Hutchinson equation

Almost parallel to McKendrick, Hutchinson [215], a well-known ecologist, proposed a time-delayed version for the logistic growth equation, where the nonlinear term was delayed in time. The diffusive Hutchinson equation, also known as the delayed Fisher equation. 

J. Hutchinson and R.J. Farris, Rev. Sci. Instr., in press.). These quantities are then related to the mass/length using the wave equations for a flexible string which yields ... 

When the intrinsic kinetics are nonlinear, some interesting problems arise that are best discussed first with a discrete description. A possible assumption for the kinetics is that they are independent, that is, that the rate at which component 7 disappears depends only on the concentration of component I itself (this assumption is obviously correct in the first-order case). The difficulties associated with the assumption of independence are best illustrated by considering the case of parallel th order reactions, which has been analyzed by Luss and Hutchinson (1971), who write the kinetic equation for component / as -dcj/dt = kjc", 1=1, 2,. . ., N, where Cj is the (dimensional) concentration of component / at time t. The total initial concentration C(0) is , and this is certainly finite. Now consider the following special, but perfectly legitimate case. The value of C(0) is fixed, and the initial concentrations of all reactants are equal, so that C/(0) = C(0)/N. Furthermore, all the k/ s are equal to each other, k/ = k. One now obtains, for the initial rate of decrease of the overall concentration ... 

Hutchinson, H. P., D. J. Jackson, and W. Morton, Equation Oriented Flowsheet Simulation, Design and Optimization, Proc. Eur. Fed. Chem. Eng, Conf. Comput. Appl. Chem. Eng., Paris, April 1983 The Development of an Equation-Oriented Flowsheet Simulation and Optimization Package, Comput. Chem. Eng., v. 10, p. 19 (1986). 

Hutchinson et al. [82] have shown that the decomposition of each individual particle can be regarded as fitting the contracting volume equation (3.3), written in the form ... 

The factors governing the slow thermal decompositions of inorganic azides have been discussed by Fox and Hutchinson [18]. They draw attention to the interest shown in early studies for fitting of kinetic results to rate equations based on nucleation and growth models. Support of kinetic interpretations by microscopic observations (e g., [21]), contributed significantly towards establishment of the role of the active, advancing interface in solid state reactions. The kinetic characteristics of some of the metal azides are summarized in Table 11.1. 

Hutchinson BR, Galpin PF, Raithby GD (1988) Apphcation of Additive Correction Multigrid to the Coupled Fluid Flow Equations. Numerical Heat Transfer 13 133-147... 

Nakane and Hutchinson have further shown that the aldol step in this cyclization is stereoselective as well as regioselective. Treatment of (39) with Hunig s base, acetic anhydride and 4-(A, A -dimethyl-amino)pyridine (DMAP), followed by sodium borohydride reduction of the intermediate P-hydroxy aldehyde, gives diol (40) in 48% yield no diastereomeric diols were detected (equation 102). 2i... 

From the rate equations associated to (3.84) the quantity T = A + Bi+B2 is constant, so that the resource A = T—Bi B2 = R(B 1, B2) satisfies the hypothesis and the dynamics is of the form (3.83) with R linear in IR. B2. Since they are a particular case of (3.79) where = aiT — di, r2 = a2T — d2, an = a 2 = a 1, 021 = a22 = a2, and since (3.80) does not hold generally, we know that there will be no coexistence except in the non-generic case in which d /a = d2/a2. One of the competitors will disappear. We can show that the same competitive exclusion (Hardin, 1960 Hutchinson, 1961) behavior holds also in the more general case (3.83) with arbitrary R(Bi,B2). To this end, let us choose constants ci, C2 such that Ciai + c2a2 — 0 and define a = Cidi + c2d2, that can be made positive by adequate choice of C and C2. Eq. (3.83) implies that... 

Integration of this equation from e = 0 to s = e and correspondingly from So = 0 to o = So for two cases with strain-hardening exponents Al = 0.1 and 0.4, both for an initial imperfection factor tj = 0.005, gives results shown in Figs. 10.4(a) and (b) for the ratio s/so as a function of the increasing strain eo/N in the perfect bar (Hutchinson and Neal 1977). These results illustrate the process of strain localization in the bar and allow us to assess the influence of the strain-rate sensitivity. 

The residual stress in the coating is accounted for, but the model is not entirely valid to describe the stresses when some plastic deformation occurs. These key studies were further developed to take into account the elastic stress distribution and residual stress in the coating, which resulted in an improved equation for determining the strain energy release rate (Hutchinson and Suo, 1991 Venkataraman et al., 1993) ... 

Models of structural recovery include the Kovacs-Aklonis-Hutchinson-Ramos (KAHR) model (119), Moynihan s model (120), and Ngai s coupling model (121). These models are based on work done originally by Narayanaswamy (122), incorporating the ideas of Tool (13). The models of stnictiual recovery reflect the nonlinear and nonexponential effects observed experimentally. The historical development of these equations has been detailed (7,8) only a brief description follows. The KAHR formulation (119), which is written in terms of a departure from equilibrium S rather than in terms of Tf, is conceptually easier to use when the full three-dimensional PVT surface is considered ... 

Attempting to avoid the drawbacks of the Goldwasser method (see equations (5.12) — (5.22)), Hutchinson introduced a procedure in which the equilibrium conditions are formulated in general form. This eliminates the need of adapting the scheme to specific tasks. Each reaction is considered in the general form... 

The dependence of oj on D (for D2 = 0) is shown in Figure 4.28 from Suo and Hutchinson (1990) and from numerical solution of the equations in (4.48). When the film and substrate materials have identical elastic constants, that is, when D = D2 = 0, the elasticity solution yields uj 52.1° while the solution based on the plate idealization yields oj 48.6°. The discrete points in the figure are the numerical results reported by Suo and Hutchinson (1990) and the line through these points represents the expres-... 

Various techniques, such as interaction integral [16] and direct K solution using enriched finite elements [17], have been used to calculate K2/K ratio for an interface crack. Here, we used the near-tip asymptotic displacement field solution of Hutchinson and Su [8] to calculate the K2/K ratio, and hence cp, from the displacements of the quarter-point nodes behind the crack tip. The required derivations are brought in the Appendix. An independent comparison between results from a direct K solution using enriched finite elements and those obtained by inserting the displacement results from FRANC2D into the equation derived in the Appendix were found to agree within 1 3%. 

With the exception of the unknown parameter, a>, the entire solution up to this point was determined by arguments associated with linearity, geometry, and dimensionality. The determination of co, however, requires rigorous solution of the elasticity problem. This was done by Suo and Hutchinson using integral equation methods, and a tabulation of a>, which depends on a, p, and t], appears in ref. [89]. The variation of co with these parameters for two specific cases is shown in Fig. 7, from ref. [55],... 

Baxter discussed the Wiener-Hopf factorization of the inverse structure factor for disordered fluids [182]. He was able to split OZ2 Eq. (Ill) into a pair of equations, which provide a route to determine c (R. Dixon and Hutchinson complemented Baxter s developments by stating conditions for proper R values (in simulation work with cubic boxes / co T / 2) and proposed a highly accurate minimization method to solve Baxter s equations [183]. These R proper values will be denoted by R and termed zeros hereafter. As proven by the present author in extensive calculations [96,103,155], there is always (at least) one zero when physically significant functions are analyzed. By following this procedure (BDH), except for very low densities, one normally obtains more than one R [Pg.111]

A similar equation can be derived for the enthalpy. It turned out that this modification to give the retardation time a free volume dependence, was successful in describing one-step isothermal recovery but unsuccessful in describing memory effects. Kovacs, Aklonis, Hutchinson and Ramos (1979) attributed the latter to the contributions of at least two independent relaxation mechanisms involving two or more retardation times. These authors proposed a multiparameter approach, the so-called KAHR (Kovacs-Aklonis-Hutchinson-Ramos) model. The recovery process is divided into N subprocesses, which in the case of volumetric recovery may be expressed as ... 

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