Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations

Viscous fingering experiments in Hele-Shaw cells lead to striking pattern formations which have been the subject of intense focus among the physics and applied mathematics community for many years. In recent times, much attention has been devoted to devising strategies for controlling such patterns and reducing the growth of the interfacial fingers. We continue this research by reporting on numerical simulations, based on the level set method, of a generalised Hele-Shaw model for which the geometry of the Hele-Shaw cell is altered. First, we investigate how imposing constant and time-dependent injection rates in a Hele-Shaw cell that is either standard, tapered or rotating can be used to reduce the development of viscous fingering when an inviscid fluid is injected into a viscous fluid over a finite time period. We perform a series of numerical experiments comparing the effectiveness of each strategy to determine how these non-standard Hele-Shaw configurations influence the morphological features of the inviscid-viscous fluid interface. Tapering plates in either converging or diverging directions leads to reduced metrics of viscous fingering at the final time when compared to the standard parallel configuration, especially with carefully chosen injection rates; for the rotating plate case, the effect is even more dramatic, with sufficiently large rotation rates completely stabilising the interface. Next, we illustrate how the number of non-splitting fingers can be controlled by injecting the inviscid fluid at a time-dependent rate while increasing the gap between the plates. Simulations compare well with previous experimental results for various injection rates and geometric configurations. Further, we demonstrate how the fully nonlinear dynamics of the problem affect the number of fingers that emerge and how well this number agrees with predictions from linear stability analysis

On the Existence of Bertrand-Nash Equilibrium Prices Under Logit Demand

This article presents a proof of the existence of Bertrand-Nash equilibrium prices with multi-product firms and under the Logit model of demand that does not rely on restrictive assumptions on product characteristics, firm homogeneity or symmetry, product costs, or linearity of the utility function. The proof is based on conditions for the indirect utility function, fixed-point equations derived from the first-order conditions, and a direct analysis of the second-order conditions resulting in the uniqueness of profit-maximizing prices. Several subsequent results also demonstrate that price equilibrium under the Logit model of demand cannot adequately describe multi-product pricing.Comment: 39 Page

Fixed-Point Approaches to Computing Bertrand-Nash Equilibrium Prices Under Mixed Logit Demand: A Technical Framework for Analysis and Efficient Computational Methods

This article presents a detailed technical framework for modeling with Bertrand-Nash equilibrium prices under Mixed Logit demand. Two coercive fixed-point equations provide more stable computational methods than those obtained from the literal first-order conditions. Assumptions to justify derivation and use of these equations are provided. A brief discussion of a GMRES-Newton method with hookstep globalization strategy originally due to Viswanath is also given. This article can be considered a supplement to an article by the authors forthcoming in the journal {\em Operations Research}.Comment: 57 page

An analytical approach to solution of two- point boundary condition problems in optimal guidance Summary report, May 1965 - Apr. 1966

Analytical approaches to path-adaptive guidance functions, circular orbit trajectories, and use of Fortran-compiled program

Scenarios for optimizing potato productivity in a lunar CELSS

The use of controlled ecological life support system (CELSS) in the development and growth of large-scale bases on the Moon will reduce the expense of supplying life support materials from Earth. Such systems would use plants to produce food and oxygen, remove carbon dioxide, and recycle water and minerals. In a lunar CELSS, several factors are likely to be limiting to plant productivity, including the availability of growing area, electrical power, and lamp/ballast weight for lighting systems. Several management scenarios are outlined in this discussion for the production of potatoes based on their response to irradiance, photoperiod, and carbon dioxide concentration. Management scenarios that use 12-hr photoperiods, high carbon dioxide concentrations, and movable lamp banks to alternately irradiate halves of the growing area appear to be the most efficient in terms of growing area, electrical power, and lamp weights. However, the optimal scenario will be dependent upon the relative 'costs' of each factor

Population response of triploid grass carp to declining levels of hydrilla in the Santee Cooper Reservoirs, South Carolina

Approximately 768,500 triploid grass carp ( Ctenopharyngodon idella Valenciennes) were stocked into the Santee Cooper reservoirs, South Carolina between 1989 and 1996 to control hydrilla ( Hydrilla verticillata (L.f.) Royle). Hydrilla coverage was reduced from a high of 17,272 ha during 1994 to a few ha by 1998. During 1997, 1998 and 1999, at least 98 triploid grass carp were collected yearly for population monitoring. Estimates of age, growth, and mortality, as well as population models, were used in the study to monitor triploid grass carp and predict population trends. Condition declined from that measured during a previous study in 1994. The annual mortality rate was estimated at 28% in 1997, 32% in 1998 and 39% in 1999; however, only the 1999 mortality rate was significantly different. Few (2 out of 98) of the triploid grass carp collected during 1999 were older than age 9. We expect increased mortality due to an aging population and sparse hydrilla coverage. During 1999, we estimated about 63,000 triploid grass carp system wide and project less than 3,000 fish by 2004, assuming no future stocking. management, population size Ctenopharyngodon idella, Hydrill

Two-interface and thin filament approximation in Hele--Shaw channel flow

For a viscous fluid trapped in a Hele--Shaw channel, and pushed by a pressure difference, the fluid interface is unstable due to the Saffman--Taylor instability. We consider the evolution of a fluid region of finite extent, bounded between two interfaces, in the limit the interfaces are close, that is, when the fluid region is a thin liquid filament separating two gases of different pressure. In this limit, we derive a geometric flow rule that describes the normal velocity of the filament centreline, and evolution of the filament thickness, as functions of the thickness and centreline curvature. We show that transverse flow along the filament is necessary to regularise the instability. Numerical simulation of the thin filament flow rule is shown to closely match level-set computations of the complete two-interface model, and solutions ultimately evolve to form a bubble of increasing radius and decreasing thickness

Light emitting diodes as a plant lighting source

Electroluminescence in solid materials is defined as the generation of light by the passage of an electric current through a body of solid material under an applied electric field. A specific type of electroluminescence, first noted in 1923, involves the generation of photons when electrons are passed through a p-n junction of certain solid materials (junction of a n-type semiconductor, an electron donor, and a p-type semiconductor, an electron acceptor). The development of this light emitting semiconductor technology dates back less than 30 years. During this period of time, the LED has evolved from a rare and expensive light generating device to one of the most widely used electronic components. A number of LED characteristics are of considerable importance in selecting a light source for plant lighting in a controlled environment facility. Of particular importance is the characteristic that light is generated by an LED at a rate far greater than the corresponding thermal radiation predicted by the bulk temperature of the device as defined by Plank's radiation law. This is in sharp contrast to other light sources, such as an incandescent or high intensity discharge lamp. A plant lighting system for controlled environments must provide plants with an adequate flux of photosynthetically active radiation, plus providing photons in the spectral regions that are involved in the photomorphogenic and phototropic responses that result in normal plant growth and development. Use of light sources that emit photons over a broad spectral range generally meet these two lighting requirements. Since the LED's emit over specific spectral regions, they must be carefully selected so that the levels of photsynthetically active and photomorphogenic and phototropic radiation meet these plant requirements