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 present modeling and analysis of smectic C phases of liquid crystals capable of sustaining spontaneous polarization. The layered liquid crystals are also assumed to be chiral. We study minimization of the total energy subject to electrostatic constraints. In order to determine mathematically and physically relevant boundary conditions, we appeal to the analogy between the current problem and the vorticity in fluids. We place a special emphasis on the nonlocal and self-energy effects arising from spontaneous polarization. We discuss examples pertaining to the electric field created by the liquid crystal in dielectric medium, and also to the possible role of a domain shape as an energy reduction mechanism.