Spectral Element Discretization of the Circular Driven Cavity. Part IV: The Navier-Stokes Equations

This paper deals with the spectral element discretization of the Navier-Stokes equations in a disk with discontinuous boundary data, which is known as the driven cavity problem. The numerical treatment does not involve any regularization of these data. Relying on a variational formulation in the primitive variables of velocity and pressure, we describe a discretization of these equations and derive error estimates in appropriate weighted Sobolev spaces. We propose an algorithm to solve the nonlinear discrete system and present numerical experiments to verify its efficiency.     

Convection laminaire forcée dans le sillage d'une plaque chaude placée dans un canal

In this Note, a numerical investigation of laminar convection wake above a heated plate placed in a channel is carried out. An analytical study based on a three region structure is proposed in the immediate neighbourhood of the trailing edge. Velocity and temperature at the centerline channel as well as the pressure gradient are presented in asymptotic expressions. Comparison of these results with numerical solutions enhances the analytical study. To cite this article: Z. Doulfoukar, A. Achiq, C. R. Mecanique 333 (2005).     

Computational approach for a pair of bubble coalescence process

The coalescence of bubbles has great value in mineral recovery and oil industry. In this paper, two co-axial bubbles rising in a cylinder is modelled to study the coalescence of bubbles for four computational experimental test cases. The Reynolds’ (Re) number is chosen in between 8.50 and 10, Bond number, Bo ∼4.25-50, Morton number, M 0.0125-14.7. The viscosity ratio (μ_r) and density ratio (ρ_r) of liquid to bubble are kept constant (100 and 850 respectively). It was found that the Bo number has significant effect on the coalescence process for constant Re, μ_r and ρ_r. The bubble-bubble distance over time was validated against published experimental data. The results show that VOF approach can be used to model these phenomena accurately. The surface tension was changed to alter the Bo and density of the fluids to alter the Re and M, keeping the μ_r and ρ_r the same. It was found that for lower Bo, the bubble coalesce is slower and the pocket at the lower part of the leading bubble is less concave (towards downward) which is supported by the experimental data.     

Mathematical Models for Studying the Value of Motivational Leadership in Teams

Mathematical models are presented for studying the value of leadership in a team where the members interact with each other. The models are based on a leader’s role of motivating each team member to perform closer to his/her maximum ability. These models include controllable parameters whose values reflect the amount of task interdependence among the workers as well as the motivational skill and variability in the skill of the leader. Confirming results—such as the fact that the skill level of the leader is a critical factor in the expected performance of the team—establish credibility in the models. Mathematical analysis and computer simulations are used to provide new managerial insights into the value of the leader—such as the fact that the skill of the leader can be more important than controlling the amount of interdependence among the team members and that having a choice of multiple leaders with no particular motivating skill is beneficial to the performance of small teams but not to large teams.Daniel Solow received a B.S. in Mathematics from Carnegie-Mellon, an M.S. in Operations Research from the University of California at Berkeley, and a Ph. D. in Operations Research from Stanford University. He has been a professor at Case Western Reserve University since 1978. His research interests include complex systems, discrete, linear, and nonlinear optimization. He has also developed systematic methods for teaching mathematical proofs, computer programming, and operations research.Sandy Kristin Piderit is an assistant professor of organizational behavior at the Weatherhead School of Management at Case Western Reserve University, and earned her Ph.D. from the University of Michigan. She studies the roles of relationships among coworkers on their performance and satisfaction with their work environments, and has published studies in the Academy of Management Review, the Journal of Management Studies, and Management Science.Apostolos Burnetas received a Diploma in Electrical Engineering from National Technical University in Athens, Greece, and an M.B.A. and Ph.D. in Operations Research from Rutgers University. He has been at the Department of Operations at Case Western Reserve University and is currently an Associate Professor at the Department of Mathematics at the University of Athens. His research interests include stochastic models and optimization, complex systems, and applications in queueing systems, supply chain and the interface of operations with finance.Chartchai Leenawong received a B.S. in Mathematics from Chulalongkorn University in Bangkok, an M.S. in Computer Science from the Asian Institute of Technology in Bangkok, and a Ph.D. in Operations Research from Case Western Reserve University. His research interests include mathematical modeling of complex systems as applied to business organizations. He has been a professor at King Mongkut’s Institute of Technology Ladkrabang, Bangkok, Thailand since 2002.     

Finite element approximation of the Neumann eigenvalue problem in domains with multiple cracks

^** Email: belhach{at}poncelet.univ-metz.fr^*** Email: bucur{at}math.univ-metz.fr^**** Email: jmse{at}math.univ-metz.fr We study the Neumann–Laplacian eigenvalue problem in domainswith multiple cracks. We derive a mixed variational formulationwhich holds on the whole geometric domain (including the cracks)and implements efficient finite-element discretizations forthe computation of eigenvalues. Optimal error estimates aregiven and several numerical examples are presented, confirmingthe efficiency of the method. As applications, we numericallyinvestigate the behaviour of the low eigenvalues in domainswith a large number of cracks.     

Modeling on an ecological food chain with recycling

We propose two nutrient-phytoplankton models with instantaneous and time delayed recycling, investigate the dynamics and examine the responses to model complexities. Instead of the familiar specific uptake rate and growth rate functions, we assume only that the nutrient uptake and phytoplankton growth rate functions are positive, increasing and bounded above. We use geometrical and analytical methods to find conditions for the existence of none, one, or at most two positive steady states and analyze the stability properties of each of these equilibria. With the variation of parameters, the system may lose its stability and bifurcation may occur. We study the occurrence of Hopf bifurcation and the possibility of stability switching. Numerical simulations illustrate the analytical results and provide further insight into the dynamics of the models, biological interpretations are given.     

Interaction of CO2 gas with silicone elastomer at high ambient pressures

Interaction of high-pressure CO₂ gas with a silicone elastomer, and to a lesser extent, with a nitrile rubber and a PTFE have been investigated. Sorptive dilations of the polymers were measured with the help of custom-made piezoelectric ultrasonic transducers under gas pressures of up to ca. 22 MPa at 42°C. The gas mass sorption was determined by a vibrating reed probe. For the silicone elastomer system the dilation isotherm mimics the sorption isotherm. The partial molar volume (PMV) of the absorbed CO₂ gas in the silicone elastomer has been computed. A significant drop in the PMV value is observed when the CO₂ gas becomes supercritical. In the transition region, the transmission of ultrasonic signals through the specimen indicated the formation of discrete small (estimated as about 60 μm in diameter) high density zones of CO₂ in the rubber matrix. The plasticization effects of the absorbed high pressure CO₂ gas have been identified from the interpretation of the changes in the acoustic longitudinal modulus obtained from ultrasonic transmission measurements. The effects of rapid gas decompression on the structural integrity of the various polymers have also been determined. Significant inflation of certain specimens occur toward the latter stages of the decompression cycle. The initiation and development of internal cracks or bubbles was followed by monitoring the ultrasonic signal attenuation.