In this paper we are interested with a strongly coupled system of partial differential equations that modelizes free convection in a two-dimensional bounded domain filled with a fluid saturated porous medium. This model is inspired by the one of free convection near a semi-infinite impermeable vertical flat plate embedded in a fluid saturated porous medium. We establish the existence and uniqueness of the solution for small data in some unusual spaces.

This work is a theoretical study of the mechanical behavior of two different weave types: plain and twill. Traditional methods permit the study of plain weaves but prove quite difficult for twill weaves. Indeed, the difficulties related to modeling the mechanical behavior of the twill weave are due to its very complex geometry and its nonsymmetry, which require the application of the finite element method. This method requires first a mathematical formulation of the problem and then a mesh of the basic cells of the plain and twill fabrics. The next step is to simulate shearing and tensile tests. Analyzing the results has proved to be very hard and thus demands a study of the stress field of the basic cell.

The arrangement as well as the properties and the structure of the fibres within the yarn and the yarns within the fabric generate a complex mechanism of deformation in such material. Therefore, intends to develop a theoretical model of the mechanical behaviour of the twill weave based on previous researches concerning the simplest plain weave. However, scaling up from the plain to the twill weave is not a direct transformation due to the non-symmetry of the latter. The finite element method does not require simplifying hypotheses. Thus, it is possible to simulate different stresses, to determine the fabric response and to compare the behaviour of the various structures. This simulation requires the use of a realistic meshing of the basic cell and an accurate characterisation of the physical parameters of the material that composes the basic cell. Assuming the material to be elastic, the derived and, consequently, the discreet mathematical formulations of the problem have both been solved. The coefficients from those formulas are then used in the Modulef software. For each stage of the development, uniaxial, biaxial and perpendicular to the fabric plan, tensile tests have been simulated, as well as pure shear testing. The next step consisted of computing the Tresca and Von Mises stresses within the basic cell and the micro-stress field within the basic cell components.