Keller equation

In Fig. 1.4a, an example of the radius-time curve for a stably pulsating bubble calculated by the modified Keller equation is shown for one acoustic cycle [43]. After the bubble expansion during the rarefaction phase of ultrasound, a bubble strongly collapses, which is the inertial or Rayleigh collapse. After the collapse, there is a bouncing radial motion of a bubble. In Fig. 1.4b, the calculated flux of OH... 

Most experimental data obtained to date have been interpreted on the basis of the Rubinow-Keller equation (216) for the lift force on a spinning, translating sphere in an unbounded fluid at rest (or in uniform flow) at infinity. The spin is assumed to be given by Eq. (253), whereas the velocity U appearing in Eq. (216) has usually been interpreted as the axial slip velocity. This gives rise to a lift force... 

As observed repeatedly, most investigators concerned with the radial migration problem have attempted to analyze their data on the basis of a rather broad interpretation of the Rubinow-Keller equation. In the neutrally buoyant case, i.e., UaMV - Q, this interpretation predicts [cf. Eqs. (256) and (247)] a radial velocity... 

An empirical relation, developed by J. E. Keller, equation 4.2 is for the heat transfer coefficient between a moving molten liquid and a solid. 

Rigorous error bounds are discussed for linear ordinary differential equations solved with the finite difference method by Isaacson and Keller (Ref. 107). Computer software exists to solve two-point boundary value problems. The IMSL routine DVCPR uses the finite difference method with a variable step size (Ref. 247). Finlayson (Ref. 106) gives FDRXN for reaction problems. 

Basically there are two approaches to predicting the occurrence of erosion corrosion. Practical or experience based methods typified by Keller s approach for carbon steels in wet steam. Keller developed an equation that related the erosion corrosion rate as a function of temperature, steam quality, velocity and geometric factor. In recent years this approach has... 

Procedures enabling the calculation of bifurcation and limit points for systems of nonlinear equations have been discussed, for example, by Keller (13) Heinemann et al. (14-15) and Chan (16). In particular, in the work of Heineman et al., a version of Keller s pseudo-arclength continuation method was used to calculate the multiple steady-states of a model one-step, nonadiabatic, premixed laminar flame (Heinemann et al., (14)) a premixed, nonadiabatic, hydrogen-air system (Heinemann et al., (15)). 

Figure 1 compares the conversion predicted for any reduced time I = k t with the use of Keller s theory and the above values of k /k and k2/k with experimental results obtained when polyacrylamide was exposed to 0.2N NaOH at 53 C. It may be seen that the reaction slows down at large x much more than predicted by Keller s model. In fact, this decrease of the reaction rate is even more pronounced than predicted by Keller s equations for the case where a single reacted nearest neighbor completely inhibits amide hydrolysis. We believe that this discrepancy is due to the repulsion of the catalyzing hydroxyl ions from amide residues by non-neighboring carboxylate groups. 

In this equation, liquid has been assumed as incompressible. In the following Keller and Herring equations, the liquid compressibility has been taken into account to the first order of R jc-x. where c x is the sound velocity in the liquid far from a bubble [41]. 

Spin-orbit coupling problems are of a genuine quantum nature since a priori spin is a quantity that only occurs in quantum mechanics. However, already Thomas (Thomas, 1927) had introduced a classical model for spin precession. Later, Rubinow and Keller (Rubinow and Keller, 1963) derived the Thomas precession from a WKB-like approach to the Dirac equation. They found that although the spin motion only occurs in the first semiclassical correction to the relativistic classical electron motion, it can be expressed in merely classical terms. 

Our approach is based on a systematic semiclassical study of the Dirac equation. After separating particles and anti-particles to arbitrary powers in h, a semiclassical expansion of the quantum dynamics in the Heisenberg picture is developed. To leading order this method produces classical spin-orbit dynamics for particles and anti-particles, respectively, that coincide with the findings of Rubinow and Keller Hamiltonian relativistic (anti-) particles drive a spin precession along their trajectories. A modification of that method leads to a semiclassical equivalent of the Foldy-Wouthuysen transformation resulting in relativistic quantum Hamiltonians with spin-orbit coupling. 

The adjustment of measurements to compensate for random errors involves the resolution of a constrained minimization problem, usually one of constrained least squares. Balance equations are included in the constraints these may be linear but are generally nonlinear. The objective function is usually quadratic with respect to the adjustment of measurements, and it has the covariance matrix of measurements errors as weights. Thus, this matrix is essential in the obtaining of reliable process knowledge. Some efforts have been made to estimate it from measurements (Almasy and Mah, 1984 Darouach et al., 1989 Keller et al., 1992 Chen et al., 1997). The difficulty in the estimation of this matrix is associated with the analysis of the serial and cross correlation of the data. 

K4. Keller, H. B., and Takami, H., Proc. Adv. Symp. Numerical Solutions Non-Linear Differential Equations pp. 115 -140 (1966). 

Conditions favoring dimerization have been applied independently by Keller and Tarrant102,103 as well as Henne and Postelneck104 in homologous coupling of chlorofluorocarbon iodides to generate a,cross-couple chlorofluorocarbon iodides, however, yielded mixtures and limited the utility of this reaction105. 

Such purely mathematical problems as the existence and uniqueness of solutions of parabolic partial differential equations subject to free boundary conditions will not be discussed. These questions have been fully answered in recent years by the contributions of Evans (E2), Friedman (Fo, F6, F7), Kyner (K8, K9), Miranker (M8), Miranker and Keller (M9), Rubinstein (R7, R8, R9), Sestini (S5), and others, principally by application of fixed-point theorems and Green s function techniques. Readers concerned with these aspects should consult these authors for further references. 

In many cases ordinary differential equations (ODEs) provide adequate models of chemical reactors. When partial differential equations become necessary, their discretization will again lead to large systems of ODEs. Numerical methods for the location, continuation and stability analysis of periodic and quasi-periodic trajectories of systems of coupled nonlinear ODEs (both autonomous and nonautonomous) are extensively used in this work. We are not concerned with the numerical description of deterministic chaotic trajectories where they occur, we have merely inferred them from bifurcation sequences known to lead to deterministic chaos. Extensive literature, as well as a wide choice of algorithms, is available for the numerical analysis of periodic trajectories (Keller, 1976,1977 Curry, 1979 Doedel, 1981 Seydel, 1981 Schwartz, 1983 Kubicek and Hlavacek, 1983 Aluko and Chang, 1984). 

Fig. 2 Effect of irradiance on the cellular DMSP C-ratio (molimol) in Emiliania huxleyi (closed symbols). Data are recalculated from Slezak and Herndl (2003) (diamonds) Keller and Korjeff-Bellows (1996) (squares) van Rijssel and Buma (2002) and van Rijssel and Gieskes (2002) (triangles). For comparison, data from experiments with Phaocystis antarctica are included (open symbols Stefels and van Leeuwe 1998 Stefels unpublished). Equation of the linear regression fit (E. huxleyi data only) DMSP C = 0.00002 x PAR + 0.0084 (f = 0.693)...

Equations (7.26) and (7.27) furnish limiting results for both dilute and concentrated systems. Nunan and Keller (1984) subsequently extended these numerical calculations to intermediate concentrations as well, simultaneously... 

Equation (8.22) constitutes the means whereby the configuration-specific kinematic viscosity of the suspension may be computed from the prescribed spatially periodic, microscale, kinematic viscosity data v(r) by first solving an appropriate microscale unit-cell problem. Its Lagrangian derivation differs significantly from volume-average Eulerian approaches (Zuzovsky et al, 1983 Nunan and Keller, 1984) usually employed in deriving such suspension-scale properties. 

Gravimetry can also be associated with adsorption manometry, which is a simple and safe way to study co-adsorption of two gases, provided their molar masses are sufficiently different (Keller et al, 1992). The manometric experiment provides a total amount adsorbed n°a = n° + n%, whereas the gravimetric experiment provides the total mass adsorbed m t=m° + m. Since n°Mx = m and n M1 = m" we have two unknowns, n° and n, and two equations. From these we can obtain, for instance ... 

If burial of organic matter equates to O2 release to the atmosphere over long timescales, then the oxidative weathering of ancient organic matter in sedimentary rocks equates to O2 consumption. This process has been called georesprration by some authors (Keller and Bacon, 1998). It can be represented by Equation (6), the reverse of (5) ... 

The basis of the theory of electromagnetic fields studied by geophysicists is provided by Maxwell s equations (Stratton, 1941 Zhdanov and Keller, 1994) ... 

Rizk MA, Elghobashi SE (1989) A Two-Equation Turbulence Model for Dispersed Dilute Confined Two-Phase Flows. Int J multiphase Flow 15(1) 119-133 Rubinow SI, Keller JB (1961) The transverse force on a spinning sphere moving in a viscous fluid. J Fluid Mech 11 447-459... 

The first statistical models of these interactions are the well-known Thomas-Fermi (TF) and Thomas-Fermi-Dirac (TFD) theories based on the idea of approximating the behavior of electrons by that of the uniform negatively charged gas. Some authors (Sheldon, 1955 Teller, 1962 Balazs, 1967 Firsov, 1953,1957 Townsend and Handler, 1962 Townsend and Keller, 1963 Goodisman, 1971) proved that these theories provide an adequate description of purely repulsive diatomic interactions. Abraham-son (1963, 1964) and Konowalow et al. (Konowalow, 1969 Konowalow and Zakheim, 1972) extended this region to intermediate internuclear distances, but Gombas (1949) and March (1957) showed that the Abraham-son approach is incorrect, and so the question of how adequately the TFD theory provides diatomic interactions for closed-shell atoms is still open. Here we need to note that until recently, there has existed only work by Sheldon (1955), as far as we know, in which the TFD interaction potential is actually calculated by solving the TFD equation for a series of internuclear distances (see also, Kaplan, 1982). 

More generally, the problem of closure of the Reynolds equations is treated as the problem of establishing mathematical relationships between two-point correlation moments of various order [41,260,290,492]. Keller and Fridman [221 ]... 

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