Joseph W. Wilder

NONAXISYMMETRIC AND AXISYMMETRIC CONVECTION IN PROPAGATING REACTION-DIFFUSION FRONTS

Lenses are successively and continuously fed to first precision grinding and cutting device wherein they are ground. Thereafter, the lenses are automatically conveyed to a second grinding and polishing device wherein they are further polished, and finally the lenses are automatically removed, washed and discharged from said apparatus.

Hydrodynamic Instability of Chemical Waves

We present a theory for the transition to convection for flat chemical wave fronts propagating upward. The theory is based on the hydrodynamic equations and the one‐variable reaction‐diffusion equation that describes the chemical front for the iodate–arsenous acid reaction. The reaction term involves the reaction rate constants and the chemical composition of the mixture. This allows the discussion of the effects of the different chemical variables on the transition to convection. We have studied perturbations of different wavelengths on an unbounded flat chemical front and found that for wavelengths larger than a critical wavelength (λ≳λc) the perturbations grow in time, while for smaller wavelengths the perturbations diminish. The critical wavelength depends not only on the density difference between the unreacted and reacted fluids, but also on the speed and thickness of the chemical front.We present a theory for the transition to convection for flat chemical wave fronts propagating upward. The theory is based on the hydrodynamic equations and the one‐variable reaction‐diffusion equation that describes the chemical front for the iodate–arsenous acid reaction. The reaction term involves the reaction rate constants and the chemical composition of the mixture. This allows the discussion of the effects of the different chemical variables on the transition to convection. We have studied perturbations of different wavelengths on an unbounded flat chemical front and found that for wavelengths larger than a critical wavelength (λ≳λc) the perturbations grow in time, while for smaller wavelengths the perturbations diminish. The critical wavelength depends not only on the density difference between the unreacted and reacted fluids, but also on the speed and thickness of the chemical front.

A three variable differential equation model for gypsy moth population dynamics

Abstract The dynamics of gypsy moth, Lymantria dispar (Lepidoptera: Lymantriidae), populations are extremely complex. As a result, many of the models which have been proposed to model these populations are likewise very complicated. This complexity makes analysis of the underlying dynamics difficult. In this work a model is proposed which involves only three variables: gypsy moth biomass density, foliage biomass density and natural enemy biomass density. The dynamics of this model are shown to include period doubling as a route to chaos, among other interesting nonlinear phenomena. The model also evidences similar behavior to that noted from field studies in which researchers attempted to artificially stimulate outbreaks of gypsy moths. While these attempts failed in nature and in the model, the model predicts that under certain circumstances it may be possible to stimulate these outbreaks.

Dynamics of falling raindrops

A standard undergraduate mechanics problem involves a raindrop which grows in size as it falls through a mist of suspended water droplets. Ignoring air drag, the asymptotic drop acceleration is g/7, independent of the mist density and the drop radius. Here we show that air drag overwhelms mist drag, producing drop accelerations of order 10 −3 g. Analytical solutions are facilitated by a new empirical form of the air drag coefficient C = 12R −1/2 , which agrees with experimental data on liquid drops in the Reynolds-number range 10 <R< 1000 relevant to precipitating spherical drops. Solutions including air drag are within reach of students of intermediate mechanics and nonlinear dynamics. Even without air drag, the dynamics of a raindrop falling through a stationary mist serves as an important and non-trivial application of Newton’s second law because the mass of the drop changes with time. Undergraduate mechanics students are sometimes able to solve the nonlinear dynamical equations of motion to find the deceptively simple acceleration g/ 7o f an infinitesimal-radius drop released from rest, assuming that the drop accretes all of the mist that it encounters. Dick [1] showed that drops of arbitrary initial radius and velocity approach this acceleration asymptotically. Krane [2] confirmed that inelastic collisions account for the lost mechanical energy of the falling drop. Partovi and Aston [3] included air drag in the problem, assuming a constant drag coefficient for pedagogical simplicity. The objective of this paper is to include the variations in the air drag coefficient for growing raindrops. As raindrops grow in radius from r = 0. 1m m tor = 1 mm within a cloud, their drag coefficients decrease from about C = 5 to about C = 0.5. To account for this decrease, we employ a simple but accurate empirical relationship for the dependence of the drag coefficient on the Reynolds number, which allows us to obtain simple exponential solutions for the asymptotic drop radius, speed, acceleration, and distance travelled. Because of their accuracy, these solutions closely mimic the behaviour of real raindrops, and predict the actual time required for a raindrop to fall through a cloud. Because of their simplicity, these solutions are accessible to students of intermediate mechanics and nonlinear dynamics, who benefit by this soluble yet realistic example. Our approach to the problem is couched in the language and formalism of modern nonlinear dynamics. Since air densities ρa ≈ 10 −3 gc m −3 greatly exceed the mist densities [4, 5] ρm ≈ 10 −6 gc m −3 typical of terrestrial rain clouds, air drag might be expected to play an important role in raindrop dynamics. Air drag indeed overwhelms the force of the accreting mist

Derivation of a nonlinear front evolution equation for chemical waves involving convection

Abstract The nonlinear stability of exothermic autocatalytic reaction fronts is considered using the viscous thermohydrodynamic fluid equations in the limits of infinite and zero thermal diffusivity. A nonlinear front evolution equation is derived using asymptotic analysis and is solved for an ascending front. The method used is similar to that which has been used in the derivation of a front equation for flame propagation.

Convective instability of autocatalytic reaction fronts in vertical cylinders

Linear stability analysis predicts that the onset of convection for an ascending autocatalytic reaction front in a vertical cylinder corresponds to a nonaxisymmetric mode. This mode consists of a single convective roll confined to the region near the reaction front, with fluid rising in half of the cylinder and falling in the other half. Experiments show a flat front below the onset of convection and an axisymmetric front well above the onset of convection. New experiments are called for to closely examine the onset of convection in order to test this prediction.

Effect of initial condition sensitivity and chaotic transients on predicting future outbreaks of gypsy moths

A previously validated, simple model of the population dynamics of the gypsy moth, Lymantria dispar (Lepidoptera: Lymantriidae), is used to show that initial condition sensitivity in the cases of (a) chaotic dynamics, and (b) fractal basin boundaries associated with non-chaotic attractors both severely limit the ability to predict future behaviour even for short time scales. Also demonstrated is the fact that short-term transients can cause even small discrepancies in initial population densities to lead to erroneous conclusions about future behaviour over periods of the order of a decade. This work addresses the quality of data needed to make predictions based on population data from a single year. It is demonstrated that, even when it is assumed that the model perfectly reflects the population dynamics and that the parameters are known exactly, extremely small errors in the specification of initial conditions are required to obtain reliable predictions. The model also shows that the observed dynamical difference between populations in North America (chaotic dynamics) and Europe (periodic dynamics) could be explained by the presence of chaotic transients.

A predictive model for gypsy moth population dynamics with model validation

A simple model for gypsy moth, Lymantria dispar (Lepidoptera: Lymantriidae), population dynamics is presented. Comparison with data from the Melrose Highlands study shows the model to exhibit the same qualitative and quantitative behavior as the data. Predictions are made about future outbreaks using only a portion of the field data to fit the model parameters. Comparison of these predictions with the rest of the data shows excellent agreement in both timing and magnitude of future outbreaks for times on the order of 10 years. This work represents the first model for gypsy moth population dynamics which is capable of accurate quantitative predictions over this time period.

NUMERICAL SOLUTION OF A TWO-DIMENSIONAL FLUIDIZED BED MODEL

SUMMARY The numerical solution of a model describing a two-dimensional fluidized bed is considered. The model takes the form of a hyperbolic system of conservation laws with source term, coupled with an elliptic equation for determining a streamfunction. Operator splitting is used to produce homogeneous one-dimensional hyperbolic systems and ordinary differential equations involving the source term. The one-dimensional hyperbolic problems are solved using Roe’s method with the addition of an entropy fix. The numerical procedure is second-order in time and first-order in space. Second-order-accuracy in space is obtained using flux limiting techniques. Numerical experiments which show the development of bubbles in the bed are presented. The familiar kidney-shaped bubble, observed experimentally, is found when using the method which is second-order in space. On the same mesh, the first-order method produces bubbles which are no longer kidney-shaped.

Modelling of two-dimensional spatial effects on the spread of forest pests and their management

This work reports the modelling of forest pests, using a recent model for gypsy moth (Lymantria dispar (Lepidoptera: Lymantriidae)) populations in a two-dimensional, homogeneous stand of host trees. While the underlying dynamics of the model may be either periodic or chaotic, it was found that if spraying is done, the observed dynamics appear to be chaotic. In addition, it was found that in some cases, spraying with a strong pesticide can actually benefit the pest, increasing the amount of damage done to the foliage. Under some conditions it was found that a small land-owner may not be at a disadvantage if the area surrounding his/her land is not treated with a pesticide.