The problem of the spreading of a liquid film along a solid surface: A new mathematical formulation

A new mathematical model is proposed for the spreading of a liquid film on a solid surface. The model is based on the standard lubrication approximation for gently sloping films (with the no-slip condition for the fluid at the solid surface) in the major part of the film where it is not too thin. In the remaining and relatively small regions near the contact lines it is assumed that the so-called autonomy principle holds—i.e., given the material components, the external conditions, and the velocity of the contact lines along the surface, the behavior of the fluid is identical for all films. The resulting mathematical model is formulated as a free boundary problem for the classical fourth-order equation for the film thickness. A class of self-similar solutions to this free boundary problem is considered.

Quasi-separation of the biharmonic partial differential equation

In this paper, we consider analytical and numerical solutions to the Dirichlet boundary-value problem for the biharmonic partial differential equation on a disc of finite radius in the plane. The physical interpretation of these solutions is that of the harmonic oscillations of a thin, clamped plate. For the linear, fourth-order, biharmonic partial differential equation in the plane, it is well known that the solution method of separation in polar coordinates is not possible, in general. However, in this paper, for circular domains in the plane, it is shown that a method, here called quasi-separation of variables, does lead to solutions of the partial differential equation. These solutions are products of solutions of two ordinary linear differential equations: a fourth-order radial equation and a second-order angular differential equation. To be expected, without complete separation of the polar variables, there is some restriction on the range of these solutions in comparison with the corresponding separated solutions of the second-order harmonic differential equation in the plane. Notwithstanding these restrictions, the quasi-separation method leads to solutions of the Dirichlet boundary-value problem on a disc with centre at the origin, with boundary conditions determined by the solution and its inward drawn normal taking the value 0 on the edge of the disc. One significant feature for these biharmonic boundary-value problems, in general, follows from the form of the biharmonic differential expression when represented in polar coordinates. In this form, the differential expression has a singularity at the origin, in the radial variable. This singularity translates to a singularity at the origin of the fourth-order radial separated equation; this singularity necessitates the application of a third boundary condition in order to determine a self-adjoint solution to the Dirichlet boundary-value problem. The penultimate section of the paper reports on numerical solutions to the Dirichlet boundary-value problem; these results are also presented graphically. Two specific cases are studied in detail and numerical values of the eigenvalues are compared with the results obtained in earlier studies.

Low-temperature organic Rankine cycle engine with isothermal expansion for use in desalination

Groundwater salinity is a widespread problem that contributes to the freshwater deficit of humanity. Consequently, where conventional energy supply is also lacking, organic Rankine cycle (ORC) engines are being considered as a feasible option to harness readily available low-grade heat (<180°C) to drive the desalination of the saline water via reverse osmosis (RO). However, this application is still not very well developed, and has significantly high specific energy consumption (SEC). Hence, this study explores the isothermal expansion of the ORC working fluid to achieve improved efficiency for driving a batch-RO desalination process, "DesaLink". Here, the working fluid is directly vaporized in the expansion cylinder which is heated externally by heat transfer fluid, thus obviating the need for a separate external boiler and high-pressure piping. Experimental investigations with R245fa have shown cycle efficiency of 8.8%. And it is predicted that the engine could drive DesaLink to produce 256 L of freshwater per 8 h per day, from 4000 ppm saline water, with a thermal and mechanical SEC of 2.5 and 0.36 kWh/m3, respectively, representing a significant improvement on previously reported or predicted SEC values. © 2014 © 2014 Balaban Desalination Publications. All rights reserved.

Robust solutions of bi-blend recipe optimization with quadratic constraints

Production companies use raw materials to compose end-products. They often make different products with the same raw materials. In this research, the focus lies on the production of two end-products consisting of (partly) the same raw materials as cheap as possible. Each of the products has its own demand and quality requirements consisting of quadratic constraints. The minimization of the costs, given the quadratic constraints is a global optimization problem, which can be difficult because of possible local optima. Therefore, the multi modal character of the (bi-) blend problem is investigated. Standard optimization packages (solvers) in Matlab and GAMS were tested on their ability to solve the problem. In total 20 test cases were generated and taken from literature to test solvers on their effectiveness and efficiency to solve the problem. The research also gives insight in adjusting the quadratic constraints of the problem in order to make a robust problem formulation of the bi-blend problem.

Problems in physics and mathematics contexts with middle and high school classes: vertical and horizontal comparison

My thesis falls within the framework of physics education and teaching of mathematics. The objective of this report was made possible by using geometrical (in mathematics) and qualitative (in physics) problems. We have prepared four (resp. three) open answer exercises for mathematics (resp. physics). The test batch has been selected across two different school phases: end of the middle school (third year, 8\textsuperscript{th} grade) and beginning of high school (second and third year, 10\textsuperscript{th} and 11\textsuperscript{th} grades respectively). High school students achieved the best results in almost every problem, but 10\textsuperscript{th} grade students got the best overall results. Moreover, a clear tendency to not even try qualitative problems resolution has emerged from the first collection of graphs, regardless of subject and grade. In order to improve students' problem-solving skills, it is worth to invest on vertical learning and spiral curricula. It would make sense to establish a stronger and clearer connection between physics and mathematical knowledge through an interdisciplinary approach.

Two-dimensional supersonic nonlinear Schrodinger flow past an extended obstacle

Supersonic flow of a superfluid past a slender impenetrable macroscopic obstacle is studied in the framework of the two-dimensional (2D) defocusing nonlinear Schroumldinger (NLS) equation. This problem is of fundamental importance as a dispersive analog of the corresponding classical gas-dynamics problem. Assuming the oncoming flow speed is sufficiently high, we asymptotically reduce the original boundary-value problem for a steady flow past a slender body to the one-dimensional dispersive piston problem described by the nonstationary NLS equation, in which the role of time is played by the stretched x coordinate and the piston motion curve is defined by the spatial body profile. Two steady oblique spatial dispersive shock waves (DSWs) spreading from the pointed ends of the body are generated in both half planes. These are described analytically by constructing appropriate exact solutions of the Whitham modulation equations for the front DSW and by using a generalized Bohr-Sommerfeld quantization rule for the oblique dark soliton fan in the rear DSW. We propose an extension of the traditional modulation description of DSWs to include the linear ""ship-wave"" pattern forming outside the nonlinear modulation region of the front DSW. Our analytic results are supported by direct 2D unsteady numerical simulations and are relevant to recent experiments on Bose-Einstein condensates freely expanding past obstacles.

Step size control for efficient discrete element simulation

The step size determines the accuracy of a discrete element simulation. The position and velocity updating calculation uses a pre-calculated table and hence the control of step size can not use the integration formulas for step size control. A step size control scheme for use with the table driven velocity and position calculation uses the difference between the calculation result from one big step and that from two small steps. This variable time step size method chooses the suitable time step size for each particle at each step automatically according to the conditions. Simulation using fixed time step method is compared with that of using variable time step method. The difference in computation time for the same accuracy using a variable step size (compared to the fixed step) depends on the particular problem. For a simple test case the times are roughly similar. However, the variable step size gives the required accuracy on the first run. A fixed step size may require several runs to check the simulation accuracy or a conservative step size that results in longer run times. (C) 2001 Elsevier Science Ltd. All rights reserved.

Confined-unconfined flow in a horizontal confined aquifer

Mixed confined and unconfined groundwater flow occurs in a bounded initially dry aquifer when the hydraulic head at the side boundary suddenly rises above the elevation of the aquifer's top boundary. The flow problem as modelled by the Boussinesq equation is non-trivial because of the involvement of two moving boundaries. The transformed equation (based on a similarity transformation) can, however, be dealt with more easily. Here, we present an approximate analytical solution for this flow problem. The approximate solution is compared with an 'exact' numerical solution and found to be a very accurate description for describing the mixed confined and unconfined flow in the confined aquifer. (C) 2002 Elsevier Science B.V. All rights reserved.

Perimetria de hiperacuidade preferencial no estudo do edema macular diabético

Introdução – A diabetes é uma das maiores epidemias do último século. Mais de 250 milhões de pessoas, em todo o mundo, sofrem de diabetes. Das complicações derivadas da diabetes são as principais causas de cegueira, de insuficiência renal e de amputação de membros inferiores, derivando estes, predominantemente, da disfunção vascular. Quando surge perda de pericitos na parede vascular ocorrem uma série de alterações da microcirculação que levam ao aparecimento de microaneurismas e outras alterações vasculares que possibilitam a passagem de componentes sanguíneos para o tecido retiniano adjacente que, em situação de normalidade, não ocorreriam, sendo esta uma das causas do edema macular exsudativo diabético. A perimetria de hiperacuidade preferencial (PHP) é um teste psicofísico que pretende detetar metamorfopsias na Degenerescência Macular ligada à Idade (DMI). Uma vez que o edema macular diabético (EMD) se destaca como uma das principais causas de deficiência visual e baixa visão, pretende-se verificar a eficácia do PHP no estudo do edema macular diabético, respondendo à seguinte questão: “Qual a capacidade do perímetro de hiperacuidade preferencial em detetar metamorfopsias em pacientes com edema macular diabético?“ Metodologia – Estudo quantitativo, do tipo descritivo e correlacional. Selecionou-se uma amostra de 33 pacientes, onde se analisou um total de 60 olhos. Resultados – A sensibilidade do PHP na deteção de metamorfopsias associadas ao EMD na tomografia de coerência ótica (OCT) foi de 70,6%, a especificidade foi de 11,5% e a eficiência global do teste de 45%. Comparando os resultados encontrados no PHP e no OCT, constatou-se a existência de uma correlação inversa fraca (Phi = -0,215). Conclusões – Este novo método de diagnóstico revela-se sensível, contudo pouco específico e eficaz na deteção de metamorfopsias consequentes da existência de EMD. - ABSTRACT - Introduction – Preferential hyperacuity perimeter (PHP) is a new psychophysical test, which principle is based on the detection of metamorphopsia in age-related macular degeneration (AMD). It is intended to verify its effectiveness in the study of diabetic macular edema (DME). When there is loss of pericytes in the vascular wall occur a number of microcirculatory changes that lead to the appearance of microaneurysms and other vascular changes that allow the passage of blood components to the surrounding retinal tissue than in normal situation does not occur, this being one of the causes exudative diabetic macular edema. Methodology – It was performed a quantitative study, using descriptive and correlational analysis. A sample of 33 patients was selected, and 60 eyes were analyzed. Results – The sensitivity of PHP on the detection of metamorphopsia associated to EMD was 70.6%, the specificity was 11.5% and the global efficiency of the test was 45%. It was found a weak negative correlation (Phi= -0.215) between the PHP and optical coherence tomography (OCT). Conclusions – This new method of diagnosis was sensitive, but not very specific and effective on the detection of metamorphopsia, due to the DME.

A importância das tarefas realizadas em sala de aula no desenvolvimento da resolução criativa de problemas

Relatório de Estágio apresentado à Escola Superior de Educação de Lisboa para obtenção de grau de mestre em Ensino do 1.º e do 2.º Ciclo do Ensino Básico

An approximate mathematical model for solidification of a flowing liquid in a microchannel

A mathematical model is developed to analyse the combined flow and solidification of a liquid in a small pipe or two-dimensional channel. In either case the problem reduces to solving a single equation for the position of the solidification front. Results show that for a large range of flow rates the closure time is approximately constant, and the value depends primarily on the wall temperature and channel width. However, the ice shape at closure will be very different for low and high fluxes. As the flow rate increases the closure time starts to depend on the flow rate until the closure time increases dramatically, subsequently the pipe will never close.

Evolution of foraging behaviour in response to chronic malnutrition in Drosophila melanogaster.

Chronic exposure to food of low quality may exert conflicting selection pressures on foraging behaviour. On the one hand, more active search behaviour may allow the animal to find patches with slightly better, or more, food; on the other hand, such active foraging is energetically costly, and thus may be opposed by selection for energetic efficiency. Here, we test these alternative hypotheses in Drosophila larvae. We show that populations which experimentally evolved improved tolerance to larval chronic malnutrition have shorter foraging path length than unselected control populations. A behavioural polymorphism in foraging path length (the rover-sitter polymorphism) exists in nature and is attributed to the foraging locus (for). We show that a sitter strain (for(s2)) survives better on the poor food than the rover strain (for(R)), confirming that the sitter foraging strategy is advantageous under malnutrition. Larvae of the selected and control populations did not differ in global for expression. However, a quantitative complementation test suggests that the for locus may have contributed to the adaptation to poor food in one of the selected populations, either through a change in for allele frequencies, or by interacting epistatically with alleles at other loci. Irrespective of its genetic basis, our results provide two independent lines of evidence that sitter-like foraging behaviour is favoured under chronic larval malnutrition.

Amplicon DNA melting analysis for the simultaneous detection of Brucella spp and Mycobacterium tuberculosis complex. Potential use in rapid differential diagnosis between extrapulmonary tuberculosis and focal complications of brucellosis.

Some sites of extrapulmonary tuberculosis and focal complications of brucellosis are very difficult to differentiate clinically, radiologically, and even histopathologically. Conventional microbiological methods for the diagnosis of extrapulmonary tuberculosis and complicated brucellosis not only lack adequate sensitivity, they are also time consuming, which could lead to an unfavourable prognosis. The aim of this work was to develop a multiplex real-time PCR assay based on SYBR Green I to simultaneously detect Brucella spp and Mycobacterium tuberculosis complex and evaluate the efficacy of the technique with different candidate genes. The IS711, bcsp31 and omp2a genes were used for the identification of Brucella spp and the IS6110, senX3-regX3 and cfp31 genes were targeted for the detection of the M. tuberculosis complex. As a result of the different combinations of primers, nine different reactions were evaluated. A test was defined as positive only when the gene combinations were capable of co-amplifying both pathogens in a single reaction tube and showed distinguishable melting temperatures for each microorganism. According to the melting analysis, only three combinations of amplicons (senX3-regX3+bcsp31, senX3-regX3+IS711 and IS6110+IS711) were visible. Detection limits of senX3-regX3+bcsp31 and senX3-regX3+IS711 were of 2 and 3 genome equivalents for M. tuberculosis complex and Brucella while for IS6110+IS711 they were of 200 and 300 genome equivalents, respectively. The three assays correctly identified all the samples, showing negative results for the control patients. The presence of multicopy elements and GC content were the components most influencing the efficiency of the test; this should be taken into account when designing a multiplex-based SYBR Green I assay. In conclusion, multiplex real time PCR assays based on the targets senX3-regX3+bcsp31 and senX3-regX3+IS711 using SYBR Green I are highly sensitive and reproducible. This may therefore be a practical approach for the rapid differential diagnosis between extrapulmonary tuberculosis and complicated brucellosis.

Bootstrapping pairs in Distance-Based Regression

La regressió basada en distàncies és un mètode de predicció que consisteix en dos passos: a partir de les distàncies entre observacions obtenim les variables latents, les quals passen a ser els regressors en un model lineal de mínims quadrats ordinaris. Les distàncies les calculem a partir dels predictors originals fent us d'una funció de dissimilaritats adequada. Donat que, en general, els regressors estan relacionats de manera no lineal amb la resposta, la seva selecció amb el test F usual no és possible. En aquest treball proposem una solució a aquest problema de selecció de predictors definint tests estadístics generalitzats i adaptant un mètode de bootstrap no paramètric per a l'estimació dels p-valors. Incluim un exemple numèric amb dades de l'assegurança d'automòbils.

Bootstrapping pairs in Distance-Based Regression

La regressió basada en distàncies és un mètode de predicció que consisteix en dos passos: a partir de les distàncies entre observacions obtenim les variables latents, les quals passen a ser els regressors en un model lineal de mínims quadrats ordinaris. Les distàncies les calculem a partir dels predictors originals fent us d'una funció de dissimilaritats adequada. Donat que, en general, els regressors estan relacionats de manera no lineal amb la resposta, la seva selecció amb el test F usual no és possible. En aquest treball proposem una solució a aquest problema de selecció de predictors definint tests estadístics generalitzats i adaptant un mètode de bootstrap no paramètric per a l'estimació dels p-valors. Incluim un exemple numèric amb dades de l'assegurança d'automòbils.