Microwave heating of ceramic composites

The microwave heating of a ceramic composite is modelled and analysed. The composite consists of many small ceramic particles embedded in a ceramic cement. The composite is assumed to be well insulated, and each particle is assumed to be in imperfect thermal contact with the surrounding cement. Based on these two assumptions an asymptotic theory exploiting the small Blot number and small non-dimensional contact conductance is developed. Our asymptotic theory yields a set of nonlinear partial differential equations which govern the temperature in the composite. These are reduced to a set of coupled nonlinear ordinary differential equations in which the surface area of each particle enters as a parameter. Recent experiments with such composites have shown that the steady-state temperature of the composite is strongly dependent upon the radii of the embedded particles. Our model captures this effect. In fact, our analysis shows that the assumption of imperfect thermal contact between the particles and the ceramic cement is essential for this trend to be established

Diffusive and wavelike phenomena in thermal processing of materials

Contemporary materials science abounds with novel processing methods. Devices such as lasers, microwave sources, and electron beam guns, provide unprecedented control over the deposition of energy within a material. The modern materials scientist has the ability to deposit energy volumetrically, to precisely control the location of energy deposition within a material, and to deposit energy in extremely short intervals of time. While making possible numerous thermal processing methods, these devices also push the limits of our understanding of the response of materials to energy deposition. In order to optimize and control these processing methods, it becomes necessary to further our understanding of this response. Here, we investigate several problems, motivated by the study of thermal processing methods, whose analyses further our understanding of these new parameter regimes. First, we consider two classes of problems arising in microwave processing of ceramics. These problems are characterized by volumetric energy deposition and a weak coupling between thermal diffusion and electromagnetic wave propagation. Next, we investigate a sequence of problems motivated by and arising in the study of an electron beam joining process. These problems are characterized by rapid volumetric energy deposition and a strong coupling between thermal diffusion and elastic wave propagation

Curve fitting with the Bubble Board

The bubble board is a device to create simultaneously 56 identical soap bubbles. Students study the relation between time and the number of remaining bubbles for different concentrations of glycerin and use linear, exponential, and logistic decay models to fit the data

Approximations in Canonical Electrostatic MEMS Models

Abstract. The mathematical modeling and analysis of electrostatically actuated micro-and nanoelectromechanical systems (MEMS and NEMS) has typically relied upon simplified electrostatic field approximations to facilitate the analysis. Usually, the small aspect ratio of typical MEMS and NEMS devices is used to simplify Laplace's equation. Terms small in this aspect ratio are ignored. Unfortunately, such an approximation is not uniformly valid in the spatial variables. Here, this approximation is revisited and a uniformly valid asymptotic theory for a general "drum shaped" electrostatically actuated device is presented. The structure of the solution set for the standard non-uniformly valid theory is reviewed and new numerical results for several domain shapes presented. The effect of retaining typically ignored terms on the solution set of the standard theory is explored

Toward Integration: From Quantitative Biology to Mathbio-Biomath?

In response to the call of BIO2010 for integrating quantitative skills into undergraduate biology education, 30 Howard Hughes Medical Institute (HHMI) Program Directors at the 2006 HHMI Program Directors Meeting established a consortium to investigate, implement, develop, and disseminate best practices resulting from the integration of math and biology. With the assistance of an HHMI-funded mini-grant, led by Karl Joplin of East Tennessee State University, and support in institutional HHMI grants at Emory and University of Delaware, these institutions held a series of summer institutes and workshops to document progress toward and address the challenges of implementing a more quantitative approach to undergraduate biology education. This report summarizes the results of the four summer institutes (2007–2010). The group developed four draft white papers, a wiki site, and a listserv. One major outcome of these meetings is this issue of CBE—Life Sciences Education, which resulted from proposals at our 2008 meeting and a January 2009 planning session. Many of the papers in this issue emerged from or were influenced by these meetings

Vertical integration of research and education in the mathematical sciences - VIGRE: VIGRE/GT: vertical integration of research & education at Georgia Tech

Issued as final reportNational Science Foundation (U.S.