Mathematical Analysis of Some Partial Differential Equations with Applications

In the first part of this dissertation, we produce and study a generalized mathematical model of solid combustion. Our generalized model encompasses two special cases from the literature: a case of negligible heat diffusion in the product, for example, when the burnt product is a foam-like substance; and another case in which diffusivities in the reactant and product are assumed equal. In addition to that, our model pinpoints the dynamics in a range of settings, in which the diffusivity ratio between the burned and unburned materials varies between 0 and 1. The dynamics of temperature distribution and interfacial front propagation in this generalized solid combustion model are studied through both asymptotic and numerical analyses. For asymptotic analysis, we first analyze the linear instability of a basic solution to the generalized model. We then focus on the weakly nonlinear case where a small perturbation of a neutrally stable parameter is taken so that the linearized problem is marginally unstable. Multiple scale expansion method is used to obtain an asymptotic solution for large time by modulating the most linearly unstable mode. On the other hand, we integrate numerically the exact problem by the Crank-Nicolson method. Since the numerical solutions are very sensitive to the derivative interfacial jump condition, we integrate the partial differential equation to obtain an integral-differential equation as an alternative condition. The result system of nonlinear algebraic equations is then solved by the Newton’s method, taking advantage of the sparse structure of the Jacobian matrix. By a comparison of our asymptotic and numerical solutions, we show that our asymptotic solution captures the marginally unstable behaviors of the solution for a range of model parameters. Using the numerical solutions, we also delineate the role of the diffusivity ratio between the burned and unburned materials. We find that for a representative set of this parameter values, the solution is stabilized by increasing the temperature ratio between the temperature of the fresh mixture and the adiabatic temperature of the combustion products. This trend is quite linear when a parameter related to the activation energy is close to the stability threshold. Farther from this threshold, the behavior is more nonlinear as expected. Finally, for small values of the temperature ratio, we find that the solution is stabilized by increasing the diffusivity ratio. This stabilizing effect does not persist as the temperature ratio increases. Competing effects produce a “cross-over” phenomenon when the temperature ratio increases beyond about 0.2. In the second part, we study the existence and decay rate of a transmission problem for the plate vibration equation with a memory condition on one part of the boundary. From the physical point of view, the memory effect described by our integral boundary condition can be caused by the interaction of our domain with another viscoelastic element on one part of the boundary. In fact, the three different boundary conditions in our problem formulation imply that our domain is composed of two different materials with one condition imposed on the interface and two other conditions on the inner and outer boundaries, respectively. These transmission problems are interesting not only from the point of view of PDE general theory, but also due to their application in mechanics. For our mathematical analysis, we first prove the global existence of weak solution by using Faedo-Galerkin’s method and compactness arguments. Then, without imposing zero initial conditions on one part of the boundary, two explicit decay rate results are established under two different assumptions of the resolvent kernels. Both of these decay results allow a wider class of relaxation functions and initial data, and thus generalize some previous results existing in the literature

Aircraft engine hot section technology: An overview of the HOST Project

NASA sponsored the Turbine Engine Hot Section (HOST) project to address the need for improved durability in advanced aircraft engine combustors and turbines. Analytical and experimental activities aimed at more accurate prediction of the aerothermal environment, the thermomechanical loads, the material behavior and structural responses to loads, and life predictions for cyclic high temperature operation were conducted from 1980 to 1987. The project involved representatives from six engineering disciplines who are spread across three work disciplines - industry, academia, and NASA. The HOST project not only initiated and sponsored 70 major activities, but also was the keystone in joining the multiple disciplines and work sectors to focus on critical research needs. A broad overview of the project is given along with initial indications of the project's impact

Hydrodynamic Analysis of Binary Immiscible Metallurgical Flow in a Novel Mixing Process: Rheomixing

This paper presents a hydrodynamic analysis of binary immiscible metallurgical flow by a numerical simulation of the rheomixing process. The concept of multi-controll is proposed for classifying complex processes and identifying individual processes in an immiscible alloy system in order to perform simulations. A brief review of fabrication methods for immiscible alloys is given, and fluid flow aspects of a novel fabrication method – rheomixing by twin-screw extruder (TSE) are analysed. Fundamental hydrodynamic micro-mechanisms in a TSE are simulated by a piecewise linear (PLIC) volume-of-fluid (VOF) method coupled with the continuum surface force (CFS) algorithm. This revealed that continuous reorientation in the TSE process could produce fine droplets and the best mixing efficiency. It is verified that TSE is a better mixing device than single screw extruder (SSE) and can achieve finer droplets. Numerical results show good qualitative agreement with experimental results. It is concluded that rheomixing by a TSE can be successfully employed for casting immiscible engineering alloys due to its unique characteristics of reorientation and surface renewal

THERMAL AND MECHANICAL RESPONSES OF FIBER REINFORCED POLYMER COMPOSITES UNDER ONE-SIDED FIRE EXPOSURE

This research investigated the thermal and mechanical responses of fiber reinforced polymer (FRP) composites in fire. The research focused on thermal decomposition and heat transfer, deformation, delamination, and structural integrity of FRP composites. The research was undertaken by thermal and fire testing, and fire dynamics and finite element modeling. To simplify the modeling of the decomposition of FRP composites, an infinite-rate pyrolysis model was incorporated into heat transfer modeling to predict the thermal response of the composite panels under one sided heating. The thermal prediction by the infinite-rate model was compared to the finite-rate model, in which the decomposition was described by Arrhenius equation, and was validated with both bench and intermediate scale fire tests. A concept of shift temperature was introduced into the heat transfer to account for the effect of heating rate on the decomposition temperature. With temperature results given by the heat transfer model, a simplified plane strain model was proposed to predict the mechanical response of FRP composites. Based on a bilinear traction-separation law, cohesive elements in commercial finite element software ABAQUS were incorporated in the mechanical model to consider the effect of delamination for sandwich panels. In order to evaluate the effect of heat flux of a composite's own flame on its thermal response and fire properties, two-layer flame geometry was proposed to predict the effect of flame heat flux on the thermal response of char-forming materials. The total flame heat flux in a typical cone test was estimated based on general turbulent flame temperature and combustible gas temperature. All prediction results were validated with experimental data. It was demonstrated that (1) the modeling of decomposition reaction using the infinite-rate model required less input parameters, (2) a material's own flame had significant influent on its fire reaction properties at the beginning of flaming combustion, (3) the plane-strain model was capable of predicting deformation and time-to-failure with a good accuracy, and (4) cohesive elements can be used to model the delamination of sandwich FRP panels in fire

Development of an Analytical Model to Determine the Heat Fluxes to a Structural Element Due to a Travelling Fire

peer reviewedThe term “travelling fire” is used to label fires which burn locally and move across the floor over a period of time in large compartments. Through experimental and numerical campaigns and while observing the tragic travelling fire events, it became clear that such fires imply a transient heating of the surrounding structure. The necessity to better characterize the thermal impact generated on the structure by a travelling fire motivated the development of an analytical model allowing to capture, in a simple manner, the multidimensional transient heating of a structure considering the effect of the ventilation. This paper first presents the basic assumptions of a new analytical model which is based on the virtual solid flame concept; a comparison of the steel temperatures measured during a travelling fire test in a steel-framed building with the ones obtained analytically is then presented. The limitations inherent to the analyticity of the model are also discussed. This paper suggests that the developed analytical model can allow for both an acceptable representation of the travelling fire in terms of fire spread and steel temperatures while not being computationally demanding, making it potentially desirable for pre-design