The goal of this study is to provide analytical and numerical assessments to a fluid flow based on an Eyring–Powell viscosity term and a Darcy–Forchheimer law in a porous media. The analysis provides results about regularity, existence and uniqueness of solutions. Travelling wave solutions are explored, supported by the Geometric Perturbation Theory to build profiles in the proximity of the equation critical points. Finally, a numerical routine is provided as a baseline for the validity of the analytical approach presented for low Reynolds numbers typical in a porous medium.
The analysis in the present paper provides insights into the Liouville-type results for an Eyring-Powell fluid considered as having an incompressible and unsteady flow. The gradients in the spatial distributions of the initial data are assumed to be globally (in the sense of energy) bounded. Under this condition, solutions to the Eyring-Powell fluid equations are regular and bounded under the L2 norm. Additionally, a numerical assessment is provided to show the mentioned regularity of solutions in the travelling wave domain. This exercise serves as a validation of the analytical approach firstly introduced.
This work provides an analytical approach to characterize and determine solutions to a porous medium system of equations with views in applications to invasive-invaded biological dynamics. Firstly, the existence and uniqueness of solutions are proved. Afterwards, profiles of solutions are obtained making use of the self-similar structure that permits showing the existence of a diffusive front. The solutions are then studied within the Travelling Waves (TW) domain showing the existence of potential and exponential profiles in the stable connection that converges to the stationary solutions in which the invasive species predominates. The TW profiles are shown to exist based on the geometry perturbation theory together with an analytical-topological argument in the phase plane. The finding of an exponential decaying rate (related with the advection and diffusion parameters) in the invaded species TW is not trivial in the nonlinear diffusion case and reflects the existence of a TW trajectory governed by the invaded species runaway (in the direction of the advection) and the diffusion (acting in a finite speed front or support).
<abstract><p>The intention along the presented analysis is to explore existence, uniqueness, regularity of solutions and travelling waves profiles to a Darcy-Forchheimer fluid flow formulated with a non-linear diffusion. Such formulation is the main novelty of the present study and requires the introduction of an appropriate mathematical treatment to deal with the introduced degenerate diffusivity. Firstly, the analysis on existence, regularity and uniqueness is shown upon definition of an appropriate test function. Afterwards, the problem is formulated within the travelling wave domain and analyzed close the critical points with the Geometric Perturbation Theory. Based on this theory, exact and asymptotic travelling wave profiles are obtained. In addition, the Geometric Perturbation Theory is used to provide evidences of the normal hyperbolicity in the involved manifolds that are used to get the associated travelling wave solutions. The main finding, which is not trivial in the non-linear diffusion case, is related with the existence of an exponential profile along the travelling frame. Eventually, a numerical exercise is introduced to validate the analytical solutions obtained.</p></abstract>
The purpose of the present study is to obtain regularity results and existence topics regarding an Eyring–Powell fluid. The geometry under study is given by a semi-infinite conduct with a rectangular cross section of dimensions L×H. Starting from the initial velocity profiles (u10,u20) in xy-planes, the fluid flows along the z-axis subjected to a constant magnetic field and Dirichlet boundary conditions. The global existence is shown in different cases. First, the initial conditions are considered to be squared-integrable; this is the Lebesgue space (u10,u20)∈L2(Ω), Ω=[0,L]×[0,H]×(0,∞). Afterward, the results are extended for (u10,u20)∈Lp(Ω), p>2. Lastly, the existence criteria are obtained when (u10,u20)∈H1(Ω). A physical interpretation of the obtained bounds is provided, showing the rheological effects of shear thinningand shear thickening in Eyring–Powell fluids.
RResumenComo contrapunto al monumentalismo tradicional en la enseñanza de las obras matemáticas, la teoría antropológica de lo didáctico (TAD) propone el paradigma del cuestionamiento del mundo, siendo los recorridos de estudio e investigación (REI) los dispositivos que propone para implementarlo en las instituciones escolares. Sin embargo, esta implantación del nuevo paradigma encuentra restricciones, en particular de índole metodológica. En el presente trabajo presentamos una propuesta para posibilitar la supervivencia de los REI en una institución con una metodología basada en el aprendizaje cooperativo, y analizamos cómo los REI y el aprendizaje cooperativo pueden complementarse.Palabras-clave: Teoría Antropológica de lo didáctico, Recorridos de estudio e investigación, aprendizaje cooperativo.AbstractThe Anthropological Theory of the Didactic proposes the paradigm of questioning the world as the counterpoint for the traditional monumentalism when teaching mathematical works. In order to introduce the new paradigm in school institutions, the TAD proposes the study and research paths (SRP). Nevertheless, there exist restrictions which complicates the implementation of this new paradigm, particularly of methodological nature. In this paper, we present a proposal which makes it possible the survival of the SRP in an institution whose methodology is based on cooperative learning, and we analyse how SRP and cooperative learning can complement each other.Keywords: Anthropological Theory of didactics, Study and research tours, cooperative learning.
En este trabajo teórico se abordan las transformaciones y adaptaciones que se dieron en España en los años 60 para adaptarse al cambio de filosofía que supuso la Matemática Moderna. El cambio en las bases epistemológicas sobre la enseñanza de las Matemáticas se produjo de forma general en el contexto internacional. Sin embargo, el proceso de Transposición Didáctica fue diferente en cada país. El objetivo de nuestra investigación es analizar las restricciones transpositivas que se vivieron en España en los años 60 y ejemplificar el cambio de epistemología a través de textos escolares y disposiciones legales de ese momento histórico. Para llevar a cabo nuestro análisis y reflexión se ha hecho uso de dos herramientas propuestas por la Teoría Antropológica de lo Didáctico: la Transposición Didáctica y los Niveles de Codeterminación. Adicionalmente a estas herramientas se ha realizado una revisión bibliográfica de la legislación y un análisis de distintos materiales escolares del período en cuestión.
This paper addresses the mathematical education received during the pre-university stage based on the teaching-learning processes experienced by 225 students from the master’s degree in Teacher Training of Secondary, Baccalaureate, and FP and the Degree of Teacher of Primary Education of the Madrid Open University (UDIMA). For collecting the required information, a computerized questionnaire designed by the authors of this work and validated by the Ethics Committee of the Madrid Open University (UDIMA), has been used. The results of our study reveal the preservice teachers' memories about mathematics during the Primary and Secondary stages. Traditional teaching models, based on the repetition of calculation procedures, are the majority compared to other active teaching models. It is observed that a progressive increase in the methodologies supported by solving complex problems, detecting a moderate influence of the legislative changes produced in Spain in 2006. The mastery of classical teaching models and the moderate work around complex problems detected in pre-university education can be major constraints when developing new competency-based legislative approaches.
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