Some free boundary problems in potential flow regime usinga based level set method

Recent advances in the field of fluid mechanics with moving fronts are linked to the use of Level Set Methods, a versatile mathematical technique to follow free boundaries which undergo topological changes. A challenging class of problems in this context are those related to the solution of a partial differential equation posed on a moving domain, in which the boundary condition for the PDE solver has to be obtained from a partial differential equation defined on the front. This is the case of potential flow models with moving boundaries. Moreover the fluid front will possibly be carrying some material substance which will diffuse in the front and be advected by the front velocity, as for example the use of surfactants to lower surface tension. We present a Level Set based methodology to embed this partial differential equations defined on the front in a complete Eulerian framework, fully avoiding the tracking of fluid particles and its known limitations. To show the advantages of this approach in the field of Fluid Mechanics we present in this work one particular application: the numerical approximation of a potential flow model to simulate the evolution and breaking of a solitary wave propagating over a slopping bottom and compare the level set based algorithm with previous front tracking models

On the development of kink-bands: A case study in the Westasturian-Leonese Zone (Variscan belt, NW Spain)

A field analysis of kink bands developed in slates from three areas (Grandas, Boal and Luarca areas) of the Westasturian-Leonese Zone (Iberian Variscan belt) is presented. The analysis of the main parameters that characterize the geometry of the studied kink bands shows that those of the Grandas and Luarca areas exhibit a different evolution than those of the Boal area. In this latter area, the interlimb angle of the kink bands has lower values than those developed in the former areas and it involves rotation of the foliation inside and outside the band. In the areas with higher bulk shortening associated with the development of kink bands, chevron folds formed by juxtaposition of kink bands. Slip between folia and their rotation was probably the dominant mechanism in the formation of the kink bands, as deduced from the different values of the angle between the kink plane and the foliation inside (φK) and outside (φ) the band, and the occurrence of fractures along the kink planes and small steps between folia cross-cutting these fractures planes. The fractures along the kink planes prevented subsequent hinge migration. Geometrical analysis of kink bands formed by slip between folia and their rotation provides an estimation of the changes in area and thickness, and the strain inside the kink band. For angles of folia rotation ψ < 50°, the ratio between the strain ellipse axes is < 3 inside the band; this ratio is almost independent of the orientation of the kink planes with respect to the foliation outside the band (angle φ)

Total bulk strain in flattened parallel folds

A new simple method is proposed to analyse the total bulk shortening in flattened parallel folds. Application of the method requires determining first the amount of flattening using any of the available techniques or by using a new method proposed in this paper. Then, some shape information must be obtained from the fold, such as the interlimb angle or the aspect ratio, as well as the eccentricity of the conic section that gives the best fit to the midline of the folded layer. The latter parameter can be obtained using the program “Fold Profiler”. Finally, the flattening and the shape data are used to obtain the total bulk shortening of the folded layer. A computer code is provided that performs these calculations. The method does not consider the initial layer shortening prior to buckling and therefore gives an estimate of the minimum bulk shortening associated with the fold. The strain data obtained with this method are not point values from selected locations inside the folded structure, but an overall evaluation of its bulk strain that can be advantageously used in the regional interpretation. The analysis is two-dimensional and only considers deformation in the fold profile plane. The study is completed with an example of the application of the method to natural folds

Saw-tooth structures and curved veins related to folds in the south-central Pyrenees(Spain)

Two generation of folds (F1 and F2) and associated structures, developed in the Eocene turbidites of the south-central Pyrenees, are analyzed in this paper. F1 folds are close, have sub-horizontal axes and southwards vergence. They have an associated cleavage S1. Competent layers were folded by layer-parallel shortening, tangential longitudinal strain, some possible flexural flow and an obliquely superimposed homogeneous strain due mainly to simple shear. Flexural slip is also an important mechanism in the whole multilayer. F2 folds are gentle and scarce; they fold the S1 cleavage. Among the structures associated with F1 folds, there are sets of veins with curved form in the competent layers. The displacement of each vein gave rise usually to a step in the layer boundary, so that a set of veins produces a structure that is named “saw-tooth structure”. The veins initiated as small faults that made flexural slip difficult and gave rise to a concentration of stress on the steps, leading to an opening of the fractures and a propagation of them along a curved path, as suggested by a simple mechanical model. This propagation agrees with finite element models developed by other authors

On The Connection Between Dirac And Ricatti Equations

We analyse the behaviour of the Dirac equation in d = 1 + 1 with Lorentz scalar potential. As the system is known to provide a physical realization of supersymmetric quantum mechanics, we take advantage of the factorization method in order to enlarge the restricted class of solvable problems. To be precise, it suffices to integrate a Ricatti equation to construct one-parameter families of solvable potentials. To illustrate the procedure in a simple but relevant context, we resort to a model which has proved useful in showing the phenomenon of fermion number fractionalization. 1. Introduction. When solving the Dirac equation in d = 1 + 1 with Lorentz scalar potential, the underlying supersymmetric structure is crucial. As the system provides a physical realization of supersymmetric quantum mechanics (susy qm henceforth), the problem reduces itself to a pair of Schrodinger-like hamiltonians related by means of supersymmetry. In doing so, both operators share identical energy spectra u..

Spontaneous Symmetry Breaking Phenomena with Non-Equivalent Vacua

We study the existence of bidimensional bosonic models for which the spontaneous symmetry breaking phenomenon yields, at classical level, non-equivalent vacua. Once we introduce the concept of vacuum manifold V , defined in terms of the vacuum field configurations, the behavior of the models at issue can be analyzed by means of the so-called classical moduli space M c (V). The aforementioned structure allows us to classify the kink-like excitations into loops and links. To be precise, the loops interpolate smoothly between equivalent minima of the classical potential while the links connect vacua located at different points in M c (V). Although exact results are very hard to come by, we resort to models nice enough to provide us with the solitary waves in closed form. 1. Introduction. Since the mid-seventies the search for classical solutions of the corresponding non-linear equations has been one of the most successful tools in both classical and quantum field theories. Unless other..

The hinge lines of non-cylindrical folds

Hinge lines are loci of high curvature points on folded surfaces. They are significant geometrical features of geological folds, and the arrangement of hinge lines constructed for the surface serves to characterize important aspects of the fold pattern. Since the current definition of hinge line is only appropriate for cylindrical folds, we propose a new definition for use with folds of general shape. Like the concept of ridge lines used in differential geometry, the new definition uses the lines of curvature (principal curvature trajectories) as a reference frame for comparing curvatures across the surface. A hinge line passes through points of extreme principal curvature magnitude observed along the corresponding principal curvature trajectory. Two types of hinge lines are defined and methods for constructing hinge lines are suggested