions turn out to be the most useful for the description of this process.