Complete Numerical Solutions Research Articles

Application of nonlinear sensitivity analysis to enzyme mechanisms

Computer programs were developed to permit the routine application of nonlinear sensitivity analysis to general complex enzyme mechanisms. Complete numerical solutions to the stiff differential equations which describe such mechanisms permit simulation of the full time-course development of all concentrations, including substrate, product, and all intermediate species. Alternatively, the method could be used for steady-state initial rate studies if desired. The sensitivity of all concentrations at any specific time during the reaction (or for any set of initial conditions) to variations in the rate constants can be determined. These variations in the rate constants can span orders of magnitude if desired. The method also describes the effect of coupling between rate constants. This permits the rate constants of an arbitrary model to be rank-ordered in terms of their importance to the time development of concentration variation of any desired species. The method also identifies which pairs of rate constants are most strongly coupled. The method should be very useful in experimental design, model testing, and reduction, and in the determination of the reliability of rate constants derived for a particular model from rate data. In order to illustrate the method it was applied to several published enzyme mechanisms, the most complex of which involved eight species. By examining the sensitivities it was possible to determine the most important pathways of reaction and to understand the origin of bursts and lags in product production. The method also permitted reduction of the most complex model to simpler models which showed essentially the same behavior.

Computation of wave fields in the ocean around an Isaland

AbstractA linear wave equation correct to first order in bed slope is used to calculate the wave field in the sea around an idealized island. This is of circular cylindrical shape and is situated on a paraboloidal shoal in an ocean of constant depth (Figure 1). The sides of the island are assumed fully reflecting. The incident waves are plane and periodic. Wave periods up to 30 min are investigated, and the Coriolis force is neglected. The solution of the wave equation is represented by a finite Fourier series, and a large number of very accurate numerical computations are carried through. The results appear partly in figures showing amplitude and phase angle curves (in some cases extending to the water area of constant depth outside the shoal), partly in figures showing amplitude vs wave period in fixed points. Comparison with solutions to the linearized long‐wave equation is made, and the validity range of the corresponding shallow water theory is given. The influence of the shoal is studied by investigating the wave field around an island in an ocean of constant depth. New criteria are given for the applicability of a geometrical optics approach (i. e. refraction). Complete numerical refraction solutions for points at the shoreline (corresponding to many wave orthogonals ending at the point) for shallows water waves, as for the general case, demonstrate the inadequacy of this approach for long‐period waves (seismic seawaves: tsunamis). All non‐linear effects, including dissipation, are excluded.

Boundary layer over spinning blunt body of revolution at incidence including Magnus forces

An incompressible laminar flow over a spinning blunt-body at incidence is investigated. The approach follows strictly the three-dimensional boundary layer theory, and the lack of initial profiles is readily resolved. The rule of the dependence zone is satisfied with the Krause scheme, and complete numerical solutions are obtained for an ellipsoid of revolution at 6° incidence and two spin rates. Spinning causes asymmetry which, in turn, introduces the Magnus force. The asymmetry is most pronounced in crossflow, but is also noticeable in the skin friction and displacement thickness of the meridional flow. A variety of crossflow profiles are determined as are the streamline patterns in the cross- and meridional-plane which are especially useful in visualizing the flow structure. Detailed distribution of skin friction, displacement thickness, and centrifugal pressure are presented. A negative crossflow displacement thickness is found to be physically meaningful. The Magnus forces due to the crossflow skin friction and the centrifugal pressure are determined; these two forces partly compensate for each other. At lower spin rate, the frictional force is larger, resulting in a positive Magnus force. At high spin rate, the opposite is obtained. At high incidence (30°) the present boundary layer calculations could be carried out in the longitudinal direction, only up to the beginning of an open separation.

Diffusion theory for crystal growth at arbitrary solute concentration

The influence of the solute concentration upon the kinetics of crystal growth is studied for various cases with equilibrium and nonequilibrium boundary conditions, constant cooling rate, and temperature-dependent diffusion coefficient. A perturbation expansion is used to calculate first-order corrections in the low concentration region. Complete numerical solutions for nonequilibrium crystal–liquid interface conditions are obtained by a finite difference method. The influence of the various parameters upon the time-dependent growth rates is discussed.

Mechanistic details of the Belousov–Zhabotinskii oscillations

AbstractA detailed chemical mechanism for the cerium catalyzed bromination and oxidation of malonic acid by bromate has been developed, and complete numerical solutions have been obtained for the temporal behavior of the concentrations of the intermediate species. The many points of similarity with the experimentally observed initial decay, long quiescence, and abrupt transition to sustained oscillation establish confidence in the essential validity of the proposed mechanism. Close examination of the time dependence of these species concentrations as well as the magnitude of the several reaction channels permits the clarification of the switching mechanism causing theoscillatory behavior of this system. The results appear to exhibit limit cycle behavior, in accordance with theoretical predictions.

The Optimum Factory Town

This paper studies the determination of the optimum size and arrangement of a monocentric city. The major consideration is the trade-off' between economies of scale in production in the central business district and the diseconomies of congestion in commuter transport. Planned optimum allocations and their decentralized implementation by suitable taxes and transfers are discussed. Rent and density profiles and land use are solved in closedform, and complete numerical solutions are obtained for particular parameter values. It is shown that negative exponential rent and density profiles may not be very good approximations, and that cities of more than a million inhabitants are dificult to justify in the,framework considered. * The general conceptual framework of the theory of monocentric cities is too well-known to need another airing.' In this paper I offer improvement on earlier work at several specific points. I shall indicate these by comparing this paper with some recent ones. The most serious defect common to several of these papers (e.g. those of Solow and of Oron, Pines, and Sheshinski2) is their assumption of nonincreasing returns to scale in production in the central business district. This destroys all rationale for organizing production in a city, as dispersed production will save transport costs without losing output. To preserve the urban structure, we must then impose the condition that all output has to be shipped to a point market place, which is an arbitrary and unsatisfactory way out. The role of increasing returns in understanding urban problems is clearly stated by Koopmans,3 and general theorems relating city size to increasing returns are proved by Mirrlees and Starrett.4 I shall be concerned with more specific cases which can yield approximate numerical results,

Burning Rate Development in a Closed Vessel of Arbitrary Shape and Variable Volume, for Variable but Uniform Pressure

An expression is derived to compute the mass burning rate in a closed vessel for which the variable volume and pressure during the process of combustion are known. The solution presented will be satisfactory for applications where the effects of heat transfer and variable chemical reaction during the combustion process can be neglected, and may serve as a basis of comparison for more complete numerical solutions.

On the Limit Analysis of Stability of Slopes

ABSTRACT The upper bound theorem of the generalized theory of perfect plasticity is applied to obtain complete numerical solutions for the critical height of an embankment. A rotational discontinuity mechanism (logarithmic spiral) is assumed in the analysis. The analysis includes the existing solutions as a special case, so that it may be considered a generalization of previous solutions.

A hydrodynamic theory of desalination by electrodialysis

A hydrodynamic theory of demineralization by electrodialysis has been developed for a multichannel system with steady laminar flow between plane, parallel membranes. The modeling of the system is found to be governed by four basic similarity parameters: (i) a dimensionless applied potential, (ii) the product of the channel aspect ratio and the inverse Péclet number, (iii) the ratio of brine and dialysate inlet concentrations, and (iv) a parameter measuring membrane resistance. For sufficiently long channels it is shown that there are two distinct regions: a “developing” region where the concentration diffusion layers are growing, and a “developed” region where the diffusion layers fill the channel. Parabolic and uniform velocity profiles are considered and self-consistent solutions are derived for the distributions of salt concentration, electric field and current density in the system, as well as for the total current. An integral method of solution is used. In the limits of low and high polarization analytic solutions are obtained which when matched at their point of equality closely approximate the complete numerical solutions. It is found that under a wide range of operating conditions, the solution for the total current is represented by the empirical formulaI^=[1−exp(−Ψ^3)]1/3,where Î andΨ^are, respectively, a dimensionless current and potential embodying the four similarity parameters mentioned. Comparison is made of the calculated limiting total current with experiment.

On the wind-driven circulation in an ocean with bottom topography

The influence of bottom topography on the wind-driven ocean circulation is investi gated for a homogeneous, s-plane model with an imposed, steady wind stress curl. The linear model studied by Munk (1950) and the non-linear model studied by Carrier & Robinson (1962) are special cases as is the simple model proposed by Warren (1963) foi the description of Gulf Stream meanders. Although the effects of density stratification are neglected, inertial and frictional effects are included and solutions determined for various kinds of topography. Simple analytic models are examined and found to be useful in describing limited regions of the ocean basin. In particular the concept of an interior regime, in which nonlinear and frictional terms are unimportant, can be extended to the case of variable depth. In regions of strong inertial flow over topography, the simple meander model investigated by Rossby (1940) has some success in predicting the path of the stream. In both models the importance of isopleths of constant F/H , where F is the Coriolis parameter and H is the depth, is demonstrated by showing that there is a strong tendency for streamlines to follow or meander about such curves. Complete numerical solutions are found for the finite difference vorticity equation by an efficient relaxation scheme developed for this purpose. Results show that the response of the ocean may be strongly controlled by the shape of the ocean bottom and that such gross features of the ocean circulation as Gulf Stream separation, meandering, and variable transport may be related to topograpic effects. DOI: 10.1111/j.2153-3490.1967.tb01510.x

Extended Space-Charge Theory in Low-Pressure Thermionic Converters

The problem of space-charge limited currents in a diode with plane-parallel electrodes is considered for the case where space-charge limitations are counteracted by the introduction of ions at the cathode. The ``no-collision'' approximation is used, in which inelastic and short-range elastic collisions are neglected, and each particle interacts with the Coulomb field of all the other particles. The electrostatic potential is determined, in a self-consistent manner, from Poisson's equation. The equations are simplified by transforming to reduced variables in a manner described by Langmuir and others. The first integrations of Poisson's equation are given for the six most important cases. The first of these has been reported by P. L. Auer, but rigorous analysis of the remaining five cases has not been previously reported. Analytical solution of the first-order differential equations derived is impossible. Numerical solutions have been obtained for illustrative purposes on the IBM-704 computer. More complete numerical solutions are being compiled.